Finite Group Databasebeta

← #481232

Subgroups — #481232

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60306C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#33479C211yesmaximal, characteristic, complemented
4#11632C211yesmaximal, characteristic, complemented
5#11631C211yesmaximal, characteristic, complemented
6#11630C211yesmaximal, characteristic, complemented
7#11624C211yesmaximal, characteristic, complemented
8#11623C211yesmaximal, characteristic, complemented
9#11622C211yesmaximal, characteristic, complemented
10#11616C211yesmaximal, characteristic, complemented
11#11615C211yesmaximal, characteristic, complemented
12#11614C211yesmaximal, characteristic, complemented
13#11584C211yesmaximal, characteristic, complemented
14#11583C211yesmaximal, characteristic, complemented
15#11582C211yesmaximal, characteristic, complemented
16#3460C411yescharacteristic
17#2993—21no—
18#2993C411yescharacteristic
19#2993C411yescharacteristic
20#2989C411yescharacteristic
21#2989C411yescharacteristic
22#2981—21no—
23#2981C411yescharacteristic
24#2981C411yescharacteristic
25#2976—21no—
26#2976—21no—
27#2975C411yescharacteristic
28#2966—21no—
29#2966C2211yescharacteristic
30#2966C2211yescharacteristic
31C8×#41C2211yescharacteristic
32C8×#41C2211yescharacteristic
33#2931—21no—
34#2931C2211yescharacteristic
35#2931C2211yescharacteristic
36#2929—21no—
37#2929—21no—
38C2×C8×D4C2211yescharacteristic
39#2183—21no—
40#2183—21no—
41#2169—21no—
42#2169—21no—
43#2168—21no—
44#2168—21no—
45#2168—21no—
46#2168—21no—
47#2163C2211yescharacteristic
48#2163C2211yescharacteristic, complemented
49#2162C2211yescharacteristic
50#2162C2211yescharacteristic, complemented
51#2162C2211yescharacteristic, complemented
52#2162C2211yescharacteristic, complemented
53#2150C2211yescharacteristic
54#2150C2211yescharacteristic, complemented
55C2×C4⋊C16C2211yescharacteristic, complemented
56C4×M32C2211yescharacteristic
57C4×M32C2211yescharacteristic, complemented
58C2×C4×C16C2211yescharacteristic, complemented
59Q8⋊C16C2211yescharacteristic, complemented
60Q8⋊C16C2211yescharacteristic, complemented
61Q8⋊C16C2211yescharacteristic, complemented
62Q8⋊C16C2211yescharacteristic, complemented
63D4⋊C16C2211yescharacteristic
64D4⋊C16C2211yescharacteristic
65D4⋊C16C2211yescharacteristic
66D4⋊C16C2211yescharacteristic
67D4⋊C16C2211yescharacteristic
68D4⋊C16C2211yescharacteristic
69D4⋊C16C2211yescharacteristic
70D4⋊C16C2211yescharacteristic
71D4⋊C16C2211yescharacteristic, complemented
72D4⋊C16C2211yescharacteristic, complemented
73D4⋊C16C2211yescharacteristic, complemented
74D4⋊C16C2211yescharacteristic, complemented
75C2×#131—21no—
76C2×#131—21no—
77C23×C8—21no—
78#544C811yescharacteristic
79#542—21no—
80#542C811yescharacteristic
81#542C811yescharacteristic
82#540—21no—
83#540C811yescharacteristic
84#540C811yescharacteristic
85#538—21no—
86#538—21no—
87#537C811yescharacteristic
88#535C811yescharacteristic
89#535C811yescharacteristic
90#532—21no—
91#532C2×C411yescharacteristic
92#532C2×C411yescharacteristic
93#529—21no—
94#529C2×C411yescharacteristic
95#529C2×C411yescharacteristic
96#525—21no—
97#525—21no—
98C4×#41C2×C411yescharacteristic
99C4×#41C2×C411yescharacteristic
100C2×C4×D4C2×C411yescharacteristic
101C2×M32—21no—
102C2×M32—21no—
103C2×M32—21no—
104C2×M32—21no—
105C22×C16—21no—
106C22×C16—21no—
107#446—21no—
108#446C2×C411yescharacteristic
109#446C2×C411yescharacteristic
110C8×Q8C2×C411yescharacteristic
111C8×Q8D411yescharacteristic
112C8×Q8D411yescharacteristic
113C8×Q8C2311yescharacteristic
114#436—21no—
115#436—21no—
116#436—21no—
117#436—21no—
118#436—21no—
119#436—21no—
120#436—21no—
121#436C2×C411yescharacteristic
122#436C2×C411yescharacteristic
123#436C2×C411yescharacteristic
124#436C2×C411yescharacteristic
125#436C2×C411yescharacteristic
126#436C2×C411yescharacteristic
127#435—21no—
128#435—21no—
129#435—21no—
130#435—21no—
131#435—21no—
132#435—21no—
133#435—21no—
134#435—21no—
135#435—21no—
136#435—21no—
137#435—21no—
138#435—21no—
139C8×D4—21no—
140C8×D4—21no—
141C8×D4—21no—
142C8×D4—21no—
143C8×D4—21no—
144C8×D4—21no—
145C8×D4—21no—
146C8×D4—21no—
147C8×D4C2×C411yescharacteristic
148C8×D4C2×C411yescharacteristic
149C8×D4C2×C411yescharacteristic
150C8×D4D411yescharacteristic
151C8×D4D411yescharacteristic
152C8×D4D411yescharacteristic
153C8×D4D411yescharacteristic
154C8×D4D411yescharacteristic
155C8×D4D411yescharacteristic
156C8×D4C2311yescharacteristic
157C8×D4C2311yescharacteristic
158C8×D4C2311yescharacteristic
159#433—21no—
160#433—21no—
161#433—21no—
162#433—21no—
163#432—21no—
164#432C2×C411yescharacteristic
165#432C2×C411yescharacteristic
166#431—21no—
167#431C2×C411yescharacteristic
168#431C2×C411yescharacteristic
169#431C2×C411yescharacteristic
170#431C2×C411yescharacteristic
171#424—21no—
172#424—21no—
173#423—21no—
174#423C2×C411yescharacteristic
175#423C2×C411yescharacteristic
176C2×C4⋊C8C2×C411yescharacteristic
177#408—21no—
178#408—21no—
179#408—21no—
180#408—21no—
181#408—21no—
182#408—21no—
183#407—21no—
184#407—21no—
185C2×C22⋊C8—21no—
186#405—21no—
187#405—21no—
188C2×C4×C8—21no—
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C4⋊C16—21no—
193C4⋊C16—21no—
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C4⋊C16C2311yescharacteristic
201C4⋊C16C2311yescharacteristic
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205C22⋊C16—21no—
206C22⋊C16—21no—
207C22⋊C16—21no—
208#346—21no—
209#346—21no—
210#346—21no—
211#346—21no—
212C4×C16—21no—
213C4×C16—21no—
214C4×C16C2311yescharacteristic
215C4×C16C2311yescharacteristic
216C4×C16C2311yescharacteristic
217C4×C16C2311yescharacteristic
218C2×#41—21no—
219C2×#41—21no—
220C2×#41C2×C811yescharacteristic
221C2×#41C2×C811yescharacteristic
222C22×D4C2×C811yescharacteristic
223C23×C4—21no—
224#131—81no—
225#131—81no—
226#131—81no—
227#131—81no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—81no—
244C22×C8—81no—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8C2×C811yescharacteristic
253C4⋊Q8C2×C811yescharacteristic
254#127—21no—
255#127C2×C811yescharacteristic
256#127C2×C811yescharacteristic
257#126—21no—
258#126—21no—
259#126—21no—
260#126—21no—
261#125—21no—
262#125C2×C811yescharacteristic
263#125C2×C811yescharacteristic
264#124—21no—
265#124—21no—
266#124—21no—
267#124—21no—
268#124—21no—
269#124C2×C811yescharacteristic
270#124C2×C811yescharacteristic
271#123—21no—
272#123—21no—
273#123—21no—
274#123—21no—
275#123—21no—
276#123—21no—
277#123—21no—
278#123—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8C2×C811yescharacteristic
285C22⋊Q8C2×C811yescharacteristic
286#121—21no—
287#121—21no—
288#121—21no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296#121—21no—
297#121C2×C811yescharacteristic
298#121C2×C811yescharacteristic
299#121C2×C811yescharacteristic
300#121C2×C811yescharacteristic
301#121C2×C811yescharacteristic
302#121C2×C811yescharacteristic
303#120—21no—
304#120—21no—
305#120—21no—
306#120—21no—
307C4×Q8C22⋊C411yescharacteristic
308C4×Q8C2×C811yescharacteristic
309C4×Q8M1611yescharacteristic
310C4×Q8C22×C411yescharacteristic
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4C22⋊C411yescharacteristic
320C4×D4C22⋊C411yescharacteristic
321C4×D4C22⋊C411yescharacteristic
322C4×D4C2×C811yescharacteristic
323C4×D4C2×C811yescharacteristic
324C4×D4C2×C811yescharacteristic
325C4×D4M1611yescharacteristic
326C4×D4M1611yescharacteristic
327C4×D4M1611yescharacteristic
328C4×D4C22×C411yescharacteristic
329C4×D4C22×C411yescharacteristic
330C4×D4C22×C411yescharacteristic
331#117—21no—
332#117—21no—
333#117—21no—
334#117C2×C811yescharacteristic
335#117C2×C811yescharacteristic
336#117C2×C811yescharacteristic
337#117C2×C811yescharacteristic
338C2×C4⋊C4C2×C811yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42—21no—
341C2×C42C22×C411yescharacteristic
342C2×C42C22×C411yescharacteristic
343C2×C42C22×C411yescharacteristic
344M32—42no—
345M32—42no—
346M32—81no—
347M32—81no—
348C2×C16—21no—
349C2×C16—21no—
350C2×C16—21no—
351C2×C16—21no—
352C2×C16—21no—
353C2×C16—21no—
354C2×C16—21no—
355C2×C16—21no—
356C2×C16—42no—
357C2×C16—81no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8C22⋊C411yescharacteristic
366C4⋊C8C22⋊C411yescharacteristic
367C4⋊C8C22⋊C411yescharacteristic
368C4⋊C8C22⋊C411yescharacteristic
369C4⋊C8C22⋊C411yescharacteristic
370C4⋊C8C22⋊C411yescharacteristic
371C4⋊C8C22×C411yescharacteristic
372C4⋊C8C22×C411yescharacteristic
373C4⋊C8C22×C411yescharacteristic
374C4⋊C8C22×C411yescharacteristic
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380C22⋊C8—21no—
381C22⋊C8—21no—
382C22⋊C8—21no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—21no—
387C22⋊C8—21no—
388C22⋊C8—21no—
389C22⋊C8—21no—
390C22⋊C8C22⋊C411yescharacteristic
391C22⋊C8C22⋊C411yescharacteristic
392C22⋊C8C22⋊C411yescharacteristic
393C22⋊C8C22⋊C411yescharacteristic
394C22⋊C8C22⋊C411yescharacteristic
395C22⋊C8C22⋊C411yescharacteristic
396#97—21no—
397#97—21no—
398#97—21no—
399#97—21no—
400C4×C8—21no—
401C4×C8—21no—
402C4×C8—21no—
403C4×C8C22×C411yescharacteristic
404C4×C8C22×C411yescharacteristic
405C4×C8C22×C411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2×D411yescharacteristic
410C4×C8C2×D411yescharacteristic
411C4×C8C2×D411yescharacteristic
412C4×C8C2411yescharacteristic
413C24—21no—
414#41—42no—
415#41—42no—
416#41—81no—
417#41—81no—
418#41—81no—
419#41—81no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8C22⋊C811yescharacteristic
423C2×Q8C22×C811yescharacteristic
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—42no—
437C2×D4C22⋊C811yescharacteristic
438C2×D4C22⋊C811yescharacteristic
439C2×D4C22⋊C811yescharacteristic
440C2×D4C22×C811yescharacteristic
441C2×D4C22×C811yescharacteristic
442C2×D4C22×C811yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C22×C811yescharacteristic
457C22×C4C22×C811yescharacteristic
458C22×C4C22×C811yescharacteristic
459C22×C4C2×M1611yescharacteristic
460C22×C4C2×M1611yescharacteristic
461C22×C4C2×M1611yescharacteristic
462M16—81no—
463M16—81no—
464M16—81no—
465M16—81no—
466M16—81no—
467M16—81no—
468M16—81no—
469M16—81no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—81no—
490C2×C8—81no—
491C2×C8—81no—
492C2×C8—81no—
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C2×C22⋊C411yescharacteristic
496C2×C8C2×C22⋊C411yescharacteristic
497C2×C8C2×C22⋊C411yescharacteristic
498C2×C8C2×C22⋊C411yescharacteristic
499C2×C8C22×D411yescharacteristic
500C2×C8C22×D411yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4C22⋊C811yescharacteristic
513C4⋊C4C22⋊C811yescharacteristic
514C4⋊C4C22⋊C811yescharacteristic
515C4⋊C4C22⋊C811yescharacteristic
516C4⋊C4C22⋊C811yescharacteristic
517C4⋊C4C22⋊C811yescharacteristic
518C4⋊C4C22×C811yescharacteristic
519C4⋊C4C22×C811yescharacteristic
520C4⋊C4C22×C811yescharacteristic
521C4⋊C4C22×C811yescharacteristic
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4C22⋊C811yescharacteristic
542C22⋊C4C22⋊C811yescharacteristic
543C22⋊C4C22⋊C811yescharacteristic
544C22⋊C4C22⋊C811yescharacteristic
545C22⋊C4C22⋊C811yescharacteristic
546C22⋊C4C22⋊C811yescharacteristic
547C42—21no—
548C42—21no—
549C42—21no—
550C42C2×C22⋊C411yescharacteristic
551C42C2×C22⋊C411yescharacteristic
552C42C2×C22⋊C411yescharacteristic
553C42C22×C811yescharacteristic
554C42C22×C811yescharacteristic
555C42C22×C811yescharacteristic
556C42C2×M1611yescharacteristic
557C42C2×M1611yescharacteristic
558C42C2×M1611yescharacteristic
559C42C23×C411yescharacteristic
560C16—22no—
561C16—41no—
562C16—41no—
563C16—41no—
564C16—41no—
565C16—41no—
566C16—41no—
567C16—41no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—42no—
572C23—81no—
573C23—81no—
574C23C2×C22⋊C811yescharacteristic
575C23C2×C22⋊C811yescharacteristic
576C23C2×C22⋊C811yescharacteristic
577Q8—22no—
578Q8—41no—
579Q8—81no—
580Q8—81no—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—42no—
588D4—42no—
589D4—81no—
590D4—81no—
591D4—81no—
592D4—81no—
593D4—81no—
594D4—81no—
595D4—81no—
596D4—81no—
597D4—81no—
598D4—81no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4C2×C22⋊C811yescharacteristic
628C2×C4C2×C22⋊C811yescharacteristic
629C2×C4C2×C22⋊C811yescharacteristic
630C2×C4C2×C22⋊C811yescharacteristic
631C2×C4C2×C22⋊C811yescharacteristic
632C2×C4C2×C22⋊C811yescharacteristic
633C2×C4C2×C22⋊C811yescharacteristic
634C2×C4C2×C22⋊C811yescharacteristic
635C2×C4C2×C22⋊C811yescharacteristic
636C2×C4C22×C22⋊C411yescharacteristic
637C2×C4C23×C811yescharacteristic
638C2×C4C22×M1611yescharacteristic
639C8—21no—
640C8—21no—
641C8—21no—
642C8—21no—
643C8—21no—
644C8—21no—
645C8—21no—
646C8—41no—
647C8—41no—
648C8—41no—
649C8—41no—
650C8#57611yescharacteristic
651C8#57711yescharacteristic
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—84no—
671C22C22×C22⋊C811yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4#289711yescharacteristic
688C4#290111yescharacteristic
689C4#340712yes—
690C2—41no—
691C2—41no—
692C2—41no—
693C2—81no—
694C2—81no—
695C2—81no—
696C2#1743811yescharacteristic
697C2#3333311yescharacteristic
698C2#3334011yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C22×C22⋊C8
prime radical#3460C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #33503C2semidirect
3. #33479C2semidirect
4. #11632C2semidirect
5. #11631C2semidirect
6. #11630C2semidirect
7. #11624C2semidirect
8. #11623C2semidirect
9. #11622C2semidirect
10. #11616C2semidirect
11. #11615C2semidirect
12. #11614C2semidirect
13. #11584C2semidirect
14. #11583C2semidirect
15. #11582C2semidirect
48. #2163C22semidirect
50. #2162C22semidirect
51. #2162C22semidirect
52. #2162C22semidirect
54. #2150C22semidirect
55. C2×C4⋊C16C22semidirect
57. C4×M32C22semidirect
58. C2×C4×C16C22semidirect
59. Q8⋊C16C22semidirect
60. Q8⋊C16C22semidirect
61. Q8⋊C16C22semidirect
62. Q8⋊C16C22semidirect
71. D4⋊C16C22semidirect
72. D4⋊C16C22semidirect
73. D4⋊C16C22semidirect
74. D4⋊C16C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#603062
#335032
#334792
#116322
#116312
#116302
#116242
#116232
#116222
#116162
#116152
#116142
#115842
#115832
#115822