Finite Group Databasebeta

← #481213

Subgroups — #481213

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60218C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic, complemented
3#33420C211yesmaximal, characteristic, complemented
4#11639C211yesmaximal, characteristic, complemented
5#11637C211yesmaximal, characteristic, complemented
6#11632C211yesmaximal, characteristic, complemented
7#11630C211yesmaximal, characteristic, complemented
8#11624C211yesmaximal, characteristic, complemented
9#11623C211yesmaximal, characteristic, complemented
10#11622C211yesmaximal, characteristic, complemented
11#11621C211yesmaximal, characteristic, complemented
12#11616C211yesmaximal, characteristic, complemented
13#11615C211yesmaximal, characteristic, complemented
14#11614C211yesmaximal, characteristic, complemented
15#11613C211yesmaximal, characteristic, complemented
16#3480C411yescharacteristic
17#2971C411yescharacteristic
18#2971C411yescharacteristic
19#2969—21no—
20#2969—21no—
21#2969C411yescharacteristic
22#2969C411yescharacteristic
23#2966—21no—
24#2966—21no—
25#2966C2211yescharacteristic
26#2966C2211yescharacteristic
27C8×#41C2211yescharacteristic
28C8×#41C2211yescharacteristic
29Q8×M16C411yescharacteristic
30C2×C8×Q8C2211yescharacteristic
31#2935—21no—
32#2935C411yescharacteristic
33#2931—21no—
34#2931C2211yescharacteristic
35#2925—21no—
36#2925C411yescharacteristic
37C4×#131—21no—
38C4×#131C2211yescharacteristic
39#2164—21no—
40#2164—21no—
41#2164—21no—
42#2164—21no—
43#2163C2211yescharacteristic
44#2163C2211yescharacteristic
45#2162C2211yescharacteristic
46#2162C2211yescharacteristic
47#2162C2211yescharacteristic
48#2162C2211yescharacteristic
49#2162C2211yescharacteristic
50#2162C2211yescharacteristic
51#2151—21no—
52#2150C2211yescharacteristic
53C2×C4⋊C16C2211yescharacteristic
54#2113—21no—
55#2113—21no—
56#2108—21no—
57C4×M32C2211yescharacteristic
58C2×C4×C16C2211yescharacteristic
59Q8⋊C16C2211yescharacteristic
60Q8⋊C16C2211yescharacteristic
61Q8⋊C16C2211yescharacteristic
62Q8⋊C16C2211yescharacteristic
63Q8⋊C16C2211yescharacteristic
64Q8⋊C16C2211yescharacteristic
65Q8⋊C16C2211yescharacteristic
66Q8⋊C16C2211yescharacteristic
67D4⋊C16C2211yescharacteristic
68D4⋊C16C2211yescharacteristic
69D4⋊C16C2211yescharacteristic
70D4⋊C16C2211yescharacteristic
71D4⋊C16C2211yescharacteristic
72D4⋊C16C2211yescharacteristic
73D4⋊C16C2211yescharacteristic
74D4⋊C16C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77C2×#131D411yescharacteristic
78C2×#131D411yescharacteristic
79#559—21no—
80#559C811yescharacteristic
81#557C811yescharacteristic
82#556—21no—
83#556C811yescharacteristic
84#553—21no—
85#553—21no—
86#553C811yescharacteristic
87#553C811yescharacteristic
88#550C811yescharacteristic
89#548C811yescharacteristic
90#548C811yescharacteristic
91#532—21no—
92#532C2×C411yescharacteristic
93#529—21no—
94#529—21no—
95#529C2×C411yescharacteristic
96#529C2×C411yescharacteristic
97C4×#41—21no—
98C4×#41C2×C411yescharacteristic
99C4×#41C2×C411yescharacteristic
100C4×#41C2×C411yescharacteristic
101C2×C4×Q8C2×C411yescharacteristic
102C2×M32—21no—
103C2×M32—21no—
104C22×C16—21no—
105C22×C16—21no—
106#446—21no—
107#446—21no—
108#446C2×C411yescharacteristic
109#446C2×C411yescharacteristic
110C8×Q8—21no—
111C8×Q8—21no—
112C8×Q8C2×C411yescharacteristic
113C8×Q8C2×C411yescharacteristic
114C8×Q8C2×C411yescharacteristic
115C8×Q8C2×C411yescharacteristic
116C8×Q8C2311yescharacteristic
117C8×Q8C2311yescharacteristic
118#436—21no—
119#436—21no—
120#436—21no—
121#436—21no—
122#436—21no—
123#436—21no—
124#436—21no—
125#436C2×C411yescharacteristic
126#436C2×C411yescharacteristic
127#436C2×C411yescharacteristic
128#436C2×C411yescharacteristic
129#435—21no—
130#435—21no—
131#435—21no—
132#435—21no—
133#435—21no—
134#435—21no—
135C8×D4—21no—
136C8×D4—21no—
137C8×D4—21no—
138C8×D4—21no—
139C8×D4—21no—
140C8×D4—21no—
141C8×D4—21no—
142C8×D4C2×C411yescharacteristic
143C8×D4C2×C411yescharacteristic
144C8×D4C2311yescharacteristic
145C8×D4C2311yescharacteristic
146#433—21no—
147#433—21no—
148#433—21no—
149#433—21no—
150#433—21no—
151#433—21no—
152#432—21no—
153#432—21no—
154#432C2×C411yescharacteristic
155#432C2×C411yescharacteristic
156#431—21no—
157#431—21no—
158#431C2×C411yescharacteristic
159#431C2×C411yescharacteristic
160#423—21no—
161#423C2×C411yescharacteristic
162C2×C4⋊C8—21no—
163C2×C4⋊C8C2×C411yescharacteristic
164#408—21no—
165#408—21no—
166#408—21no—
167#405—21no—
168#405—21no—
169#405—21no—
170#405—21no—
171#405—21no—
172#405D411yescharacteristic
173#405D411yescharacteristic
174#405D411yescharacteristic
175#405D411yescharacteristic
176#405D411yescharacteristic
177#405D411yescharacteristic
178C4×M16—21no—
179C4×M16—21no—
180C4×M16—21no—
181C4×M16C2×C411yescharacteristic
182C4×M16C2×C411yescharacteristic
183C4×M16C2×C411yescharacteristic
184C2×C4×C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8—21no—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2311yescharacteristic
190C4⋊C16—21no—
191C4⋊C16—21no—
192C4⋊C16—21no—
193C4⋊C16—21no—
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16C2311yescharacteristic
197C4⋊C16C2311yescharacteristic
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C22⋊C16—21no—
201C22⋊C16—21no—
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—
211C4×C16—21no—
212C4×C16—21no—
213C4×C16—21no—
214C4×C16C2311yescharacteristic
215C4×C16C2311yescharacteristic
216C4×C16C2311yescharacteristic
217C4×C16C2311yescharacteristic
218C2×#41—21no—
219C2×#41C22⋊C411yescharacteristic
220C2×#41C2×C811yescharacteristic
221C2×#41C2×C811yescharacteristic
222C22×Q8C2×C811yescharacteristic
223#131—41no—
224#131—41no—
225#131—42no—
226#131—42no—
227#131—42no—
228#131—44no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16C22⋊C411yescharacteristic
235C2×M16C22⋊C411yescharacteristic
236C2×M16C22⋊C411yescharacteristic
237C2×M16C22⋊C411yescharacteristic
238C2×M16C22⋊C411yescharacteristic
239C2×M16C22⋊C411yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8C2×C811yescharacteristic
253#127—21no—
254#127C2×C811yescharacteristic
255#126—21no—
256#126—21no—
257#126—21no—
258#126—21no—
259#126—21no—
260#126—21no—
261#125—21no—
262#125—21no—
263#125C2×C811yescharacteristic
264#125C2×C811yescharacteristic
265#124—21no—
266#124—21no—
267#124—21no—
268#124—21no—
269#124—21no—
270#124C2×C811yescharacteristic
271#124C2×C811yescharacteristic
272#123—21no—
273#123—21no—
274#123—21no—
275#123—21no—
276#123—21no—
277#123—21no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8C2×C811yescharacteristic
281C22⋊Q8C2×C811yescharacteristic
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289#121C2×C811yescharacteristic
290#121C2×C811yescharacteristic
291#121C2×C811yescharacteristic
292#121C2×C811yescharacteristic
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8C2×C811yescharacteristic
298C4×Q8C2×C811yescharacteristic
299C4×Q8C2×C811yescharacteristic
300C4×Q8C2×C811yescharacteristic
301C4×Q8C2×C811yescharacteristic
302C4×Q8M1611yescharacteristic
303C4×Q8C22×C411yescharacteristic
304C4×Q8C22×C411yescharacteristic
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C2×C811yescharacteristic
319C4×D4C2×C811yescharacteristic
320C4×D4C2×C811yescharacteristic
321C4×D4C2×C811yescharacteristic
322C4×D4C2×C811yescharacteristic
323C4×D4M1611yescharacteristic
324C4×D4M1611yescharacteristic
325C4×D4M1611yescharacteristic
326C4×D4C22×C411yescharacteristic
327C4×D4C22×C411yescharacteristic
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117C22⋊C411yescharacteristic
333#117C22⋊C411yescharacteristic
334#117C22⋊C411yescharacteristic
335#117C2×C811yescharacteristic
336#117C2×C811yescharacteristic
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2×C811yescharacteristic
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C22×C411yescharacteristic
343C2×C42C22×C411yescharacteristic
344C2×C42C22×C411yescharacteristic
345M32—42no—
346M32—42no—
347C2×C16—21no—
348C2×C16—21no—
349C2×C16—21no—
350C2×C16—21no—
351C2×C16—21no—
352C2×C16—21no—
353C2×C16—21no—
354C2×C16—21no—
355C2×C16—42no—
356C2×C16—42no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8C22×C411yescharacteristic
368C4⋊C8C22×C411yescharacteristic
369C4⋊C8C22×C411yescharacteristic
370C4⋊C8C22×C411yescharacteristic
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380C22⋊C8—21no—
381C22⋊C8—21no—
382C22⋊C8—21no—
383#97—21no—
384#97—21no—
385#97—21no—
386#97—21no—
387#97C22⋊C411yescharacteristic
388#97C22⋊C411yescharacteristic
389#97C22⋊C411yescharacteristic
390#97C22⋊C411yescharacteristic
391#97C22⋊C411yescharacteristic
392#97C22⋊C411yescharacteristic
393C4×C8—21no—
394C4×C8—21no—
395C4×C8—21no—
396C4×C8—21no—
397C4×C8—21no—
398C4×C8—21no—
399C4×C8—21no—
400C4×C8—21no—
401C4×C8C22×C411yescharacteristic
402C4×C8C22×C411yescharacteristic
403C4×C8C22×C411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2×D411yescharacteristic
410C4×C8C2411yescharacteristic
411#41—22no—
412#41—22no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417#41—42no—
418#41—42no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8C22⋊C811yescharacteristic
424C2×Q8C22×C811yescharacteristic
425C2×Q8C22×C811yescharacteristic
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×D4C22⋊C811yescharacteristic
435C2×D4C22⋊C811yescharacteristic
436C2×D4C22⋊C811yescharacteristic
437C2×D4C22×C811yescharacteristic
438C2×D4C22×C811yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C22×C811yescharacteristic
451C22×C4C22×C811yescharacteristic
452C22×C4C22×C811yescharacteristic
453C22×C4C2×M1611yescharacteristic
454C22×C4C2×M1611yescharacteristic
455C22×C4C2×M1611yescharacteristic
456M16—22no—
457M16—22no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—42no—
463M16—42no—
464M16—42no—
465M16—42no—
466M16—42no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—42no—
487C2×C8C2×C22⋊C411yescharacteristic
488C2×C8C2×C22⋊C411yescharacteristic
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C2×C22⋊C411yescharacteristic
493C2×C8C22×D411yescharacteristic
494C2×C8C22×D411yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
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⋊C4C22⋊C811yescharacteristic
510C4⋊C4C22⋊C811yescharacteristic
511C4⋊C4C22⋊C811yescharacteristic
512C4⋊C4C22⋊C811yescharacteristic
513C4⋊C4C22⋊C811yescharacteristic
514C4⋊C4C22⋊C811yescharacteristic
515C4⋊C4C22×C811yescharacteristic
516C4⋊C4C22×C811yescharacteristic
517C4⋊C4C22×C811yescharacteristic
518C4⋊C4C22×C811yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
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⋊C4C22⋊C811yescharacteristic
536C22⋊C4C22⋊C811yescharacteristic
537C22⋊C4C22⋊C811yescharacteristic
538C22⋊C4C22⋊C811yescharacteristic
539C22⋊C4C22⋊C811yescharacteristic
540C22⋊C4C22⋊C811yescharacteristic
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42C2×C22⋊C411yescharacteristic
550C42C2×C22⋊C411yescharacteristic
551C42C2×C22⋊C411yescharacteristic
552C42C22×C811yescharacteristic
553C42C22×C811yescharacteristic
554C42C22×C811yescharacteristic
555C42C2×M1611yescharacteristic
556C42C2×M1611yescharacteristic
557C42C2×M1611yescharacteristic
558C42C23×C411yescharacteristic
559C16—41no—
560C16—41no—
561C16—41no—
562C16—41no—
563C16—41no—
564C16—41no—
565C16—41no—
566C16—41no—
567C23—21no—
568C23—21no—
569C23C2×C22⋊C811yescharacteristic
570C23C2×C22⋊C811yescharacteristic
571C23C2×C22⋊C811yescharacteristic
572Q8—21no—
573Q8—21no—
574Q8—21no—
575Q8—41no—
576Q8—42no—
577Q8—42no—
578Q8#50412yes—
579D4—21no—
580D4—21no—
581D4—21no—
582D4—21no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—81no—
592D4—81no—
593D4—81no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
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—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4C2×C22⋊C811yescharacteristic
626C2×C4C2×C22⋊C811yescharacteristic
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×C4C22×C22⋊C411yescharacteristic
635C2×C4C23×C811yescharacteristic
636C2×C4C22×M1611yescharacteristic
637C8—21no—
638C8—21no—
639C8—21no—
640C8—21no—
641C8—21no—
642C8—21no—
643C8—41no—
644C8—41no—
645C8—41no—
646C8—41no—
647C8#42112yes—
648C8#57511yescharacteristic
649C8#57511yescharacteristic
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22C22×C22⋊C811yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—22no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4#289611yescharacteristic
683C4C2×#42111yescharacteristic
684C4C2×#50411yescharacteristic
685C4#340711yescharacteristic
686C2—41no—
687C2—41no—
688C2—41no—
689C2—81no—
690C2—81no—
691C2#1743711yescharacteristic
692C2#3333411yescharacteristic
693C2#3334011yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C22×C22⋊C8
prime radical#3480C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33423C2semidirect
3. #33420C2semidirect
4. #11639C2semidirect
5. #11637C2semidirect
6. #11632C2semidirect
7. #11630C2semidirect
8. #11624C2semidirect
9. #11623C2semidirect
10. #11622C2semidirect
11. #11621C2semidirect
12. #11616C2semidirect
13. #11615C2semidirect
14. #11614C2semidirect
15. #11613C2semidirect

Maximal subgroups

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

subgroupindex
#602182
#334232
#334202
#116392
#116372
#116322
#116302
#116242
#116232
#116222
#116212
#116162
#116152
#116142
#116132