Finite Group Databasebeta

← #376837

Subgroups — #376837

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2962C211yesmaximal, characteristic
2#23551C211yesmaximal, characteristic
3C2×#1945C211yesmaximal, characteristic
4#21147C211yesmaximal, characteristic
5C2×#1922C211yesmaximal, characteristic
6#11386C211yesmaximal, characteristic
7#11329C211yesmaximal, characteristic
8#11221C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic
10#10909C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic
12#10887C211yesmaximal, characteristic
13#10886C211yesmaximal, characteristic
14#10876C211yesmaximal, characteristic
15#10706C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17#2962—24no—
18#2962—24no—
19C2×#446—21no—
20C2×#446—21no—
21C2×#446—21no—
22C2×C8×Q8—21no—
23C22×#108C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×C4×M16C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2296C2211yescharacteristic
37#2059C411yescharacteristic, complemented
38#2059C411yescharacteristic, complemented
39#2059C411yescharacteristic, complemented
40#2059C411yescharacteristic, complemented
41#2058C411yescharacteristic, complemented
42#2058C411yescharacteristic, complemented
43#2057C411yescharacteristic, complemented
44#2057C411yescharacteristic, complemented
45#1945—24no—
46#1940—21no—
47#1940—21no—
48#1939—21no—
49#1939—21no—
50#1922—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1735C2211yescharacteristic
55#1735C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C2×#339C2211yescharacteristic
64#1729C2211yescharacteristic
65#1728C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C2×#336C2211yescharacteristic
70C4×#95C2211yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79#533—22no—
80#533—22no—
81#533—22no—
82#533—22no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×C4⋊Q8C2×C411yescharacteristic
85C2×#125C2×C411yescharacteristic
86C2×#125C2×C411yescharacteristic
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#446—28no—
99#446—28no—
100#446—28no—
101C8×Q8—28no—
102C2×#108—22no—
103C2×#108—22no—
104C2×#108—41no—
105#427—22no—
106#427—22no—
107#427—22no—
108#427—22no—
109C2×#107C2×C411yescharacteristic
110C2×#107C2×C411yescharacteristic
111C2×#107C2×C411yescharacteristic
112C2×#107C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117#424—22no—
118#424—22no—
119#424—22no—
120#424—22no—
121#424—22no—
122#424—22no—
123#424—41no—
124#424D414yes—
125C2×C4⋊C8—22no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8D411yescharacteristic
136C2×C4⋊C8D411yescharacteristic
137C2×C4⋊C8D411yescharacteristic
138C2×C4⋊C8D411yescharacteristic
139#405—22no—
140#405—22no—
141#405—22no—
142#405—22no—
143#405—41no—
144#405D414yes—
145C4×M16—22no—
146C4×M16—22no—
147C2×#97—21no—
148C2×#97—21no—
149C2×#97—21no—
150C2×#97—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154C2×C4×C8—22no—
155C2×C4×C8—21no—
156C2×C4×C8—22no—
157C2×C4×C8—41no—
158#389C2×C411yescharacteristic
159#389C2×C411yescharacteristic
160#389C2×C411yescharacteristic
161#389C2×C411yescharacteristic
162#385—21no—
163#385—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C2×#95—21no—
167C2×#95—21no—
168#341—22no—
169#341—41no—
170#341Q814yescomplemented
171#340C2×C411yescharacteristic
172#340C2×C411yescharacteristic
173#340C2×C411yescharacteristic
174#340C2×C411yescharacteristic
175#340C2×C411yescharacteristic
176#340C2×C411yescharacteristic
177#340C2×C411yescharacteristic
178#340C2×C411yescharacteristic
179#339—22no—
180#339—22no—
181#339—22no—
182#339—22no—
183#339—22no—
184#339—22no—
185#339—22no—
186#339Q814yescomplemented
187#336—22no—
188#336—21no—
189#336—21no—
190#336—22no—
191#336—21no—
192#336—21no—
193#336—41no—
194#336C2×C411yescharacteristic
195#336C2×C411yescharacteristic
196#336C2×C411yescharacteristic
197#336C2×C411yescharacteristic
198C22×Q8—21no—
199C22×Q8—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4#4111yescharacteristic
203C23×C4#4111yescharacteristic
204C23×C4C2411yescharacteristic
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16C2×D412yes—
225C2×M16C2×D412yes—
226C2×M16C2×Q812yes—
227C2×M16C2×Q812yes—
228C2×M16C2×Q812yes—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8C22×C411yescharacteristic
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D412yes—
260C22×C8C2×D412yes—
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×Q812yes—
264C22×C8#4112yes—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269#125—22no—
270#125—22no—
271#125—22no—
272#125—22no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117C2×D412yes—
292#117C2×D412yes—
293#117C2×Q812yes—
294#117#4112yes—
295#117#4112yes—
296#117#4112yes—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C4⋊C4C22×C411yescharacteristic
311C2×C4⋊C4C22×C411yescharacteristic
312C2×C4⋊C4C22×C411yescharacteristic
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C22×C411yescharacteristic
339C2×C42C22×C411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×Q812yes—
343#108—44no—
344#107—22no—
345#107—22no—
346#107—22no—
347#107—22no—
348#106—22no—
349#106—22no—
350#106—22no—
351#106—22no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—42no—
361C4⋊C8—42no—
362C4⋊C8—42no—
363C4⋊C8—42no—
364#97—24no—
365#97—24no—
366#97—24no—
367#97—42no—
368C4×C8—24no—
369C4×C8—24no—
370C4×C8—24no—
371C4×C8—42no—
372C4×C8—44no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—42no—
379C24C2×#4111yescharacteristic
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16C4⋊Q814yes—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8C4×Q812yes—
499C2×C8C4×Q812yes—
500C2×C8#12012yes—
501C2×C8#12012yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4C4×D412yes—
523C4⋊C4C4×D412yes—
524C4⋊C4C4×D412yes—
525C4⋊C4C4×D412yes—
526C4⋊C4C4×Q812yes—
527C4⋊C4C4×Q812yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23C2×C4×D411yescharacteristic
565C23#52511yescharacteristic
566C23#52511yescharacteristic
567C23#55211yescharacteristic
568C23#55411yescharacteristic
569C23#55611yescharacteristic
570C23#55711yescharacteristic
571Q8—48no—
572Q8—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4#44312yes—
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×Q811yescharacteristic
630C2×C4C2×C4×Q811yescharacteristic
631C2×C4#51911yescharacteristic
632C2×C4#52011yescharacteristic
633C2×C4C2×#12011yescharacteristic
634C2×C4#52511yescharacteristic
635C2×C4C2×C4⋊Q811yescharacteristic
636C2×C4#53211yescharacteristic
637C2×C4#53311yescharacteristic
638C2×C4#53311yescharacteristic
639C2×C4#53611yescharacteristic
640C2×C4#53711yescharacteristic
641C2×C4#54111yescharacteristic
642C2×C4#54211yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54711yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4D4×Q811yescharacteristic
648C2×C4D4×Q811yescharacteristic
649C2×C4D4×Q811yescharacteristic
650C2×C4D4×Q811yescharacteristic
651C2×C4#55211yescharacteristic
652C2×C4#55411yescharacteristic
653C2×C4#55611yescharacteristic
654C2×C4#55711yescharacteristic
655C8—22no—
656C8—22no—
657C8—22no—
658C8—22no—
659C8—24no—
660C8—41no—
661C8—41no—
662C8—41no—
663C8—41no—
664C8—42no—
665C8—42no—
666C8—81no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22C4×#12011yescharacteristic
680C22#236011yescharacteristic
681C22#236511yescharacteristic
682C22#246211yescharacteristic
683C22#246511yescharacteristic
684C22#258411yescharacteristic
685C22#260411yescharacteristic
686C22C2×#44311yescharacteristic
687C22#295511yescharacteristic
688C22#295611yescharacteristic
689C22#295711yescharacteristic
690C4—22no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—42no—
697C4—41no—
698C4—41no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4C4×C4⋊Q811yescharacteristic
705C4#232311yescharacteristic
706C4#236011yescharacteristic
707C4#236511yescharacteristic
708C4#243011yescharacteristic
709C4#243411yescharacteristic
710C4#246211yescharacteristic
711C4#246511yescharacteristic
712C2—21no—
713C2—21no—
714C2—21no—
715C2—21no—
716C2#1418611yescharacteristic
717C2#2146611yescharacteristic
718C2#2146811yescharacteristic
719C2#2298211yescharacteristic
720C2#2298411yescharacteristic
721C2#2317011yescharacteristic
722C2#2317211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #2059C4semidirect
38. #2059C4semidirect
39. #2059C4semidirect
40. #2059C4semidirect
41. #2058C4semidirect
42. #2058C4semidirect
43. #2057C4semidirect
44. #2057C4semidirect
170. #341Q8semidirect
186. #339Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#235512
C2×#19452
#211472
C2×#19222
#113862
#113292
#112212
#109112
#109092
#108872
#108862
#108762
#107062