Finite Group Databasebeta

← #373192

Subgroups — #373192

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2#21098C211yesmaximal, characteristic
3C2×#1937C211yesmaximal, characteristic
4C2×#1861C211yesmaximal, characteristic
5#20852C211yesmaximal, characteristic
6C2×#1772C211yesmaximal, characteristic, complemented
7#20422C211yesmaximal, characteristic
8#11341C211yesmaximal, characteristic, complemented
9#11241C211yesmaximal, characteristic, complemented
10#11104C211yesmaximal, characteristic
11#11080C211yesmaximal, characteristic
12#10898C211yesmaximal, characteristic, complemented
13#10898C211yesmaximal, characteristic, complemented
14#10894C211yesmaximal, characteristic, complemented
15#10844C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2905C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30C2×#408C2211yescharacteristic
31#2873C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#1950—24no—
36#1938—22no—
37#1937—22no—
38#1937C414yescomplemented
39#1937C414yescomplemented
40#1861—24no—
41#1855—24no—
42#1772—24no—
43#1765—22no—
44#1764—22no—
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49#1738C2211yescharacteristic
50#1738C2211yescharacteristic
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#341C2211yescharacteristic
56#1733C2211yescharacteristic
57#1733C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C22×#41C2311yescharacteristic
62#568—22no—
63#568—22no—
64#568—22no—
65#568—41no—
66C2×#131—22no—
67C2×#131—41no—
68C22×M16—22no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#430—22no—
84#430—22no—
85#430—22no—
86#430—41no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108D414yes—
92#427—22no—
93#427—22no—
94C2×#107—21no—
95C2×#106—21no—
96#424—21no—
97#424—21no—
98#424—22no—
99#424—41no—
100#424C2×C414yes—
101#423—22no—
102#423—22no—
103#423—22no—
104#423—22no—
105#423—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—41no—
108C2×C4⋊C8—41no—
109#421—22no—
110#421—22no—
111#421—22no—
112#421—41no—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104C2×C412yes—
118C2×#104C2×C412yes—
119#416C2×C414yes—
120#416D412yes—
121#416D412yes—
122C2×#103Q812yes—
123C2×#102Q812yes—
124#408—21no—
125#408—21no—
126#408—22no—
127#408—41no—
128#408C2×C414yes—
129#407—22no—
130C2×C22⋊C8—22no—
131#405—22no—
132#405—22no—
133#405—22no—
134#405—41no—
135#405—41no—
136#405—41no—
137C4×M16—22no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—22no—
141C2×C4×C8—22no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2×C412yes—
145C2×C4×C8C2×C412yes—
146#344—22no—
147#344—22no—
148#344—22no—
149#344—41no—
150#341—22no—
151#341—22no—
152#341—22no—
153#341—21no—
154#341—21no—
155#341—22no—
156#341—21no—
157#341—21no—
158#341C2×C414yes—
159#339—22no—
160#339—22no—
161#339—22no—
162#339C2×C412yes—
163#339C2×C412yes—
164#339D414yes—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41C22×C412yes—
176C2×#41C2×D412yes—
177C22×Q8C2×Q811yescharacteristic
178C22×D4—21no—
179C22×D4C2×Q811yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C2411yescharacteristic
183#131—28no—
184#131—44no—
185#131—44no—
186#131—44no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
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×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—42no—
228C2×M16—44no—
229C2×M16—44no—
230C2×M16C22×C412yes—
231C2×M16C22×C412yes—
232C2×M16C2×D412yes—
233C2×M16C2×D412yes—
234C2×M16C2×D412yes—
235C2×M16#4112yes—
236C2×M16#4112yes—
237C2×M16#4112yes—
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×C8—21no—
254C22×C8—42no—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D412yes—
260C22×C8C2×Q812yes—
261C22×C8#4111yescharacteristic
262C22×C8#4111yescharacteristic
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117C22×C412yes—
272#117C2×D412yes—
273#117C2×D412yes—
274#117#4112yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C2×Q812yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291#108—28no—
292#108—44no—
293#108—44no—
294#107—24no—
295#106—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—44no—
299C4⋊C8—44no—
300C4⋊C8—44no—
301#104—28no—
302#104—24no—
303#104—24no—
304#104—44no—
305#103C4⋊C414yes—
306#103C4⋊C414yes—
307#102C4⋊C414yes—
308#102C4⋊C414yes—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—44no—
312#97—42no—
313#97—42no—
314#97—42no—
315C4×C8—24no—
316C4×C8—24no—
317C4×C8—42no—
318C4×C8—42no—
319C4×C8—42no—
320C24—21no—
321C242−1+411yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C2×C4⋊C412yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4C2×C4⋊C412yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+411yescharacteristic
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16—48no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16#12114yes—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—48no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8C2×C4⋊C412yes—
460C2×C8C2×C4⋊C412yes—
461C2×C8C4×D412yes—
462C2×C8C4×D412yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×C4⋊C414yes—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42C4×D412yes—
486C42C4×D412yes—
487C23—22no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—82no—
492C23#52011yescharacteristic
493C23#53911yescharacteristic
494C23#54111yescharacteristic
495C23#54711yescharacteristic
496C23#54711yescharacteristic
497C23D4×Q811yescharacteristic
498C23#55211yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—41no—
502Q8—41no—
503D4—22no—
504D4—22no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—41no—
510D4—41no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4C22×C4⋊C411yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4C4×#4111yescharacteristic
535C2×C4#52011yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4#52511yescharacteristic
538C2×C4#52511yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52911yescharacteristic
541C2×C4#53211yescharacteristic
542C2×C4#53311yescharacteristic
543C2×C4#53311yescharacteristic
544C2×C4#53811yescharacteristic
545C2×C4#54211yescharacteristic
546C2×C4D4211yescharacteristic
547C2×C4D4211yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#54711yescharacteristic
551C2×C4#54811yescharacteristic
552C2×C4D4×Q811yescharacteristic
553C2×C4#55111yescharacteristic
554C2×C4#55411yescharacteristic
555C2×C4#55411yescharacteristic
556C2×C4#55611yescharacteristic
557C2×C4#55911yescharacteristic
558C8—22no—
559C8—22no—
560C8—24no—
561C8—22no—
562C8—22no—
563C8—42no—
564C8—42no—
565C8—42no—
566C8—42no—
567C8—42no—
568C8—81no—
569C8—81no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—44no—
579C22—161no—
580C22—161no—
581C22#233411yescharacteristic
582C22D4×C4⋊C411yescharacteristic
583C22#235911yescharacteristic
584C22#242911yescharacteristic
585C22#245511yescharacteristic
586C22#247811yescharacteristic
587C22#255611yescharacteristic
588C4—24no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4#229611yescharacteristic
595C4C4×#12111yescharacteristic
596C4D4×C4⋊C411yescharacteristic
597C4#236711yescharacteristic
598C4#242411yescharacteristic
599C4#246111yescharacteristic
600C4#246511yescharacteristic
601C4#258911yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—82no—
609C2#1411211yescharacteristic
610C2#2269412yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21178C2semidirect
6. C2×#1772C2semidirect
8. #11341C2semidirect
9. #11241C2semidirect
12. #10898C2semidirect
13. #10898C2semidirect
14. #10894C2semidirect
15. #10844C2semidirect
38. #1937C4semidirect
39. #1937C4semidirect

Maximal subgroups

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

subgroupindex
#211782
#210982
C2×#19372
C2×#18612
#208522
C2×#17722
#204222
#113412
#112412
#111042
#110802
#108982
#108942
#108442