Finite Group Databasebeta

← #229299

Subgroups — #229299

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23190C211yesmaximal, characteristic, complemented
2#23093C211yesmaximal, characteristic
3#23060C211yesmaximal, characteristic, complemented
4#22658C211yesmaximal, characteristic, complemented
5C2×#1665C211yesmaximal, characteristic
6#9842C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9564C211yesmaximal, characteristic, complemented
10#8984C211yesmaximal, characteristic, complemented
11#8980C211yesmaximal, characteristic
12#8923C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic, complemented
14#8284C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#440—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#427—21no—
22#2366C2211yescharacteristic
23#2365C2211yescharacteristic
24#2354C2211yescharacteristic
25#2354C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1982C2211yescharacteristic
29#1981C2211yescharacteristic
30#1973—21no—
31#1970—21no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1934—21no—
39#1925—21no—
40#1925—21no—
41#1918—21no—
42#1880—21no—
43#1879—21no—
44#1877—21no—
45#1854—21no—
46#1854—21no—
47#1845C2211yescharacteristic
48#1808—21no—
49C4×#107—21no—
50C4×#107C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56#1665C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×D8—21no—
71C23×C8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#117—21no—
80#492—22no—
81#492C2×C414yes—
82#474—22no—
83#474C2×C414yescomplemented
84#459—22no—
85#459C2×C414yes—
86#440—44no—
87C8×D4—22no—
88C8×D4C2×C414yescomplemented
89#427—44no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107D411yescharacteristic
96C2×#107D411yescharacteristic
97C2×#106—21no—
98C2×#106—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—41no—
112C2×#102D411yescharacteristic
113C2×#102D411yescharacteristic
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#388—41no—
128#388—41no—
129#387—41no—
130#386—41no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#334C2×C414yescomplemented
156#332C2×C414yescomplemented
157#331C2×C414yes—
158D4⋊C8—22no—
159#322—22no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—41no—
167C2×SD16—24no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×SD16—81no—
172C2×SD16—81no—
173C2×D8—24no—
174C2×D8—81no—
175C2×D8—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—24no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192#127—21no—
193#127—21no—
194#127—21no—
195#127C22×C412yes—
196C4×Q8—42no—
197C4×Q8—81no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—81no—
203C4×D4—81no—
204C4×D4C22×C412yes—
205C4×D4#4112yes—
206#117—42no—
207#117—42no—
208#117—81no—
209#117—81no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C2411yescharacteristic
241#107—22no—
242#107—24no—
243#107—42no—
244#107—42no—
245#107—81no—
246#107—81no—
247#107D814yescomplemented
248#106—42no—
249#106—42no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8C22×C412yes—
256C4⋊C8C22×C412yes—
257C4⋊C8C22×C412yes—
258#103—42no—
259#103—81no—
260#102—24no—
261#102—24no—
262#102—41no—
263#102—41no—
264#102—42no—
265#102—42no—
266#102—81no—
267#102—81no—
268#102—82no—
269C22⋊C8—24no—
270#97—42no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—42no—
276C4×C8—81no—
277C4×C8C22×C412yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—41no—
283#95—41no—
284#95—41no—
285#95—41no—
286C24—21no—
287C24—41no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4C4×D412yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—161no—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338SD16—48no—
339SD16—84no—
340D8—84no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—48no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8C4×D412yes—
366C2×C8C4×D412yes—
367C2×C8C2×D812yes—
368C2×C8#13512yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—82no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4C4×D412yes—
396C4⋊C4C2×D812yes—
397C4⋊C4C2×D812yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—161no—
405C22⋊C4—161no—
406C22⋊C4—161no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—82no—
414C42—82no—
415C42—82no—
416C42C23×C411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C422−1+411yescharacteristic
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23D4211yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—81no—
438Q8—81no—
439D4—22no—
440D4—22no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—82no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—81no—
452D4—161no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—164no—
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4#52011yescharacteristic
493C2×C4#54711yescharacteristic
494C2×C4#54811yescharacteristic
495C2×C4#55311yescharacteristic
496C2×C4C22×D811yescharacteristic
497C2×C4C2×#13511yescharacteristic
498C2×C4#57511yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57711yescharacteristic
504C8—22no—
505C8—41no—
506C8—41no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—81no—
511C8—81no—
512C8C4×D814yes—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—161no—
524C22—161no—
525C22—161no—
526C22—161no—
527C22#236111yescharacteristic
528C22#326711yescharacteristic
529C22#327111yescharacteristic
530C22D4×D811yescharacteristic
531C22#328411yescharacteristic
532C22#328811yescharacteristic
533C22#328911yescharacteristic
534C4—21no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4—82no—
544C4—161no—
545C4—161no—
546C4C2×C4×D811yescharacteristic
547C4#294311yescharacteristic
548C4#294611yescharacteristic
549C4#294911yescharacteristic
550C4#334811yescharacteristic
551C4#335011yescharacteristic
552C4#335811yescharacteristic
553C4#336111yescharacteristic
554C2—82no—
555C2—162no—
556C2#1975811yescharacteristic
557C2#1976211yescharacteristic
558C2#1985411yescharacteristic
559C2#1985811yescharacteristic
560C2#2299511yescharacteristic
561C2#2303311yescharacteristic
562C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23190C2semidirect
3. #23060C2semidirect
4. #22658C2semidirect
6. #9842C2semidirect
7. #9620C2semidirect
8. #9565C2semidirect
9. #9564C2semidirect
10. #8984C2semidirect
12. #8923C2semidirect
13. #8921C2semidirect
14. #8284C2semidirect
15. #8202C2semidirect
56. #1665C4semidirect
83. #474C2×C4semidirect
88. C8×D4C2×C4semidirect
155. #334C2×C4semidirect
156. #332C2×C4semidirect
247. #107D8semidirect

Maximal subgroups

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

subgroupindex
#231902
#230932
#230602
#226582
C2×#16652
#98422
#96202
#95652
#95642
#89842
#89802
#89232
#89212
#82842
#82022