Finite Group Databasebeta

← #501378

Subgroups — #501378

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic, complemented
2C22×C22⋊C16C211yesmaximal, characteristic
3#32931C211yesmaximal, characteristic
4#20658C211yesmaximal, characteristic
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic, complemented
10#13060C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12868C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15#12834C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24#2848—21no—
25#2280—21no—
26#2237—24no—
27C2×#367C2211yescharacteristic, complemented
28C2×#367C2211yescharacteristic, complemented
29C2×#366C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31C2×#358C2211yescharacteristic, complemented
32C2×#358C2211yescharacteristic, complemented
33C2×#358C2211yescharacteristic, complemented
34C2×#358C2211yescharacteristic, complemented
35C2×C22⋊C16—24no—
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#1892—21no—
41#1892—21no—
42#1892—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1807—21no—
51#1807—21no—
52#1735—21no—
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61C24×C4D411yescharacteristic
62C22×Q16C2311yescharacteristic
63C23×C8—22no—
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—22no—
67C23×C8—41no—
68C23×C8—41no—
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C16—22no—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16—22no—
84C22×C16—41no—
85C22×C16—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#467—22no—
91#467—22no—
92#467—22no—
93#467—22no—
94#467—41no—
95#467—41no—
96#427—22no—
97#427—22no—
98#427—41no—
99#427—41no—
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#106—21no—
108C2×#106—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112#385—41no—
113#385—41no—
114#385—41no—
115#381—41no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#379—41no—
124#379—41no—
125C4×C22⋊C4—41no—
126C4×C22⋊C4—41no—
127#367—22no—
128#367—41no—
129#366—22no—
130#366—41no—
131#358—22no—
132#358—22no—
133#358—41no—
134#358—41no—
135C22⋊C16—22no—
136C22⋊C16—22no—
137C22⋊C16—22no—
138C22⋊C16D414yes—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25D811yescharacteristic
147C22×Q8—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4D811yescharacteristic
155C23×C4D811yescharacteristic
156C23×C4D811yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8—41no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D412yes—
196C22×C8C2×D411yescharacteristic
197C22×C8#4112yes—
198C22×C8#4112yes—
199C22×C8#4112yes—
200C22×C8C2411yescharacteristic
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
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⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
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×C42—41no—
234C2×C42—41no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—44no—
246C2×C16—44no—
247C2×C16C2×D412yes—
248C2×C16C2×D412yes—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#106—42no—
264#106—42no—
265#103—42no—
266#103—42no—
267#103—42no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24C2×D811yescharacteristic
283C24C2×D811yescharacteristic
284C24C2×D811yescharacteristic
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4#12112yes—
327C22×C4#12312yes—
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4#13512yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334Q16—41no—
335Q16—41no—
336Q16—41no—
337Q16—41no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8#12112yes—
361C2×C8#12312yes—
362C2×C8C22×D411yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8C2×#4111yescharacteristic
365C2×C8C2×#4111yescharacteristic
366C2×C8H(5,𝔽2)11yescharacteristic
367C2×C8H(5,𝔽2)11yescharacteristic
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C42—82no—
398C42—82no—
399C16—22no—
400C16—22no—
401C16—41no—
402C16—41no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—28no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23#46612yes—
427C23#48012yes—
428C23#50812yes—
429C23#50812yes—
430C23#51012yes—
431C23C22×D811yescharacteristic
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4#46612yes—
483C2×C4#48012yes—
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4#53411yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C2×C4#57511yescharacteristic
490C8—22no—
491C8—22no—
492C8—41no—
493C8#54611yescharacteristic
494C8#54611yescharacteristic
495C8#55111yescharacteristic
496C8#55111yescharacteristic
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—416no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22#223714yescomplemented
519C22C2×#46611yescharacteristic
520C22C2×#48011yescharacteristic
521C22#318611yescharacteristic
522C22C2×#50811yescharacteristic
523C22C2×#50811yescharacteristic
524C22C2×#51011yescharacteristic
525C22#341411yescharacteristic
526C4—22no—
527C4—22no—
528C4—41no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4—161no—
536C4—161no—
537C4#241211yescharacteristic
538C4#305611yescharacteristic
539C4#309111yescharacteristic
540C4#319011yescharacteristic
541C2—24no—
542C2—42no—
543C2—42no—
544C2#2455011yescharacteristic
545C2#3360011yescharacteristic
546C2#3360611yescharacteristic
547C2C2×#223711yescharacteristic
548C2#3364511yescharacteristic
549C2#3375611yescharacteristic
550C2#3375711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2237C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13383C2semidirect
8. #13382C2semidirect
9. #13060C2semidirect
10. #13060C2semidirect
11. #12994C2semidirect
12. #12921C2semidirect
13. #12868C2semidirect
14. #12835C2semidirect
15. #12834C2semidirect
27. C2×#367C22semidirect
28. C2×#367C22semidirect
29. C2×#366C22semidirect
30. C2×#366C22semidirect
31. C2×#358C22semidirect
32. C2×#358C22semidirect
33. C2×#358C22semidirect
34. C2×#358C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1380C22semidirect
518. C22#2237semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C22×C22⋊C162
#329312
#206582
#134942
#134922
#133832
#133822
#130602
#129942
#129212
#128682
#128352
#128342