Finite Group Databasebeta

← #390950

Subgroups — #390950

Order 512.

755 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23622C211yesmaximal, characteristic
2#11368C211yesmaximal, characteristic, complemented
3#11363C211yesmaximal, characteristic, complemented
4#11363C211yesmaximal, characteristic, complemented
5#11254C211yesmaximal, characteristic, complemented
6#11137C211yesmaximal, characteristic
7#10961C211yesmaximal, characteristic
8#10941C211yesmaximal, characteristic
9#10906C211yesmaximal, characteristic, complemented
10#10902C211yesmaximal, characteristic, complemented
11#10898C211yesmaximal, characteristic, complemented
12#10815C211yesmaximal, characteristic, complemented
13#10771C211yesmaximal, characteristic
14#10753C211yesmaximal, characteristic
15#10744C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C2×#430C2211yescharacteristic
21C2×#413C2211yescharacteristic
22C2×#413C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25#2879C2211yescharacteristic
26C2×#408C2211yescharacteristic
27#2873C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2070C414yescomplemented
30#2069C414yescomplemented
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37#1742C2211yescharacteristic
38#1742C2211yescharacteristic
39#1742C2211yescharacteristic
40#1738C2211yescharacteristic
41#1738C2211yescharacteristic
42#1738C2211yescharacteristic
43#1738C2211yescharacteristic
44#1738C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#338C2211yescharacteristic
50C2×#338C2211yescharacteristic
51C2×#338C2211yescharacteristic
52#1729C2211yescharacteristic
53C22×#41C2311yescharacteristic
54#568—22no—
55#568—22no—
56#568—41no—
57#568—41no—
58C2×#131—41no—
59C2×#131—41no—
60C22×M16—22no—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75#430—41no—
76#430—41no—
77#430—41no—
78#430—41no—
79#430—41no—
80#430C2×C414yes—
81#430C2×C414yes—
82#430C2×C414yes—
83C2×#108—22no—
84C2×#108—41no—
85C2×#108—41no—
86C2×#108—41no—
87#413—41no—
88#413—41no—
89#413—41no—
90#413—41no—
91#413—41no—
92#413—41no—
93C2×#101C2×C412yes—
94C2×#101C2×C412yes—
95C2×#101C2×C412yes—
96C2×#100C2×C412yes—
97C2×#100C2×C412yes—
98C2×#100C2×C412yes—
99#410—41no—
100#410—41no—
101C2×#99Q814yes—
102#408—22no—
103#408—21no—
104#408—21no—
105#408—41no—
106#408—41no—
107#407—22no—
108C2×C22⋊C8—22no—
109#405—41no—
110#405—41no—
111#405—41no—
112#405—41no—
113C4×M16—41no—
114C2×#97—21no—
115C2×C4×C8—21no—
116#344—22no—
117#344—22no—
118#344—41no—
119#344—41no—
120#344—41no—
121#344—41no—
122#343—22no—
123#343—41no—
124#343—41no—
125#341—22no—
126#341—21no—
127#341—21no—
128#341—41no—
129#341—41no—
130#341—41no—
131#341—41no—
132#341—41no—
133#341—41no—
134#341—41no—
135#338—22no—
136#338—41no—
137#338C2×C414yes—
138#336—41no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C22×Q8C22×C411yescharacteristic
152C22×D4C22×C411yescharacteristic
153C22×D4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156#131—44no—
157#131—44no—
158#131—44no—
159#131—44no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
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—44no—
218C2×M16—44no—
219C2×M16—42no—
220C2×M16C22×C412yes—
221C2×M16C22×C412yes—
222C2×M16C22×C412yes—
223C2×M16C22×C412yes—
224C2×M16C22×C412yes—
225C2×M16C22×C412yes—
226C2×M16#4112yes—
227C2×M16#4112yes—
228C2×M16#4112yes—
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—42no—
247C22×C8#4111yescharacteristic
248C22×C8#4111yescharacteristic
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C22⋊C4C2×Q812yes—
260C2×C22⋊C4C2×Q812yes—
261C2×C42—21no—
262C2×C42—21no—
263#108—44no—
264#108—44no—
265#108—44no—
266#108—44no—
267#101—42no—
268#101—42no—
269#101—42no—
270#100—42no—
271#100—42no—
272#100—42no—
273#99—44no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—44no—
277#97—42no—
278#97—42no—
279C4×C8—42no—
280C4×C8—42no—
281C24C4×Q812yes—
282C24C22×Q811yescharacteristic
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4C4×Q812yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C42−1+411yescharacteristic
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—48no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—48no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—81no—
445C23—82no—
446C23C2×C4×Q811yescharacteristic
447C23#54311yescharacteristic
448C23#54411yescharacteristic
449C23#55411yescharacteristic
450C23#55411yescharacteristic
451C23#55411yescharacteristic
452C23#55411yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4C4×#4111yescharacteristic
477C2×C4C4×#4111yescharacteristic
478C2×C4#51811yescharacteristic
479C2×C4#51811yescharacteristic
480C2×C4#51811yescharacteristic
481C2×C4#52011yescharacteristic
482C2×C4#53811yescharacteristic
483C2×C4#53811yescharacteristic
484C2×C4#53811yescharacteristic
485C2×C4#53811yescharacteristic
486C2×C4#53811yescharacteristic
487C2×C4#53911yescharacteristic
488C2×C4#54011yescharacteristic
489C2×C4#54011yescharacteristic
490C2×C4#54011yescharacteristic
491C2×C4#54011yescharacteristic
492C2×C4#55111yescharacteristic
493C2×C4#55111yescharacteristic
494C2×C4#55111yescharacteristic
495C2×C4#55111yescharacteristic
496C2×C4#55111yescharacteristic
497C2×C4#55211yescharacteristic
498C2×C4#55311yescharacteristic
499C2×C4#55611yescharacteristic
500C2×C4#56011yescharacteristic
501C2×C4#56011yescharacteristic
502C2×C4#56211yescharacteristic
503C2×C4#56211yescharacteristic
504C8—42no—
505C8—42no—
506C8—42no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—41no—
513C8—41no—
514C8—42no—
515C8—81no—
516C8—81no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22#232911yescharacteristic
529C22#236811yescharacteristic
530C22#236811yescharacteristic
531C22#274811yescharacteristic
532C22#276811yescharacteristic
533C22#278011yescharacteristic
534C22#278811yescharacteristic
535C4—42no—
536C4—82no—
537C4—82no—
538C4#233311yescharacteristic
539C4#233311yescharacteristic
540C4#237311yescharacteristic
541C4#237711yescharacteristic
542C4#273511yescharacteristic
543C4#274011yescharacteristic
544C4#274311yescharacteristic
545C4#274911yescharacteristic
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—82no—
554C2#1459111yescharacteristic
subgroupG/H
centerC4#2749
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14591
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11368C2semidirect
3. #11363C2semidirect
4. #11363C2semidirect
5. #11254C2semidirect
9. #10906C2semidirect
10. #10902C2semidirect
11. #10898C2semidirect
12. #10815C2semidirect
29. #2070C4semidirect
30. #2069C4semidirect

Maximal subgroups

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

subgroupindex
#236222
#113682
#113632
#112542
#111372
#109612
#109412
#109062
#109022
#108982
#108152
#107712
#107532
#107442