Finite Group Databasebeta

← #390342

Subgroups — #390342

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23652C211yesmaximal, characteristic
2#21184C211yesmaximal, characteristic, complemented
3#21071C211yesmaximal, characteristic
4#11367C211yesmaximal, characteristic, complemented
5#11282C211yesmaximal, characteristic, complemented
6#11146C211yesmaximal, characteristic
7#11131C211yesmaximal, characteristic
8#11060C211yesmaximal, characteristic
9#10951C211yesmaximal, characteristic
10#10941C211yesmaximal, characteristic
11#10919C211yesmaximal, characteristic, complemented
12#10905C211yesmaximal, characteristic, complemented
13#10905C211yesmaximal, characteristic, complemented
14#10901C211yesmaximal, characteristic, complemented
15#10831C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C22×#104C2211yescharacteristic
22#2896C2211yescharacteristic
23#2896C2211yescharacteristic
24C2×#413C2211yescharacteristic
25C2×#410C2211yescharacteristic
26#2879C2211yescharacteristic
27#2873C2211yescharacteristic
28#2873C2211yescharacteristic
29#2873C2211yescharacteristic
30#2296C2211yescharacteristic
31#2082C414yescomplemented
32#2081C414yescomplemented
33#1953—24no—
34#1929—24no—
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43#1742C2211yescharacteristic
44#1740C2211yescharacteristic
45#1733C2211yescharacteristic
46#1733C2211yescharacteristic
47#1733C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53#1728C2211yescharacteristic
54#1728C2211yescharacteristic
55C22×#41C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#430—22no—
71#430—22no—
72#430—22no—
73#430—22no—
74#430—41no—
75#430—41no—
76#430—41no—
77#430C2×C414yes—
78C2×#108—22no—
79C2×#108—22no—
80C2×#108—41no—
81C2×#108D414yes—
82#421—41no—
83#421—41no—
84C2×#104—22no—
85C2×#104—22no—
86C2×#104—41no—
87C2×#104—41no—
88#419C2×C412yes—
89#419C2×C412yes—
90#418C2×C412yes—
91#418C2×C412yes—
92#417—22no—
93#417—41no—
94#416—22no—
95#416—41no—
96#413—41no—
97#413—41no—
98C2×#101C2×C412yes—
99C2×#100C2×C412yes—
100#410—22no—
101#410—41no—
102C2×#99—22no—
103#408—22no—
104#408—41no—
105#407—22no—
106#405—22no—
107#405—22no—
108#405—22no—
109#405—41no—
110#405—41no—
111#405—41no—
112C4×M16—41no—
113C4×M16—41no—
114C4×M16—41no—
115#389—41no—
116#385—22no—
117C4×C4⋊C4—41no—
118#344—22no—
119#344—22no—
120#344—22no—
121#344—22no—
122#344—41no—
123#344—41no—
124#344C2×C414yes—
125#344C2×C414yes—
126#343—41no—
127#342—22no—
128#339—22no—
129#339—22no—
130#339—41no—
131#339—41no—
132#339—41no—
133#339—41no—
134H(3,ℤ/4)—22no—
135H(3,ℤ/4)C2×C414yes—
136#336—22no—
137#336—41no—
138C2×#41—21no—
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—
150C22×Q8C22×C411yescharacteristic
151C22×D4—21no—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2411yescharacteristic
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
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×M16C22×C412yes—
196C2×M16C22×C412yes—
197C2×M16C22×C412yes—
198C2×M16C22×C412yes—
199C2×M16C2×D412yes—
200C2×M16C2×D412yes—
201C2×M16#4112yes—
202C2×M16#4112yes—
203C2×M16#4112yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C2×D412yes—
212#117—21no—
213#117—21no—
214#117—21no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117C22×C412yes—
230#117C22×C412yes—
231#117#4112yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4#4112yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254#108—28no—
255#108—44no—
256#108—44no—
257#108—44no—
258#104—44no—
259#104—44no—
260#103—24no—
261#103—42no—
262#102—24no—
263#102—42no—
264#101—42no—
265#100—42no—
266#99—44no—
267C22⋊C8—44no—
268#97—42no—
269#97—42no—
270#97—42no—
271C4×C8—42no—
272C4×C8—42no—
273C4×C8—42no—
274#95—42no—
275C24—21no—
276C24C2×#4111yescharacteristic
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
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—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—82no—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
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×C42−1+411yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
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—22no—
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—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8C4×D414yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—84no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—81no—
436C23—161no—
437C23C4×#4111yescharacteristic
438C23#53811yescharacteristic
439C23#54211yescharacteristic
440C23#54811yescharacteristic
441C23#55111yescharacteristic
442C23#55611yescharacteristic
443C23#55911yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
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—82no—
476C2×C4—81no—
477C2×C4—161no—
478C2×C4—161no—
479C2×C4—161no—
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C4×#4111yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#52011yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4#53411yescharacteristic
487C2×C4#53611yescharacteristic
488C2×C4#53811yescharacteristic
489C2×C4#53911yescharacteristic
490C2×C4#53911yescharacteristic
491C2×C4#53911yescharacteristic
492C2×C4#53911yescharacteristic
493C2×C4#54011yescharacteristic
494C2×C4#54111yescharacteristic
495C2×C4#54211yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#55011yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55311yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55511yescharacteristic
503C2×C4#55611yescharacteristic
504C2×C4#56011yescharacteristic
505C2×C4#56211yescharacteristic
506C2×C4#56211yescharacteristic
507C2×C4#56311yescharacteristic
508C8—24no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—42no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—82no—
524C22—81no—
525C22—161no—
526C22—161no—
527C22#233511yescharacteristic
528C22#236711yescharacteristic
529C22#237311yescharacteristic
530C22#271611yescharacteristic
531C22#273011yescharacteristic
532C22#279611yescharacteristic
533C22#280811yescharacteristic
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—81no—
540C4—161no—
541C4#232211yescharacteristic
542C4#233411yescharacteristic
543C4#236311yescharacteristic
544C4#237611yescharacteristic
545C4#268811yescharacteristic
546C4#269411yescharacteristic
547C4#277711yescharacteristic
548C4#278311yescharacteristic
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—161no—
557C2#1456911yescharacteristic
subgroupG/H
centerC4#2688
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14569
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. #21184C2semidirect
4. #11367C2semidirect
5. #11282C2semidirect
11. #10919C2semidirect
12. #10905C2semidirect
13. #10905C2semidirect
14. #10901C2semidirect
15. #10831C2semidirect
31. #2082C4semidirect
32. #2081C4semidirect

Maximal subgroups

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

subgroupindex
#236522
#211842
#210712
#113672
#112822
#111462
#111312
#110602
#109512
#109412
#109192
#109052
#109012
#108312