Finite Group Databasebeta

← #388004

Subgroups — #388004

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20713C211yesmaximal, characteristic
2#11320C211yesmaximal, characteristic, complemented
3#11302C211yesmaximal, characteristic, complemented
4#11284C211yesmaximal, characteristic, complemented
5#11273C211yesmaximal, characteristic, complemented
6#11138C211yesmaximal, characteristic
7#11137C211yesmaximal, characteristic
8#10961C211yesmaximal, characteristic
9#10941C211yesmaximal, characteristic
10#10905C211yesmaximal, characteristic, complemented
11#10900C211yesmaximal, characteristic, complemented
12#10828C211yesmaximal, characteristic, complemented
13#10804C211yesmaximal, characteristic, complemented
14#10752C211yesmaximal, characteristic
15#10743C211yesmaximal, characteristic
16#3428C2211yescharacteristic
17C2×#430C2211yescharacteristic
18#2911C2211yescharacteristic
19C2×#413C2211yescharacteristic
20C2×#413C2211yescharacteristic
21C2×#410C2211yescharacteristic
22C2×#410C2211yescharacteristic
23#2879C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#1830C414yescomplemented
28#1829C414yescomplemented
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37#1742C2211yescharacteristic
38#1738C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43#1733C2211yescharacteristic
44#1733C2211yescharacteristic
45#1733C2211yescharacteristic
46#1733C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52#1729C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62#520—41no—
63#520—41no—
64#519C2×C412yes—
65#518—22no—
66#518C2×C412yes—
67C4×#41—41no—
68C4×#41—41no—
69C2×#117—22no—
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76#430—41no—
77#430—41no—
78C2×#108—22no—
79#428—22no—
80#428—41no—
81#427—41no—
82#413—41no—
83#413—41no—
84#413—41no—
85#413—41no—
86C2×#101C2×C412yes—
87C2×#101C2×C412yes—
88C2×#100C2×C412yes—
89C2×#100C2×C412yes—
90#410—22no—
91#410—22no—
92#410—41no—
93#410—41no—
94C2×#99—22no—
95C2×#99—22no—
96#408—22no—
97#408—21no—
98#408—21no—
99#408—41no—
100#408—41no—
101#407—22no—
102C2×C22⋊C8—22no—
103#405—22no—
104#405—41no—
105#405—41no—
106#405—41no—
107C4×M16—41no—
108C2×#97—21no—
109C2×C4×C8—21no—
110#344—22no—
111#344—22no—
112#344—41no—
113#344—41no—
114#344C2×C414yes—
115#344C2×C414yes—
116#344C2×C414yes—
117#344C2×C414yes—
118#343—41no—
119#341—21no—
120#341—21no—
121#341—41no—
122#341—41no—
123#340—41no—
124#340—41no—
125#339—41no—
126#339—41no—
127#339—41no—
128#339—41no—
129#339—41no—
130#339—41no—
131H(3,ℤ/4)—22no—
132H(3,ℤ/4)—41no—
133H(3,ℤ/4)—41no—
134#336—41no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
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—
147C22×Q8C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C22×D4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
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×M16C22×C412yes—
190C2×M16C22×C412yes—
191C2×M16C22×C412yes—
192C2×M16C22×C412yes—
193C2×M16#4112yes—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8#4111yescharacteristic
208C22×C8#4111yescharacteristic
209C4×Q8—42no—
210C4×Q8—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—44no—
214C4×D4—44no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—24no—
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#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—42no—
237#117—42no—
238#117C22×C412yes—
239#117C22×C412yes—
240#117#4112yes—
241#117#4112yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—42no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—42no—
263#108—44no—
264#107—42no—
265#106—42no—
266#101—42no—
267#101—42no—
268#100—42no—
269#100—42no—
270#99—44no—
271#99—44no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—44no—
275#97—42no—
276#97—42no—
277C4×C8—42no—
278C4×C8—42no—
279C24#11712yes—
280C24C2×#4111yescharacteristic
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—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4#11712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
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×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—44no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—44no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—81no—
445C23—82no—
446C23C2×#11711yescharacteristic
447C23#53811yescharacteristic
448C23#53811yescharacteristic
449C23#55111yescharacteristic
450C23#55111yescharacteristic
451C23#55211yescharacteristic
452C23#55211yescharacteristic
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—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—161no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—164no—
484C2×C4—161no—
485C2×C4C4×#4111yescharacteristic
486C2×C4C4×#4111yescharacteristic
487C2×C4#51811yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#52011yescharacteristic
491C2×C4#53811yescharacteristic
492C2×C4#53811yescharacteristic
493C2×C4#53811yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53911yescharacteristic
496C2×C4#53911yescharacteristic
497C2×C4#54011yescharacteristic
498C2×C4#54011yescharacteristic
499C2×C4#54211yescharacteristic
500C2×C4#54211yescharacteristic
501C2×C4#55111yescharacteristic
502C2×C4#55111yescharacteristic
503C2×C4#55111yescharacteristic
504C2×C4#55211yescharacteristic
505C2×C4#55211yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55311yescharacteristic
508C2×C4#55911yescharacteristic
509C2×C4#56011yescharacteristic
510C2×C4#56211yescharacteristic
511C2×C4#56211yescharacteristic
512C2×C4#56311yescharacteristic
513C8—42no—
514C8—42no—
515C8—42no—
516C8—42no—
517C8—41no—
518C8—41no—
519C8—42no—
520C8—81no—
521C8—81no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22#231211yescharacteristic
534C22#234311yescharacteristic
535C22#234311yescharacteristic
536C22#274311yescharacteristic
537C22#274911yescharacteristic
538C22#275111yescharacteristic
539C22#276511yescharacteristic
540C4—42no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—161no—
546C4—161no—
547C4#233511yescharacteristic
548C4#233511yescharacteristic
549C4#237411yescharacteristic
550C4#237611yescharacteristic
551C4#274311yescharacteristic
552C4#274911yescharacteristic
553C4#275111yescharacteristic
554C4#276511yescharacteristic
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—82no—
563C2#1450711yescharacteristic
subgroupG/H
centerC4#2749
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14507
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. #11320C2semidirect
3. #11302C2semidirect
4. #11284C2semidirect
5. #11273C2semidirect
10. #10905C2semidirect
11. #10900C2semidirect
12. #10828C2semidirect
13. #10804C2semidirect
27. #1830C4semidirect
28. #1829C4semidirect

Maximal subgroups

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

subgroupindex
#207132
#113202
#113022
#112842
#112732
#111382
#111372
#109612
#109412
#109052
#109002
#108282
#108042
#107522
#107432