Finite Group Databasebeta

← #388027

Subgroups — #388027

Order 512.

745 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#21098C211yesmaximal, characteristic
2#11356C211yesmaximal, characteristic, complemented
3#11309C211yesmaximal, characteristic, complemented
4#11305C211yesmaximal, characteristic, complemented
5#11278C211yesmaximal, characteristic, complemented
6#11176C211yesmaximal, characteristic
7#11110C211yesmaximal, characteristic
8#10951C211yesmaximal, characteristic
9#10917C211yesmaximal, characteristic, complemented
10#10902C211yesmaximal, characteristic, complemented
11#10817C211yesmaximal, characteristic, complemented
12#10803C211yesmaximal, characteristic, complemented
13#10787C211yesmaximal, characteristic
14#10782C211yesmaximal, characteristic
15#10757C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18#2905C2211yescharacteristic
19C2×#421C2211yescharacteristic
20#2896C2211yescharacteristic
21#2896C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#410C2211yescharacteristic
24#2879C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#1938C414yescomplemented
28#1937C414yescomplemented
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35#1742C2211yescharacteristic
36#1742C2211yescharacteristic
37#1742C2211yescharacteristic
38#1740C2211yescharacteristic
39#1740C2211yescharacteristic
40#1738C2211yescharacteristic
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1735C2211yescharacteristic
44#1733C2211yescharacteristic
45#1733C2211yescharacteristic
46#1733C2211yescharacteristic
47C2×H(3,ℤ/4)C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53C22×#41C2311yescharacteristic
54#568—22no—
55#568—22no—
56#568—22no—
57#568—41no—
58C2×#131—41no—
59C2×#131—41no—
60C22×M16—41no—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75#430—22no—
76#430—41no—
77C2×#108—41no—
78#424—41no—
79#424C2×C414yes—
80#423—41no—
81#421—41no—
82#421—41no—
83C2×#104C2×C414yes—
84#419—21no—
85#419—21no—
86#418—21no—
87#418—21no—
88#417—22no—
89#417—41no—
90#416—22no—
91#416—41no—
92#416—41no—
93#416C2×C412yes—
94#416C2×C412yes—
95C2×#103—21no—
96C2×#102—21no—
97#410—22no—
98#410—41no—
99C2×#99—22no—
100#408—41no—
101#408C2×C414yes—
102#407—41no—
103#405—41no—
104#405—41no—
105#405—41no—
106#405—41no—
107C4×M16—41no—
108C2×#97C2×C412yes—
109C2×C4×C8C2×C412yes—
110#344—22no—
111#344—41no—
112#344—41no—
113#344—41no—
114#344—41no—
115#344—41no—
116#343—22no—
117#343—22no—
118#343—41no—
119#342—22no—
120#342—22no—
121#341—41no—
122#341—41no—
123#341C2×C412yes—
124#341C2×C412yes—
125#340—41no—
126#339—41no—
127#339—41no—
128#339C2×C414yes—
129H(3,ℤ/4)—22no—
130H(3,ℤ/4)—22no—
131H(3,ℤ/4)—41no—
132H(3,ℤ/4)—41no—
133#336—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×#41C22×C412yes—
146C2×#41#4112yes—
147C22×Q8#4111yescharacteristic
148C22×D4—21no—
149C22×D4#4111yescharacteristic
150C23×C4—21no—
151C23×C4C2411yescharacteristic
152#131—44no—
153#131—44no—
154#131—44no—
155#131—44no—
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—44no—
194C2×M16—44no—
195C2×M16—42no—
196C2×M16C22×C412yes—
197C2×M16C22×C412yes—
198C2×M16#4112yes—
199C2×M16#4112yes—
200C2×M16#4112yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—42no—
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C412yes—
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#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117C22×C412yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4#4112yes—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C22×C412yes—
259#108—44no—
260C4⋊C8—44no—
261#104—44no—
262#103—22no—
263#103—22no—
264#103—42no—
265#103—42no—
266#102—22no—
267#102—22no—
268#102—42no—
269#102—42no—
270#99—44no—
271C22⋊C8—44no—
272#97—42no—
273#97—42no—
274C4×C8—42no—
275C4×C8—42no—
276C24—21no—
277C24C2×#4111yescharacteristic
278#41—22no—
279#41—22no—
280#41—22no—
281#41—22no—
282#41—22no—
283#41—22no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8#11712yes—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4#11712yes—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
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×C4C23×C411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
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×C42−1+411yescharacteristic
338C22×C42−1+411yescharacteristic
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—48no—
350M16—41no—
351M16—41no—
352M16—41no—
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—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—48no—
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—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—81no—
440C23—161no—
441C23C4×#4111yescharacteristic
442C23#53911yescharacteristic
443C23#54211yescharacteristic
444C23#55111yescharacteristic
445C23#55311yescharacteristic
446C23#55311yescharacteristic
447C23#55911yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—41no—
451Q8—41no—
452D4—22no—
453D4—22no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—41no—
459D4—41no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—161no—
479C2×C4C2×#11711yescharacteristic
480C2×C4C4×#4111yescharacteristic
481C2×C4#51811yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#52011yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4#53811yescharacteristic
486C2×C4#53811yescharacteristic
487C2×C4#53811yescharacteristic
488C2×C4#53811yescharacteristic
489C2×C4#53811yescharacteristic
490C2×C4#53911yescharacteristic
491C2×C4#53911yescharacteristic
492C2×C4#54111yescharacteristic
493C2×C4#54211yescharacteristic
494C2×C4#54211yescharacteristic
495C2×C4#55111yescharacteristic
496C2×C4#55111yescharacteristic
497C2×C4#55111yescharacteristic
498C2×C4#55111yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#56011yescharacteristic
504C2×C4#56011yescharacteristic
505C2×C4#56211yescharacteristic
506C2×C4#56211yescharacteristic
507C8—41no—
508C8—41no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—81no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—161no—
527C22#233411yescharacteristic
528C22#235311yescharacteristic
529C22#237411yescharacteristic
530C22#274611yescharacteristic
531C22#275111yescharacteristic
532C22#277911yescharacteristic
533C22#279611yescharacteristic
534C4—42no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—161no—
539C4—161no—
540C4—161no—
541C4#231811yescharacteristic
542C4#233711yescharacteristic
543C4#234311yescharacteristic
544C4#237711yescharacteristic
545C4#273511yescharacteristic
546C4#273911yescharacteristic
547C4#274711yescharacteristic
548C4#277811yescharacteristic
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—82no—
557C2#1450911yescharacteristic
subgroupG/H
centerC4#2377
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14509
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. #11356C2semidirect
3. #11309C2semidirect
4. #11305C2semidirect
5. #11278C2semidirect
9. #10917C2semidirect
10. #10902C2semidirect
11. #10817C2semidirect
12. #10803C2semidirect
27. #1938C4semidirect
28. #1937C4semidirect

Maximal subgroups

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

subgroupindex
#210982
#113562
#113092
#113052
#112782
#111762
#111102
#109512
#109172
#109022
#108172
#108032
#107872
#107822
#107572