Finite Group Databasebeta

← #365750

Subgroups — #365750

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2C2×#1980C211yesmaximal, characteristic
3#21200C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic, complemented
5#13911C211yesmaximal, characteristic, complemented
6#11079C214yesmaximal, complemented
7#10851C211yesmaximal, characteristic, complemented
8#10781C214yesmaximal, complemented
9C4×#343C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520C2212yes—
12C22×#104C2214yes—
13C2×#417C2214yes—
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2367—22no—
17#2366—21no—
18#2365—21no—
19#2354—21no—
20#2352—22no—
21D4×C4⋊C4—21no—
22#2325—22no—
23#2311C2212yes—
24C4×#121—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1980C418yescomplemented
31#1959—22no—
32#1957—21no—
33#1956—21no—
34#1836—22no—
35#1834—21no—
36#1834—21no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39#1736C2214yes—
40C2×#339C2214yes—
41#1442—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C22×#41C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47#532—22no—
48#532—22no—
49#532C2×C414yes—
50#532Q814yes—
51C2×C4⋊Q8D411yescharacteristic
52C2×#127D411yescharacteristic
53C2×#124D412yes—
54C2×#121—21no—
55C2×#121—21no—
56#520—28no—
57#520—44no—
58C2×C4×Q8—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117—22no—
65C2×#117C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42D412yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#104—44no—
77C2×#104—44no—
78C2×#104C2×C4116yescomplemented
79#417—28no—
80#417—44no—
81C2×#103—24no—
82C2×#102—24no—
83C4×M16—22no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16C2×C414yescomplemented
87C2×#97—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#391—22no—
92#390—22no—
93#390—22no—
94#389—22no—
95#388—24no—
96#388—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#385—24no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384D411yescharacteristic
106#384D411yescharacteristic
107#382—44no—
108#382D412yes—
109#380—21no—
110#380—21no—
111#380—22no—
112#380—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C414yescomplemented
122C4×C22⋊C4—24no—
123C2×#95—21no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95D412yes—
127#343—24no—
128#343—24no—
129#340—24no—
130#340—24no—
131#339—44no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—24no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C2×#41C22×C414yes—
139C2×#41C2×Q814yes—
140C22×Q8C2×D412yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×D412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16C22×C414yes—
165C2×M16#4114yes—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—22no—
169C4⋊Q8C4⋊C414yes—
170#127—22no—
171#127C4⋊C414yes—
172#124—24no—
173#124C4⋊C418yes—
174#121—24no—
175#121—24no—
176#121—24no—
177#121—24no—
178C4×Q8—48no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—84no—
187#117—24no—
188#117—24no—
189#117—48no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235C2×C42C22×C414yes—
236C2×C42C22×C412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×Q812yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244#104—416no—
245#103—48no—
246#102—48no—
247#97—24no—
248#97—42no—
249C4×C8—24no—
250C4×C8—42no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—48no—
259#95—84no—
260C24—22no—
261C24C22×D411yescharacteristic
262#41—28no—
263#41—28no—
264#41—28no—
265#41—28no—
266#41—44no—
267#41—44no—
268#41—44no—
269#41—44no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8C2×C4⋊C414yes—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—81no—
288C2×D4—81no—
289C2×D4C2×C4⋊C414yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4C22⋊Q812yes—
332C22×C4C22⋊Q812yes—
333C22×C4#12414yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)12yes—
344M16—24no—
345M16—24no—
346M16—24no—
347M16—24no—
348M16—42no—
349M16—42no—
350M16—42no—
351M16—42no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—28no—
386C42—48no—
387C42—44no—
388C42—44no—
389C42—44no—
390C42—82no—
391C42C2×C4⋊C412yes—
392C42C2×C4⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42C22⋊Q812yes—
396C42C22⋊Q812yes—
397C42#12312yes—
398C42#12312yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23C2×C4×D411yescharacteristic
411C23C2×#12011yescharacteristic
412C23C2×C22⋊Q811yescharacteristic
413C23C2×#12411yescharacteristic
414C23#53211yescharacteristic
415C23#53511yescharacteristic
416C23#53511yescharacteristic
417Q8—28no—
418Q8—44no—
419D4—28no—
420D4—44no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—48no—
425D4—44no—
426D4—84no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—168no—
471C2×C4#38512yes—
472C2×C4#38512yes—
473C2×C4#38512yes—
474C2×C4#38512yes—
475C2×C4#45412yes—
476C2×C4#45512yes—
477C2×C4C22×C4⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C4×#4112yes—
480C2×C4#51812yes—
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4#52912yes—
484C2×C4#54012yes—
485C2×C4#54612yes—
486C2×C4#54612yes—
487C2×C4#54612yes—
488C2×C4#55012yes—
489C2×C4#55312yes—
490C2×C4#55412yes—
491C2×C4#55412yes—
492C2×C4#55512yes—
493C8—44no—
494C8—44no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22C2×#38511yescharacteristic
515C22C4×#12411yescharacteristic
516C22#232811yescharacteristic
517C22#243111yescharacteristic
518C22#243511yescharacteristic
519C22#256311yescharacteristic
520C22#256611yescharacteristic
521C22C2×#45511yescharacteristic
522C22C2×#45411yescharacteristic
523C22#302111yescharacteristic
524C22#302211yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—44no—
529C4—44no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—162no—
535C4#186014yes—
536C4#234912yes—
537C4#236312yes—
538C4#242012yes—
539C4#245412yes—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—84no—
545C2#1397911yescharacteristic
546C2C2×#186011yescharacteristic
547C2#2086511yescharacteristic
548C2#2562111yescharacteristic
549C2#2562411yescharacteristic
550C2#2586911yescharacteristic
551C2#2587211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20764C2semidirect
5. #13911C2semidirect
6. #11079C2semidirect
7. #10851C2semidirect
8. #10781C2semidirect
9. C4×#343C2semidirect
30. #1980C4semidirect
78. C2×#104C2×C4semidirect
86. C4×M16C2×C4semidirect
121. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
C2×#19802
#212002
#207642
#139112
#110792
#108512
#107812
C4×#3432