Finite Group Databasebeta

← #365751

Subgroups — #365751

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 557 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#13910C211yesmaximal, characteristic, complemented
6#11078C214yesmaximal, 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—
25C2×#389—22no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1980C418yescomplemented
32#1959—22no—
33#1957—21no—
34#1956—21no—
35#1836—22no—
36#1834—21no—
37#1834—21no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40#1736C2214yes—
41C2×#339C2214yes—
42#1444—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—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42D412yes—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#104—44no—
78C2×#104—44no—
79C2×#104C2×C4116yescomplemented
80#417—28no—
81#417—44no—
82C2×#103—24no—
83C2×#102—24no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16C2×C414yescomplemented
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#391—22no—
93#390—22no—
94#390—22no—
95#389—22no—
96#389—48no—
97#388—24no—
98#388—24no—
99#387—24no—
100#386—24no—
101#386—24no—
102#385—24no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384—44no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#382D412yes—
111#380—21no—
112#380—21no—
113#380—22no—
114#380—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C414yescomplemented
124C4×C22⋊C4—24no—
125C2×#95—21no—
126C2×#95—24no—
127C2×#95D412yes—
128#343—24no—
129#343—24no—
130#340—24no—
131#340—24no—
132#339—44no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41C22×C414yes—
140C2×#41C2×Q814yes—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C2×D412yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
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×M16—22no—
165C2×M16C22×C414yes—
166C2×M16#4114yes—
167C22×C8—22no—
168C22×C8—22no—
169C4⋊Q8—22no—
170C4⋊Q8C4⋊C414yescomplemented
171#127—22no—
172#127C4⋊C414yescomplemented
173#124—24no—
174#124C4⋊C418yescomplemented
175#121—24no—
176#121—24no—
177#121—24no—
178#121—24no—
179C4×Q8—48no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—48no—
187C4×D4—84no—
188#117—24no—
189#117—24no—
190#117—48no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—84no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—44no—
239C2×C42C22×C414yes—
240C2×C42C22×C412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×Q812yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248#104—416no—
249#103—48no—
250#102—48no—
251#97—24no—
252#97—42no—
253C4×C8—24no—
254C4×C8—42no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—48no—
263C24—22no—
264C24C22×D411yescharacteristic
265#41—28no—
266#41—28no—
267#41—28no—
268#41—28no—
269#41—44no—
270#41—44no—
271#41—44no—
272#41—44no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8C2×C4⋊C414yes—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4C2×C4⋊C414yes—
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—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12414yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)12yes—
347M16—24no—
348M16—24no—
349M16—24no—
350M16—24no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—216no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—816no—
381C4⋊C4—82no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—48no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C42—28no—
390C42—48no—
391C42—44no—
392C42—44no—
393C42—44no—
394C42—82no—
395C42C2×C4⋊C412yes—
396C42C2×C4⋊C412yes—
397C42C4×D412yes—
398C42C4×D412yes—
399C42C22⋊Q812yes—
400C42C22⋊Q812yes—
401C42#12312yes—
402C42#12312yes—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—24no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C2×C4×D411yescharacteristic
415C23C2×#12011yescharacteristic
416C23C2×C22⋊Q811yescharacteristic
417C23C2×#12411yescharacteristic
418C23#53211yescharacteristic
419C23#53511yescharacteristic
420C23#53511yescharacteristic
421Q8—28no—
422Q8—44no—
423D4—28no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—44no—
428D4—48no—
429D4—44no—
430D4—84no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
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—48no—
455C2×C4—48no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—168no—
475C2×C4#38512yes—
476C2×C4#38512yes—
477C2×C4#38512yes—
478C2×C4#38512yes—
479C2×C4#45312yes—
480C2×C4#45612yes—
481C2×C4C22×C4⋊C411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C4×#4112yes—
484C2×C4#51812yes—
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4#52912yes—
488C2×C4#54012yes—
489C2×C4#54612yes—
490C2×C4#54612yes—
491C2×C4#54612yes—
492C2×C4#55012yes—
493C2×C4#55312yes—
494C2×C4#55412yes—
495C2×C4#55412yes—
496C2×C4#55512yes—
497C8—44no—
498C8—44no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×#38511yescharacteristic
519C22C4×#12411yescharacteristic
520C22#232811yescharacteristic
521C22#243111yescharacteristic
522C22#243511yescharacteristic
523C22#256311yescharacteristic
524C22#256611yescharacteristic
525C22C2×#45311yescharacteristic
526C22C2×#45611yescharacteristic
527C22#302111yescharacteristic
528C22#302211yescharacteristic
529C4—22no—
530C4—22no—
531C4—22no—
532C4—48no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—162no—
539C4#185914yes—
540C4#234912yes—
541C4#236312yes—
542C4#242012yes—
543C4#245412yes—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—84no—
549C2#1397911yescharacteristic
550C2C2×#185911yescharacteristic
551C2#2086511yescharacteristic
552C2#2562211yescharacteristic
553C2#2562311yescharacteristic
554C2#2587011yescharacteristic
555C2#2587111yescharacteristic
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. #13910C2semidirect
6. #11078C2semidirect
7. #10851C2semidirect
8. #10781C2semidirect
9. C4×#343C2semidirect
31. #1980C4semidirect
79. C2×#104C2×C4semidirect
87. C4×M16C2×C4semidirect
123. C4×C4⋊C4C2×C4semidirect
170. C4⋊Q8C4⋊C4semidirect
172. #127C4⋊C4semidirect
174. #124C4⋊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
#139102
#110782
#108512
#107812
C4×#3432