Finite Group Databasebeta

← #380987

Subgroups — #380987

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11228C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2629—24no—
19#2629C418yescomplemented
20C2×#398—22no—
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25#1798—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C4⋊#95—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—22no—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—22no—
47C2×#125—42no—
48C2×#125—42no—
49C2×#125C2×C414yescomplemented
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117C2×C414yes—
56C2×#117D414yes—
57C2×#117Q814yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4Q814yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85#398—216no—
86#396—216no—
87#389—24no—
88#389C2×C418yescomplemented
89#384—22no—
90#384C2×C414yescomplemented
91#380—24no—
92#380C2×C418yes—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—22no—
109C25C2×D411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×Q812yes—
131C23×C4C2×Q812yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149#125—48no—
150#125—48no—
151#117—48no—
152#117—48no—
153#117—48no—
154#117—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C22⋊C418yes—
194C2×C4⋊C4C22×C414yes—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4C2×Q812yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4C22⋊C418yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C4⋊C8—48no—
221C22⋊C8—48no—
222#95—28no—
223#95—28no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24C2×C22⋊C412yes—
237C24C22⋊Q812yes—
238C24C22⋊Q812yes—
239C24C22×D411yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—48no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4C2×C22⋊C414yes—
308C22×C4C2×C22⋊C414yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C4×Q814yes—
311C22×C4C4×Q814yes—
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12414yes—
315C22×C4C4⋊Q814yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C22×Q811yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C42−1+412yes—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—416no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—88no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—48no—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23#39114yes—
392C23C22×C22⋊C411yescharacteristic
393C23C2×C22⋊Q811yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23#53411yescharacteristic
396C23#53411yescharacteristic
397C23#53511yescharacteristic
398C23#53611yescharacteristic
399C23#57612yes—
400C23#57712yes—
401C23#57712yes—
402C23#57812yes—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—816no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39114yes—
463C2×C4C2×C4×Q811yescharacteristic
464C2×C4C2×C4×Q811yescharacteristic
465C2×C4#51812yes—
466C2×C4#52012yes—
467C2×C4C2×#12411yescharacteristic
468C2×C4C2×C4⋊Q811yescharacteristic
469C2×C4#53412yes—
470C2×C4#53712yes—
471C2×C4#53812yes—
472C2×C4#53812yes—
473C2×C4#53912yes—
474C2×C4#53912yes—
475C2×C4#54312yes—
476C2×C4#54312yes—
477C2×C4#54412yes—
478C2×C4#54412yes—
479C8—82no—
480C8—82no—
481C22—28no—
482C22—24no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22C2×#39111yescharacteristic
493C22#230811yescharacteristic
494C22#231111yescharacteristic
495C22#262611yescharacteristic
496C22#262611yescharacteristic
497C22#262711yescharacteristic
498C22#262711yescharacteristic
499C22#289712yes—
500C22#289812yes—
501C22#308112yes—
502C22#308212yes—
503C22#308312yes—
504C22#308412yes—
505C22#319812yes—
506C22#320112yes—
507C22#320312yes—
508C22#320412yes—
509C22#322011yescharacteristic
510C22#322111yescharacteristic
511C22#322211yescharacteristic
512C22#322311yescharacteristic
513C22#324112yes—
514C22#324212yes—
515C4—24no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—84no—
524C4#232912yes—
525C4#232912yes—
526C4#263912yes—
527C4#265912yes—
528C2—44no—
529C2#1428111yescharacteristic
530C2#2131512yes—
531C2#2177212yes—
532C2#2187211yescharacteristic
533C2#2187311yescharacteristic
534C2#2996211yescharacteristic
535C2#2996311yescharacteristic
536C2#2996411yescharacteristic
537C2#2996511yescharacteristic
538C2#2997812yes—
539C2#2997912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11385C2semidirect
5. #11385C2semidirect
6. #11373C2semidirect
7. #11371C2semidirect
8. #11228C2semidirect
9. #11227C2semidirect
19. #2629C4semidirect
49. C2×#125C2×C4semidirect
88. #389C2×C4semidirect
90. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#17982
#137952
#113852
#113732
#113712
#112282
#112272
#107312
#107302