Finite Group Databasebeta

← #4035353

Subgroups — #4035353

Order 512.

4320 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2668C213yesmaximal, complemented
2C2×#2430C213yesmaximal, complemented
3#17106C213yesmaximal, complemented
4#15948C213yesmaximal, complemented
5#15467C211yesmaximal, characteristic, complemented
6#15384C213yesmaximal, complemented
7#15312C216yesmaximal, complemented
8#14737C216yesmaximal, complemented
9#14533C213yesmaximal, complemented
10C23×#41C2211yescharacteristic, complemented
11C22×#123C2213yes—
12C22×#121C2216yes—
13C22×#120C2213yes—
14#2668—224no—
15#2430—224no—
16C2×#399C2213yes—
17C2×#397C2216yes—
18C2×#397C2213yescomplemented
19C2×#395C2213yescomplemented
20C2×#394C2216yes—
21C2×#394C2213yescomplemented
22C2×#393C2213yes—
23C2×#392C2213yes—
24C2×#386C2216yes—
25C2×#386C2216yes—
26C2×#386C2216yes—
27C2×#386C2216yescomplemented
28C2×#385C2216yescomplemented
29C2×#384C2211yescharacteristic, complemented
30C2×#379C2213yes—
31C2×#379C2211yescharacteristic
32C22×#95C2213yescomplemented
33#1445C2213yescomplemented
34#1444C2216yescomplemented
35#1443C2212yescomplemented
36#1443C2216yescomplemented
37#1442C2216yescomplemented
38#1440C2213yes—
39#1440C2213yes—
40#1440C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1438C2216yes—
44#1438C2213yes—
45#1437C2213yes—
46#1437C2213yescomplemented
47C22⋊#95C2216yescomplemented
48C22⋊#95C2216yescomplemented
49C26C2311yescharacteristic
50C22×#41—212no—
51C22×#41—42no—
52C22×#41—46no—
53C23×Q8C2311yescharacteristic
54C23×D4C2313yes—
55C23×D4C2313yes—
56C23×D4C2316yes—
57C24×C4C2313yes—
58C24×C4C2316yes—
59C2×#123—212no—
60C2×#123—46no—
61C2×#123—46no—
62C2×#121—224no—
63C2×#121—224no—
64C2×#121—412no—
65C2×#120—224no—
66C2×#120—46no—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2313yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2316yes—
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2313yes—
81C22×C22⋊C4C2313yes—
82C22×C22⋊C4C2316yes—
83C22×C42C2311yescharacteristic
84#399—212no—
85#397—224no—
86#397—46no—
87#395—212no—
88#394—212no—
89#394—224no—
90#393—212no—
91#392D4124yescomplemented
92#386—224no—
93#386—224no—
94#386—224no—
95#386—224no—
96#385—224no—
97#384—42no—
98#379—212no—
99#379—42no—
100C2×#95—212no—
101C2×#95C2313yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2313yes—
107C2×#95C2316yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C2×#95C2312yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2313yes—
114C25—26no—
115C25—43no—
116C25—46no—
117C25—42no—
118C25—41no—
119C25C2413yes—
120C25C2413yes—
121C2×#41—448no—
122C2×#41—448no—
123C2×#41—416no—
124C22×Q8—26no—
125C22×Q8—41no—
126C22×Q8—43no—
127C22×D4—212no—
128C22×D4—212no—
129C22×D4—212no—
130C22×D4—212no—
131C22×D4—26no—
132C22×D4—212no—
133C22×D4—26no—
134C22×D4—43no—
135C22×D4—43no—
136C22×D4—46no—
137C22×D4—43no—
138C22×D4—46no—
139C22×D4—46no—
140C22×D4—46no—
141C22×D4—46no—
142C22×D4C2×D4112yes—
143C22×D4C2×D4124yes—
144C23×C4—26no—
145C23×C4—26no—
146C23×C4—26no—
147C23×C4—212no—
148C23×C4—212no—
149C23×C4—212no—
150C23×C4—46no—
151C23×C4—46no—
152C23×C4—46no—
153C23×C4—46no—
154C23×C4—46no—
155C23×C4—46no—
156C23×C4C2413yes—
157C23×C4C2416yes—
158C23×C4C2413yes—
159C23×C4C2413yes—
160C23×C4C2413yes—
161C23×C4C2416yes—
162C23×C4C2411yescharacteristic
163#123—448no—
164#121—496no—
165#120—448no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—212no—
168C2×C4⋊C4—26no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—46no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—43no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—43no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—212no—
180C2×C22⋊C4—212no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—26no—
183C2×C22⋊C4—212no—
184C2×C22⋊C4—26no—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—26no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—46no—
198C2×C22⋊C4—43no—
199C2×C22⋊C4—43no—
200C2×C22⋊C4—43no—
201C2×C22⋊C4—43no—
202C2×C22⋊C4—46no—
203C2×C22⋊C4—46no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—46no—
206C2×C22⋊C4—46no—
207C2×C22⋊C4—46no—
208C2×C22⋊C4—43no—
209C2×C22⋊C4C2×D4112yes—
210C2×C22⋊C4C2×D4124yes—
211C2×C42—41no—
212C2×C42—42no—
213#95—212no—
214#95—26no—
215#95—212no—
216#95—212no—
217#95—212no—
218#95—26no—
219#95—212no—
220#95—46no—
221#95—46no—
222#95—42no—
223#95—43no—
224#95—46no—
225#95C2×D4112yes—
226C24—23no—
227C24—23no—
228C24—26no—
229C24—26no—
230C24—26no—
231C24—26no—
232C24—26no—
233C24—26no—
234C24—23no—
235C24—23no—
236C24—26no—
237C24—26no—
238C24—26no—
239C24—26no—
240C24—26no—
241C24—412no—
242C24—412no—
243C24—44no—
244C24—86no—
245C24—812no—
246C24—86no—
247C24—82no—
248C24—86no—
249C24—812no—
250C24—812no—
251C24C22×D416yes—
252C24C22×D4112yes—
253C24H(5,𝔽2)16yes—
254C24C2511yescharacteristic
255#41—4128no—
256C2×Q8—412no—
257C2×Q8—412no—
258C2×Q8—44no—
259C2×D4—248no—
260C2×D4—248no—
261C2×D4—412no—
262C2×D4—424no—
263C2×D4—424no—
264C2×D4—412no—
265C2×D4—424no—
266C2×D4—412no—
267C2×D4—424no—
268C2×D4—412no—
269C2×D4—412no—
270C2×D4—812no—
271C2×D4—812no—
272C2×D4—812no—
273C2×D4—812no—
274C2×D4—812no—
275C2×D4—86no—
276C22×C4—224no—
277C22×C4—26no—
278C22×C4—23no—
279C22×C4—26no—
280C22×C4—23no—
281C22×C4—23no—
282C22×C4—26no—
283C22×C4—23no—
284C22×C4—23no—
285C22×C4—23no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—23no—
295C22×C4—26no—
296C22×C4—26no—
297C22×C4—23no—
298C22×C4—23no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—26no—
308C22×C4—26no—
309C22×C4—26no—
310C22×C4—424no—
311C22×C4—424no—
312C22×C4—424no—
313C22×C4—424no—
314C22×C4—412no—
315C22×C4—412no—
316C22×C4—812no—
317C22×C4—812no—
318C22×C4—812no—
319C22×C4—812no—
320C22×C4—812no—
321C22×C4C22×D4112yes—
322C22×C4C22×D416yes—
323C22×C4C22×D416yes—
324C22×C4H(5,𝔽2)12yes—
325C4⋊C4—424no—
326C4⋊C4—412no—
327C4⋊C4—412no—
328C4⋊C4—86no—
329C22⋊C4—424no—
330C22⋊C4—412no—
331C22⋊C4—412no—
332C22⋊C4—424no—
333C22⋊C4—424no—
334C22⋊C4—412no—
335C22⋊C4—412no—
336C22⋊C4—424no—
337C22⋊C4—812no—
338C22⋊C4—812no—
339C22⋊C4—86no—
340C22⋊C4—86no—
341C42—44no—
342C23—212no—
343C23—26no—
344C23—212no—
345C23—212no—
346C23—212no—
347C23—212no—
348C23—212no—
349C23—26no—
350C23—212no—
351C23—212no—
352C23—212no—
353C23—26no—
354C23—26no—
355C23—212no—
356C23—26no—
357C23—46no—
358C23—43no—
359C23—43no—
360C23—43no—
361C23—46no—
362C23—43no—
363C23—46no—
364C23—46no—
365C23—46no—
366C23—46no—
367C23—43no—
368C23—46no—
369C23—46no—
370C23—46no—
371C23—46no—
372C23—46no—
373C23—43no—
374C23—43no—
375C23—43no—
376C23—46no—
377C23—46no—
378C23—46no—
379C23—46no—
380C23—46no—
381C23—46no—
382C23—46no—
383C23—88no—
384C23—848no—
385C23—824no—
386C23—824no—
387C23—816no—
388C23D42112yes—
389C23C23×D413yes—
390C23C2×H(5,𝔽2)13yes—
391C23C2×H(5,𝔽2)11yescharacteristic
392C23#58512yes—
393C23#58516yes—
394Q8—416no—
395D4—448no—
396D4—848no—
397D4—824no—
398C2×C4—212no—
399C2×C4—212no—
400C2×C4—26no—
401C2×C4—26no—
402C2×C4—26no—
403C2×C4—212no—
404C2×C4—212no—
405C2×C4—212no—
406C2×C4—448no—
407C2×C4—43no—
408C2×C4—43no—
409C2×C4—46no—
410C2×C4—46no—
411C2×C4—46no—
412C2×C4—43no—
413C2×C4—41no—
414C2×C4—43no—
415C2×C4—43no—
416C2×C4—46no—
417C2×C4—46no—
418C2×C4—46no—
419C2×C4—46no—
420C2×C4—43no—
421C2×C4—46no—
422C2×C4—43no—
423C2×C4—43no—
424C2×C4—46no—
425C2×C4—43no—
426C2×C4—46no—
427C2×C4—46no—
428C2×C4—46no—
429C2×C4—46no—
430C2×C4—43no—
431C2×C4—43no—
432C2×C4—46no—
433C2×C4—46no—
434C2×C4—46no—
435C2×C4—46no—
436C2×C4—848no—
437C22—212no—
438C22—224no—
439C22—412no—
440C22—46no—
441C22—412no—
442C22—46no—
443C22—412no—
444C22—412no—
445C22—412no—
446C22—46no—
447C22—46no—
448C22—46no—
449C22—412no—
450C22—412no—
451C22—832no—
452C22—86no—
453C22—86no—
454C22—86no—
455C22—86no—
456C22C2×D4213yes—
457C22#348413yes—
458C22#348513yes—
459C22#348516yes—
460C22#350013yes—
461C22#352911yescharacteristic
462C22#353211yescharacteristic
463C22#355113yes—
464C22#356016yes—
465C22#356016yes—
466C4—42no—
467C4—46no—
468C4—46no—
469C4—412no—
470C4—83no—
471C4—83no—
472C4—86no—
473C2—412no—
474C2—86no—
475C2#4349216yes—
476C2#4627613yes—
477C2#4642713yes—
478C2#5766912yes—
479C2#5899011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2668C2semidirect
2. C2×#2430C2semidirect
3. #17106C2semidirect
4. #15948C2semidirect
5. #15467C2semidirect
6. #15384C2semidirect
7. #15312C2semidirect
8. #14737C2semidirect
9. #14533C2semidirect
10. C23×#41C22semidirect
18. C2×#397C22semidirect
19. C2×#395C22semidirect
21. C2×#394C22semidirect
27. C2×#386C22semidirect
28. C2×#385C22semidirect
29. C2×#384C22semidirect
32. C22×#95C22semidirect
33. #1445C22semidirect
34. #1444C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
46. #1437C22semidirect
47. C22⋊#95C22semidirect
48. C22⋊#95C22semidirect
91. #392D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26682
C2×#24302
#171062
#159482
#154672
#153842
#153122
#147372
#145332