Finite Group Databasebeta

← #5022209

Subgroups — #5022209

Order 512.

2560 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16246C213yesmaximal, complemented
2#15948C213yesmaximal
3#15802C216yesmaximal, complemented
4#15669C213yesmaximal, complemented
5#15480C213yesmaximal, complemented
6#15471C216yesmaximal, complemented
7#14706C213yesmaximal, complemented
8#14652C213yesmaximal, complemented
9#13827C211yesmaximal, characteristic, complemented
10C22×#123C2213yes—
11C22×#120C2211yescharacteristic
12C23×C42C2211yescharacteristic, complemented
13C2×#399C2211yescharacteristic, complemented
14C2×#397C2213yes—
15C2×#397C2216yes—
16C2×#396C2212yescomplemented
17C2×#386C2216yes—
18C2×#382C2213yescomplemented
19C2×#381C2213yescomplemented
20C2×#379C2213yes—
21C2×#379C2213yes—
22#1445C2213yescomplemented
23#1443C2216yescomplemented
24#1442C2216yescomplemented
25#1442C2216yescomplemented
26#1441C2216yescomplemented
27#1440C2213yes—
28#1440C2213yes—
29#1439C2216yes—
30#1439C2216yes—
31#1439C2216yes—
32#1439C2216yes—
33#1439C2216yes—
34#1438C2216yes—
35#1438C2213yes—
36#1437C2213yes—
37#1437C2213yes—
38#1437C2216yes—
39#1437C2213yes—
40#1437C2216yes—
41#1437C2213yescomplemented
42#1436C2213yescomplemented
43#1436C2216yescomplemented
44#1436C2216yescomplemented
45#1436C2213yescomplemented
46C22⋊#95C2216yescomplemented
47C26C2311yescharacteristic
48C23×D4C2313yes—
49C24×C4C2313yes—
50C2×#123—212no—
51C2×#123—46no—
52C2×#123—46no—
53C2×#120—42no—
54C2×#120—44no—
55C22×C4⋊C4C2313yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2316yes—
59C22×C22⋊C4C2313yes—
60C22×C22⋊C4C2313yes—
61C22×C22⋊C4C2316yes—
62C22×C22⋊C4C2313yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2313yes—
68C22×C22⋊C4C2313yes—
69C22×C22⋊C4C2313yes—
70C22×C42—42no—
71C22×C42—44no—
72C22×C42C2311yescharacteristic
73C22×C42C2313yes—
74C22×C42C2313yes—
75#399—42no—
76#397—212no—
77#397—412no—
78#396—44no—
79#386—412no—
80#382—46no—
81#381—46no—
82#379—212no—
83#379—46no—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2313yes—
88C2×#95C2312yes—
89C2×#95C2316yes—
90C2×#95C2311yescharacteristic
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2313yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C25—26no—
101C25—41no—
102C25—42no—
103C25—46no—
104C25—43no—
105C25C2413yes—
106C25C2411yescharacteristic
107C22×D4—26no—
108C22×D4—26no—
109C22×D4—212no—
110C22×D4—43no—
111C22×D4—46no—
112C23×C4—26no—
113C23×C4—26no—
114C23×C4—26no—
115C23×C4—212no—
116C23×C4—46no—
117C23×C4C2413yes—
118C23×C4C2416yes—
119C23×C4C2416yes—
120C23×C4C2413yes—
121C23×C4C2413yes—
122C23×C4C2413yes—
123C23×C4C2413yes—
124#123—448no—
125#120—416no—
126C2×C4⋊C4—26no—
127C2×C4⋊C4—26no—
128C2×C4⋊C4—26no—
129C2×C4⋊C4—26no—
130C2×C4⋊C4—43no—
131C2×C4⋊C4—43no—
132C2×C4⋊C4—43no—
133C2×C4⋊C4—46no—
134C2×C4⋊C4—43no—
135C2×C4⋊C4—46no—
136C2×C4⋊C4—46no—
137C2×C4⋊C4—43no—
138C2×C22⋊C4—26no—
139C2×C22⋊C4—26no—
140C2×C22⋊C4—26no—
141C2×C22⋊C4—26no—
142C2×C22⋊C4—26no—
143C2×C22⋊C4—212no—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—46no—
149C2×C22⋊C4—43no—
150C2×C22⋊C4—43no—
151C2×C22⋊C4—43no—
152C2×C22⋊C4—43no—
153C2×C22⋊C4—43no—
154C2×C22⋊C4—46no—
155C2×C22⋊C4—46no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—43no—
162C2×C22⋊C4—46no—
163C2×C22⋊C4—46no—
164C2×C22⋊C4—46no—
165C2×C22⋊C4—46no—
166C2×C22⋊C4—46no—
167C2×C22⋊C4#41112yes—
168C2×C42—43no—
169C2×C42—42no—
170C2×C42—46no—
171C2×C42—41no—
172C2×C42—43no—
173C2×C42—46no—
174C2×C42—416no—
175#95—26no—
176#95—212no—
177#95—212no—
178#95—212no—
179#95—46no—
180#95—46no—
181#95—46no—
182#95—46no—
183#95—43no—
184#95—43no—
185#95—46no—
186#95—46no—
187#95—46no—
188#95—42no—
189#95—46no—
190#95—41no—
191C24—23no—
192C24—23no—
193C24—26no—
194C24—28no—
195C24—26no—
196C24—26no—
197C24—23no—
198C24—23no—
199C24—26no—
200C24—26no—
201C24—26no—
202C24—26no—
203C24—44no—
204C24—412no—
205C24—86no—
206C24—86no—
207C24—812no—
208C24—812no—
209C24—86no—
210C24—812no—
211C24—86no—
212C24C2×#4116yes—
213C24H(5,𝔽2)12yes—
214C24H(5,𝔽2)14yes—
215C24C2511yescharacteristic
216C2×D4—412no—
217C2×D4—424no—
218C2×D4—412no—
219C2×D4—86no—
220C2×D4—812no—
221C22×C4—224no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—424no—
259C22×C4—424no—
260C22×C4—412no—
261C22×C4—412no—
262C22×C4C2×#4116yes—
263C22×C4C2×#4116yes—
264C22×C4H(5,𝔽2)16yes—
265C4⋊C4—412no—
266C4⋊C4—412no—
267C4⋊C4—86no—
268C4⋊C4—812no—
269C22⋊C4—412no—
270C22⋊C4—412no—
271C22⋊C4—412no—
272C22⋊C4—412no—
273C22⋊C4—812no—
274C22⋊C4—812no—
275C22⋊C4—812no—
276C22⋊C4—86no—
277C22⋊C4—812no—
278C22⋊C4—812no—
279C22⋊C4—86no—
280C42—412no—
281C42—44no—
282C42—412no—
283C23—26no—
284C23—212no—
285C23—26no—
286C23—212no—
287C23—26no—
288C23—26no—
289C23—26no—
290C23—212no—
291C23—212no—
292C23—41no—
293C23—43no—
294C23—46no—
295C23—43no—
296C23—43no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—46no—
301C23—46no—
302C23—46no—
303C23—43no—
304C23—46no—
305C23—43no—
306C23—43no—
307C23—46no—
308C23—46no—
309C23—46no—
310C23—46no—
311C23—43no—
312C23—46no—
313C23—816no—
314C23—88no—
315C23—824no—
316C23—848no—
317C23—824no—
318C23H(3,𝔽4)14yes—
319C23C22×#4113yes—
320C23C2×H(5,𝔽2)11yescharacteristic
321C23C2×H(5,𝔽2)13yes—
322C23C2×H(5,𝔽2)12yes—
323C23#58516yes—
324D4—824no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—26no—
328C2×C4—212no—
329C2×C4—212no—
330C2×C4—212no—
331C2×C4—448no—
332C2×C4—46no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—46no—
337C2×C4—43no—
338C2×C4—43no—
339C2×C4—46no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—43no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—43no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—46no—
353C2×C4—46no—
354C2×C4—43no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—43no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C22—212no—
366C22—412no—
367C22—44no—
368C22—42no—
369C22—46no—
370C22—46no—
371C22—412no—
372C22—412no—
373C22—46no—
374C22—46no—
375C22—832no—
376C22—86no—
377C22—86no—
378C22—86no—
379C22—86no—
380C22#344411yescharacteristic
381C22#349013yes—
382C22#350813yes—
383C22#351316yes—
384C22#352116yes—
385C22C2×H(3,𝔽4)11yescharacteristic
386C22#352913yes—
387C22#353313yes—
388C22#353416yes—
389C22#355113yes—
390C4—46no—
391C4—46no—
392C4—46no—
393C4—83no—
394C4—86no—
395C4—86no—
396C4—83no—
397C2—44no—
398C2—86no—
399C2#3541512yes—
400C2#3543811yescharacteristic
401C2#4556813yes—
402C2#4610216yes—
403C2#4627113yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16246C2semidirect
3. #15802C2semidirect
4. #15669C2semidirect
5. #15480C2semidirect
6. #15471C2semidirect
7. #14706C2semidirect
8. #14652C2semidirect
9. #13827C2semidirect
12. C23×C42C22semidirect
13. C2×#399C22semidirect
16. C2×#396C22semidirect
18. C2×#382C22semidirect
19. C2×#381C22semidirect
22. #1445C22semidirect
23. #1443C22semidirect
24. #1442C22semidirect
25. #1442C22semidirect
26. #1441C22semidirect
41. #1437C22semidirect
42. #1436C22semidirect
43. #1436C22semidirect
44. #1436C22semidirect
45. #1436C22semidirect
46. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#162462
#159482
#158022
#156692
#154802
#154712
#147062
#146522
#138272