Finite Group Databasebeta

← #4030606

Subgroups — #4030606

Order 512.

2758 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16666C213yesmaximal, complemented
2#16297C211yesmaximal, characteristic
3#15996C216yesmaximal, complemented
4#15484C216yesmaximal, complemented
5#14916C213yesmaximal, complemented
6#14508C213yesmaximal, complemented
7#14172C213yesmaximal, complemented
8#14127C213yesmaximal
9#13903C213yesmaximal
10C22×#125C2213yescomplemented
11C22×C4×Q8C2211yescharacteristic, complemented
12C23×C22⋊C4C2213yes—
13C2×#401C2212yescomplemented
14C2×#393C2213yes—
15C2×#385C2216yes—
16C2×#385C2216yes—
17C2×#381C2213yes—
18C2×#381C2213yes—
19C2×#380C2213yes—
20C2×#379C2213yes—
21C2×#379C2211yescharacteristic
22C2×C4×C22⋊C4C2213yes—
23#1446C2213yescomplemented
24#1445C2216yescomplemented
25#1444C2211yescharacteristic, complemented
26#1444C2213yescomplemented
27#1443C2216yescomplemented
28#1442C2213yescomplemented
29#1442C2216yescomplemented
30#1441C2216yescomplemented
31#1441C2216yescomplemented
32#1439C2213yes—
33#1439C2216yes—
34#1439C2213yes—
35#1439C2216yes—
36#1439C2216yes—
37#1439C2213yes—
38#1439C2216yes—
39#1437C2216yes—
40#1437C2216yes—
41#1436C2216yescomplemented
42#1436C2216yescomplemented
43#1434C2213yescomplemented
44#1434C2213yescomplemented
45C22⋊#95C2216yes—
46C22⋊#95C2216yes—
47C26C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C24×C4C2313yes—
50C24×C4C2313yes—
51C2×#125—46no—
52C2×#125—412no—
53C2×C4×Q8—46no—
54C22×C4⋊C4C2313yes—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2316yes—
58C22×C4⋊C4C2313yes—
59C22×C22⋊C4—212no—
60C22×C22⋊C4—46no—
61C22×C22⋊C4—46no—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2313yes—
68C22×C22⋊C4C2313yes—
69C22×C22⋊C4C2313yes—
70C22×C42C2313yes—
71C22×C42C2313yes—
72#401—44no—
73#393—212no—
74#385—412no—
75#385—412no—
76#381—212no—
77#381—46no—
78#380—46no—
79#379—212no—
80#379—42no—
81C4×C22⋊C4—46no—
82C2×#95C2313yes—
83C2×#95C2316yes—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2316yes—
88C2×#95C2312yes—
89C2×#95C2313yes—
90C2×#95C2313yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2311yescharacteristic
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C25—26no—
101C25—26no—
102C25—43no—
103C25—43no—
104C25—41no—
105C25—41no—
106C25#4114yes—
107C25C2413yes—
108C22×Q8—26no—
109C22×Q8—43no—
110C22×Q8—41no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4—26no—
114C23×C4—26no—
115C23×C4—26no—
116C23×C4—26no—
117C23×C4—43no—
118C23×C4—43no—
119C23×C4—43no—
120C23×C4—43no—
121C23×C4—46no—
122C23×C4—46no—
123C23×C4C2416yes—
124C23×C4C2413yes—
125C23×C4C2416yes—
126C23×C4C2413yes—
127C23×C4C2413yes—
128C23×C4C2411yescharacteristic
129C23×C4C2413yes—
130C23×C4C2413yes—
131#125—448no—
132C4×Q8—416no—
133C2×C4⋊C4—212no—
134C2×C4⋊C4—212no—
135C2×C4⋊C4—212no—
136C2×C4⋊C4—212no—
137C2×C4⋊C4—212no—
138C2×C4⋊C4—43no—
139C2×C4⋊C4—43no—
140C2×C4⋊C4—46no—
141C2×C4⋊C4—46no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—43no—
144C2×C4⋊C4—46no—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—26no—
150C2×C22⋊C4—212no—
151C2×C22⋊C4—212no—
152C2×C22⋊C4—212no—
153C2×C22⋊C4—212no—
154C2×C22⋊C4—26no—
155C2×C22⋊C4—26no—
156C2×C22⋊C4—26no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—43no—
160C2×C22⋊C4—43no—
161C2×C22⋊C4—43no—
162C2×C22⋊C4—46no—
163C2×C22⋊C4—46no—
164C2×C22⋊C4—448no—
165C2×C22⋊C4—43no—
166C2×C22⋊C4—43no—
167C2×C22⋊C4—43no—
168C2×C22⋊C4—43no—
169C2×C22⋊C4#41112yes—
170C2×C42—26no—
171C2×C42—26no—
172C2×C42—46no—
173C2×C42—46no—
174#95—212no—
175#95—212no—
176#95—212no—
177#95—26no—
178#95—212no—
179#95—41no—
180#95—43no—
181#95—43no—
182#95—43no—
183#95—43no—
184#95—46no—
185#95—46no—
186#95—46no—
187#95—46no—
188#95—46no—
189#95—46no—
190#95—42no—
191#95—46no—
192C24—26no—
193C24—23no—
194C24—26no—
195C24—26no—
196C24—23no—
197C24—23no—
198C24—26no—
199C24—23no—
200C24—224no—
201C24—412no—
202C24—44no—
203C24—412no—
204C24—424no—
205C24—412no—
206C24—86no—
207C24—86no—
208C24—86no—
209C24—86no—
210C24—86no—
211C24—82no—
212C24C2×#4116yes—
213C24C2×#4116yes—
214C242−1+416yes—
215C24C2511yescharacteristic
216C2×Q8—412no—
217C2×Q8—412no—
218C2×Q8—44no—
219C22×C4—26no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—224no—
238C22×C4—26no—
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—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—412no—
258C22×C4—412no—
259C22×C4—412no—
260C22×C4—86no—
261C22×C4—86no—
262C22×C4—812no—
263C22×C4—812no—
264C22×C4—86no—
265C22×C4—86no—
266C22×C4—86no—
267C22×C4—86no—
268C22×C4C2×#4116yes—
269C22×C4C2×#4116yes—
270C22×C4H(5,𝔽2)12yes—
271C22×C4H(5,𝔽2)16yes—
272C22×C4H(5,𝔽2)16yes—
273C4⋊C4—412no—
274C4⋊C4—424no—
275C4⋊C4—412no—
276C4⋊C4—424no—
277C4⋊C4—412no—
278C22⋊C4—412no—
279C22⋊C4—424no—
280C22⋊C4—424no—
281C22⋊C4—412no—
282C22⋊C4—424no—
283C22⋊C4—424no—
284C22⋊C4—86no—
285C22⋊C4—86no—
286C42—412no—
287C42—412no—
288C23—26no—
289C23—212no—
290C23—212no—
291C23—212no—
292C23—26no—
293C23—26no—
294C23—26no—
295C23—212no—
296C23—212no—
297C23—26no—
298C23—212no—
299C23—26no—
300C23—448no—
301C23—43no—
302C23—46no—
303C23—46no—
304C23—46no—
305C23—46no—
306C23—43no—
307C23—43no—
308C23—46no—
309C23—43no—
310C23—46no—
311C23—824no—
312C23—824no—
313C23—88no—
314C23—88no—
315C23—88no—
316C23—824no—
317C23#552112yes—
318C23C22×#4111yescharacteristic
319C23C22×#4113yes—
320C23C2×H(5,𝔽2)11yescharacteristic
321C23C2×H(5,𝔽2)13yes—
322C23C2×H(5,𝔽2)13yes—
323C23C2×2−1+413yes—
324C23#58511yescharacteristic
325Q8—416no—
326C2×C4—212no—
327C2×C4—26no—
328C2×C4—26no—
329C2×C4—26no—
330C2×C4—26no—
331C2×C4—212no—
332C2×C4—26no—
333C2×C4—26no—
334C2×C4—212no—
335C2×C4—26no—
336C2×C4—43no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—46no—
340C2×C4—41no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—46no—
344C2×C4—43no—
345C2×C4—43no—
346C2×C4—43no—
347C2×C4—43no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—43no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—43no—
363C2×C4—46no—
364C2×C4—43no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—824no—
369C2×C4—824no—
370C22—212no—
371C22—224no—
372C22—412no—
373C22—46no—
374C22—46no—
375C22—412no—
376C22—46no—
377C22—46no—
378C22—46no—
379C22—46no—
380C22—412no—
381C22—832no—
382C22#345911yescharacteristic
383C22C2×#55213yes—
384C22#348113yes—
385C22#349013yes—
386C22#350513yes—
387C22#350813yes—
388C22#352016yes—
389C22#353213yes—
390C22#353211yescharacteristic
391C22#353613yes—
392C22#353813yes—
393C22#356113yes—
394C4—42no—
395C4—46no—
396C4—46no—
397C4—46no—
398C4—46no—
399C4—86no—
400C4—83no—
401C4—86no—
402C2—412no—
403C2#3954013yes—
404C2#4437813yes—
405C2#4699611yescharacteristic
406C2#5169113yes—
407C2#5898711yescharacteristic
408C2#5928513yes—
409C2#6003711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16666C2semidirect
3. #15996C2semidirect
4. #15484C2semidirect
5. #14916C2semidirect
6. #14508C2semidirect
7. #14172C2semidirect
10. C22×#125C22semidirect
11. C22×C4×Q8C22semidirect
13. C2×#401C22semidirect
23. #1446C22semidirect
24. #1445C22semidirect
25. #1444C22semidirect
26. #1444C22semidirect
27. #1443C22semidirect
28. #1442C22semidirect
29. #1442C22semidirect
30. #1441C22semidirect
31. #1441C22semidirect
41. #1436C22semidirect
42. #1436C22semidirect
43. #1434C22semidirect
44. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#166662
#162972
#159962
#154842
#149162
#145082
#141722
#141272
#139032