Finite Group Databasebeta

← #616216

Subgroups — #616216

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C216yesmaximal
2#15153C216yesmaximal, complemented
3#15150C216yesmaximal
4#14712C211yesmaximal, characteristic, complemented
5#14712C211yesmaximal, characteristic, complemented
6#14692C211yesmaximal, characteristic, complemented
7#14686C211yesmaximal, characteristic
8#14686C211yesmaximal, characteristic
9#14049C216yesmaximal, complemented
10C4×#384C211yesmaximal, characteristic, complemented
11C42×C22⋊C4C211yesmaximal, characteristic
12C22×#126C2214yes—
13C23×C42C2211yescharacteristic
14#2496—248no—
15C2×#400C22112yes—
16C2×#397C22112yes—
17C2×#389C2216yes—
18C2×#389C2216yes—
19C2×#387C22112yes—
20C2×#384C2216yes—
21C2×#384C2216yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380C2216yes—
27C2×#380C2216yes—
28C2×C4×C4⋊C4C2216yes—
29C2×C4×C22⋊C4C2213yes—
30C2×C4×C22⋊C4C2213yes—
31C2×C43C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1445C22124yes—
34C4⋊#95C2216yes—
35C4⋊#95C2216yes—
36C4⋊#95C2216yes—
37C4⋊#95C2216yes—
38C4⋊#95C2216yes—
39C4×#95C2213yes—
40C4×#95C2213yes—
41C24×C4C2313yes—
42C2×#126—248no—
43C22×C4⋊C4C23112yes—
44C22×C4⋊C4C2314yes—
45C22×C4⋊C4C2314yes—
46C22×C4⋊C4C2314yes—
47C22×C4⋊C4C2316yes—
48C22×C4⋊C4C2316yes—
49C22×C4⋊C4C2316yes—
50C22×C4⋊C4C2316yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2313yes—
54C22×C22⋊C4C2311yescharacteristic
55C22×C42—24no—
56C22×C42—24no—
57C22×C42—24no—
58C22×C42C2313yes—
59C22×C42C2314yes—
60C22×C42C2313yes—
61C22×C42C2313yes—
62C22×C42C2313yes—
63C22×C42C2313yes—
64C22×C42C2311yescharacteristic
65#400—248no—
66#397—248no—
67#389—224no—
68#389—224no—
69#387—248no—
70#384—224no—
71#384—224no—
72#384—24no—
73#384—24no—
74#384—24no—
75#384—24no—
76#380—224no—
77#380—224no—
78C4×C4⋊C4—224no—
79C4×C22⋊C4—212no—
80C4×C22⋊C4D4124yes—
81C2×#95C23112yes—
82C2×#95C23112yes—
83C2×#95C23112yes—
84C2×#95C23112yes—
85C2×#95C23112yes—
86C2×#95C2316yes—
87C2×#95C2313yes—
88C2×#95C2313yes—
89C43—24no—
90C43—24no—
91C25C2411yescharacteristic
92C23×C4—26no—
93C23×C4—26no—
94C23×C4—26no—
95C23×C4—26no—
96C23×C4—26no—
97C23×C4—26no—
98C23×C4C2×D4112yes—
99C23×C4C2414yes—
100C23×C4C2414yes—
101C23×C4C2414yes—
102C23×C4C2414yes—
103C23×C4C2413yes—
104C23×C4C2413yes—
105C23×C4C2411yescharacteristic
106C23×C4C2411yescharacteristic
107C23×C4C2413yes—
108C23×C4C2413yes—
109#126—464no—
110C2×C4⋊C4—224no—
111C2×C4⋊C4—224no—
112C2×C4⋊C4—224no—
113C2×C4⋊C4—212no—
114C2×C4⋊C4—212no—
115C2×C4⋊C4—212no—
116C2×C4⋊C4—212no—
117C2×C4⋊C4—224no—
118C2×C4⋊C4—248no—
119C2×C4⋊C4—212no—
120C2×C4⋊C4—212no—
121C2×C4⋊C4—212no—
122C2×C4⋊C4—212no—
123C2×C4⋊C4—212no—
124C2×C4⋊C4—212no—
125C2×C4⋊C4—212no—
126C2×C4⋊C4—212no—
127C2×C22⋊C4—212no—
128C2×C22⋊C4—224no—
129C2×C22⋊C4—224no—
130C2×C22⋊C4C2×D4112yes—
131C2×C22⋊C4C2×D4112yes—
132C2×C42—26no—
133C2×C42—26no—
134C2×C42—26no—
135C2×C42—26no—
136C2×C42—26no—
137C2×C42—26no—
138C2×C42—224no—
139C2×C42—26no—
140C2×C42—26no—
141C2×C42—26no—
142C2×C42—26no—
143C2×C42—26no—
144C2×C42—26no—
145C2×C42—416no—
146C2×C42C2×D4112yes—
147C2×C42C2×D4112yes—
148C2×C42#41112yes—
149C2×C42#4114yes—
150C2×C42#4114yes—
151C2×C42#4114yes—
152#95—224no—
153#95—224no—
154#95—224no—
155#95—224no—
156#95—224no—
157#95—212no—
158#95C2×D4112yes—
159#95C2×D4112yes—
160C24—23no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—23no—
165C24—23no—
166C24C22×D416yes—
167C24C2511yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—23no—
171C22×C4—23no—
172C22×C4—23no—
173C22×C4—23no—
174C22×C4—26no—
175C22×C4—23no—
176C22×C4—26no—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—26no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—224no—
186C22×C4—224no—
187C22×C4—224no—
188C22×C4—212no—
189C22×C4—212no—
190C22×C4—212no—
191C22×C4—212no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—212no—
195C22×C4—212no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4—412no—
201C22×C4—412no—
202C22×C4—412no—
203C22×C4—412no—
204C22×C4C22×D416yes—
205C22×C4C22×D416yes—
206C22×C4C22×D416yes—
207C22×C4C22×D416yes—
208C22×C4C22×D416yes—
209C22×C4C22×D416yes—
210C22×C4C2×#41112yes—
211C22×C4C2×#4116yes—
212C22×C4C2×#4116yes—
213C22×C4C2×#4116yes—
214C22×C4C2×#4116yes—
215C22×C42−1+416yes—
216C22×C42−1+416yes—
217C4⋊C4—416no—
218C4⋊C4—416no—
219C4⋊C4—416no—
220C4⋊C4—448no—
221C4⋊C4—424no—
222C4⋊C4—424no—
223C4⋊C4—424no—
224C4⋊C4—424no—
225C22⋊C4—224no—
226C22⋊C4—416no—
227C22⋊C4—416no—
228C22⋊C4#127116yes—
229C42—224no—
230C42—224no—
231C42—224no—
232C42—412no—
233C42—416no—
234C42—412no—
235C42#126116yescomplemented
236C23—26no—
237C23—26no—
238C23—26no—
239C23—26no—
240C23—26no—
241C23—26no—
242C23—41no—
243C23—43no—
244C23—43no—
245C23—43no—
246C23—46no—
247C23C2×#12714yes—
248C23C23×D413yes—
249C23C22×#4111yescharacteristic
250C23C22×#4111yescharacteristic
251C23C22×#4111yescharacteristic
252C23C22×#4113yes—
253C23C2×2−1+413yes—
254C23C2×2−1+413yes—
255C2×C4—26no—
256C2×C4—26no—
257C2×C4—26no—
258C2×C4—212no—
259C2×C4—26no—
260C2×C4—26no—
261C2×C4—26no—
262C2×C4—212no—
263C2×C4—26no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—43no—
270C2×C4—43no—
271C2×C4—43no—
272C2×C4—46no—
273C2×C4—43no—
274C2×C4—46no—
275C2×C4—448no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—412no—
281C2×C4—412no—
282C2×C4—412no—
283C2×C4—412no—
284C2×C4—412no—
285C2×C4—412no—
286C2×C4—412no—
287C2×C4—412no—
288C2×C4C2×#126112yes—
289C2×C4C2×#12714yes—
290C2×C4C2×#12714yes—
291C2×C4#532112yes—
292C2×C4#532112yes—
293C2×C4#541112yes—
294C2×C4#547112yes—
295C2×C4#547112yes—
296C2×C4#552112yes—
297C2×C4#556112yes—
298C22—24no—
299C22—24no—
300C22—24no—
301C22—46no—
302C22—46no—
303C22—46no—
304C22—46no—
305C22C22×#12611yescharacteristic
306C22C22×#12711yescharacteristic
307C22C2×#53213yes—
308C22C2×#53213yes—
309C22C2×#54113yes—
310C22C2×#54713yes—
311C22C2×#54713yes—
312C22C2×#55213yes—
313C22C2×#55613yes—
314C22#349913yes—
315C22#349913yes—
316C22#350916yes—
317C4—24no—
318C4—24no—
319C4—212no—
320C4—46no—
321C4—46no—
322C4—84no—
323C4—84no—
324C4—84no—
325C4—84no—
326C4#2496124yescomplemented
327C2—44no—
328C2C2×#249613yes—
329C2#3531913yes—
330C2#3617911yescharacteristic
331C2#3617911yescharacteristic
332C2#3618113yes—
333C2#3618813yes—
334C2#3664411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #15153C2semidirect
4. #14712C2semidirect
5. #14712C2semidirect
6. #14692C2semidirect
9. #14049C2semidirect
10. C4×#384C2semidirect
235. C42#126semidirect
326. C4#2496semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
#151532
#151502
#147122
#146922
#146862
#140492
C4×#3842
C42×C22⋊C42