Finite Group Databasebeta

← #6374596

Subgroups — #6374596

Order 512.

2880 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2#60124C212yesmaximal
3#21883C218yesmaximal, complemented
4#21830C212yesmaximal, complemented
5#21799C212yesmaximal, complemented
6#21753C214yesmaximal
7#21429C218yesmaximal, complemented
8#20727C214yesmaximal, complemented
9C2×#541—216no—
10C2×#541C418yes—
11C2×#541C2214yes—
12C2×#541C2214yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#126C414yes—
15C22×#125C414yes—
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×#117C2211yescharacteristic
20#2924—216no—
21#2921C2214yes—
22#2921C2214yes—
23#2920C2214yes—
24#2920C2214yes—
25#2905C2214yes—
26C2×#407C2214yes—
27#2873C2214yes—
28#1998C22116yescomplemented
29#1966C22116yescomplemented
30#1901C22132yes—
31#1821C2218yes—
32#1801C22116yescomplemented
33#1784C2218yes—
34C4×#101C22116yescomplemented
35C23×Q8C2312yes—
36C24×C4C2311yescharacteristic
37C22×M16C2318yes—
38#541—264no—
39#541—216no—
40#541—216no—
41#541—216no—
42#541—216no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8C2×C414yes—
45C2×C4⋊Q8C2314yes—
46C2×#126—216no—
47C2×#126—216no—
48C2×#126C2×C4116yes—
49C2×#126C2×C4116yes—
50C2×#125—216no—
51C2×#125—216no—
52C2×#125—28no—
53C2×#125C2×C4116yes—
54C2×#125C2×C4116yes—
55C2×#125C2×C414yes—
56C2×#125C2314yes—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8C2×C418yes—
60C2×C22⋊Q8C2318yes—
61C2×C4×Q8—28no—
62C2×C4×Q8—28no—
63C2×C4×Q8C2×C414yes—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8C2314yes—
66C2×C4×Q8C2314yes—
67C2×#117C2×C414yes—
68C2×#117D4116yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4C2×C414yes—
73C22×C4⋊C4C2×C414yes—
74C22×C4⋊C4C2×C414yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4C2×C414yes—
78C22×C22⋊C4C2×C412yes—
79C22×C42C2×C414yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#433—216no—
83#433—216no—
84#432—28no—
85#432—28no—
86#431—28no—
87#431—28no—
88#424—216no—
89#423C2318yes—
90#423C2318yes—
91C2×#101C23132yes—
92#407—28no—
93#407C2318yes—
94#407C2318yes—
95C2×C22⋊C8—28no—
96#405—216no—
97C4×M16C2318yes—
98C4×M16C2318yes—
99#343C23132yes—
100C25C22×C411yescharacteristic
101C22×Q8—24no—
102C22×Q8—24no—
103C22×Q8—48no—
104C22×Q8C22×C414yes—
105C22×Q8C2414yes—
106C23×C4—44no—
107C23×C4C22×C414yes—
108C23×C4C22×C414yes—
109C23×C4C22×C412yes—
110C23×C4C22×C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C412yes—
113C23×C4C2×D418yes—
114C23×C4C2411yescharacteristic
115C23×C4C2412yes—
116C2×M16—28no—
117C2×M16—416no—
118C2×M16C24116yes—
119C22×C8—28no—
120C4⋊Q8—216no—
121C4⋊Q8—28no—
122C4⋊Q8—28no—
123#126—264no—
124#126—264no—
125#125—264no—
126#125—264no—
127#125—216no—
128#125—28no—
129#125—28no—
130C22⋊Q8—232no—
131C22⋊Q8—216no—
132C22⋊Q8—216no—
133C4×Q8—216no—
134C4×Q8—216no—
135C4×Q8—216no—
136C4×Q8—216no—
137#117C22⋊C4132yes—
138#117C2×D418yes—
139#117C2×D418yes—
140#117C2×D418yes—
141#117C2×D418yes—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4C22⋊C418yes—
153C2×C4⋊C4C22⋊C418yes—
154C2×C4⋊C4C22×C418yes—
155C2×C4⋊C4C22×C418yes—
156C2×C4⋊C4C22×C4116yes—
157C2×C4⋊C4C22×C4116yes—
158C2×C4⋊C4C22×C4116yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C2414yes—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4C22⋊C418yes—
166C2×C22⋊C4C22⋊C418yes—
167C2×C22⋊C4C22×C4116yes—
168C2×C22⋊C4C22×C414yes—
169C2×C22⋊C4C22×C414yes—
170C2×C42—28no—
171C2×C42—28no—
172C2×C42C22×C418yes—
173C2×C42C22×C418yes—
174C2×C42C22×C412yes—
175C2×C42C22×C412yes—
176C2×C42C2×D418yes—
177C2×C42C2×D418yes—
178C2×C42C2411yescharacteristic
179C2×C42C2411yescharacteristic
180C2×C42C2412yes—
181C4⋊C8—216no—
182C4⋊C8—216no—
183#101—432no—
184C22⋊C8—216no—
185C22⋊C8—216no—
186#97—28no—
187#97—28no—
188C4×C8—28no—
189C4×C8—28no—
190C24—42no—
191C24—44no—
192C24C2×C22⋊C414yes—
193C24C23×C411yescharacteristic
194C24C23×C412yes—
195C2×Q8—28no—
196C2×Q8—28no—
197C2×Q8—216no—
198C2×Q8—48no—
199C2×Q8—416no—
200C2×Q8—416no—
201C22×C4—28no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—416no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C2×C22⋊C414yes—
217C22×C4C2×C22⋊C414yes—
218C22×C4C2×C22⋊C414yes—
219C22×C4C2×C22⋊C418yes—
220C22×C4C23×C418yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C23×C412yes—
223C22×C4C23×C418yes—
224C22×C4C23×C412yes—
225C22×C4C23×C412yes—
226C22×C4C23×C414yes—
227C22×C4C22×D414yes—
228C22×C4C22×D414yes—
229C22×C4C22×D414yes—
230C22×C4C2511yescharacteristic
231M16—232no—
232M16—416no—
233C2×C8—216no—
234C2×C8—416no—
235C4⋊C4—28no—
236C4⋊C4—216no—
237C4⋊C4—216no—
238C4⋊C4—232no—
239C4⋊C4—232no—
240C4⋊C4—232no—
241C4⋊C4—28no—
242C4⋊C4C2×C22⋊C4116yes—
243C4⋊C4C2×C22⋊C4116yes—
244C22⋊C4—232no—
245C22⋊C4C2×C22⋊C4116yes—
246C22⋊C4C2×C22⋊C4116yes—
247C42—216no—
248C42—216no—
249C42C2×C22⋊C418yes—
250C42C2×C22⋊C418yes—
251C42C22×D414yes—
252C42C22×D414yes—
253C42C22×D414yes—
254C42C22×D414yes—
255C23—24no—
256C23—22no—
257C23—28no—
258C23—24no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—88no—
263C23C22×C22⋊C414yes—
264C23C22×C22⋊C412yes—
265C23C24×C411yescharacteristic
266Q8—416no—
267Q8—416no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—216no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4C22×C22⋊C414yes—
291C2×C4C22×C22⋊C414yes—
292C2×C4C22×C22⋊C412yes—
293C2×C4C22×C22⋊C418yes—
294C2×C4C22×C22⋊C414yes—
295C2×C4C23×D412yes—
296C2×C4#58514yes—
297C8—416no—
298C22—28no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—28no—
303C22—24no—
304C22—88no—
305C22—84no—
306C22#288914yes—
307C22C23×C22⋊C411yescharacteristic
308C22#342812yes—
309C4—24no—
310C4—24no—
311C4—28no—
312C4—24no—
313C4—22no—
314C4—48no—
315C4—48no—
316C4#344914yes—
317C2—24no—
318C2—22no—
319C2—44no—
320C2#3404411yescharacteristic
321C2C2×#288912yes—
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21883C2semidirect
4. #21830C2semidirect
5. #21799C2semidirect
7. #21429C2semidirect
8. #20727C2semidirect
28. #1998C22semidirect
29. #1966C22semidirect
32. #1801C22semidirect
34. C4×#101C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
#601242
#218832
#218302
#217992
#217532
#214292
#207272