Finite Group Databasebeta

← #184987

Subgroups — #184987

Order 512.

2154 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2#17449C214yesmaximal, complemented
3#17357C212yesmaximal
4#8327C218yesmaximal, complemented
5C2×#541—28no—
6C2×#541—28no—
7C2×#541—24no—
8C2×#541C414yes—
9C2×#541C2214yes—
10C22×C4⋊Q8C411yescharacteristic
11C22×#126—22no—
12C22×#125—22no—
13C22×#125C411yescharacteristic
14C22×C22⋊Q8C412yes—
15C22×C4×Q8C2212yes—
16C22×#117C2211yescharacteristic
17#2920C2214yes—
18#1491—28no—
19#1487—28no—
20#1479C22116yescomplemented
21#1464—24no—
22#1463—22no—
23#1462—22no—
24#1458C2218yes—
25C23×Q8C2×C412yes—
26C24×C4C2×C411yescharacteristic
27#541—232no—
28#541—232no—
29#541—232no—
30#541—232no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8C2×C414yes—
35C2×#126—216no—
36C2×#126—216no—
37C2×#126—28no—
38C2×#126—28no—
39C2×#125—216no—
40C2×#125—216no—
41C2×#125—28no—
42C2×#125—24no—
43C2×#125—24no—
44C2×#125—28no—
45C2×#125—22no—
46C2×#125C2×C414yes—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8C2×C418yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—22no—
57C2×C4×Q8C2×C414yes—
58C2×C4×Q8D418yes—
59C2×C4×Q8D418yes—
60C2×C4×Q8C2314yes—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117C2×C412yes—
64C2×#117D414yes—
65C2×#117C2312yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4D412yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—22no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—24no—
79#431—24no—
80#431C2318yes—
81#423—24no—
82#407—24no—
83C4×M16—24no—
84Q8⋊C8—232no—
85#324—28no—
86#323—28no—
87C25C22⋊C411yescharacteristic
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—48no—
92C22×Q8C22⋊C414yes—
93C22×Q8C22⋊C414yes—
94C22×Q8C22×C414yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C412yes—
103C23×C4C22×C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C2×M16—44no—
107C4⋊Q8—28no—
108C4⋊Q8—28no—
109C4⋊Q8—216no—
110#126—264no—
111#126—264no—
112#125—264no—
113#125—264no—
114#125—28no—
115#125—28no—
116#125—216no—
117C22⋊Q8—216no—
118C22⋊Q8—216no—
119C22⋊Q8—216no—
120C22⋊Q8—216no—
121C4×Q8—28no—
122C4×Q8—28no—
123C4×Q8—28no—
124C4×Q8—28no—
125C4×Q8—28no—
126C4×Q8—28no—
127C4×Q8C22⋊C4116yes—
128C4×Q8C2×D4116yes—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4C22⋊C414yes—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C414yes—
162C2×C22⋊C4C2×D414yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—28no—
167C2×C42—28no—
168C2×C42—24no—
169C2×C42C22⋊C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D414yes—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—28no—
177C4⋊C8—44no—
178C22⋊C8—28no—
179C22⋊C8—44no—
180#97—44no—
181C4×C8—28no—
182C24—22no—
183C24—42no—
184C24—44no—
185C24C2×C22⋊C411yescharacteristic
186C24C2×C22⋊C412yes—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—216no—
191C2×Q8—48no—
192C2×Q8—48no—
193C2×Q8—48no—
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8C2×C22⋊C418yes—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—44no—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C414yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4#12014yes—
226C22×C4#12014yes—
227C22×C4C23×C411yescharacteristic
228C22×C4C22×D412yes—
229M16—88no—
230C2×C8—48no—
231C4⋊C4—232no—
232C4⋊C4—232no—
233C4⋊C4—232no—
234C4⋊C4—24no—
235C4⋊C4—216no—
236C4⋊C4—28no—
237C4⋊C4—216no—
238C4⋊C4—28no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4C2×C22⋊C418yes—
242C22⋊C4—232no—
243C22⋊C4—28no—
244C22⋊C4—28no—
245C42—24no—
246C42—216no—
247C42—216no—
248C42—22no—
249C42—22no—
250C42C2×C22⋊C414yes—
251C42#12014yes—
252C42#12014yes—
253C42C22×D414yes—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—24no—
258C23—44no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—88no—
263C23#37912yes—
264C23C22×C22⋊C411yescharacteristic
265Q8—216no—
266Q8—48no—
267Q8—48no—
268Q8—48no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4#37914yes—
295C2×C4#37911yescharacteristic
296C2×C4#37911yescharacteristic
297C2×C4C22×C22⋊C412yes—
298C2×C4C2×#12014yes—
299C2×C4#56814yes—
300C8—88no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—42no—
306C22—44no—
307C22—84no—
308C22—88no—
309C22#179614yes—
310C22C2×#37911yescharacteristic
311C22#287912yes—
312C4—21no—
313C4—21no—
314C4—24no—
315C4—24no—
316C4—42no—
317C4—48no—
318C4—44no—
319C4—48no—
320C4#287914yes—
321C2—24no—
322C2—22no—
323C2—82no—
324C2#2058011yescharacteristic
325C2C2×#179612yes—
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17449C2semidirect
4. #8327C2semidirect
20. #1479C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
#174492
#173572
#83272