Finite Group Databasebeta

← #496403

Subgroups — #496403

Order 512.

942 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C2×#2568C211yesmaximal, characteristic
2#20727C212yesmaximal
3#13301C218yesmaximal, complemented
4#12860C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7#2920—22no—
8#2568—22no—
9#2568—22no—
10#2568C414yes—
11#2568C2214yes—
12C2×#400—22no—
13C2×#391—22no—
14C2×#391C2212yes—
15C2×#382C412yes—
16C2×C4×C4⋊C4C411yescharacteristic
17#1821—22no—
18#1801—22no—
19#1801C2214yes—
20#1784C2214yes—
21C4×#101—22no—
22C4×#101C2214yes—
23#1394C22116yes—
24C23×Q8—21no—
25C23×Q8C2311yescharacteristic
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—41no—
28C2×C4⋊Q8—42no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8C2×C414yes—
31C2×C4×Q8—41no—
32C2×C4×Q8—42no—
33C2×C4×Q8—41no—
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8D412yes—
36C2×C4×Q8D412yes—
37C2×C4×Q8C2312yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D412yes—
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45C22×C42D412yes—
46#432—42no—
47#432—42no—
48#431—42no—
49#431—42no—
50#423—24no—
51C2×#101—24no—
52C2×#101C2318yes—
53#407—24no—
54C4×M16—24no—
55#400—24no—
56#400—24no—
57#400—24no—
58#400—24no—
59#391—24no—
60#391—24no—
61#391—24no—
62#391—24no—
63#391—24no—
64#391C2×C414yes—
65#391C2314yes—
66#382—22no—
67#382—22no—
68#382C2×C418yes—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2×C414yes—
71C2×#95—21no—
72C2×#95—24no—
73C2×#95—24no—
74#343—24no—
75#343—24no—
76C22×Q8—24no—
77C22×Q8—24no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—42no—
81C22×Q8—48no—
82C22×Q8—42no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8C22×C411yescharacteristic
86C22×Q8C2×D412yes—
87C22×Q8C2411yescharacteristic
88C23×C4—22no—
89C23×C4—22no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—28no—
96C4⋊Q8—44no—
97C4⋊Q8—44no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8—42no—
105C4×Q8—42no—
106C4×Q8—42no—
107C4×Q8—42no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22×C414yes—
125C2×C4⋊C4C22×C412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C42—21no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—44no—
137C2×C42C22⋊C412yes—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C412yes—
140C2×C42C22×C412yes—
141C2×C42C22×C414yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C4⋊C8—44no—
145C4⋊C8—44no—
146#101—216no—
147#101—48no—
148C22⋊C8—44no—
149C22⋊C8—44no—
150#97—44no—
151C4×C8—44no—
152#95—24no—
153#95—28no—
154#95—28no—
155#95—28no—
156#95—28no—
157C24C2×C22⋊C411yescharacteristic
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—48no—
167C2×Q8—48no—
168C2×Q8—416no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—84no—
172C2×Q8—88no—
173C2×Q8—84no—
174C2×Q82−1+414yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—48no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#9911yescharacteristic
199C22×C4#9912yes—
200C22×C4#9911yescharacteristic
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C23×C411yescharacteristic
208C22×C4C22×D412yes—
209M16—48no—
210M16—48no—
211C2×C8—48no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C4⋊C4—44no—
222C4⋊C4—48no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—44no—
236C42—44no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23C2×#9911yescharacteristic
242C23C2×#9911yescharacteristic
243C23C22×C22⋊C411yescharacteristic
244Q8—44no—
245Q8—48no—
246Q8—44no—
247Q8—88no—
248Q8—88no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—88no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4#35514yes—
284C2×C4C2×#9912yes—
285C2×C4C2×#9912yes—
286C2×C4#41012yes—
287C2×C4#51912yes—
288C2×C4#53614yes—
289C8—88no—
290C22—24no—
291C22—24no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22C2×#35512yes—
299C22#213512yes—
300C22#231011yescharacteristic
301C22C22×#9911yescharacteristic
302C22C2×#41011yescharacteristic
303C4—24no—
304C4—22no—
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4—84no—
313C4C2×#35514yes—
314C2—21no—
315C2—21no—
316C2—22no—
317C2—22no—
318C2#2182511yescharacteristic
319C2C22×#35511yescharacteristic
320C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13301C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#207272
#133012
#128602