Finite Group Databasebeta

← #480751

Subgroups — #480751

Order 512.

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

normal subgroups only

This group has 914 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#17365C212yesmaximal
3#11573C218yesmaximal, complemented
4#11511C214yesmaximal
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×#391C412yes—
15C2×#382C2212yes—
16C2×C4×C4⋊C4C411yescharacteristic
17#1467—22no—
18#1467C2214yes—
19#1463—22no—
20#1463C2214yes—
21#1461—22no—
22#1461C2214yes—
23#1324C22116yes—
24C23×Q8—21no—
25C23×Q8C2×C411yescharacteristic
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—41no—
28C2×C4⋊Q8—42no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8C2×C414yes—
31C2×C4×Q8—42no—
32C2×C4×Q8—41no—
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⋊C4D412yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45C22×C42D412yes—
46#432—24no—
47#432—42no—
48#431—24no—
49#431—42no—
50#423—42no—
51#407—24no—
52C4×M16—42no—
53#400—24no—
54#400—24no—
55#400—24no—
56#400—24no—
57#391—22no—
58#391—22no—
59#391—24no—
60#391—24no—
61#391—24no—
62#391—24no—
63#391C2×C418yes—
64#382—24no—
65#382C2×C414yes—
66#382C2314yes—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2×C414yes—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95—24no—
72#324—24no—
73#324—24no—
74#324—24no—
75#324C2318yes—
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×Q8C22⋊C411yescharacteristic
87C22×Q8C22×C412yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—44no—
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—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4C22×C414yes—
123C2×C4⋊C4C22×C411yescharacteristic
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2411yescharacteristic
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×C414yes—
141C2×C42C22×C412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C4⋊C8—44no—
145C4⋊C8—44no—
146C22⋊C8—28no—
147C22⋊C8—44no—
148#97—44no—
149C4×C8—44no—
150#95—24no—
151#95—28no—
152#95—28no—
153#95—28no—
154#95—28no—
155C24C2×C22⋊C411yescharacteristic
156C2×Q8—24no—
157C2×Q8—22no—
158C2×Q8—24no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—48no—
165C2×Q8—48no—
166C2×Q8—416no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—84no—
170C2×Q8—88no—
171C2×Q8—84no—
172C2×Q8#10114yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#9911yescharacteristic
197C22×C4#9912yes—
198C22×C4#9911yescharacteristic
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C23×C411yescharacteristic
206C22×C4C22×D412yes—
207M16—88no—
208C2×C8—48no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C42—22no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—28no—
233C42—44no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23C2×#9911yescharacteristic
239C23C2×#9911yescharacteristic
240C23C22×C22⋊C411yescharacteristic
241Q8—44no—
242Q8—48no—
243Q8—44no—
244Q8—88no—
245Q8—88no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—82no—
277C2×C4—88no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4#35514yes—
281C2×C4C2×#9912yes—
282C2×C4C2×#9912yes—
283C2×C4C2×#10112yes—
284C2×C4C2×#10114yes—
285C2×C4#56812yes—
286C8—88no—
287C22—24no—
288C22—24no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22#212912yes—
296C22C2×#35512yes—
297C22#287911yescharacteristic
298C22C22×#9911yescharacteristic
299C22C22×#10111yescharacteristic
300C4—22no—
301C4—22no—
302C4—22no—
303C4—24no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—84no—
309C4—84no—
310C4C2×#35514yes—
311C2—21no—
312C2—21no—
313C2—22no—
314C2—22no—
315C2#1739711yescharacteristic
316C2C2×#212911yescharacteristic
317C2C22×#35511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11573C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#173652
#115732
#115112