Finite Group Databasebeta

← #115659

Subgroups — #115659

Order 512.

1006 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2567C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#21799C211yesmaximal, characteristic
4#17449C212yesmaximal
5#7950C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2567—24no—
11#2567—24no—
12C2×#395—22no—
13C2×#391C412yes—
14C2×#389—22no—
15C2×#384C412yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1997—24no—
18#1966—24no—
19#1821—22no—
20#1784—22no—
21#1491—24no—
22#1487—24no—
23#1479—24no—
24#1285—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8D412yes—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8D414yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C42—21no—
46C22×C42C2×C412yes—
47C22×C42C2311yescharacteristic
48#432—22no—
49#432—22no—
50#432—22no—
51#432—22no—
52#431—22no—
53#431—22no—
54#431—22no—
55#431—22no—
56#423—22no—
57#423D414yes—
58C2×#101—28no—
59#407—22no—
60#407—22no—
61C4×M16—22no—
62C4×M16D414yes—
63#395—216no—
64#391—24no—
65#391—24no—
66#389—216no—
67#384—24no—
68#384—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71C2×#95—24no—
72C2×#95—21no—
73#343—28no—
74Q8⋊C8—216no—
75C22×Q8—24no—
76C22×Q8—21no—
77C22×Q8—22no—
78C22×Q8—24no—
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—22no—
83C22×Q8—42no—
84C22×Q8—84no—
85C22×Q8C22⋊C412yes—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C22⋊C414yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—24no—
94C2×M16—24no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4×Q8—24no—
104C4×Q8—24no—
105C4×Q8—24no—
106C4×Q8—24no—
107C4×Q8—42no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×Q8—42no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—216no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—88no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42—41no—
136C2×C42—44no—
137C2×C42C22⋊C414yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42#4112yes—
143C2×C42#4112yes—
144C4⋊C8—24no—
145C4⋊C8—24no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#101—88no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#97—24no—
154#97—42no—
155C4×C8—24no—
156C4×C8—42no—
157#95—216no—
158#95—24no—
159C24C2×C22⋊C411yescharacteristic
160C2×Q8—24no—
161C2×Q8—22no—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—48no—
165C2×Q8—48no—
166C2×Q8—42no—
167C2×Q8—48no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—84no—
172C2×Q8—88no—
173C2×Q8—88no—
174C2×Q8#10114yescomplemented
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C4×D412yes—
199C22×C4C4×D412yes—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4#12012yes—
203C22×C4#12011yescharacteristic
204C22×C4#12112yes—
205C22×C4#12112yes—
206C22×C4#12311yescharacteristic
207C22×C4#12311yescharacteristic
208M16—48no—
209M16—48no—
210C2×C8—44no—
211C2×C8—44no—
212C4⋊C4—216no—
213C4⋊C4—24no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—816no—
222C42—44no—
223C42—44no—
224C42—44no—
225C42—44no—
226C42#12012yes—
227C42#12112yes—
228C42#12112yes—
229C42#12112yes—
230C42#12112yes—
231C42#12312yes—
232C42#12412yes—
233C42#12712yes—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23#37911yescharacteristic
239C23#38611yescharacteristic
240C23#38611yescharacteristic
241Q8—28no—
242Q8—48no—
243Q8—44no—
244Q8—88no—
245Q8—88no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—28no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—416no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4#38612yes—
275C2×C4#39012yes—
276C2×C4#39212yes—
277C2×C4#39212yes—
278C2×C4#39412yes—
279C2×C4#39412yes—
280C2×C4C2×#10112yes—
281C2×C4#41312yes—
282C2×C4#45614yes—
283C8—84no—
284C8—84no—
285C22—24no—
286C22—24no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22#143811yescharacteristic
294C22#178411yescharacteristic
295C22#178511yescharacteristic
296C22#188612yes—
297C22#191112yes—
298C4—22no—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—48no—
303C4—44no—
304C4—84no—
305C4—82no—
306C4#190114yes—
307C4#200514yes—
308C2—21no—
309C2—22no—
310C2—22no—
311C2—21no—
312C2#1096811yescharacteristic
313C2#1097811yescharacteristic
314C2#1098911yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
174. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#218292
#217992
#174492
#79502