Finite Group Databasebeta

← #115657

Subgroups — #115657

Order 512.

1066 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1066 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×#2546C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17449C212yesmaximal
5#7950C212yesmaximal
6C22×C4⋊Q8C412yes—
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2546—24no—
12#2546—24no—
13C2×#398—22no—
14C2×#391C412yes—
15C2×#389—22no—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1997—24no—
18#1997—24no—
19#1821—22no—
20#1821—22no—
21#1491—24no—
22#1487—24no—
23#1479—24no—
24#1285—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8C2×C412yes—
27C2×C4⋊Q8—24no—
28C2×C4⋊Q8—24no—
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—48no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8D414yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D412yes—
48C22×C42—21no—
49C22×C42C2×C412yes—
50C22×C42C2311yescharacteristic
51#432—22no—
52#432—22no—
53#432—22no—
54#432—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#423—22no—
60#423D414yes—
61#407—22no—
62#407—22no—
63C4×M16—22no—
64C4×M16D414yes—
65#398—216no—
66#391—24no—
67#391—24no—
68#389—216no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71C2×#95—24no—
72C2×#95—21no—
73#343—28no—
74#343—28no—
75Q8⋊C8—216no—
76C22×Q8—24no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—24no—
81C22×Q8—24no—
82C22×Q8—22no—
83C22×Q8—22no—
84C22×Q8—42no—
85C22×Q8—44no—
86C22×Q8—44no—
87C22×Q8C22⋊C412yes—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4C22⋊C414yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—24no—
97C4⋊Q8—416no—
98C4⋊Q8—48no—
99C4⋊Q8—48no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—24no—
111C4×Q8—24no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—216no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—416no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—41no—
139C2×C42—42no—
140C2×C42—41no—
141C2×C42—44no—
142C2×C42C22⋊C414yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#97—24no—
158#97—42no—
159C4×C8—24no—
160C4×C8—42no—
161#95—216no—
162#95—24no—
163C24C2×C22⋊C411yescharacteristic
164C2×Q8—24no—
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—48no—
169C2×Q8—48no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—88no—
179C2×Q8—88no—
180C2×Q8#10014yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—28no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—44no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C4×D412yes—
206C22×C4C4×D412yes—
207C22×C4#12011yescharacteristic
208C22×C4#12012yes—
209C22×C4#12012yes—
210C22×C4#12011yescharacteristic
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215M16—48no—
216M16—48no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—216no—
220C4⋊C4—24no—
221C4⋊C4—432no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C42—44no—
230C42—44no—
231C42—44no—
232C42—44no—
233C42#12012yes—
234C42#12112yes—
235C42#12112yes—
236C42#12112yes—
237C42#12112yes—
238C42#12312yes—
239C42#12412yes—
240C42#12712yes—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23#37911yescharacteristic
246C23#38611yescharacteristic
247C23#38611yescharacteristic
248Q8—28no—
249Q8—48no—
250Q8—44no—
251Q8—88no—
252Q8—88no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—416no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4#38612yes—
283C2×C4#39012yes—
284C2×C4#39212yes—
285C2×C4#39212yes—
286C2×C4#39412yes—
287C2×C4#39412yes—
288C2×C4C2×#10012yes—
289C2×C4#41312yes—
290C2×C4#45614yes—
291C8—84no—
292C8—84no—
293C22—24no—
294C22—24no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22#143811yescharacteristic
302C22#178311yescharacteristic
303C22#178511yescharacteristic
304C22#188612yes—
305C22#191212yes—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—48no—
311C4—44no—
312C4—84no—
313C4—82no—
314C4#190014yes—
315C4#200514yes—
316C2—21no—
317C2—22no—
318C2—22no—
319C2—21no—
320C2#1096711yescharacteristic
321C2#1097811yescharacteristic
322C2#1099011yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#25462
#218292
#174492
#79502