Finite Group Databasebeta

← #96969

Subgroups — #96969

Order 512.

744 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 134 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
1#34271C211yesmaximal, characteristic
2#11346C211yesmaximal, characteristic
3#8701C211yesmaximal, characteristic
4#7579C212yesmaximal
5#7579C212yesmaximal
6C2×#544C411yescharacteristic
7#2370—22no—
8#2370—22no—
9#2369—22no—
10#2365—22no—
11#2329—22no—
12#2316C411yescharacteristic
13C4×C22⋊Q8—22no—
14#2296C2212yes—
15#2274C412yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1817C2212yes—
18C2×#336—22no—
19C2×#336C2211yescharacteristic
20C2×#336C2211yescharacteristic
21#1280—22no—
22#1271—22no—
23#1271—22no—
24#1271—22no—
25#1271—22no—
26C23×C8—21no—
27C23×C8—21no—
28#544—24no—
29#544—22no—
30#544C814yescomplemented
31C2×C4⋊Q8—22no—
32C2×#125—22no—
33C2×C22⋊Q8—22no—
34C2×C4×Q8—22no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—22no—
42#391—24no—
43#391—22no—
44#389—24no—
45#389—22no—
46#389—22no—
47#389—22no—
48#389D412yes—
49#389D412yes—
50#389D412yes—
51#389D412yes—
52#387—24no—
53#387—24no—
54#387—24no—
55#385—22no—
56#385—22no—
57#380—22no—
58C4×C4⋊C4—24no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C22⋊C4—24no—
69C2×#95D412yes—
70#336—22no—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—44no—
75C22×Q8—22no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—42no—
92C4⋊Q8—28no—
93C4⋊Q8—28no—
94#125—28no—
95#125—28no—
96C22⋊Q8—28no—
97C22⋊Q8—28no—
98C4×Q8—82no—
99C4×Q8—82no—
100#117—22no—
101#117—22no—
102#117—44no—
103#117C2×C814yes—
104#117M1614yes—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C2×C812yes—
122C2×C4⋊C4M1612yes—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—24no—
129C2×C42—24no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C22⋊C8—44no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—22no—
145C24C2×C22⋊C411yescharacteristic
146C2×Q8—28no—
147C2×Q8—82no—
148C2×Q8—82no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4C22⋊C814yes—
172C22×C4#10114yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4#12012yes—
175C22×C4#12011yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4C22×C811yescharacteristic
178C22×C4C2×M1612yes—
179C22×C4C2×M1611yescharacteristic
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—28no—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4C22⋊C814yes—
211C4⋊C4C22⋊C814yes—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—44no—
215C42—24no—
216C42—24no—
217C42—42no—
218C42—42no—
219C42—44no—
220C42—42no—
221C42—42no—
222C42—82no—
223C42—84no—
224C42—84no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23#37911yescharacteristic
230C23C2×C22⋊C811yescharacteristic
231C23#40711yescharacteristic
232C23#41012yes—
233C23C2×#10112yes—
234Q8—88no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4C2×C22⋊C812yes—
275C2×C4#40712yes—
276C2×C4#45311yescharacteristic
277C2×C4#45411yescharacteristic
278C2×C4#45511yescharacteristic
279C2×C4#45611yescharacteristic
280C8—82no—
281C8—82no—
282C8—162no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22#146112yes—
292C22#146712yes—
293C22#177911yescharacteristic
294C22#178211yescharacteristic
295C22#178411yescharacteristic
296C22#179211yescharacteristic
297C22#179311yescharacteristic
298C22#179411yescharacteristic
299C22#179411yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—44no—
309C4—44no—
310C4—44no—
311C4—82no—
312C4—82no—
313C2—22no—
314C2—22no—
315C2#830311yescharacteristic
316C2#831811yescharacteristic
317C2#831911yescharacteristic
318C2#845411yescharacteristic
319C2#845711yescharacteristic
320C2#846011yescharacteristic
321C2#846311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #544C8semidirect

Maximal subgroups

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

subgroupindex
#342712
#113462
#87012
#75792