Finite Group Databasebeta

← #124994

Subgroups — #124994

Order 512.

666 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 82 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#14159C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7131C212yesmaximal, complemented
4#7104C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—22no—
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1436—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C2×C22⋊Q8—22no—
21C2×C22⋊Q8—22no—
22C2×C22⋊Q8—42no—
23C2×C22⋊Q8—41no—
24C2×C22⋊Q8—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C42—21no—
30C2×C22⋊C8—24no—
31C2×C22⋊C8—42no—
32C2×C22⋊C8—42no—
33C4×C22⋊C4—42no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—41no—
39C2×#95—41no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95D412yes—
44C2×#95C2311yescharacteristic
45C25C22⋊C411yescharacteristic
46C22×Q8—21no—
47C22×Q8—41no—
48C22×Q8—41no—
49C22×Q8—82no—
50C22×Q8M1612yes—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—42no—
64C23×C4—44no—
65C23×C4—82no—
66C23×C4—82no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—44no—
72C22⋊Q8—44no—
73C22⋊Q8—44no—
74C22⋊Q8—44no—
75C22⋊Q8—44no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4M1612yes—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—42no—
91C2×C42—82no—
92C22⋊C8—44no—
93C22⋊C8—44no—
94C22⋊C8—84no—
95C22⋊C8—84no—
96#95—41no—
97#95—42no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—44no—
102#95—44no—
103#95—82no—
104#95—82no—
105C24—22no—
106C24—22no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—22no—
111C24—42no—
112C24#9911yescharacteristic
113C24#9911yescharacteristic
114C24C2×C22⋊C411yescharacteristic
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—82no—
118C2×Q8—82no—
119C2×Q8—84no—
120C2×Q8—84no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—84no—
171C22×C4—88no—
172C22×C4—88no—
173C22×C4#10012yes—
174C22×C4C2×M1611yescharacteristic
175C22×C4C2×M1611yescharacteristic
176C22×C4#13514yes—
177C2×C8—84no—
178C2×C8—84no—
179C2×C8—84no—
180C4⋊C4—42no—
181C4⋊C4—48no—
182C4⋊C4—42no—
183C22⋊C4—48no—
184C22⋊C4—44no—
185C22⋊C4—44no—
186C42—84no—
187C23—22no—
188C23—21no—
189C23—21no—
190C23—24no—
191C23—24no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—41no—
208C23—42no—
209C23#35412yes—
210C23#35612yes—
211C23#40711yescharacteristic
212C23C2×#9911yescharacteristic
213C23C2×#10011yescharacteristic
214C23#41612yes—
215C23#42112yes—
216Q8—84no—
217Q8—84no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—88no—
251C2×C4—88no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C8—164no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—22no—
277C22—44no—
278C22—44no—
279C22—48no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#146211yescharacteristic
287C22#148812yes—
288C22#151612yes—
289C22#153611yescharacteristic
290C22#154311yescharacteristic
291C22#212112yes—
292C22C2×#35411yescharacteristic
293C22C2×#35611yescharacteristic
294C4—44no—
295C4—42no—
296C4—41no—
297C4—41no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—82no—
302C4—82no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2—42no—
307C2—42no—
308C2#764511yescharacteristic
309C2#1150211yescharacteristic
310C2#1150611yescharacteristic
311C2#1184512yes—
312C2#1185412yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7131C2semidirect
4. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#141592
#86982
#71312
#71042