Finite Group Databasebeta

← #127380

Subgroups — #127380

Order 512.

777 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 136 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#21786C211yesmaximal, characteristic
2#20722C212yesmaximal
3#7237C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C2×#431C411yescharacteristic, complemented
7#2280—22no—
8#1954—22no—
9#1954C412yescomplemented
10#1954C2212yes—
11#1819C414yescomplemented
12#1799—22no—
13#1799C2214yes—
14#1781—22no—
15#1779—22no—
16C4×#99—22no—
17C4×#99C414yescomplemented
18C23×D4D411yescharacteristic
19C24×C4D412yes—
20C2×C4×D4—42no—
21C2×C4×D4—42no—
22C2×C4×D4C2×C412yes—
23C2×C4×D4C2×C411yescharacteristic
24C2×C4×D4D412yes—
25C2×C4×D4D411yescharacteristic
26C2×C4×D4Q811yescharacteristic
27C2×C4×D4C2311yescharacteristic
28C2×#117—42no—
29C22×C4⋊C4Q811yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C42C2×C411yescharacteristic
32#431—42no—
33#431—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C412yes—
36C2×#99—24no—
37C2×#99—24no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—24no—
42C2×C4×C8—21no—
43C2×C4×C8C2×C412yes—
44#381—42no—
45#380—24no—
46#380—42no—
47#379—24no—
48C4×C22⋊C4—24no—
49C4×C22⋊C4—42no—
50#342—24no—
51#342C2×C418yes—
52C25C22⋊C412yes—
53C22×D4—24no—
54C22×D4—42no—
55C22×D4—42no—
56C22×D4C22⋊C412yes—
57C22×D4C4⋊C411yescharacteristic
58C22×D4C2×D411yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—84no—
64C23×C4C4211yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C4⋊C411yescharacteristic
69C23×C4C4⋊C412yes—
70C23×C4C4⋊C412yes—
71C23×C4C2×D412yes—
72C22×C8—22no—
73C22×C8—22no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—41no—
77C4×D4—41no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—41no—
83C4×D4—41no—
84#117—88no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4C4⋊C412yes—
88C2×C4⋊C4C4⋊C411yescharacteristic
89C2×C4⋊C4C2×Q811yescharacteristic
90C2×C22⋊C4—28no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4C4212yes—
96C2×C22⋊C4C22⋊C412yes—
97C2×C22⋊C4C4⋊C412yes—
98C2×C22⋊C4C22×C412yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—44no—
102C2×C42C4211yescharacteristic
103C2×C42C22⋊C411yescharacteristic
104C2×C42C4⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106C4⋊C8—42no—
107C4⋊C8—42no—
108#99—48no—
109#99—48no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—84no—
113C22⋊C8—84no—
114C4×C8—42no—
115C4×C8—42no—
116#95—44no—
117#95—44no—
118C24—22no—
119C24—22no—
120C24—24no—
121C24—42no—
122C24—82no—
123C24—82no—
124C24#9512yes—
125C24#9511yescharacteristic
126C24C2×C22⋊C412yes—
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4—42no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—416no—
136C2×D4#13514yes—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—48no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—48no—
156C22×C4—84no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4—81no—
163C22×C4#9511yescharacteristic
164C22×C4#9511yescharacteristic
165C22×C4#9512yes—
166C22×C4#9511yescharacteristic
167C22×C4#9912yes—
168C22×C4#9912yes—
169C22×C4C2×C4211yescharacteristic
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×C4⋊C411yescharacteristic
172C22×C4C2×C4⋊C412yes—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—42no—
177C2×C8—42no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—88no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—88no—
189C22⋊C4—88no—
190C42—22no—
191C42—22no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—88no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—82no—
215C23—82no—
216C23—88no—
217C23—88no—
218C23#34212yes—
219C23C2×#9511yescharacteristic
220C23#41614yes—
221D4—44no—
222D4—48no—
223D4—44no—
224C2×C4—28no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—88no—
253C2×C4—88no—
254C2×C4—88no—
255C2×C4H(3,ℤ/4)12yes—
256C2×C4#34212yes—
257C2×C4C2×#9911yescharacteristic
258C2×C4C2×#9911yescharacteristic
259C2×C4#42712yes—
260C8—82no—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—24no—
268C22—44no—
269C22—42no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22—82no—
279C22—88no—
280C22—88no—
281C22C2×H(3,ℤ/4)11yescharacteristic
282C22#173511yescharacteristic
283C22C2×#34211yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—42no—
292C4—168no—
293C4#212712yes—
294C2—21no—
295C2—21no—
296C2—44no—
297C2—42no—
298C2—82no—
299C2#798211yescharacteristic
300C2#1257711yescharacteristic
301C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7237C2semidirect
5. C2×#431C4semidirect
6. C2×#431C4semidirect
9. #1954C4semidirect
11. #1819C4semidirect
17. C4×#99C4semidirect

Maximal subgroups

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

subgroupindex
#217862
#207222
#72372