Finite Group Databasebeta

← #128050

Subgroups — #128050

Order 512.

860 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 130 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#20722C212yesmaximal
2#14345C211yesmaximal, characteristic
3#7307C214yesmaximal
4C22×C4×D4C2211yescharacteristic
5C2×#383C411yescharacteristic, complemented
6C2×#382C411yescharacteristic, complemented
7#2280—22no—
8#1819—22no—
9#1799—22no—
10#1799C414yescomplemented
11#1781C2214yes—
12C4×#99—22no—
13C4×#99C414yescomplemented
14#1445C412yescomplemented
15#1439—22no—
16#1438C2212yes—
17C22⋊#95—22no—
18C23×D4C2311yescharacteristic
19C24×C4D412yes—
20C2×C4×D4—42no—
21C2×C4×D4—42no—
22C2×C4×D4C2×C412yes—
23C2×C4×D4C2×C412yes—
24C2×C4×D4D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×C4×D4Q811yescharacteristic
27C2×C4×D4Q811yescharacteristic
28C2×#117—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—24no—
32C22×C22⋊C4D412yes—
33C22×C42C2×C411yescharacteristic
34C2×#99—24no—
35C2×#99C2×C418yes—
36#383—42no—
37#382—42no—
38#381—42no—
39#380—24no—
40#380—42no—
41#379—24no—
42C4×C22⋊C4—24no—
43C4×C22⋊C4—42no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—24no—
48C2×#95C2×C412yes—
49C2×#95C2×C412yes—
50#342—24no—
51#342—24no—
52C25C2×D412yes—
53C22×D4—41no—
54C22×D4—41no—
55C22×D4—44no—
56C22×D4C22×C412yes—
57C22×D4C2×D411yescharacteristic
58C22×D4C2×Q811yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—84no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C4⋊C412yes—
72C23×C4C4⋊C412yes—
73C23×C4C22×C411yescharacteristic
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—42no—
82#117—88no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4C4212yes—
89C2×C4⋊C4C4⋊C411yescharacteristic
90C2×C4⋊C4C4⋊C411yescharacteristic
91C2×C22⋊C4—28no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—48no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—84no—
98C2×C22⋊C4—84no—
99C2×C22⋊C4C22⋊C412yes—
100C2×C22⋊C4C22⋊C412yes—
101C2×C22⋊C4C4⋊C412yes—
102C2×C22⋊C4C4⋊C412yes—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—44no—
106C2×C42C4211yescharacteristic
107C2×C42C4211yescharacteristic
108C2×C42C4⋊C411yescharacteristic
109C2×C42C4⋊C411yescharacteristic
110#99—48no—
111#99—48no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—44no—
116#95—44no—
117#95—42no—
118#95—42no—
119#95—84no—
120C24—24no—
121C24—22no—
122C24—22no—
123C24—42no—
124C24—42no—
125C24—44no—
126C24—42no—
127C24C2×C22⋊C411yescharacteristic
128C24C2×C22⋊C412yes—
129C24C2×C4⋊C412yes—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—42no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—48no—
138C2×D4—48no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
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—22no—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—48no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—84no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—84no—
176C22×C4—84no—
177C22×C4#9511yescharacteristic
178C22×C4#9511yescharacteristic
179C22×C4#9512yes—
180C22×C4#9512yes—
181C22×C4#9511yescharacteristic
182C22×C4#9511yescharacteristic
183C22×C4#9912yes—
184C22×C4#9912yes—
185C22×C4C2×C4211yescharacteristic
186C22×C4C2×C4⋊C411yescharacteristic
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—84no—
191C4⋊C4—88no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—84no—
199C22⋊C4—84no—
200C22⋊C4—816no—
201C22⋊C4—88no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—22no—
206C42—88no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—48no—
223C23—48no—
224C23—44no—
225C23—42no—
226C23—42no—
227C23—82no—
228C23—82no—
229C23—84no—
230C23—84no—
231C23C2×#9511yescharacteristic
232C23C2×#9911yescharacteristic
233C23C2×#9911yescharacteristic
234C23#41014yes—
235D4—44no—
236D4—48no—
237D4—44no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—88no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—88no—
279C2×C4—88no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#34212yes—
283C2×C4#34212yes—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—82no—
301C22—84no—
302C22—88no—
303C22—84no—
304C22—82no—
305C22C2×#34211yescharacteristic
306C22#174012yes—
307C22#212712yes—
308C22#212811yescharacteristic
309C22#212911yescharacteristic
310C4—22no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—162no—
320C4—168no—
321C2—21no—
322C2—22no—
323C2—22no—
324C2—21no—
325C2—44no—
326C2—82no—
327C2#810011yescharacteristic
328C2#1258211yescharacteristic
329C2#1258311yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#383C4semidirect
6. C2×#382C4semidirect
10. #1799C4semidirect
13. C4×#99C4semidirect
14. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#143452
#73072