Finite Group Databasebeta

← #122235

Subgroups — #122235

Order 512.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#20685C211yesmaximal, characteristic
2#11418C211yesmaximal, characteristic
3C2×#1314C211yesmaximal, characteristic
4#7055C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C2×#407C411yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2112—21no—
9C2×C22⋊C16—21no—
10#1818—21no—
11#1818—21no—
12#1817C412yes—
13#1817C2212yes—
14#1729—21no—
15#1729—21no—
16#1314—21no—
17#1314—21no—
18#1314C2212yes—
19#1314C2212yes—
20C24×C4C2×C411yescharacteristic
21C22×M16—21no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×#117—21no—
25C2×#117—21no—
26C2×#117—21no—
27C2×#117—21no—
28C2×#117—41no—
29C2×#117—41no—
30C2×#117—41no—
31C2×#117—41no—
32C22×C4⋊C4C811yescharacteristic
33C22×C4⋊C4C811yescharacteristic
34C22×C22⋊C4C812yes—
35C22×C42C2×C412yes—
36C2×M32—41no—
37C22×C16—41no—
38#407—41no—
39#407—41no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43C2×C22⋊C8C2×C412yes—
44C2×C22⋊C8C2×C411yescharacteristic
45C2×C22⋊C8D411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C22⋊C8C2311yescharacteristic
48C22⋊C16—22no—
49C22⋊C16—22no—
50C22⋊C16—41no—
51C22⋊C16—41no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56C25C2×C811yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C2×C812yes—
66C23×C4C2×C811yescharacteristic
67C23×C4C2×C812yes—
68C23×C4M1611yescharacteristic
69C23×C4M1611yescharacteristic
70C23×C4C22×C411yescharacteristic
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86#117—42no—
87#117—42no—
88#117—42no—
89#117—42no—
90#117—42no—
91#117—42no—
92#117—41no—
93#117—41no—
94#117—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—42no—
111C2×C42—42no—
112M32—82no—
113C2×C16—42no—
114C2×C16—42no—
115C2×C16—82no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C24—21no—
123C24—21no—
124C24—22no—
125C24—41no—
126C24—42no—
127C24—81no—
128C24C22⋊C811yescharacteristic
129C24C22⋊C811yescharacteristic
130C24C22×C811yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—82no—
163C22×C4C22⋊C811yescharacteristic
164C22×C4C22⋊C811yescharacteristic
165C22×C4#9911yescharacteristic
166C22×C4#9911yescharacteristic
167C22×C4#10011yescharacteristic
168C22×C4#10011yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C2×M1611yescharacteristic
171M16—82no—
172M16—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—81no—
187C2×C8—81no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C22⋊C4—44no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—44no—
198C42—42no—
199C42—42no—
200C16—82no—
201C16—82no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—42no—
212C23—42no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—81no—
220C23—81no—
221C23—84no—
222C23—84no—
223C23#32311yescharacteristic
224C23#32311yescharacteristic
225C23C2×C22⋊C811yescharacteristic
226C23#50412yes—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4#32311yescharacteristic
258C2×C4#32311yescharacteristic
259C2×C4#35212yes—
260C2×C4#35212yes—
261C2×C4C2×#9911yescharacteristic
262C2×C4C2×#10011yescharacteristic
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—82no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—84no—
286C22—81no—
287C22—81no—
288C22—84no—
289C22#131712yes—
290C22C2×#32311yescharacteristic
291C22#211311yescharacteristic
292C22#211511yescharacteristic
293C4—21no—
294C4—21no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4—82no—
299C4#131712yes—
300C4C2×#35211yescharacteristic
301C4C2×#35211yescharacteristic
302C2—21no—
303C2—21no—
304C2—22no—
305C2—41no—
306C2—42no—
307C2—81no—
308C2#1141911yescharacteristic
309C2C2×#131711yescharacteristic
310C2#1143611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#323
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7055C2semidirect

Maximal subgroups

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

subgroupindex
#206852
#114182
C2×#13142
#70552