Finite Group Databasebeta

← #108257

Subgroups — #108257

Order 512.

1758 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1758 conjugacy classes of subgroups — too many to draw. Only its 202 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#34174C211yesmaximal, characteristic
2C2×#1782C211yesmaximal, characteristic
3#20580C211yesmaximal, characteristic
4#7943C214yesmaximal, complemented
5C23×#41C2211yescharacteristic
6C2×#519C412yescomplemented
7C22×#99—22no—
8C2×#407—22no—
9C2×#407C2212yes—
10#2341—28no—
11#2309—22no—
12#2309C412yescomplemented
13#2280—22no—
14#2280C2212yes—
15C2×#381C2212yes—
16#1782—22no—
17#1782C414yescomplemented
18#1780—22no—
19#1780C414yescomplemented
20#1284—28no—
21C22×#41—24no—
22C22×#41—24no—
23C22×#41—48no—
24C22×#41C2×C412yes—
25C22×#41C2×C412yes—
26C22×#41D412yes—
27C22×#41Q812yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C23×D4D412yes—
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34#519—48no—
35C2×C4×Q8—28no—
36C2×#117—24no—
37C2×#117C2×C414yes—
38C22×C22⋊C4—22no—
39C2×#99—28no—
40C2×#99—48no—
41#407—24no—
42#407—24no—
43#407—48no—
44C2×C22⋊C8—28no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2×C414yes—
47#390—216no—
48#390—28no—
49#381—22no—
50#381—24no—
51#381—22no—
52#381C2×C414yes—
53#380—28no—
54#380—24no—
55#379—24no—
56#379—22no—
57C4×C22⋊C4—28no—
58C4×C22⋊C4D418yes—
59C2×#95—24no—
60C25C22⋊C412yes—
61C25C2×D411yescharacteristic
62C2×#41—28no—
63C2×#41—28no—
64C2×#41—28no—
65C2×#41—28no—
66C2×#41—48no—
67C2×#41—48no—
68C2×#41—48no—
69C2×#41—48no—
70C2×#41—432no—
71C2×#41—432no—
72C22×Q8—28no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8C4212yes—
76C22×Q8C4⋊C412yes—
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—28no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4—44no—
83C22×D4—44no—
84C22×D4C4212yes—
85C22×D4C22⋊C414yes—
86C22×D4C4⋊C414yes—
87C22×D4C4⋊C412yes—
88C23×C4—28no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4C22⋊C414yes—
95C23×C4C22⋊C412yes—
96C23×C4C4⋊C414yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D414yes—
103C23×C4C2×Q811yescharacteristic
104C23×C4#4114yes—
105C2×M16—48no—
106C2×M16—42no—
107C2×M16—42no—
108C22×C8—24no—
109C4×Q8—432no—
110#117—48no—
111#117—48no—
112#117—48no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—216no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—48no—
120C2×C22⋊C4—42no—
121C2×C42—28no—
122C2×C42C22⋊C418yes—
123#99—416no—
124C22⋊C8—28no—
125C22⋊C8—416no—
126C22⋊C8—44no—
127#95—28no—
128#95—28no—
129#95—44no—
130C24—24no—
131C24—22no—
132C24—22no—
133C24—24no—
134C24—24no—
135C24—82no—
136C24—84no—
137C24#9514yes—
138C24C2×C22⋊C411yescharacteristic
139C24C2×C22⋊C411yescharacteristic
140C24C2×C4⋊C411yescharacteristic
141C24#12012yes—
142C24#12312yes—
143#41—432no—
144#41—432no—
145#41—432no—
146#41—432no—
147C2×Q8—216no—
148C2×Q8—216no—
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—24no—
152C2×Q8—24no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—28no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—416no—
164C2×D4—416no—
165C2×D4—48no—
166C2×D4—48no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—48no—
183C22×C4—42no—
184C22×C4—48no—
185C22×C4—416no—
186C22×C4—416no—
187C22×C4—44no—
188C22×C4—84no—
189C22×C4—88no—
190C22×C4#9514yes—
191C22×C4#9918yes—
192C22×C4C2×C4211yescharacteristic
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C4⋊C411yescharacteristic
196C22×C4C2×C4⋊C411yescharacteristic
197C22×C4#11712yes—
198C22×C4C4×D414yes—
199C22×C4C4×D414yes—
200C22×C4#12012yes—
201C22×C4#12114yes—
202C22×C4C22⋊Q814yes—
203C22×C4#12312yes—
204M16—84no—
205M16—84no—
206C2×C8—48no—
207C2×C8—44no—
208C4⋊C4—416no—
209C4⋊C4—48no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—88no—
213C22⋊C4—816no—
214C22⋊C4—84no—
215C42—48no—
216C42—416no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—44no—
225C23—42no—
226C23—44no—
227C23—42no—
228C23—42no—
229C23—48no—
230C23—48no—
231C23—44no—
232C23—44no—
233C23—84no—
234C23—88no—
235C23—88no—
236C23—88no—
237C23C2×#9511yescharacteristic
238C23#37911yescharacteristic
239C23#38111yescharacteristic
240C23#38512yes—
241C23#38612yes—
242C23C2×#9914yes—
243Q8—216no—
244Q8—216no—
245D4—416no—
246D4—416no—
247D4—48no—
248D4—48no—
249C2×C4—24no—
250C2×C4—28no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—28no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—416no—
266C2×C4—416no—
267C2×C4—416no—
268C2×C4—48no—
269C2×C4—84no—
270C2×C4—88no—
271C2×C4—168no—
272C2×C4#34218yes—
273C2×C4C4×C22⋊C412yes—
274C2×C4#38012yes—
275C2×C4#38512yes—
276C2×C4#38612yes—
277C8—84no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—24no—
284C22—24no—
285C22—44no—
286C22—48no—
287C22—48no—
288C22—44no—
289C22—48no—
290C22—84no—
291C22—82no—
292C22—82no—
293C22—84no—
294C22—88no—
295C22—88no—
296C22C22⋊#9511yescharacteristic
297C22C2×#34212yes—
298C22#178112yes—
299C22#181912yes—
300C4—22no—
301C4—22no—
302C4—28no—
303C4—22no—
304C4—22no—
305C4—88no—
306C4—164no—
307C2—22no—
308C2—22no—
309C2—22no—
310C2—42no—
311C2—42no—
312C2—84no—
313C2#1072511yescharacteristic
314C2#1074312yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7943C2semidirect
6. C2×#519C4semidirect
12. #2309C4semidirect
17. #1782C4semidirect
19. #1780C4semidirect

Maximal subgroups

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

subgroupindex
#341742
C2×#17822
#205802
#79432