Finite Group Databasebeta

← #124549

Subgroups — #124549

Order 512.

1334 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C2×#1446C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7131C214yesmaximal, complemented
4C24×Q8C411yescharacteristic
5C22×#95—22no—
6C22×#95C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—22no—
9#1446—22no—
10#1446C412yes—
11#1446C2212yes—
12#1280—22no—
13#1280C2214yes—
14C23×Q8—21no—
15C23×Q8—22no—
16C23×Q8—48no—
17C23×Q8—41no—
18C23×Q8—41no—
19C23×Q8—42no—
20C23×Q8C814yescomplemented
21C23×Q8C2×C412yes—
22C24×C4—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D412yes—
26C2×C22⋊C8—24no—
27C2×C22⋊C8—42no—
28C2×C22⋊C8—42no—
29C2×#95—21no—
30C2×#95—24no—
31C2×#95—24no—
32C2×#95—41no—
33C2×#95—41no—
34C2×#95—42no—
35C2×#95—48no—
36C2×#95C2×C412yes—
37C2×#95C2×C411yescharacteristic
38C2×#95D412yes—
39C2×#95C2311yescharacteristic
40C25C22⋊C411yescharacteristic
41C22×Q8—24no—
42C22×Q8—22no—
43C22×Q8—416no—
44C22×Q8—48no—
45C22×Q8—48no—
46C22×Q8—416no—
47C22×Q8—44no—
48C22×Q8—44no—
49C22×Q8—42no—
50C22×Q8—42no—
51C22×Q8—44no—
52C22×Q8—44no—
53C22×Q8—42no—
54C22×Q8—42no—
55C22×Q8—432no—
56C22×Q8—416no—
57C22×Q8—48no—
58C22×Q8—48no—
59C22×Q8C2×C814yes—
60C23×C4—28no—
61C23×C4—24no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—24no—
71C23×C4—42no—
72C23×C4—44no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—42no—
76C23×C4—44no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4—42no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4—42no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C2×C812yes—
86C23×C4M1612yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D412yes—
89C22×C8—44no—
90C22⋊C8—44no—
91C22⋊C8—44no—
92C22⋊C8—84no—
93C22⋊C8—84no—
94#95—41no—
95#95—42no—
96#95—41no—
97#95—42no—
98#95—48no—
99#95—48no—
100#95—42no—
101C24—24no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24C22⋊C812yes—
107C24#9911yescharacteristic
108C24#9911yescharacteristic
109C24#10012yes—
110C24C2×C22⋊C411yescharacteristic
111C2×Q8—48no—
112C2×Q8—416no—
113C2×Q8—48no—
114C2×Q8—48no—
115C2×Q8—44no—
116C2×Q8—44no—
117C2×Q8—432no—
118C2×Q8—416no—
119C2×Q8—416no—
120C2×Q8—432no—
121C2×Q8—416no—
122C2×Q8—432no—
123C2×Q8—48no—
124C2×Q8—416no—
125C2×Q8—416no—
126C2×Q8—48no—
127C22×C4—28no—
128C22×C4—216no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—48no—
147C22×C4—416no—
148C22×C4—48no—
149C22×C4—44no—
150C22×C4—48no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—416no—
168C22×C4—416no—
169C22×C4—48no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—88no—
178C22×C4—88no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4C22⋊C812yes—
182C22×C4C22×C811yescharacteristic
183C22×C4C2×M1611yescharacteristic
184C22×C4#13714yes—
185C2×C8—84no—
186C2×C8—84no—
187C2×C8—84no—
188C23—28no—
189C23—28no—
190C23—28no—
191C23—28no—
192C23—22no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23#32312yes—
208C23#32312yes—
209C23AΣL(1,𝔽16)14yes—
210C23#35412yes—
211C23#35612yes—
212C23C2×C22⋊C811yescharacteristic
213C23C2×#9911yescharacteristic
214C23C2×#10011yescharacteristic
215C23#41912yes—
216C23#42112yes—
217Q8—48no—
218Q8—48no—
219Q8—432no—
220Q8—416no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—416no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—416no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—48no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—88no—
254C2×C4—88no—
255C2×C4—816no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C8—164no—
261C22—28no—
262C22—28no—
263C22—216no—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—24no—
270C22—24no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—21no—
276C22—22no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22#131614yes—
281C22#131914yes—
282C22C2×#32311yescharacteristic
283C22#147912yes—
284C22#151112yes—
285C22#153311yescharacteristic
286C22#155811yescharacteristic
287C22C2×AΣL(1,𝔽16)12yes—
288C22C2×#35411yescharacteristic
289C22C2×#35611yescharacteristic
290C4—44no—
291C4—48no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—84no—
296C4—82no—
297C2—21no—
298C2—21no—
299C2—24no—
300C2—24no—
301C2—22no—
302C2#759511yescharacteristic
303C2C2×#131611yescharacteristic
304C2C2×#131911yescharacteristic
305C2#1180812yes—
306C2#1181512yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×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
20. C23×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#14462
#86982
#71312