Finite Group Databasebeta

← #374027

Subgroups — #374027

Order 512.

2593 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 2593 conjugacy classes of subgroups — too many to draw. Only its 354 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×#2614C211yesmaximal, characteristic
2C2×#2310C212yesmaximal, complemented
3C2×#1899C214yesmaximal
4#11197C212yesmaximal, complemented
5#10966C212yesmaximal
6#10736C214yesmaximal, complemented
7C24×Q8C2211yescharacteristic
8C22×#123C2212yes—
9C22×#123C2211yescharacteristic
10C22×#117C2214yes—
11C22×#99C2214yes—
12#2614—28no—
13C2×#395C414yescomplemented
14C2×#393C414yescomplemented
15#2310—216no—
16C2×#391—28no—
17C2×#381C2214yes—
18C2×#381C2214yes—
19C2×#381C2211yescharacteristic
20C2×#381C2212yes—
21#1899—216no—
22C2×#342C2218yes—
23C2×#342C2214yes—
24C23×Q8—28no—
25C23×Q8—28no—
26C23×Q8—44no—
27C23×Q8C2×C414yes—
28C23×Q8D418yes—
29C23×D4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2312yes—
32C2×#123—24no—
33C2×#123—24no—
34C2×#123—24no—
35C2×#123—28no—
36C2×#123—28no—
37C2×#123—44no—
38C2×#117—216no—
39C2×#117—28no—
40C2×#117—28no—
41C2×#117—48no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4C2×C414yes—
44C22×C4⋊C4C2×C412yes—
45C22×C22⋊C4C2×C414yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2318yes—
50C22×C42—24no—
51C2×#99—216no—
52C2×#99—44no—
53#395—216no—
54#393—216no—
55#391—264no—
56#381—28no—
57#381—24no—
58#381—24no—
59#381—24no—
60#381—216no—
61#381D4116yes—
62C2×#95—28no—
63C2×#95—28no—
64C2×#95C2×C414yes—
65C2×#95C2×C418yes—
66#342—216no—
67#342—44no—
68C25C22×C411yescharacteristic
69C25C2411yescharacteristic
70C22×Q8—232no—
71C22×Q8—216no—
72C22×Q8—216no—
73C22×Q8—216no—
74C22×Q8—216no—
75C22×Q8—48no—
76C22×Q8—48no—
77C22×Q8—48no—
78C22×Q8—416no—
79C22×Q8—432no—
80C22×Q8—416no—
81C22×Q8C22⋊C4116yes—
82C22×D4—22no—
83C22×D4—24no—
84C22×D4—24no—
85C22×D4—24no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—28no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C412yes—
95C23×C4C22×C414yes—
96C23×C4C22×C414yes—
97C23×C4C22×C412yes—
98C23×C4C2×D414yes—
99C23×C4C2×D414yes—
100C23×C4C2×D414yes—
101C23×C4C2×D418yes—
102C23×C4#4118yes—
103#123—416no—
104#123—48no—
105#123—48no—
106#123—48no—
107#123—48no—
108#117—432no—
109#117—432no—
110C2×C4⋊C4—216no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—216no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—48no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—216no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4C2×D4116yes—
133C2×C42—216no—
134C2×C42—48no—
135C2×C42—48no—
136#99—416no—
137#95—232no—
138#95—28no—
139#95—216no—
140#95—232no—
141C24—21no—
142C24—24no—
143C24—22no—
144C24—44no—
145C24—44no—
146C24C23×C411yescharacteristic
147C24C22×D412yes—
148C24C22×D414yes—
149C24C2×#4114yes—
150C24H(5,𝔽2)14yes—
151C2×Q8—264no—
152C2×Q8—416no—
153C2×Q8—464no—
154C2×Q8—432no—
155C2×Q8—432no—
156C2×Q8—432no—
157C2×Q8—432no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—42no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—48no—
164C2×D4—44no—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—216no—
174C22×C4—216no—
175C22×C4—216no—
176C22×C4—216no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—216no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—432no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—44no—
198C22×C4—48no—
199C22×C4C2×C22⋊C414yes—
200C22×C4C2×C22⋊C418yes—
201C22×C4C4×D4116yes—
202C22×C4#12018yes—
203C22×C4#12118yes—
204C22×C4#12118yes—
205C22×C4#12318yes—
206C4⋊C4—432no—
207C4⋊C4—48no—
208C4⋊C4—416no—
209C22⋊C4—216no—
210C22⋊C4—216no—
211C22⋊C4—48no—
212C22⋊C4—42no—
213C22⋊C4—44no—
214C22⋊C4—42no—
215C22⋊C4—44no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—416no—
219C42—432no—
220C23—28no—
221C23—24no—
222C23—48no—
223C23—48no—
224C23—48no—
225C23—41no—
226C23—44no—
227C23—42no—
228C23—44no—
229C23—84no—
230C23—88no—
231C23C22×C22⋊C411yescharacteristic
232C23C2×C4×D414yes—
233C23#51812yes—
234C23C2×#12012yes—
235C23C2×#12114yes—
236C23C2×#12312yes—
237C23#53414yes—
238C23D4214yes—
239C23#54618yes—
240C23#55114yes—
241Q8—464no—
242D4—84no—
243D4—84no—
244C2×C4—28no—
245C2×C4—232no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—216no—
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—216no—
252C2×C4—28no—
253C2×C4—24no—
254C2×C4—28no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—432no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—416no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—816no—
268C2×C4—816no—
269C2×C4—84no—
270C2×C4—88no—
271C2×C4#386116yes—
272C22—28no—
273C22—24no—
274C22—24no—
275C22—28no—
276C22—28no—
277C22—24no—
278C22—44no—
279C22—48no—
280C22—44no—
281C22—81no—
282C22—82no—
283C22—84no—
284C22—164no—
285C22C2×#38612yes—
286C22#230811yescharacteristic
287C22#235814yes—
288C22#241014yes—
289C22#241214yes—
290C22#288414yes—
291C4—216no—
292C4—44no—
293C4—816no—
294C4—81no—
295C4—82no—
296C4—84no—
297C2—24no—
298C2—24no—
299C2—24no—
300C2—161no—
301C2#1413411yescharacteristic
302C2#2098614yes—
303C2#2175112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×Q8 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2310C2semidirect
4. #11197C2semidirect
6. #10736C2semidirect
13. C2×#395C4semidirect
14. C2×#393C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C2×#23102
C2×#18992
#111972
#109662
#107362