Finite Group Databasebeta

← #391693

Subgroups — #391693

Order 512.

2653 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 2653 conjugacy classes of subgroups — too many to draw. Only its 250 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×#2849C211yesmaximal, characteristic
2#14681C212yesmaximal, complemented
3#11197C214yesmaximal, complemented
4#11196C212yesmaximal, complemented
5#10966C214yesmaximal, complemented
6#10965C212yesmaximal
7C25×C4C2211yescharacteristic
8C22×#123C2212yes—
9C22×#120C2211yescharacteristic
10C22×#99C2214yes—
11#2849—28no—
12C2×#386C414yescomplemented
13C2×#385C414yescomplemented
14C2×#381C2214yes—
15C2×#381C2214yes—
16C2×#381C2212yes—
17C2×#379C2214yes—
18C2×#379C2211yescharacteristic
19C2×#342C2218yes—
20C2×#342C2214yes—
21C22⋊#95—28no—
22C26C2311yescharacteristic
23C23×D4C2×C412yes—
24C23×D4C2311yescharacteristic
25C24×C4—48no—
26C24×C4—44no—
27C24×C4C2×C414yes—
28C24×C4D418yes—
29C24×C4C2312yes—
30C2×#123—24no—
31C2×#123—24no—
32C2×#123—48no—
33C2×#120—24no—
34C2×#120—44no—
35C22×C4⋊C4C2×C414yes—
36C22×C22⋊C4—28no—
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2×C414yes—
39C22×C22⋊C4C2×C414yes—
40C22×C22⋊C4C2318yes—
41C22×C22⋊C4C2311yescharacteristic
42C22×C22⋊C4C2312yes—
43C2×#99—48no—
44C2×#99—44no—
45#386—28no—
46#386—28no—
47#385—28no—
48#385—28no—
49#381—24no—
50#381—24no—
51#381—48no—
52#381—48no—
53#379—24no—
54#379—48no—
55C2×#95—28no—
56C2×#95—28no—
57C2×#95C2×C418yes—
58#342—44no—
59#342—48no—
60C25—24no—
61C25—28no—
62C25—48no—
63C25C22×C412yes—
64C25C22×C411yescharacteristic
65C25C2×D414yes—
66C25C2411yescharacteristic
67C22×D4—24no—
68C22×D4—22no—
69C22×D4—24no—
70C22×D4—28no—
71C22×D4—44no—
72C22×D4—48no—
73C23×C4—28no—
74C23×C4—28no—
75C23×C4—28no—
76C23×C4—28no—
77C23×C4—416no—
78C23×C4—48no—
79C23×C4—416no—
80C23×C4—416no—
81C23×C4—416no—
82C23×C4—416no—
83C23×C4—432no—
84C23×C4—416no—
85C23×C4C22⋊C418yes—
86C23×C4C22⋊C418yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C414yes—
89C23×C4C22×C414yes—
90C23×C4C22×C412yes—
91C23×C4C2×D414yes—
92C23×C4C2×D414yes—
93#123—416no—
94#123—416no—
95#120—416no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—48no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—28no—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—48no—
116C2×C22⋊C4—432no—
117C2×C22⋊C4—416no—
118#99—416no—
119#95—216no—
120#95—416no—
121#95—416no—
122C24—24no—
123C24—28no—
124C24—216no—
125C24—28no—
126C24—28no—
127C24—28no—
128C24—28no—
129C24—21no—
130C24—24no—
131C24—22no—
132C24—44no—
133C24—48no—
134C24—44no—
135C24—464no—
136C24—432no—
137C24—44no—
138C24—44no—
139C24—48no—
140C24—44no—
141C24C2×C22⋊C414yes—
142C24C23×C411yescharacteristic
143C24C22×D412yes—
144C24H(5,𝔽2)14yes—
145C24H(5,𝔽2)18yes—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—44no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—48no—
153C2×D4—44no—
154C2×D4—48no—
155C2×D4—816no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—28no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4—28no—
163C22×C4—28no—
164C22×C4—28no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—28no—
171C22×C4—24no—
172C22×C4—416no—
173C22×C4—416no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—464no—
177C22×C4—4128no—
178C22×C4—48no—
179C22×C4—416no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—432no—
184C22×C4—48no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4C2×C22⋊C418yes—
188C4⋊C4—416no—
189C22⋊C4—416no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—832no—
201C23—28no—
202C23—24no—
203C23—232no—
204C23—216no—
205C23—24no—
206C23—28no—
207C23—24no—
208C23—28no—
209C23—48no—
210C23—416no—
211C23—432no—
212C23—432no—
213C23—432no—
214C23—48no—
215C23—416no—
216C23—432no—
217C23—464no—
218C23—432no—
219C23—48no—
220C23—416no—
221C23—42no—
222C23—41no—
223C23—44no—
224C23—44no—
225C23—84no—
226C23—88no—
227C23#41018yes—
228C23C22×C22⋊C411yescharacteristic
229C23#51812yes—
230C23#51814yes—
231C23#53414yes—
232C23#53414yes—
233C23#53414yes—
234C23#56014yes—
235C23#56018yes—
236C23H(3,𝔽4)14yes—
237D4—816no—
238D4—88no—
239C2×C4—28no—
240C2×C4—28no—
241C2×C4—28no—
242C2×C4—432no—
243C2×C4—432no—
244C2×C4—416no—
245C2×C4—416no—
246C2×C4—416no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—48no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—88no—
255C2×C4—84no—
256C2×C4—816no—
257C2×C4—816no—
258C2×C4—816no—
259C2×C4—88no—
260C22—28no—
261C22—24no—
262C22—28no—
263C22—28no—
264C22—28no—
265C22—28no—
266C22—216no—
267C22—44no—
268C22—48no—
269C22—44no—
270C22—464no—
271C22—432no—
272C22—44no—
273C22—44no—
274C22—48no—
275C22—81no—
276C22—82no—
277C22—84no—
278C22—164no—
279C22#230811yescharacteristic
280C22#230812yes—
281C22#237514yes—
282C22#281218yes—
283C22C2×#41014yes—
284C4—48no—
285C4—81no—
286C4—82no—
287C4—84no—
288C4—168no—
289C2—28no—
290C2—24no—
291C2—48no—
292C2—161no—
293C2#1465711yescharacteristic
294C2#2174712yes—
295C2#2174814yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14681C2semidirect
3. #11197C2semidirect
4. #11196C2semidirect
5. #10966C2semidirect
12. C2×#386C4semidirect
13. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
#146812
#111972
#111962
#109662
#109652