Finite Group Databasebeta

← #479828

Subgroups — #479828

Order 512.

1029 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 154 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#11287C214yesmaximal, complemented
2#11089C214yesmaximal, complemented
3#11089C214yesmaximal, complemented
4#10980C212yesmaximal
5#10980C211yesmaximal, characteristic
6C22×#121C2211yescharacteristic
7C22×#99C2212yes—
8C22×#99C2214yes—
9C22×#99C2214yes—
10C22×#99C2212yes—
11#2315C2214yes—
12C2×#342C2214yes—
13C2×#342C2214yes—
14C2×#342C2214yes—
15C22⋊#95C2212yes—
16C22⋊#95C2214yes—
17C23×D4Q811yescharacteristic
18C23×D4C2312yes—
19C24×C4C2311yescharacteristic
20C2×#121—44no—
21C2×#121—42no—
22C22×C4⋊C4Q811yescharacteristic
23C22×C22⋊C4Q812yes—
24C22×C22⋊C4C2314yes—
25C22×C22⋊C4C2314yes—
26C22×C22⋊C4C2314yes—
27C2×#99—28no—
28C2×#99—28no—
29C2×#99—42no—
30C2×#99—44no—
31C2×#99—44no—
32C2×#99—44no—
33C2×#99—44no—
34C2×#99—44no—
35C2×#99—44no—
36C2×#99—42no—
37#385—44no—
38#385—44no—
39#380—44no—
40C2×#95—44no—
41C2×#95—42no—
42#342—28no—
43#342—44no—
44#342—44no—
45C25C2×Q812yes—
46C25C2411yescharacteristic
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—24no—
54C22×D4—24no—
55C22×D4—41no—
56C22×D4—42no—
57C22×D4—82no—
58C23×C4—24no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—82no—
64C23×C4C2×Q812yes—
65C23×C4C2×Q811yescharacteristic
66C23×C4C2×Q811yescharacteristic
67#121—416no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4—44no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4#4118yes—
93#99—48no—
94#99—416no—
95#99—48no—
96#99—416no—
97#95—48no—
98#95—88no—
99#95—84no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—24no—
104C24—24no—
105C24—81no—
106C24—81no—
107C24—82no—
108C24—82no—
109C24C22×Q811yescharacteristic
110C24C2×#4114yes—
111C24H(5,𝔽2)14yes—
112C24H(5,𝔽2)12yes—
113C24H(5,𝔽2)14yes—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—42no—
126C2×D4—44no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—82no—
132C2×D4—84no—
133C2×D4—82no—
134C2×D4—84no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—28no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—48no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—81no—
158C22×C4—82no—
159C22×C4—84no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4—84no—
165C22×C4—84no—
166C4⋊C4—44no—
167C4⋊C4—164no—
168C4⋊C4—164no—
169C22⋊C4—28no—
170C22⋊C4—28no—
171C22⋊C4—48no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—44no—
182C23—42no—
183C23—44no—
184C23—41no—
185C23—42no—
186C23—44no—
187C23—41no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—44no—
192C23—44no—
193C23—82no—
194C23—82no—
195C23—82no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23—84no—
200C23—84no—
201C23—164no—
202C23—164no—
203C23#53814yes—
204C23#54014yes—
205C23#54014yes—
206C23#54311yescharacteristic
207C23#54312yes—
208C23#55112yes—
209C23#55314yes—
210C23#55414yes—
211C23#56012yes—
212C23#56014yes—
213C23H(3,𝔽4)14yes—
214D4—88no—
215D4—84no—
216D4—84no—
217D4—84no—
218D4—84no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—48no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—88no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—88no—
237C2×C4—84no—
238C2×C4—84no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—88no—
242C2×C4—88no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—164no—
246C2×C4—164no—
247C2×C4—164no—
248C22—21no—
249C22—22no—
250C22—24no—
251C22—24no—
252C22—21no—
253C22—22no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—44no—
258C22—42no—
259C22—44no—
260C22—44no—
261C22—42no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—162no—
266C22—162no—
267C22—164no—
268C22—164no—
269C22#274812yes—
270C22#275314yes—
271C22#278014yes—
272C22#282814yes—
273C22#284011yescharacteristic
274C4—42no—
275C4—88no—
276C4—81no—
277C4—82no—
278C4—164no—
279C4—164no—
280C2—24no—
281C2—24no—
282C2—22no—
283C2—24no—
284C2—162no—
285C2#1727111yescharacteristic
286C2#3287311yescharacteristic
287C2#3287411yescharacteristic
subgroupG/H
center, socleC22#2840
commutator, Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #11287C2semidirect
2. #11089C2semidirect
3. #11089C2semidirect

Maximal subgroups

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

subgroupindex
#112872
#110892
#109802