Finite Group Databasebeta

← #479724

Subgroups — #479724

Order 512.

954 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 216 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#15195C212yesmaximal
2#11324C214yesmaximal, complemented
3#11316C214yesmaximal, complemented
4#11147C211yesmaximal, characteristic
5#11096C214yesmaximal
6C22×#107C2214yes—
7C22×C22⋊C8C2212yes—
8C2×#384C2212yes—
9C2×#380C2214yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×#340C2214yes—
12C2×#340C2214yes—
13C2×#340C2214yes—
14C2×#336C2214yes—
15#1442—22no—
16#1442—22no—
17#1439—22no—
18#1439—22no—
19C4⋊#95C2212yes—
20C4⋊#95C2214yes—
21C24×C4C2311yescharacteristic
22C23×C8C2314yes—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2314yes—
26C22×C4⋊C4C2314yes—
27C22×C22⋊C4—24no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42C2312yes—
31C2×#107—44no—
32C2×#107—44no—
33C2×#107—44no—
34C2×C22⋊C8—28no—
35C2×C22⋊C8—42no—
36#384—42no—
37#384—42no—
38#380—44no—
39#380—44no—
40C4×C22⋊C4—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—24no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95D412yes—
47#340—28no—
48#340—44no—
49#340—44no—
50#336—28no—
51C25C2×D411yescharacteristic
52C23×C4—24no—
53C23×C4—22no—
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×D412yes—
60C23×C4C2×D412yes—
61C23×C4C2411yescharacteristic
62C22×C8—24no—
63C22×C8—24no—
64C22×C8—24no—
65C22×C8—24no—
66C22×C8—24no—
67C22×C8—24no—
68C22×C8#4118yes—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—84no—
84C2×C4⋊C4—84no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—48no—
90C2×C22⋊C4—84no—
91C2×C22⋊C4—84no—
92C2×C22⋊C4Q1614yes—
93C2×C42—24no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—24no—
97#107—416no—
98C22⋊C8—48no—
99#95—42no—
100#95—44no—
101#95—48no—
102#95—44no—
103#95—84no—
104#95—84no—
105C24—21no—
106C24—22no—
107C24—24no—
108C24—22no—
109C24—21no—
110C24—24no—
111C24C2×Q1612yes—
112C24C22×D411yescharacteristic
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4C2×Q1614yes—
147C22×C4#13714yes—
148C22×C4C2×#4114yes—
149C22×C4H(5,𝔽2)14yes—
150C22×C4H(5,𝔽2)12yes—
151C22×C42−1+414yes—
152C2×C8—28no—
153C2×C8—28no—
154C2×C8—28no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—88no—
168C4⋊C4—88no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C22⋊C4—816no—
172C42—44no—
173C42—42no—
174C42—42no—
175C23—28no—
176C23—22no—
177C23—24no—
178C23—44no—
179C23—44no—
180C23—44no—
181C23—41no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—42no—
186C23—44no—
187C23—41no—
188C23#53411yescharacteristic
189C23#53512yes—
190C23#54714yes—
191C23C22×Q1611yescharacteristic
192C23C2×#13712yes—
193C23#57511yescharacteristic
194C23#57614yes—
195C2×C4—28no—
196C2×C4—28no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4#53914yes—
225C2×C4#54214yes—
226C2×C4#54214yes—
227C2×C4#55112yes—
228C2×C4#55314yes—
229C2×C4#56014yes—
230C2×C4#56312yes—
231C2×C4#56314yes—
232C8—48no—
233C22—44no—
234C22—48no—
235C22—42no—
236C22—82no—
237C22—84no—
238C22—81no—
239C22—81no—
240C22#271712yes—
241C22#271814yes—
242C22#281511yescharacteristic
243C22#318911yescharacteristic
244C22#319511yescharacteristic
245C22#319912yes—
246C22#320912yes—
247C22#321612yes—
248C22#322014yes—
249C22#333814yes—
250C22#334614yes—
251C22#334714yes—
252C22#335714yes—
253C4—44no—
254C4—44no—
255C4—81no—
256C4—164no—
257C4—164no—
258C4#275714yes—
259C4#283814yes—
260C2—82no—
261C2#1722911yescharacteristic
262C2#3226112yes—
263C2#3226812yes—
264C2#3228414yes—
265C2#3229514yes—
266C2#3276211yescharacteristic
267C2#3276411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11324C2semidirect
3. #11316C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#113242
#113162
#111472
#110962