Finite Group Databasebeta

← #391738

Subgroups — #391738

Order 512.

1079 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1079 conjugacy classes of subgroups — too many to draw. Only its 234 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#35672C211yesmaximal, characteristic
2#20639C212yesmaximal, complemented
3#11206C214yesmaximal, complemented
4#11168C212yesmaximal, complemented
5#11071C214yesmaximal
6#10968C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C2×#519C2211yescharacteristic
9#2905C2214yes—
10C22×#101C2214yes—
11C2×#407C2214yes—
12#2484C414yes—
13#2482C414yes—
14#2423—24no—
15#2421—24no—
16#2317C2212yes—
17#2310C2211yescharacteristic
18#2280C2212yes—
19#1801—24no—
20#1801—24no—
21C2×#344C2218yes—
22#1742C2214yes—
23C2×#343C2214yes—
24C23×Q8C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16C2318yes—
27C2×C22⋊Q8—24no—
28C2×C22⋊Q8—44no—
29#519—44no—
30C2×C4×Q8D414yes—
31C2×C4×Q8D414yes—
32C2×#117C2×C414yes—
33C2×#117C2312yes—
34C2×#117C2312yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4C2×C412yes—
38#424—48no—
39#423—28no—
40C2×#101—28no—
41C2×#101—48no—
42#407—48no—
43C2×C22⋊C8—44no—
44#400—24no—
45#399—24no—
46#398—24no—
47#397—24no—
48#391—24no—
49#389—24no—
50#388—28no—
51#387—28no—
52#385C2×C418yes—
53#384—22no—
54#382—22no—
55#381C2×C414yes—
56#381C2×C414yes—
57#380—22no—
58#380—22no—
59#379C2×C414yes—
60C4×C22⋊C4—24no—
61#344—48no—
62#343—24no—
63#343—24no—
64#343—44no—
65C25C22×C411yescharacteristic
66C22×Q8—24no—
67C22×Q8—24no—
68C22×Q8—24no—
69C22×Q8C2×D414yes—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4C22×C411yescharacteristic
75C23×C4C22×C412yes—
76C23×C4C22×C412yes—
77C23×C4C2411yescharacteristic
78C2×M16—28no—
79C2×M16—28no—
80C2×M16—28no—
81C2×M16—28no—
82C2×M16—28no—
83C2×M16—28no—
84C22×C8—28no—
85C22⋊Q8—416no—
86C4×Q8—48no—
87C4×Q8—48no—
88#117—28no—
89#117—24no—
90#117—24no—
91#117—48no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4C22⋊C414yes—
99C2×C4⋊C4C22⋊C414yes—
100C2×C4⋊C4C2×D412yes—
101C2×C4⋊C4C2×D412yes—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4C22×C414yes—
107C2×C22⋊C4C22×C414yes—
108C2×C42—22no—
109C2×C42C22⋊C418yes—
110C2×C42C2×D414yes—
111C4⋊C8—416no—
112#101—416no—
113C22⋊C8—416no—
114#95—24no—
115#95—24no—
116#95—24no—
117#95—24no—
118#95—28no—
119C24—21no—
120C24—22no—
121C24—24no—
122C24—42no—
123C24—81no—
124C24C23×C411yescharacteristic
125C2×Q8—28no—
126C2×Q8—24no—
127C2×Q8—24no—
128C2×Q8—28no—
129C2×Q8—44no—
130C2×Q8—48no—
131C2×Q8—44no—
132C22×C4—24no—
133C22×C4—28no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—48no—
146C22×C4—44no—
147C22×C4—82no—
148C22×C4C2×C22⋊C414yes—
149C22×C4C2×C22⋊C414yes—
150C22×C4C2×C22⋊C414yes—
151C22×C4C22×D412yes—
152C22×C4H(5,𝔽2)14yes—
153C22×C4H(5,𝔽2)14yes—
154C22×C42−1+414yes—
155M16—48no—
156M16—48no—
157M16—48no—
158M16—48no—
159C2×C8—48no—
160C2×C8—48no—
161C2×C8—48no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—88no—
170C22⋊C4—28no—
171C22⋊C4—28no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—84no—
175C42—28no—
176C42—48no—
177C42—42no—
178C42—42no—
179C23—22no—
180C23—24no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—42no—
186C23—44no—
187C23—41no—
188C23—42no—
189C23—44no—
190C23—88no—
191C23C22×C22⋊C411yescharacteristic
192C23#51812yes—
193C23#52014yes—
194Q8—48no—
195Q8—48no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—28no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4#53414yes—
215C2×C4#53512yes—
216C2×C4#53612yes—
217C2×C4#53714yes—
218C2×C4#56014yes—
219C2×C4#56218yes—
220C2×C4#56314yes—
221C8—88no—
222C22—21no—
223C22—22no—
224C22—21no—
225C22—22no—
226C22—44no—
227C22—44no—
228C22—48no—
229C22—42no—
230C22—82no—
231C22—84no—
232C22—81no—
233C22—81no—
234C22#230811yescharacteristic
235C22#231112yes—
236C22#237614yes—
237C4—44no—
238C4—44no—
239C4—44no—
240C4—84no—
241C4—82no—
242C4—81no—
243C4#281314yes—
244C4#281414yes—
245C2—22no—
246C2—24no—
247C2—82no—
248C2#1465811yescharacteristic
249C2#2178011yescharacteristic
250C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20639C2semidirect
3. #11206C2semidirect
4. #11168C2semidirect

Maximal subgroups

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

subgroupindex
#356722
#206392
#112062
#111682
#110712
#109682