Finite Group Databasebeta

← #101305

Subgroups — #101305

Order 512.

499 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 86 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#23629C211yesmaximal, characteristic
2#14489C211yesmaximal, characteristic
3#8716C212yesmaximal
4#7595C211yesmaximal, characteristic
5#7037C212yesmaximal
6C2×#419—21no—
7C2×#391C2211yescharacteristic
8#2285C2211yescharacteristic
9#2074—22no—
10#2072C414yescomplemented
11#1817C2212yes—
12#1736—21no—
13#1728C2211yescharacteristic
14#1446—21no—
15#1443—22no—
16#1436—21no—
17#1434—22no—
18#1434C2212yes—
19#1294—22no—
20#1294—22no—
21#1272—22no—
22C23×Q8D411yescharacteristic
23C22×M16—21no—
24C2×#117—21no—
25C22×C4⋊C4D411yescharacteristic
26C22×C42—21no—
27C22×C42C2311yescharacteristic
28#419—44no—
29C2×#103—24no—
30C2×C22⋊C8—22no—
31#391—22no—
32#391C2×C412yes—
33#391D412yes—
34#384C2×C412yes—
35#382—21no—
36#380—22no—
37C4×C4⋊C4—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95D412yes—
47C2×#95D412yes—
48#340—24no—
49#336—21no—
50#336—22no—
51#336—22no—
52#336C2×C412yes—
53C22×Q8—21no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—42no—
59C22×Q8C22⋊C412yes—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D412yes—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—22no—
71C22×C8—42no—
72#117—42no—
73#117—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4C22⋊C412yes—
81C2×C22⋊C4—41no—
82C2×C42—42no—
83C2×C42—44no—
84C2×C42—41no—
85C2×C42—81no—
86C2×C42C22×C411yescharacteristic
87C2×C42C2×D411yescharacteristic
88C2×C42#4112yes—
89#103—48no—
90C22⋊C8—42no—
91C22⋊C8—42no—
92#95—24no—
93#95—24no—
94#95—42no—
95#95—42no—
96#95—44no—
97#95—44no—
98#95—42no—
99#95—42no—
100#95—42no—
101#95—81no—
102#95—81no—
103#95—81no—
104#95—82no—
105C24#12011yescharacteristic
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8—44no—
111C2×Q8—44no—
112C2×Q8—42no—
113C2×Q8—81no—
114C2×Q8—81no—
115C2×Q8—82no—
116C2×Q8—82no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C4×D412yes—
151C22×C4#12112yes—
152C22×C4#12311yescharacteristic
153C22×C4#13512yes—
154C22×C4AGL(1,ℤ/8)12yes—
155M16—82no—
156M16—82no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8—82no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—81no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C22⋊C4—82no—
170C42—41no—
171C42—41no—
172C42—42no—
173C42—84no—
174C42—82no—
175C23—22no—
176C23—22no—
177C23#38611yescharacteristic
178C23#44912yes—
179C23#45511yescharacteristic
180C23#45511yescharacteristic
181C23#45812yes—
182Q8—44no—
183Q8—82no—
184Q8—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#41611yescharacteristic
229C2×C4#41811yescharacteristic
230C2×C4#48212yes—
231C8—82no—
232C8—162no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—22no—
239C22—21no—
240C22—21no—
241C22#161212yes—
242C22#161212yes—
243C22#187511yescharacteristic
244C22#188511yescharacteristic
245C22#189011yescharacteristic
246C22#190512yes—
247C22#190612yes—
248C4—21no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—84no—
254C4—82no—
255C4—82no—
256C4—162no—
257C4—162no—
258C2—22no—
259C2—22no—
260C2#902811yescharacteristic
261C2#918212yes—
262C2#942611yescharacteristic
263C2#943911yescharacteristic
264C2#952612yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2072C4semidirect

Maximal subgroups

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

subgroupindex
#236292
#144892
#87162
#75952
#70372