Finite Group Databasebeta

← #94101

Subgroups — #94101

Order 512.

366 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1325C212yesmaximal
2#10841C211yesmaximal, characteristic
3C22×C4×C8C411yescharacteristic
4C2×#384C411yescharacteristic
5C2×#384C2211yescharacteristic
6C2×#340—22no—
7C2×#324—22no—
8#1325—22no—
9#1325C414yescomplemented
10C23×C8—21no—
11C22×C4⋊C4—21no—
12C22×C4⋊C4—41no—
13C22×C4⋊C4—41no—
14C22×C4⋊C4D412yes—
15C22×C42C2×C411yescharacteristic
16C2×C22⋊C8—42no—
17C2×C4×C8—21no—
18C2×C4×C8—41no—
19C2×C4×C8—41no—
20C2×C4×C8C812yescomplemented
21#384—21no—
22#384—41no—
23#384—41no—
24#384C812yescomplemented
25#384C2×C412yes—
26#384D411yescharacteristic
27#384Q811yescharacteristic
28#340—44no—
29#324—24no—
30#324—42no—
31C23×C4—21no—
32C23×C4—22no—
33C23×C4—41no—
34C23×C4C22⋊C411yescharacteristic
35C22×C8—22no—
36C22×C8—22no—
37C22×C8—22no—
38C22×C8—41no—
39C22×C8—41no—
40C22×C8—41no—
41C22×C8—41no—
42C22×C8—82no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—82no—
56C2×C4⋊C4—82no—
57C2×C4⋊C4—82no—
58C2×C4⋊C4—82no—
59C2×C4⋊C4C22⋊C412yes—
60C2×C4⋊C4C4⋊C412yes—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C4211yescharacteristic
65C2×C42C4⋊C411yescharacteristic
66C2×C42C2×C811yescharacteristic
67C2×C42C2×C811yescharacteristic
68C2×C42M1611yescharacteristic
69C2×C42M1611yescharacteristic
70C22⋊C8—44no—
71C22⋊C8—82no—
72C4×C8—42no—
73C4×C8—42no—
74C24#9911yescharacteristic
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—82no—
95C22×C4—82no—
96C22×C4—82no—
97C22×C4—82no—
98C22×C4—81no—
99C22×C4—81no—
100C22×C4#9511yescharacteristic
101C22×C4C22⋊C811yescharacteristic
102C22×C4C22⋊C811yescharacteristic
103C22×C4#9911yescharacteristic
104C22×C4#10011yescharacteristic
105C22×C4#10111yescharacteristic
106C2×C8—24no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—42no—
112C2×C8—42no—
113C2×C8—84no—
114C2×C8—162no—
115C2×C8—162no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—44no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C4⋊C4—84no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C4⋊C4—84no—
125C4⋊C4—84no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—21no—
132C42C4×C811yescharacteristic
133C42#9711yescharacteristic
134C42C4⋊C811yescharacteristic
135C42C4⋊C811yescharacteristic
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23#32311yescharacteristic
141C23#32411yescharacteristic
142C23#34211yescharacteristic
143C23#35311yescharacteristic
144C23#35411yescharacteristic
145C23#35511yescharacteristic
146C23#35611yescharacteristic
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—82no—
183C2×C4—162no—
184C2×C4—162no—
185C2×C4—161no—
186C2×C4—161no—
187C2×C4#32311yescharacteristic
188C2×C4#32411yescharacteristic
189C2×C4#33611yescharacteristic
190C2×C4#34311yescharacteristic
191C8—44no—
192C8—164no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22#128011yescharacteristic
206C22#131811yescharacteristic
207C22#131911yescharacteristic
208C22#132411yescharacteristic
209C22#132511yescharacteristic
210C22#139311yescharacteristic
211C22#139411yescharacteristic
212C22#140811yescharacteristic
213C22#140911yescharacteristic
214C22#141011yescharacteristic
215C22#141111yescharacteristic
216C4—22no—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—82no—
225C4—82no—
226C4—162no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2#718911yescharacteristic
232C2#735411yescharacteristic
233C2#735511yescharacteristic
234C2#737011yescharacteristic
235C2#737111yescharacteristic
236C2#749811yescharacteristic
237C2#749911yescharacteristic
subgroupG/H
center, socleC23#353
commutatorC42C4×C8
Frattini, cosocleC2×#384C22
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1325C4semidirect
20. C2×C4×C8C8semidirect
24. #384C8semidirect

Maximal subgroups

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

subgroupindex
C2×#13252
#108412