Finite Group Databasebeta

← #94183

Subgroups — #94183

Order 512.

352 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 48 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#12860C211yesmaximal, characteristic
2#11506C211yesmaximal, characteristic
3#10928C211yesmaximal, characteristic
4C22×#103—21no—
5C22×#97C411yescharacteristic
6C2×#391C2211yescharacteristic
7#1784—21no—
8C2×#340—21no—
9#1463—21no—
10C4⋊#95C411yescharacteristic
11#1394—21no—
12#1394C412yescomplemented
13#1319—21no—
14#1319C412yescomplemented
15C23×Q8D411yescharacteristic
16C23×C8—21no—
17C22×C4⋊C4—21no—
18C22×C4⋊C4D411yescharacteristic
19C22×C42C2×C411yescharacteristic
20C2×#103—24no—
21C2×#103—42no—
22C2×#103—42no—
23C2×#101—22no—
24C2×#101—41no—
25#407—41no—
26#407—41no—
27C2×#97—41no—
28C2×#97—42no—
29#391—41no—
30#391C2×C411yescharacteristic
31#391C2×C411yescharacteristic
32#391D411yescharacteristic
33#391Q811yescharacteristic
34C2×#95—41no—
35C2×#95—41no—
36#340—42no—
37#324—22no—
38#324—41no—
39C22×Q8—21no—
40C22×Q8—22no—
41C22×Q8—41no—
42C22×Q8—41no—
43C22×Q8—41no—
44C22×Q8—41no—
45C22×Q8—41no—
46C22×Q8C22⋊C411yescharacteristic
47C22×Q8C4⋊C411yescharacteristic
48C23×C4—21no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—41no—
52C23×C4C22⋊C411yescharacteristic
53C2×M16—42no—
54C2×M16—81no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C22⋊C411yescharacteristic
72C2×C4⋊C4C4⋊C411yescharacteristic
73C2×C42—21no—
74C2×C42—41no—
75C2×C42—41no—
76C2×C42C4211yescharacteristic
77C2×C42C4⋊C411yescharacteristic
78#103—48no—
79#101—42no—
80#101—42no—
81#101—81no—
82#101—81no—
83C22⋊C8—42no—
84C22⋊C8—81no—
85C22⋊C8—81no—
86C22⋊C8—81no—
87#97—42no—
88#97—42no—
89#95—41no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—82no—
94C24#9911yescharacteristic
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—42no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—81no—
110C2×Q8—81no—
111C2×Q8—81no—
112C2×Q8—81no—
113C2×Q8—81no—
114C2×Q8#10412yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—82no—
143C22×C4—81no—
144C22×C4#9511yescharacteristic
145C22×C4#9911yescharacteristic
146M16—82no—
147M16—82no—
148M16—161no—
149M16—161no—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—82no—
158C2×C8—82no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—41no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—22no—
175C23#34211yescharacteristic
176C23#35311yescharacteristic
177C23#35411yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—81no—
181Q8—82no—
182Q8—82no—
183Q8—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—84no—
220C2×C4—161no—
221C2×C4—162no—
222C2×C4—161no—
223C2×C4—161no—
224C2×C4H(3,ℤ/4)11yescharacteristic
225C2×C4#33911yescharacteristic
226C2×C4AΣL(1,𝔽16)11yescharacteristic
227C2×C4#35211yescharacteristic
228C8—44no—
229C8—162no—
230C8—162no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22#128411yescharacteristic
242C22#139011yescharacteristic
243C22#139311yescharacteristic
244C22#140611yescharacteristic
245C22#140711yescharacteristic
246C22#141312yes—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—162no—
258C2—21no—
259C2—21no—
260C2—21no—
261C2—21no—
262C2—22no—
263C2#721311yescharacteristic
264C2#750511yescharacteristic
265C2#750811yescharacteristic
subgroupG/H
center, socleC22#1393
commutatorC2×C42C42
Frattini, cosocleC2×#391C22
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ C2×#391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1394C4semidirect
14. #1319C4semidirect

Maximal subgroups

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

subgroupindex
#128602
#115062
#109282