Finite Group Databasebeta

← #94190

Subgroups — #94190

Order 512.

348 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 46 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#10957C211yesmaximal, characteristic
3#7643C211yesmaximal, characteristic
4C2×#419—21no—
5C2×#391C2211yescharacteristic
6#2274C411yescharacteristic
7#1797—21no—
8#1784—21no—
9C2×#343C411yescharacteristic
10C2×#339—21no—
11#1394—21no—
12#1394C412yescomplemented
13#1272—21no—
14#1272C412yescomplemented
15C23×Q8D411yescharacteristic
16C22×M16—21no—
17C2×#117—21no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C42D411yescharacteristic
20C2×C4⋊C8—41no—
21#419—24no—
22#419—42no—
23C2×#103—42no—
24C2×#101—22no—
25C2×#101—41no—
26#407—41no—
27C2×C22⋊C8—41no—
28#391—41no—
29#391C2×C411yescharacteristic
30#391C2×C411yescharacteristic
31#391D411yescharacteristic
32#391Q811yescharacteristic
33C4×C4⋊C4—42no—
34C4×C22⋊C4—41no—
35C2×#95—41no—
36#343—41no—
37#339—42no—
38#336—22no—
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×C4C22⋊C411yescharacteristic
52C2×M16—22no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—42no—
59C22×C8—41no—
60C22×C8—42no—
61#117—22no—
62#117—22no—
63#117—41no—
64#117—41no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C4211yescharacteristic
70C2×C4⋊C4C4⋊C411yescharacteristic
71C2×C22⋊C4—41no—
72C2×C42—21no—
73C2×C42—41no—
74C2×C42—41no—
75C2×C42—41no—
76C2×C42—42no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C4⋊C411yescharacteristic
79C4⋊C8—81no—
80C4⋊C8—81no—
81#103—48no—
82#101—42no—
83#101—42no—
84#101—81no—
85#101—81no—
86C22⋊C8—82no—
87C22⋊C8—81no—
88C22⋊C8—81no—
89#95—41no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—42no—
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—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4—82no—
139C22×C4#9511yescharacteristic
140C22×C4#9911yescharacteristic
141M16—42no—
142M16—42no—
143M16—81no—
144M16—81no—
145M16—82no—
146M16—82no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8—82no—
154C4⋊C4—24no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—41no—
162C42—41no—
163C42—42no—
164C42—42no—
165C42—41no—
166C42—41no—
167C42—162no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—41no—
172C23#34211yescharacteristic
173C23AΣL(1,𝔽16)11yescharacteristic
174C23#35211yescharacteristic
175Q8—42no—
176Q8—42no—
177Q8—81no—
178Q8—82no—
179Q8—82no—
180Q8—81no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
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—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—44no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—161no—
219C2×C4H(3,ℤ/4)11yescharacteristic
220C2×C4#33911yescharacteristic
221C2×C4#35311yescharacteristic
222C2×C4#35411yescharacteristic
223C8—82no—
224C8—162no—
225C8—162no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22#128411yescharacteristic
236C22#139011yescharacteristic
237C22#139311yescharacteristic
238C22#140512yes—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—42no—
246C4—42no—
247C4—84no—
248C4—81no—
249C4—81no—
250C4—81no—
251C2—21no—
252C2—22no—
253C2—21no—
254C2—41no—
255C2#721311yescharacteristic
256C2#749111yescharacteristic
257C2#749511yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC2×C4⋊C4C42
Frattini, cosocleC2×#391C22
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ 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. #1272C4semidirect

Maximal subgroups

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

subgroupindex
#128602
#109572
#76432