Finite Group Databasebeta

← #150312

Subgroups — #150312

Order 512.

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

normal subgroups only

This group has 405 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×#2061C211yesmaximal, characteristic
2C2×#1324C211yesmaximal, characteristic
3#8030C211yesmaximal, characteristic
4#7442C214yesmaximal, complemented
5C2×#544C411yescharacteristic
6C2×#432—21no—
7C2×#428—21no—
8C2×#382C2211yescharacteristic
9#2061—21no—
10#2061—21no—
11#2061C412yes—
12#2061C2212yes—
13C2×#343C411yescharacteristic
14C2×#339—21no—
15C2×#324—21no—
16#1324—21no—
17#1324C2212yes—
18#1295—21no—
19#1295C2212yes—
20C22×M16—21no—
21#544—21no—
22#544—41no—
23#544—42no—
24#544C2×C412yes—
25C2×C4⋊Q8—41no—
26C2×#125—41no—
27C2×C22⋊Q8—41no—
28C2×#117—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42D411yescharacteristic
32#432—24no—
33#432—42no—
34#428—22no—
35#428—22no—
36#428—42no—
37C2×#107—41no—
38C2×#106—41no—
39C2×C4⋊C8—41no—
40C2×C22⋊C8—41no—
41C2×C22⋊C8—41no—
42C2×#97—22no—
43#382—41no—
44#382C2×C411yescharacteristic
45#382D411yescharacteristic
46#382D411yescharacteristic
47#382C2311yescharacteristic
48C2×#95—41no—
49#343—21no—
50#343C2×C412yes—
51#339—22no—
52#339—22no—
53#324—22no—
54#324—41no—
55C22×Q8—41no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—24no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—42no—
64C22×C8—41no—
65C22×C8—42no—
66C22×C8—81no—
67C4⋊Q8—42no—
68C4⋊Q8—42no—
69C4⋊Q8—44no—
70#125—42no—
71#125—42no—
72#125—44no—
73C22⋊Q8—42no—
74C22⋊Q8—42no—
75C22⋊Q8—42no—
76C22⋊Q8—42no—
77#117—22no—
78#117—22no—
79#117—41no—
80#117—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C22⋊C411yescharacteristic
93C2×C4⋊C4C22×C411yescharacteristic
94C2×C4⋊C4C2×D411yescharacteristic
95C2×C22⋊C4—41no—
96C2×C42—21no—
97C2×C42—22no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42SD1612yes—
101C2×C42Q1612yes—
102C2×C42C2×D411yescharacteristic
103#107—42no—
104#107—42no—
105#106—42no—
106#106—42no—
107C4⋊C8—44no—
108C22⋊C8—44no—
109C22⋊C8—42no—
110C22⋊C8—81no—
111#97—24no—
112#97—42no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117C24#9911yescharacteristic
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—162no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4#9911yescharacteristic
143C22×C4#10312yes—
144C22×C4#10412yes—
145C22×C4C2×C22⋊C411yescharacteristic
146M16—24no—
147M16—44no—
148M16—42no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8—162no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—81no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—22no—
173C42—22no—
174C42—21no—
175C42—21no—
176C42—42no—
177C42—42no—
178C42#10312yes—
179C42C2×SD1611yescharacteristic
180C42C2×Q1611yescharacteristic
181C23—41no—
182C23—41no—
183C23#32812yes—
184C23AΣL(1,𝔽16)11yescharacteristic
185C23AΣL(1,𝔽16)11yescharacteristic
186C23#35612yes—
187C23C2×#9911yescharacteristic
188Q8—162no—
189Q8—162no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—161no—
222C2×C4—161no—
223C2×C4#32812yes—
224C2×C4C2×#10311yescharacteristic
225C2×C4C2×#10411yescharacteristic
226C8—44no—
227C8—84no—
228C8—162no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—42no—
237C22#135412yes—
238C22#135412yes—
239C22C2×#32811yescharacteristic
240C22C2×AΣL(1,𝔽16)11yescharacteristic
241C22C2×#35611yescharacteristic
242C4—21no—
243C4—22no—
244C4—21no—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—81no—
249C4—82no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—82no—
254C2—21no—
255C2—22no—
256C2—22no—
257C2—21no—
258C2C2×#135411yescharacteristic
259C2#1356511yescharacteristic
260C2#1356911yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7442C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20612
C2×#13242
#80302
#74422