Finite Group Databasebeta

← #141987

Subgroups — #141987

Order 512.

392 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 88 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#21304C211yesmaximal, characteristic
2#12790C211yesmaximal, characteristic
3#12608C211yesmaximal, characteristic
4#7193C212yesmaximal
5#7192C212yesmaximal
6C2×#533C411yescharacteristic
7C2×#424C411yescharacteristic
8C2×#405C2211yescharacteristic
9C2×#364—21no—
10C2×#349C2212yes—
11C2×#349C2212yes—
12C2×#347—21no—
13#1985—21no—
14#1985—21no—
15#1984—21no—
16#1984—21no—
17#1729C411yescharacteristic
18#1729C2211yescharacteristic
19#1728C2211yescharacteristic
20C2×#336C411yescharacteristic
21#1376—21no—
22#1375—21no—
23C22×M16C2×C411yescharacteristic
24C22×M16C2311yescharacteristic
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27#533—42no—
28#533—42no—
29C2×C4⋊Q8—21no—
30C2×#125—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C2×M32—22no—
39C2×M32—22no—
40#424—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43#405—42no—
44C2×#97D411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#364—42no—
49#349—22no—
50#349—24no—
51#349Q814yescomplemented
52#347—42no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61C22×Q8—21no—
62C22×Q8—21no—
63C23×C4—21no—
64C23×C4C22×C411yescharacteristic
65C2×M16—22no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16C2×Q812yes—
71C2×M16#4112yes—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4⋊Q8—44no—
85#125—44no—
86C22⋊Q8—44no—
87C22⋊Q8—44no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92#117—22no—
93#117—22no—
94#117—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4M1611yescharacteristic
106C2×C22⋊C4M1611yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C42M1611yescharacteristic
114C2×C42M1611yescharacteristic
115M32—24no—
116M32—42no—
117M32—42no—
118M32—42no—
119C2×C16—42no—
120C2×C16—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123#97—22no—
124#97—41no—
125C4×C8—22no—
126C4×C8—41no—
127C24C2×M1611yescharacteristic
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—81no—
133C2×Q8—81no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C4×Q812yes—
149C22×C4C2×M1611yescharacteristic
150C22×C4#13112yes—
151M16—42no—
152M16—42no—
153M16—82no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—44no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—82no—
171C2×C8#10112yes—
172C2×C8C22⋊Q812yes—
173C2×C8#12411yescharacteristic
174C2×C8C4⋊Q811yescharacteristic
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—24no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C42—21no—
187C42—21no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42#9912yes—
195C42#10012yes—
196C16—44no—
197C16—82no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—41no—
202C23#40711yescharacteristic
203C23#44612yes—
204Q8—84no—
205Q8—84no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—84no—
231C2×C4#39111yescharacteristic
232C2×C4#40811yescharacteristic
233C2×C4C2×#9911yescharacteristic
234C2×C4#41011yescharacteristic
235C2×C4C2×#10011yescharacteristic
236C2×C4C2×#10111yescharacteristic
237C2×C4#44612yes—
238C8—22no—
239C8—22no—
240C8—21no—
241C8—21no—
242C8—41no—
243C8—81no—
244C8—81no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—41no—
251C22—41no—
252C22#146211yescharacteristic
253C22#146711yescharacteristic
254C22#198511yescharacteristic
255C4—22no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4#146911yescharacteristic
263C4#147211yescharacteristic
264C4#199611yescharacteristic
265C4#199711yescharacteristic
266C2—21no—
267C2—21no—
268C2—42no—
269C2#988211yescharacteristic
270C2#1153911yescharacteristic
271C2#1154011yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC22×M16C23
socleC22#1467
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
51. #349Q8semidirect

Maximal subgroups

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

subgroupindex
#213042
#127902
#126082
#71932
#71922