Finite Group Databasebeta

← #141941

Subgroups — #141941

Order 512.

404 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 106 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#21302C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C4×#349C211yesmaximal, characteristic
4#7194C214yesmaximal
5C2×#533C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2150—21no—
9C2×#349C2212yes—
10C2×#349C2212yes—
11C4×M32—21no—
12#1985—21no—
13#1985—21no—
14#1984—21no—
15#1984—21no—
16#1729C412yes—
17#1729C2212yes—
18#1377—21no—
19#1377—21no—
20C22×M16C2×C411yescharacteristic
21C22×M16C2311yescharacteristic
22C23×C8C2×C412yes—
23#533—41no—
24#533—41no—
25#533—41no—
26#533—41no—
27C2×C4⋊Q8C811yescharacteristic
28C2×C4⋊Q8C811yescharacteristic
29C2×#125C812yes—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—21no—
32C2×C4×Q8—21no—
33C2×C4×Q8—21no—
34C2×#117C2×C412yes—
35C22×C42C2×C411yescharacteristic
36C2×M32—22no—
37C2×M32—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8—41no—
42C4×M16—41no—
43C4×M16—41no—
44C2×#97D411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C4⋊C16—41no—
49C4⋊C16—41no—
50#349—22no—
51#349—24no—
52#349Q814yescomplemented
53#346—41no—
54C4×C16—41no—
55#336—22no—
56#336—22no—
57#336—22no—
58#336—22no—
59C22×Q8—21no—
60C22×Q8—21no—
61C23×C4M1611yescharacteristic
62C23×C4M1611yescharacteristic
63C23×C4C22×C411yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16C2×Q812yes—
70C2×M16#4112yes—
71C22×C8—22no—
72C22×C8—24no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83#125—42no—
84#125—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95#117—42no—
96#117—42no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2×C812yes—
102C2×C4⋊C4C2×C812yes—
103C2×C22⋊C4—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42C2×C811yescharacteristic
112C2×C42C2×C811yescharacteristic
113M32—24no—
114M32—42no—
115M32—42no—
116M32—42no—
117C2×C16—42no—
118C2×C16—42no—
119C4⋊C8—42no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122#97—41no—
123#97—41no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C24C2×M1611yescharacteristic
129C2×Q8—42no—
130C2×Q8—44no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—82no—
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—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4C22⋊C811yescharacteristic
147C22×C4C22⋊C811yescharacteristic
148C22×C4C22⋊C811yescharacteristic
149C22×C4C22⋊C811yescharacteristic
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C4×Q812yes—
152C22×C4C22×C811yescharacteristic
153C22×C4#13112yes—
154M16—42no—
155M16—42no—
156M16—44no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—84no—
170C2×C8#10112yes—
171C2×C8C22⋊Q812yes—
172C2×C8#12411yescharacteristic
173C2×C8C4⋊Q811yescharacteristic
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C22⋊C4—44no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42#9911yescharacteristic
192C42#9911yescharacteristic
193C42#10011yescharacteristic
194C42#10011yescharacteristic
195C16—44no—
196C16—82no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23C2×C22⋊C811yescharacteristic
202C23#44612yes—
203Q8—82no—
204Q8—84no—
205Q8—82no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4#32311yescharacteristic
227C2×C4#32311yescharacteristic
228C2×C4#32311yescharacteristic
229C2×C4#32311yescharacteristic
230C2×C4#39111yescharacteristic
231C2×C4#40811yescharacteristic
232C2×C4C2×#9911yescharacteristic
233C2×C4#41011yescharacteristic
234C2×C4C2×#10011yescharacteristic
235C2×C4C2×#10111yescharacteristic
236C2×C4C8×Q812yes—
237C8—22no—
238C8—22no—
239C8—21no—
240C8—21no—
241C8—41no—
242C8—82no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22C2×#32311yescharacteristic
251C22#146111yescharacteristic
252C22#198411yescharacteristic
253C4—22no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#132112yes—
262C4#146911yescharacteristic
263C4#147211yescharacteristic
264C4#199611yescharacteristic
265C4#199711yescharacteristic
266C2—21no—
267C2—21no—
268C2—42no—
269C2#987811yescharacteristic
270C2C2×#132111yescharacteristic
271C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22C2×#323
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
52. #349Q8semidirect

Maximal subgroups

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

subgroupindex
#213022
#127852
C4×#3492
#71942