Finite Group Databasebeta

← #102153

Subgroups — #102153

Order 512.

564 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 142 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#33172C211yesmaximal, characteristic
2#10862C211yesmaximal, characteristic
3#9885C214yesmaximal
4C8×C22⋊C8C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#467—21no—
7C2×#467—21no—
8C22×#107—21no—
9C22×C4×C8C2211yescharacteristic
10#2857C2212yes—
11#1969—21no—
12#1969—21no—
13#1956C412yescomplemented
14#1845C411yescharacteristic, complemented
15C4×C22⋊C8C2212yes—
16C2×#340—21no—
17C2×C82—21no—
18C4⋊#95C2212yes—
19#1294—28no—
20#1273—21no—
21C22×Q16—21no—
22C22×Q16—21no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8C2×C412yes—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C42C2311yescharacteristic
29#494—42no—
30#467—42no—
31#467—42no—
32#467—42no—
33#467—42no—
34C2×#107—24no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—42no—
38C2×#107—42no—
39C2×#103—22no—
40C2×#103—22no—
41C2×C22⋊C8D414yes—
42C2×C4×C8—22no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391—22no—
49#389—24no—
50#384C2×C412yes—
51#380—24no—
52C2×#95—24no—
53#340—42no—
54#336—22no—
55C82—42no—
56C22×Q8—22no—
57C23×C4—42no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×D412yes—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—24no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8D812yes—
76C22×C8D812yes—
77C22×C8SD1612yes—
78C22×C8SD1612yes—
79C4⋊Q8—42no—
80C4⋊Q8—42no—
81C22⋊Q8—42no—
82C22⋊Q8—42no—
83C22⋊Q8—42no—
84C22⋊Q8—42no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42#4112yes—
101#107—48no—
102#107—42no—
103#107—42no—
104#103—44no—
105#103—44no—
106C22⋊C8—24no—
107C22⋊C8Q1618yescomplemented
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112C4×C8—42no—
113C4×C8—42no—
114#95—44no—
115#95—48no—
116C24#12711yescharacteristic
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—82no—
133C22×C4—84no—
134C22×C4—82no—
135C22×C4—84no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C4×D412yes—
138C22×C4#12112yes—
139C22×C4#12411yescharacteristic
140C22×C4C2×D811yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144C22×C4C2×Q1614yes—
145C22×C4#13514yes—
146Q16—84no—
147Q16—84no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—28no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8#9912yes—
162C2×C8#10212yes—
163C2×C8#10212yes—
164C2×C8#10212yes—
165C2×C8#10212yes—
166C2×C8#10412yes—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C4⋊C4—84no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—42no—
184C23—21no—
185C23—22no—
186C23—21no—
187C23#39011yescharacteristic
188C23#49211yescharacteristic
189C23#49311yescharacteristic
190C23#49412yes—
191C23#49512yes—
192Q8—82no—
193Q8—82no—
194Q8—82no—
195Q8—82no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—84no—
213C2×C4—88no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—162no—
220C2×C4—164no—
221C2×C4—162no—
222C2×C4C2×#9911yescharacteristic
223C2×C4#41011yescharacteristic
224C2×C4C2×#10211yescharacteristic
225C2×C4C2×#10211yescharacteristic
226C2×C4C2×#10411yescharacteristic
227C2×C4#42111yescharacteristic
228C2×C4C4×Q1614yes—
229C2×C4#46714yes—
230C2×C4#48611yescharacteristic
231C2×C4#48611yescharacteristic
232C8—22no—
233C8—22no—
234C8—22no—
235C8—48no—
236C8#32714yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—22no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22#170712yes—
246C22#170911yescharacteristic
247C22#171311yescharacteristic
248C22#195711yescharacteristic
249C22#196411yescharacteristic
250C22#196912yes—
251C22#197612yes—
252C22#198011yescharacteristic
253C4—24no—
254C4—24no—
255C4—21no—
256C4—21no—
257C4—164no—
258C4—162no—
259C4—162no—
260C4C2×#32711yescharacteristic
261C4#150211yescharacteristic
262C4#151711yescharacteristic
263C4#151911yescharacteristic
264C2—21no—
265C2—22no—
266C2—21no—
267C2#968011yescharacteristic
268C2#971812yes—
269C2#975611yescharacteristic
270C2#978212yes—
271C2#982511yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
13. #1956C4semidirect
14. #1845C4semidirect
107. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#331722
#108622
#98852
C8×C22⋊C82