Finite Group Databasebeta

← #102175

Subgroups — #102175

Order 512.

508 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 110 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#33241C211yesmaximal, characteristic
2#9685C212yesmaximal, complemented
3#8274C211yesmaximal, characteristic, complemented
4C8×#97C211yesmaximal, characteristic, complemented
5#7679C212yesmaximal, complemented
6C2×#500C411yescharacteristic
7C2×#493C411yescharacteristic
8C2×#486C412yes—
9#2858C2212yes—
10#1970—21no—
11#1970—21no—
12#1970—21no—
13#1970—21no—
14#1767C2212yes—
15#1744C2211yescharacteristic
16C4×#97C2211yescharacteristic
17C42×C8C2211yescharacteristic
18C2×#335—21no—
19C2×#329—24no—
20C2×#322—21no—
21C2×C82—21no—
22#1277—21no—
23#1270—24no—
24C22×D8—21no—
25C22×D8—21no—
26C2×C4⋊Q8C2×C412yes—
27C2×#127C2×C412yes—
28C2×#124—22no—
29C2×#124—22no—
30#500—42no—
31#493—42no—
32#486—44no—
33C2×#107—21no—
34C2×#107—21no—
35C2×C4⋊C8—22no—
36C2×#102—22no—
37C2×#102—22no—
38C2×#97—21no—
39C2×#97—22no—
40C2×#97D412yes—
41C2×C4×C8—21no—
42C2×C4×C8—21no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#390—22no—
50#390—22no—
51#390—22no—
52#390—22no—
53C43C2311yescharacteristic
54#336—22no—
55#336—21no—
56#336D412yes—
57#335—24no—
58#329—48no—
59#322—24no—
60C82—24no—
61C22×Q8—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C2×D8—42no—
66C2×D8—42no—
67C2×D8—42no—
68C2×D8—42no—
69C2×D8—42no—
70C2×D8—42no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80#127—42no—
81#127—42no—
82#124—44no—
83#124—44no—
84#124—84no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42#4111yescharacteristic
97C2×C42#4111yescharacteristic
98#107—42no—
99#107—42no—
100C4⋊C8—24no—
101C4⋊C8—42no—
102#102—44no—
103#102—44no—
104#97—24no—
105#97—22no—
106#97—44no—
107#97D814yescomplemented
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—41no—
118C24—42no—
119C2×Q8—44no—
120C2×Q8—82no—
121C2×Q8—82no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—82no—
131C2×D4—82no—
132C2×D4—82no—
133C2×D4—82no—
134C22×C4—42no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4#12411yescharacteristic
141C22×C4#12711yescharacteristic
142D8—84no—
143D8—84no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—24no—
151C2×C8—24no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8#10012yes—
161C2×C8#10012yes—
162C2×C8#10412yes—
163C2×C8#10412yes—
164C2×C8#10412yes—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C22⋊C4—84no—
169C22⋊C4—84no—
170C22⋊C4—84no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—22no—
180C42—22no—
181C42—41no—
182C42C2×D812yes—
183C42#13512yes—
184C23—82no—
185C23—82no—
186C23—82no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23#39011yescharacteristic
192Q8—164no—
193D4—82no—
194D4—82no—
195D4—82no—
196D4—82no—
197D4—164no—
198D4—164no—
199C2×C4—21no—
200C2×C4—21no—
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—21no—
211C2×C4—21no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4C2×#10011yescharacteristic
216C2×C4C2×#10011yescharacteristic
217C2×C4C2×#10411yescharacteristic
218C2×C4C2×#10411yescharacteristic
219C2×C4C2×#10411yescharacteristic
220C2×C4#42111yescharacteristic
221C2×C4C4×D812yes—
222C2×C4#44312yes—
223C2×C4#46612yes—
224C2×C4#47212yes—
225C2×C4#49311yescharacteristic
226C2×C4#49511yescharacteristic
227C8—28no—
228C8—22no—
229C8—22no—
230C8—44no—
231C8—41no—
232C8—84no—
233C8#32914yescomplemented
234C22—162no—
235C22—162no—
236C22—162no—
237C22—162no—
238C22—162no—
239C22—162no—
240C22—162no—
241C22#196211yescharacteristic
242C22#196511yescharacteristic
243C22#197011yescharacteristic
244C22#197411yescharacteristic
245C22#197611yescharacteristic
246C22#197811yescharacteristic
247C22#198011yescharacteristic
248C4—22no—
249C4—22no—
250C4—22no—
251C4—22no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—162no—
256C4C2×#32911yescharacteristic
257C4#152511yescharacteristic
258C4#152711yescharacteristic
259C4#153011yescharacteristic
260C2—322no—
261C2#968511yescharacteristic
262C2#971911yescharacteristic
263C2#972411yescharacteristic
264C2#976211yescharacteristic
265C2#978711yescharacteristic
266C2#980211yescharacteristic
267C2#984411yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#493C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9685C2semidirect
3. #8274C2semidirect
4. C8×#97C2semidirect
5. #7679C2semidirect
107. #97D8semidirect
233. C8#329semidirect

Maximal subgroups

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

subgroupindex
#332412
#96852
#82742
C8×#972
#76792