Finite Group Databasebeta

← #100094

Subgroups — #100094

Order 512.

376 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 96 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#23363C211yesmaximal, characteristic
2#21726C211yesmaximal, characteristic
3#7605C211yesmaximal, characteristic
4#7604C211yesmaximal, characteristic
5#7604C211yesmaximal, characteristic
6#7019C211yesmaximal, characteristic
7#7019C211yesmaximal, characteristic
8C2×#500C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1990C411yescharacteristic
12#1990C411yescharacteristic
13#1984—22no—
14#1943—22no—
15#1916—22no—
16#1916—22no—
17#1844C411yescharacteristic
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1839C2211yescharacteristic
21#1839C2211yescharacteristic
22#1776—22no—
23C4×#97C2211yescharacteristic
24#1272—22no—
25#1271—22no—
26#1271—22no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×C4⋊Q8C2×C411yescharacteristic
29C2×C4×Q8—21no—
30C2×C4×Q8—21no—
31#500—22no—
32#500C814yescomplemented
33C2×#107—22no—
34C2×#106—22no—
35C2×C4⋊C8—22no—
36C2×#97—21no—
37C2×#97—21no—
38C2×C4×C8—22no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41#391—21no—
42#391—21no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389D412yes—
47#384C2×C411yescharacteristic
48#384C2×C411yescharacteristic
49C4×C4⋊C4—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D412yes—
53C43C2311yescharacteristic
54#340—22no—
55#340—22no—
56#336—24no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336D412yes—
68#336D412yes—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8—41no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2×C812yes—
83C4⋊Q8C2×C812yes—
84C4⋊Q8M1612yes—
85C4×Q8—81no—
86C4×Q8—81no—
87C4×Q8—82no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C22⋊C412yes—
99C2×C4⋊C4C22⋊C412yes—
100C2×C42—22no—
101C2×C42—41no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109#107—28no—
110#106—44no—
111C4⋊C8—82no—
112#97—24no—
113#97—42no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—82no—
118C4×C8C2×C812yes—
119#95—22no—
120#95—22no—
121#95—41no—
122#95—41no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—81no—
128C2×Q8—81no—
129C2×Q8—81no—
130C2×Q8—81no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4#12011yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4#12111yescharacteristic
141C22×C4#12311yescharacteristic
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8—81no—
159C2×C8—81no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—82no—
175C4⋊C4C22⋊C814yes—
176C42—22no—
177C42—41no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42—41no—
182C42—42no—
183C42—42no—
184C42—82no—
185C42C22×C811yescharacteristic
186C42C2×M1611yescharacteristic
187C42#13111yescharacteristic
188C42#13111yescharacteristic
189C23#38611yescharacteristic
190Q8—82no—
191Q8—82no—
192Q8—82no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—24no—
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—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4C2×C22⋊C811yescharacteristic
224C2×C4#40811yescharacteristic
225C2×C4C8×D411yescharacteristic
226C2×C4C8×D411yescharacteristic
227C2×C4#43511yescharacteristic
228C2×C4#43511yescharacteristic
229C2×C4#45612yes—
230C8—24no—
231C8—42no—
232C8—82no—
233C8—82no—
234C8—161no—
235C8—161no—
236C22#186911yescharacteristic
237C22#188411yescharacteristic
238C22#188611yescharacteristic
239C22#190211yescharacteristic
240C22#190311yescharacteristic
241C22#190411yescharacteristic
242C22#190511yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—22no—
248C4—21no—
249C4—21no—
250C4—44no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—82no—
256C4—161no—
257C2#895811yescharacteristic
258C2#908811yescharacteristic
259C2#908911yescharacteristic
260C2#941511yescharacteristic
261C2#941811yescharacteristic
262C2#942011yescharacteristic
263C2#942711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #500C8semidirect

Maximal subgroups

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

subgroupindex
#233632
#217262
#76052
#76042
#70192