Finite Group Databasebeta

← #99092

Subgroups — #99092

Order 512.

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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#21681C211yesmaximal, characteristic
2#8785C211yesmaximal, characteristic, complemented
3#8785C211yesmaximal, characteristic, complemented
4#7651C212yesmaximal
5#7040C212yesmaximal, complemented
6C2×#436—21no—
7C2×#436—21no—
8C2×#436C412yes—
9C4×#127C2211yescharacteristic
10#1954—21no—
11#1954—21no—
12#1954C412yes—
13#1842C2211yescharacteristic
14#1831C2212yes—
15#1831C2212yes—
16C42×C8C2211yescharacteristic
17#1277—21no—
18#1277—21no—
19#1277—21no—
20#1277—21no—
21#1270—24no—
22C22×M16—22no—
23C2×#127Q811yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×C4×D4C2×C412yes—
27#436—44no—
28#436—44no—
29#436—44no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8C2×C412yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#390—21no—
45#390—21no—
46#390D412yes—
47#384Q811yescharacteristic
48C43C2311yescharacteristic
49#336—22no—
50#336—22no—
51#336—22no—
52#336—22no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4C4⋊C412yes—
56C23×C4—22no—
57C2×M16—44no—
58C2×M16—44no—
59C2×M16—44no—
60C22×C8—22no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67#127—42no—
68C4×D4—42no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C4⋊C412yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×Q811yescharacteristic
85C2×C42#4111yescharacteristic
86C2×C42#4111yescharacteristic
87C4⋊C8—42no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C22⋊C8—44no—
95C22⋊C8—44no—
96C22⋊C8—44no—
97C4×C8—22no—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—41no—
103C24—22no—
104C2×D4—24no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4#12111yescharacteristic
124C22×C4#12111yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C22⋊Q811yescharacteristic
127M16—88no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8#10012yes—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C22⋊C4—44no—
155C42—21no—
156C42—21no—
157C42—22no—
158C42—41no—
159C42—41no—
160C42—41no—
161C42#13512yes—
162C42#13512yes—
163C42AGL(1,ℤ/8)12yes—
164C42AGL(1,ℤ/8)12yes—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23#38011yescharacteristic
173D4—84no—
174D4—84no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4C2×#10011yescharacteristic
197C2×C4#41311yescharacteristic
198C2×C4#41611yescharacteristic
199C2×C4#41611yescharacteristic
200C2×C4#41811yescharacteristic
201C2×C4#41811yescharacteristic
202C2×C4#43012yes—
203C2×C4#46312yes—
204C2×C4#46312yes—
205C2×C4#47812yes—
206C2×C4#47812yes—
207C8—22no—
208C8—41no—
209C8—41no—
210C8—41no—
211C8—82no—
212C8—162no—
213C8—162no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22#180011yescharacteristic
222C22#180411yescharacteristic
223C22#180411yescharacteristic
224C22#181111yescharacteristic
225C22#181211yescharacteristic
226C22#181411yescharacteristic
227C22#181411yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—41no—
234C4—82no—
235C4#154311yescharacteristic
236C4#155311yescharacteristic
237C4#155311yescharacteristic
238C4#155511yescharacteristic
239C2—162no—
240C2#859211yescharacteristic
241C2#862411yescharacteristic
242C2#863411yescharacteristic
243C2#866411yescharacteristic
244C2#866411yescharacteristic
245C2#867811yescharacteristic
246C2#868011yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127Q8
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. #8785C2semidirect
3. #8785C2semidirect
5. #7040C2semidirect

Maximal subgroups

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

subgroupindex
#216812
#87852
#76512
#70402