Finite Group Databasebeta

← #99088

Subgroups — #99088

Order 512.

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

normal subgroups only

This group has 434 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#21681C211yesmaximal, characteristic
2#8807C211yesmaximal, 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
17C2×#335—21no—
18C2×#334—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—42no—
30#436—42no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C412yes—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#390—21no—
46#390—21no—
47#390D412yes—
48#384Q811yescharacteristic
49C43C2311yescharacteristic
50#336—22no—
51#336—22no—
52#336—22no—
53#336—22no—
54#335—42no—
55#334—42no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4C4⋊C412yes—
59C23×C4—22no—
60C2×M16—44no—
61C2×M16—44no—
62C2×M16—44no—
63C22×C8—22no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70#127—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—41no—
74C4×D4—41no—
75C4×D4—42no—
76C4×D4—41no—
77C4×D4—42no—
78C4×D4—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C4⋊C412yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C42C22×C411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C42C2×D411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×Q811yescharacteristic
90C2×C42#4111yescharacteristic
91C2×C42#4111yescharacteristic
92C4⋊C8—42no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—41no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C22⋊C8—44no—
101C22⋊C8—44no—
102C22⋊C8—44no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—21no—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8—41no—
109C4×C8D812yes—
110C4×C8SD1612yes—
111C24—22no—
112C2×D4—24no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—42no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C2×C4⋊C411yescharacteristic
131C22×C4#11711yescharacteristic
132C22×C4#12111yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4C22⋊Q811yescharacteristic
135C22×C4C22⋊Q811yescharacteristic
136M16—88no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8#10012yes—
156C2×C8#10212yes—
157C2×C8#10212yes—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C22⋊C4—44no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—44no—
166C42—21no—
167C42—21no—
168C42—22no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42C2×D811yescharacteristic
173C42C2×SD1611yescharacteristic
174C42#13512yes—
175C42AGL(1,ℤ/8)11yescharacteristic
176C42AGL(1,ℤ/8)12yes—
177C42#13711yescharacteristic
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23#38011yescharacteristic
186D4—84no—
187D4—82no—
188D4—82no—
189D4—84no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
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—21no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4C2×#10011yescharacteristic
211C2×C4#41311yescharacteristic
212C2×C4C2×#10211yescharacteristic
213C2×C4#41611yescharacteristic
214C2×C4#41711yescharacteristic
215C2×C4#41811yescharacteristic
216C2×C4#43012yes—
217C2×C4#45911yescharacteristic
218C2×C4#46111yescharacteristic
219C2×C4#46312yes—
220C2×C4#47411yescharacteristic
221C2×C4#47611yescharacteristic
222C2×C4#47812yes—
223C8—22no—
224C8—22no—
225C8—22no—
226C8—41no—
227C8—82no—
228C8—162no—
229C8—162no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22#180011yescharacteristic
238C22#180211yescharacteristic
239C22#180411yescharacteristic
240C22#181111yescharacteristic
241C22#181211yescharacteristic
242C22#181411yescharacteristic
243C22#181411yescharacteristic
244C4—22no—
245C4—21no—
246C4—21no—
247C4—22no—
248C4—21no—
249C4—21no—
250C4—22no—
251C4—41no—
252C4—82no—
253C4#154011yescharacteristic
254C4#154511yescharacteristic
255C4#155111yescharacteristic
256C4#155711yescharacteristic
257C2—162no—
258C2#858811yescharacteristic
259C2#862411yescharacteristic
260C2#863011yescharacteristic
261C2#866211yescharacteristic
262C2#866411yescharacteristic
263C2#867611yescharacteristic
264C2#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. #8807C2semidirect
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
#88072
#87852
#76512
#70402