Finite Group Databasebeta

← #391700

Subgroups — #391700

Order 512.

2840 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 2840 conjugacy classes of subgroups — too many to draw. Only its 330 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
1C2×#2847C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic, complemented
3#14680C211yesmaximal, characteristic, complemented
4#11199C216yesmaximal, complemented
5#10969C216yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C22×#121C2213yes—
8C22×#120—21no—
9C22×#102C2214yescomplemented
10C22×C22⋊C8C2216yes—
11#2847—24no—
12#2847C418yescomplemented
13C2×#386—23no—
14C2×#380C2213yes—
15C2×#380C2216yes—
16C2×#379—21no—
17#1779—24no—
18C2×#340C22112yescomplemented
19C22⋊#95—23no—
20C26D411yescharacteristic
21C23×D4—23no—
22C23×D4C2311yescharacteristic
23C24×C4—412no—
24C24×C4C2×C414yes—
25C24×C4D413yes—
26C24×C4D414yes—
27C24×C4C2313yes—
28C23×C8C2314yes—
29C2×#121—26no—
30C2×#121—412no—
31C2×#121C2×C4112yes—
32C2×#120—28no—
33C2×#120—44no—
34C2×#120—44no—
35C22×C4⋊C4C2314yes—
36C22×C4⋊C4C2313yes—
37C22×C22⋊C4—23no—
38C22×C22⋊C4—23no—
39C22×C22⋊C4—26no—
40C2×#102—412no—
41C2×C22⋊C8—212no—
42C2×C22⋊C8—412no—
43#386—224no—
44#380—26no—
45#380—412no—
46#380C2×C4112yes—
47#379—44no—
48C2×#95—23no—
49C2×#95—26no—
50#340—412no—
51C25—21no—
52C25—46no—
53C25—46no—
54C25C22⋊C414yes—
55C25D814yes—
56C25SD1614yes—
57C25C2×D413yes—
58C22×D4—212no—
59C22×D4—21no—
60C22×D4—212no—
61C22×D4—412no—
62C22×D4—412no—
63C22×D4—46no—
64C22×D4—46no—
65C22×D4C22×C412yes—
66C23×C4—212no—
67C23×C4—212no—
68C23×C4—212no—
69C23×C4—23no—
70C23×C4—24no—
71C23×C4—424no—
72C23×C4—424no—
73C23×C4—46no—
74C23×C4—46no—
75C23×C4—46no—
76C23×C4—448no—
77C23×C4—416no—
78C23×C4C22⋊C4112yes—
79C23×C4C22×C416yes—
80C23×C4C2×D413yes—
81C23×C4C2×D416yes—
82C23×C4C2411yescharacteristic
83C22×C8—24no—
84C22×C8—212no—
85C22×C8—212no—
86#121—424no—
87#121—424no—
88#120—432no—
89C2×C4⋊C4—212no—
90C2×C4⋊C4—23no—
91C2×C4⋊C4—212no—
92C2×C4⋊C4—212no—
93C2×C4⋊C4C22×C416yes—
94C2×C22⋊C4—212no—
95C2×C22⋊C4—212no—
96C2×C22⋊C4—412no—
97C2×C22⋊C4—46no—
98C2×C22⋊C4—46no—
99C2×C22⋊C4—412no—
100C2×C22⋊C4—412no—
101C2×C22⋊C4—412no—
102#102—416no—
103C22⋊C8—424no—
104#95—212no—
105#95—412no—
106C24—26no—
107C24—26no—
108C24—26no—
109C24—26no—
110C24—26no—
111C24—26no—
112C24—22no—
113C24—412no—
114C24—412no—
115C24—424no—
116C24—424no—
117C24—416no—
118C24—448no—
119C24—46no—
120C24—41no—
121C24—41no—
122C24—46no—
123C24#102116yescomplemented
124C24C2×C22⋊C416yes—
125C24C2×D816yes—
126C24C2×SD1616yes—
127C24C22×D411yescharacteristic
128C2×D4—212no—
129C2×D4—424no—
130C2×D4—424no—
131C2×D4—424no—
132C2×D4—46no—
133C2×D4—412no—
134C2×D4—44no—
135C2×D4—824no—
136C2×D4—824no—
137C22×C4—26no—
138C22×C4—26no—
139C22×C4—26no—
140C22×C4—212no—
141C22×C4—212no—
142C22×C4—212no—
143C22×C4—212no—
144C22×C4—212no—
145C22×C4—212no—
146C22×C4—26no—
147C22×C4—412no—
148C22×C4—412no—
149C22×C4—412no—
150C22×C4—448no—
151C22×C4—464no—
152C22×C4—464no—
153C22×C4—436no—
154C22×C4—412no—
155C22×C4—46no—
156C22×C4—43no—
157C22×C4—43no—
158C22×C4—46no—
159C22×C4—412no—
160C22×C4—44no—
161C22×C4—412no—
162C22×C4C2×C22⋊C416yes—
163C22×C4C23×C411yescharacteristic
164C22×C4C22×D411yescharacteristic
165C22×C4H(5,𝔽2)112yes—
166C2×C8—412no—
167C2×C8—412no—
168C2×C8—44no—
169C4⋊C4—212no—
170C4⋊C4—46no—
171C4⋊C4—412no—
172C4⋊C4—44no—
173C22⋊C4—424no—
174C22⋊C4—424no—
175C22⋊C4—824no—
176C23—26no—
177C23—26no—
178C23—26no—
179C23—26no—
180C23—26no—
181C23—26no—
182C23—212no—
183C23—212no—
184C23—212no—
185C23—212no—
186C23—212no—
187C23—212no—
188C23—448no—
189C23—416no—
190C23—448no—
191C23—464no—
192C23—464no—
193C23—412no—
194C23—412no—
195C23—412no—
196C23—412no—
197C23—46no—
198C23—42no—
199C23—46no—
200C23—86no—
201C23—86no—
202C23—812no—
203C23—84no—
204C23C2×#102112yes—
205C23C22×C22⋊C411yescharacteristic
206C23#53416yes—
207C23C22×D811yescharacteristic
208C23C22×SD1611yescharacteristic
209C23#57516yes—
210D4—48no—
211D4—848no—
212D4—84no—
213C2×C4—26no—
214C2×C4—26no—
215C2×C4—26no—
216C2×C4—448no—
217C2×C4—412no—
218C2×C4—424no—
219C2×C4—424no—
220C2×C4—46no—
221C2×C4—46no—
222C2×C4—46no—
223C2×C4—86no—
224C2×C4—86no—
225C2×C4—812no—
226C2×C4—812no—
227C2×C4—84no—
228C2×C4#51816yes—
229C2×C4#53416yes—
230C2×C4#560112yes—
231C2×C4H(3,𝔽4)14yes—
232C8—84no—
233C22—212no—
234C22—212no—
235C22—212no—
236C22—464no—
237C22—46no—
238C22—46no—
239C22—46no—
240C22—46no—
241C22—424no—
242C22—424no—
243C22—86no—
244C22—82no—
245C22—86no—
246C22—164no—
247C22—164no—
248C22#230813yes—
249C22#281214yes—
250C22C22×#10211yescharacteristic
251C22#289613yes—
252C22#318616yes—
253C22#318716yes—
254C22#3197112yes—
255C4—46no—
256C4—86no—
257C4—164no—
258C4#237514yes—
259C4#281214yes—
260C2—412no—
261C2—162no—
262C2#1465711yescharacteristic
263C2#2175813yes—
264C2#2176613yes—
265C2#3270614yes—
266C2#3270814yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1779C2semidirect
3. #14680C2semidirect
4. #11199C2semidirect
5. #10969C2semidirect
9. C22×#102C22semidirect
12. #2847C4semidirect
18. C2×#340C22semidirect
123. C24#102semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#17792
#146802
#111992
#109692