Finite Group Databasebeta

← #122363

Subgroups — #122363

Order 512.

338 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C8×#123C211yesmaximal, characteristic
2#11488C211yesmaximal, characteristic
3#11466C211yesmaximal, characteristic
4#7064C212yesmaximal, complemented
5#7055C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C4×C8C411yescharacteristic
8C4×#123C411yescharacteristic
9#2163—21no—
10#2112—21no—
11#1923—21no—
12#1916—21no—
13#1869C411yescharacteristic
14#1850C2211yescharacteristic
15#1817C2211yescharacteristic
16C8×C4⋊C4—21no—
17C8×C22⋊C4—21no—
18C8×C22⋊C4C411yescharacteristic
19#1322—21no—
20#1314—21no—
21#1314C2212yes—
22#1314C2212yes—
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123C811yescharacteristic
27C2×C4×D4C2×C411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C2×M32—41no—
30C8×D4—41no—
31C8×D4—41no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8D411yescharacteristic
34C2×C22⋊C8—21no—
35C2×C22⋊C8C2×C411yescharacteristic
36C2×C22⋊C8D411yescharacteristic
37C2×C22⋊C8C2311yescharacteristic
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—41no—
42C2×C4×C8—41no—
43C2×C4×C8D411yescharacteristic
44#388—21no—
45#387—21no—
46#386C811yescharacteristic
47#385C811yescharacteristic
48#381C811yescharacteristic
49C4×C4⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4C2×C411yescharacteristic
52C4⋊C16—41no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—41no—
56#346—41no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×D4C2×C811yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C811yescharacteristic
65C23×C4C2×C811yescharacteristic
66C23×C4C22×C411yescharacteristic
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8—42no—
76C22×C8—41no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#123—22no—
82#123—22no—
83C4×D4—41no—
84C4×D4—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4M1611yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4C2×C811yescharacteristic
91C2×C22⋊C4C2×C811yescharacteristic
92C2×C22⋊C4C2×C811yescharacteristic
93C2×C22⋊C4M1611yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100M32—82no—
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—41no—
104C4⋊C8—42no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107C22⋊C8—42no—
108C22⋊C8—41no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—42no—
112C4×C8—42no—
113C4×C8—41no—
114C4×C8—41no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120C24C22⋊C811yescharacteristic
121C24C22×C811yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—41no—
125C2×D4—41no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C22⋊C811yescharacteristic
146C22×C4M3212yescomplemented
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—42no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—41no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—81no—
168C2×C8#9911yescharacteristic
169C2×C8#9911yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C22⋊C4—24no—
174C22⋊C4—21no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C22⋊C4—41no—
179C22⋊C4M3212yescomplemented
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C42—42no—
185C42—41no—
186C42—41no—
187C16—82no—
188C16—82no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23C2×C22⋊C811yescharacteristic
199C23C2×M3211yescharacteristic
200C23C2×M3211yescharacteristic
201D4—42no—
202D4—42no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—81no—
236C2×C4C2×#9911yescharacteristic
237C2×C4#41311yescharacteristic
238C8—21no—
239C8—41no—
240C8—41no—
241C8—41no—
242C8—42no—
243C8—81no—
244C8#35212yes—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22#145911yescharacteristic
260C22#211211yescharacteristic
261C22#211511yescharacteristic
262C4—21no—
263C4—42no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—42no—
270C4—81no—
271C4C2×#35211yescharacteristic
272C4#212011yescharacteristic
273C2—21no—
274C2—21no—
275C2—41no—
276C2—41no—
277C2—81no—
278C2#1146611yescharacteristic
279C2#1147211yescharacteristic
280C2#1148311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22#1459
prime radicalC22×D4C2×C8
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7064C2semidirect
5. #7055C2semidirect
146. C22×C4M32semidirect
179. C22⋊C4M32semidirect

Maximal subgroups

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

subgroupindex
C8×#1232
#114882
#114662
#70642
#70552