Finite Group Databasebeta

← #134880

Subgroups — #134880

Order 512.

353 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 74 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#21635C211yesmaximal, characteristic
2#11823C212yesmaximal
3#8611C211yesmaximal, characteristic
4#8608C211yesmaximal, characteristic
5#7322C212yesmaximal
6C2×C8×Q8C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8#2398C2211yescharacteristic
9#1985C411yescharacteristic, complemented
10#1955C2211yescharacteristic
11#1924—21no—
12#1924—21no—
13#1832C411yescharacteristic, complemented
14#1810—21no—
15#1807—21no—
16#1799C2212yes—
17C8×C22⋊C4—21no—
18C8×C22⋊C4—21no—
19#1726C411yescharacteristic, complemented
20#1465C2212yes—
21#1351—22no—
22#1344—22no—
23#1293—21no—
24#1289—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C8×Q8—41no—
30C8×Q8—41no—
31C8×D4—41no—
32C8×D4—41no—
33C2×#108—41no—
34C2×#106—41no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×#99—22no—
39#407—22no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8D411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×#97—21no—
45C2×#97C2×C411yescharacteristic
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8C2×C411yescharacteristic
49#400—21no—
50#397D411yescharacteristic
51#397Q811yescharacteristic
52#394—21no—
53#393D411yescharacteristic
54#393Q811yescharacteristic
55#380—22no—
56C4×C22⋊C4—21no—
57C4×C22⋊C4—21no—
58#376C2×C411yescharacteristic
59#342—22no—
60#341—41no—
61#340—41no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#324—22no—
67#323—22no—
68C22×Q8C4⋊C411yescharacteristic
69C22×D4#4111yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×M16—42no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C4×Q8—41no—
86C4×Q8—41no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4C4⋊C411yescharacteristic
92C2×C4⋊C4#4111yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4C2×D411yescharacteristic
99C2×C22⋊C4C2×Q811yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C22×C411yescharacteristic
103#108—82no—
104#106—82no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108#99—82no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#97—41no—
116#97—42no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—42no—
120C4×C8—41no—
121C4×C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—42no—
128#95C4⋊C411yescharacteristic
129#95C4⋊C411yescharacteristic
130C24#12111yescharacteristic
131C24C22⋊Q811yescharacteristic
132C2×Q8—42no—
133C2×Q8—41no—
134C2×D4—22no—
135C2×D4—41no—
136C2×D4—41no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—82no—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C2×C4⋊C411yescharacteristic
152C22×C4#11711yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4C22⋊Q811yescharacteristic
155M16—82no—
156M16—82no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8—82no—
175C2×C8#9911yescharacteristic
176C2×C8#9911yescharacteristic
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—82no—
189C22⋊C4—82no—
190C42—21no—
191C42—21no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23#38011yescharacteristic
206C23#47112yes—
207Q8—42no—
208Q8—42no—
209D4—44no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
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—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—162no—
240C2×C4—162no—
241C2×C4—162no—
242C2×C4C2×#9911yescharacteristic
243C2×C4#41011yescharacteristic
244C8—21no—
245C8—42no—
246C8—41no—
247C8—41no—
248C8—41no—
249C8—81no—
250C8—162no—
251C8#35412yes—
252C22—21no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22—81no—
264C22#179911yescharacteristic
265C22#181411yescharacteristic
266C22#181511yescharacteristic
267C4—21no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—322no—
277C4C2×#35411yescharacteristic
278C4#212611yescharacteristic
279C2—21no—
280C2—21no—
281C2—42no—
282C2—81no—
283C2#862611yescharacteristic
284C2#1279811yescharacteristic
285C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C8×Q8C4semidirect
9. #1985C4semidirect
13. #1832C4semidirect
19. #1726C4semidirect

Maximal subgroups

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

subgroupindex
#216352
#118232
#86112
#86082
#73222