Finite Group Databasebeta

← #134912

Subgroups — #134912

Order 512.

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

normal subgroups only

This group has 343 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#21567C211yesmaximal, characteristic
2#11822C212yesmaximal
3#8620C211yesmaximal, characteristic
4#8609C211yesmaximal, characteristic
5#7320C212yesmaximal
6C2×C8×D4C2211yescharacteristic
7C2×#405C411yescharacteristic, complemented
8#2395C2211yescharacteristic
9#1923—21no—
10#1916—21no—
11#1870C2211yescharacteristic
12#1851C411yescharacteristic, complemented
13#1818C411yescharacteristic, complemented
14#1808—21no—
15#1799C2212yes—
16#1771—21no—
17#1752—21no—
18#1752C411yescharacteristic, complemented
19#1568—21no—
20#1466C2212yes—
21#1352—22no—
22#1344—22no—
23#1290—21no—
24#1289—21no—
25C22×M16C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C2×#117C2×C411yescharacteristic
30C8×D4—41no—
31C8×D4—41no—
32#432—22no—
33C2×#107—41no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8D411yescharacteristic
36C2×#99—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8D411yescharacteristic
41#405—41no—
42#405—41no—
43C2×#97—21no—
44C2×#97—21no—
45C2×C4×C8—21no—
46C2×C4×C8D411yescharacteristic
47#400Q811yescharacteristic
48#399D411yescharacteristic
49#398Q811yescharacteristic
50#397—21no—
51#397D411yescharacteristic
52#394—21no—
53#380—22no—
54C4×C4⋊C4—21no—
55C4×C22⋊C4—21no—
56C4×C22⋊C4C2×C411yescharacteristic
57#342—22no—
58#340—41no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#335—41no—
65#334—41no—
66#323—22no—
67#323—22no—
68C22×D4#4111yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—42no—
72C2×M16—41no—
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×D4—41no—
86C4×D4—41no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C4⋊C411yescharacteristic
93C2×C4⋊C4C4⋊C411yescharacteristic
94C2×C4⋊C4C2×Q811yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4C2×D411yescharacteristic
101C2×C22⋊C4#4111yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2×D411yescharacteristic
105#107—82no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109#99—82no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—42no—
113C22⋊C8—41no—
114C22⋊C8—42no—
115#97—42no—
116#97—42no—
117#97—41no—
118#97—41no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—42no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—42no—
127#95C4⋊C411yescharacteristic
128#95C4⋊C411yescharacteristic
129C24#11711yescharacteristic
130C24#12111yescharacteristic
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—41no—
134C2×D4—41no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—82no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C2×C4⋊C411yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4C22⋊Q811yescharacteristic
152C22×C4C22⋊Q811yescharacteristic
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—42no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—81no—
172C2×C8—82no—
173C2×C8#9911yescharacteristic
174C2×C8#9911yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—42no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C42—21no—
192C42—21no—
193C42—41no—
194C42—41no—
195C42—41no—
196C42—41no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23#38011yescharacteristic
207C23#47211yescharacteristic
208C23#47311yescharacteristic
209D4—42no—
210D4—42no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
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—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—81no—
240C2×C4—162no—
241C2×C4—162no—
242C2×C4—162no—
243C2×C4C2×#9911yescharacteristic
244C2×C4#41011yescharacteristic
245C8—21no—
246C8—41no—
247C8—41no—
248C8—41no—
249C8—42no—
250C8—81no—
251C8—162no—
252C8#35212yes—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#179911yescharacteristic
268C22#181411yescharacteristic
269C22#181611yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—322no—
280C4C2×#35211yescharacteristic
281C4#212011yescharacteristic
282C2—21no—
283C2—21no—
284C2—41no—
285C2—41no—
286C2—81no—
287C2#862811yescharacteristic
288C2#1279411yescharacteristic
289C2#1280811yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C23×C4 ▷ 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
7. C2×#405C4semidirect
12. #1851C4semidirect
13. #1818C4semidirect
18. #1752C4semidirect

Maximal subgroups

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

subgroupindex
#215672
#118222
#86202
#86092
#73202