Finite Group Databasebeta

← #137927

Subgroups — #137927

Order 512.

391 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 100 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#33487C211yesmaximal, characteristic
2#22063C211yesmaximal, characteristic
3#11465C211yesmaximal, characteristic
4C2×#1322C212yesmaximal, complemented
5#7199C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C4×#123C411yescharacteristic
9#2168—24no—
10#2163—21no—
11#2112—21no—
12C2×C22⋊C16C2212yes—
13C2×C22⋊C16C2212yes—
14#1939—21no—
15#1923—21no—
16#1916—21no—
17#1869C411yescharacteristic
18#1850—21no—
19#1850C411yescharacteristic
20#1817C411yescharacteristic
21C8×C22⋊C4C2211yescharacteristic
22#1322—21no—
23#1322—24no—
24#1314—21no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#123C811yescharacteristic
29C2×C4×D4C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C2×M32—22no—
32C22×C16—22no—
33C22×C16—22no—
34C8×D4—22no—
35C8×D4—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8D411yescharacteristic
42C2×C22⋊C8—21no—
43C2×C22⋊C8C2×C411yescharacteristic
44C2×C22⋊C8C2×C411yescharacteristic
45C2×C22⋊C8D411yescharacteristic
46C2×C4×C8—21no—
47C2×C4×C8D411yescharacteristic
48#388—21no—
49#387—21no—
50#386C811yescharacteristic
51#385C811yescharacteristic
52#381C811yescharacteristic
53C4×C4⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2×C411yescharacteristic
56C4⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16D414yes—
62#346—22no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4C2×C811yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C2×C811yescharacteristic
71C23×C4C2×C811yescharacteristic
72C23×C4C22×C411yescharacteristic
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—81no—
79C22×C8—81no—
80C22×C8—81no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C2×D412yes—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8#4112yes—
87#123—22no—
88#123—22no—
89C4×D4—22no—
90C4×D4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4M1611yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4C2×C811yescharacteristic
97C2×C22⋊C4C2×C811yescharacteristic
98C2×C22⋊C4C2×C811yescharacteristic
99C2×C22⋊C4M1611yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—22no—
105C2×C42C22⋊C411yescharacteristic
106M32—82no—
107C2×C16—24no—
108C2×C16—42no—
109C2×C16—82no—
110C2×C16—82no—
111C4⋊C8—22no—
112C4⋊C8—41no—
113C4⋊C8—42no—
114C4⋊C8—41no—
115C4⋊C8—81no—
116C22⋊C8—22no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119C22⋊C8—81no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—81no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128C24C22⋊C811yescharacteristic
129C24C22×C811yescharacteristic
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—81no—
149C22×C4C22⋊C811yescharacteristic
150C22×C4C22⋊C811yescharacteristic
151C22×C4C22⋊C811yescharacteristic
152C22×C4M3212yescomplemented
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C4×D412yes—
155C22×C4C2×M1611yescharacteristic
156C22×C4#13112yes—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—81no—
173C2×C8—82no—
174C2×C8#12011yescharacteristic
175C2×C8#12112yes—
176C2×C8#12311yescharacteristic
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—81no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—81no—
186C22⋊C4M3212yescomplemented
187C42—21no—
188C42—21no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—42no—
194C16—44no—
195C16—82no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23C2×C22⋊C811yescharacteristic
204C23C8×D412yes—
205C23C2×M3211yescharacteristic
206C23C2×M3211yescharacteristic
207C23#50412yes—
208D4—42no—
209D4—42no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
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—82no—
241C2×C4#38611yescharacteristic
242C2×C4#40811yescharacteristic
243C2×C4#43512yes—
244C8—21no—
245C8—22no—
246C8—21no—
247C8—41no—
248C8—41no—
249C8—81no—
250C8—81no—
251C22—21no—
252C22—22no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22#186911yescharacteristic
264C22#211211yescharacteristic
265C22#211311yescharacteristic
266C22#216812yes—
267C22#216912yes—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4#191311yescharacteristic
278C4#191511yescharacteristic
279C2—21no—
280C2—21no—
281C2—22no—
282C2—22no—
283C2—81no—
284C2#910811yescharacteristic
285C2#1297711yescharacteristic
286C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1322C2semidirect
5. #7199C2semidirect
152. C22×C4M32semidirect
186. C22⋊C4M32semidirect

Maximal subgroups

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

subgroupindex
#334872
#220632
#114652
C2×#13222
#71992