Finite Group Databasebeta

← #136835

Subgroups — #136835

Order 512.

433 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 114 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#33486C211yesmaximal, characteristic
2C8×#123C211yesmaximal, characteristic
3#12844C212yesmaximal, complemented
4#11465C211yesmaximal, characteristic
5#7199C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C4×#123C411yescharacteristic
9D4×C16—24no—
10#2162—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16C2212yes—
13C2×C22⋊C16C2212yes—
14#1923—21no—
15#1916—21no—
16#1869C411yescharacteristic
17#1850C411yescharacteristic
18#1817C411yescharacteristic
19C8×C4⋊C4—21no—
20C8×C22⋊C4—21no—
21C8×C22⋊C4C2211yescharacteristic
22#1374—24no—
23#1322—21no—
24#1314—21no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#123C811yescharacteristic
29C2×C4×D4C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C16—22no—
32C22×C16—22no—
33C22×C16D414yes—
34C8×D4—22no—
35C8×D4—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8D411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×C4×C8—22no—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—22no—
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⋊C16—22no—
62C4×C16—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—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C2×D412yes—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8#4112yes—
87#123—22no—
88#123C1614yescomplemented
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
106C2×C16—24no—
107C2×C16—44no—
108C2×C16—44no—
109C2×C16—42no—
110C2×C16—82no—
111C4⋊C8—22no—
112C4⋊C8—42no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116C22⋊C8—42no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—42no—
120C4×C8—41no—
121C4×C8—42no—
122C4×C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128C24C22⋊C811yescharacteristic
129C24C22×C811yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4C2×C1612yes—
133C2×D4M3212yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C22⋊C811yescharacteristic
150C22×C4C22⋊C811yescharacteristic
151C22×C4C22⋊C811yescharacteristic
152C22×C4C2×C1612yescomplemented
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C4×D412yes—
155C22×C4C2×M1611yescharacteristic
156C22×C4#13112yes—
157C2×C8—22no—
158C2×C8—24no—
159C2×C8—22no—
160C2×C8—24no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—81no—
175C2×C8#12011yescharacteristic
176C2×C8#12112yes—
177C2×C8#12311yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C22⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—21no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4C2×C1612yescomplemented
188C42—21no—
189C42—21no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—42no—
195C16—44no—
196C16—82no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23C22⋊C1612yes—
204C23C2×C22⋊C811yescharacteristic
205C23C8×D412yes—
206C23C22×C1611yescharacteristic
207C23C2×M3211yescharacteristic
208C23#50412yes—
209D4—24no—
210D4—24no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4C22⋊C1612yes—
243C2×C4#38611yescharacteristic
244C2×C4#40811yescharacteristic
245C2×C4#43512yes—
246C8—22no—
247C8—21no—
248C8—41no—
249C8—42no—
250C8—41no—
251C8—81no—
252C8#45812yes—
253C22—21no—
254C22—22no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—42no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22#186911yescharacteristic
266C22C2×C22⋊C1611yescharacteristic
267C22#211311yescharacteristic
268C22D4×C1612yes—
269C22#216812yes—
270C4—21no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—42no—
277C4—41no—
278C4—81no—
279C4#189011yescharacteristic
280C4#189111yescharacteristic
281C2—21no—
282C2—21no—
283C2—22no—
284C2—22no—
285C2—42no—
286C2#896611yescharacteristic
287C2#1297611yescharacteristic
288C2#1298111yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12844C2semidirect
5. #7199C2semidirect
88. #123C16semidirect
152. C22×C4C2×C16semidirect
187. C22⋊C4C2×C16semidirect

Maximal subgroups

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

subgroupindex
#334862
C8×#1232
#128442
#114652
#71992