Finite Group Databasebeta

← #149796

Subgroups — #149796

Order 512.

334 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 58 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#21873C211yesmaximal, characteristic
2#11515C211yesmaximal, characteristic
3#11514C211yesmaximal, characteristic
4#7359C212yesmaximal, complemented
5#7353C212yesmaximal, complemented
6C2×#541C411yescharacteristic
7C2×#433C411yescharacteristic
8#2317C2211yescharacteristic
9#1992C411yescharacteristic
10#1991C2211yescharacteristic
11#1960C2211yescharacteristic
12#1956C411yescharacteristic
13#1873—21no—
14#1872—21no—
15#1805—21no—
16#1803—21no—
17#1464—21no—
18#1463—21no—
19#1325—21no—
20#1324—21no—
21#1324C2212yes—
22#1324C2212yes—
23#541—22no—
24#541—22no—
25#541—42no—
26C2×C4⋊Q8C2×C411yescharacteristic
27C2×#126—41no—
28C2×#125—41no—
29C2×#125C2×C411yescharacteristic
30C2×C22⋊Q8—21no—
31C2×C4×Q8—21no—
32C2×#117C2×C411yescharacteristic
33C2×#117D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35#433—42no—
36#432—41no—
37C2×C4⋊C8—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42#407—41no—
43C2×C22⋊C8—21no—
44C2×#97C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46#389—41no—
47#385—41no—
48#384C2×C411yescharacteristic
49#384D411yescharacteristic
50#382D411yescharacteristic
51#382C2311yescharacteristic
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#323—41no—
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4C22⋊C411yescharacteristic
63C2×M16—81no—
64C22×C8—21no—
65C22×C8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—41no—
68C4⋊Q8M1612yescomplemented
69#126—44no—
70#126—44no—
71#125—21no—
72#125—41no—
73#125—44no—
74#125—44no—
75#125M1612yescomplemented
76C22⋊Q8—24no—
77C22⋊Q8—42no—
78C4×Q8—24no—
79C4×Q8—42no—
80#117—22no—
81#117—21no—
82#117—21no—
83#117—81no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4C22⋊C411yescharacteristic
95C2×C4⋊C4C22⋊C411yescharacteristic
96C2×C4⋊C4C2×D411yescharacteristic
97C2×C4⋊C4C2×D411yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22×C411yescharacteristic
105C4⋊C8—42no—
106C4⋊C8—81no—
107#103—81no—
108#103—81no—
109#103—81no—
110#103—81no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—81no—
115#97—41no—
116#97—81no—
117C4×C8—41no—
118#95—41no—
119#95—41no—
120C24#9911yescharacteristic
121C2×Q8—22no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
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—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—161no—
142C22×C4#9911yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144M16—162no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—82no—
152C2×C8—82no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—161no—
169C4⋊C4—161no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—44no—
173C22⋊C4—81no—
174C42—21no—
175C42—21no—
176C42—41no—
177C42—41no—
178C42—44no—
179C42#10012yes—
180C42C2×M1611yescharacteristic
181C42C2×M1611yescharacteristic
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23C2×#9911yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—81no—
191Q8—81no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—161no—
222C2×C4#40711yescharacteristic
223C2×C4C2×#10011yescharacteristic
224C8—81no—
225C8—81no—
226C8—162no—
227C8—162no—
228C22—21no—
229C22—21no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—81no—
235C22#146211yescharacteristic
236C22#212711yescharacteristic
237C22#213311yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—82no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4—81no—
249C4—82no—
250C2—21no—
251C2—21no—
252C2—81no—
253C2#1150811yescharacteristic
254C2#1360711yescharacteristic
255C2#1360811yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2317 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7359C2semidirect
5. #7353C2semidirect
68. C4⋊Q8M16semidirect
75. #125M16semidirect

Maximal subgroups

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

subgroupindex
#218732
#115152
#115142
#73592
#73532