Finite Group Databasebeta

← #150651

Subgroups — #150651

Order 512.

284 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 52 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#29963C211yesmaximal, characteristic
2#12862C211yesmaximal, characteristic
3#11510C211yesmaximal, characteristic
4#7436C212yesmaximal, complemented
5#7429C212yesmaximal, complemented
6#2629C411yescharacteristic
7#2310C2211yescharacteristic
8#2096C411yescharacteristic
9#2095C2211yescharacteristic
10#2068—21no—
11#2066—21no—
12#2026C411yescharacteristic
13#2026C2211yescharacteristic
14#1854—21no—
15#1798C411yescharacteristic
16#1787—21no—
17#1784—21no—
18#1464—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×#117C2×C411yescharacteristic
26C2×#117D411yescharacteristic
27#432—41no—
28C2×#107—21no—
29C2×#106—21no—
30C2×C4⋊C8—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#101—22no—
36C2×#101—41no—
37#407—41no—
38C2×C22⋊C8—21no—
39#398—41no—
40#396—41no—
41#391—41no—
42#391C2×C411yescharacteristic
43#391D411yescharacteristic
44#391D411yescharacteristic
45#391C2311yescharacteristic
46#389—21no—
47#384C2×C411yescharacteristic
48#381—41no—
49#380—21no—
50#340—21no—
51#340—21no—
52#336C2×C411yescharacteristic
53#336C2×C411yescharacteristic
54#324—22no—
55#323—41no—
56C22×Q8—21no—
57C22×Q8—22no—
58C22×Q8—41no—
59C22×Q8—81no—
60C22×Q8—81no—
61C22×Q8C22⋊C411yescharacteristic
62C22×Q8C2×D411yescharacteristic
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C2×M16—42no—
66C22×C8—21no—
67C22×C8—21no—
68#125—81no—
69#117—81no—
70#117—81no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22×C411yescharacteristic
85#107—81no—
86#106—81no—
87C4⋊C8—81no—
88C4⋊C8—81no—
89#103—81no—
90#103—81no—
91#103—81no—
92#103—81no—
93#101—42no—
94#101—42no—
95#101—82no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—82no—
99#97—81no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106C24#9911yescharacteristic
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—81no—
117C2×Q8—84no—
118C2×Q8—84no—
119C2×Q8#13712yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—81no—
135C22×C4#9911yescharacteristic
136C22×C4C2×C22⋊C411yescharacteristic
137M16—82no—
138M16—82no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—82no—
146C2×C8—82no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—81no—
154C4⋊C4—82no—
155C4⋊C4—161no—
156C4⋊C4—161no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C22⋊C4—161no—
160C42—41no—
161C42—41no—
162C42—41no—
163C42—41no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23C2×#9911yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—81no—
173Q8—81no—
174Q8—84no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—84no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4#41911yescharacteristic
206C2×C4#42111yescharacteristic
207C8—81no—
208C8—81no—
209C8—162no—
210C8—162no—
211C22—21no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22#151111yescharacteristic
219C22#212111yescharacteristic
220C22#212711yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—41no—
225C4—81no—
226C4—81no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4—161no—
231C4—161no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2#1181111yescharacteristic
236C2#1360611yescharacteristic
237C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7436C2semidirect
5. #7429C2semidirect

Maximal subgroups

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

subgroupindex
#299632
#128622
#115102
#74362
#74292