Finite Group Databasebeta

← #238079

Subgroups — #238079

Order 512.

951 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 170 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#36674C211yesmaximal, characteristic, complemented
2#21747C213yesmaximal
3#10393C213yesmaximal, complemented
4#10392C213yesmaximal
5#10391C211yesmaximal, characteristic, complemented
6#10390C211yesmaximal, characteristic
7#8102C213yesmaximal, complemented
8C22×#123C2211yescharacteristic
9#2842C2213yes—
10#2841—26no—
11#2840C2211yescharacteristic
12#2396—26no—
13#2280C2213yes—
14#2280C2213yes—
15#1899—26no—
16#1899—26no—
17#1819C2213yes—
18#1819C2213yes—
19#1781C2213yes—
20#1781C2213yes—
21#1304C2213yes—
22#1304C2213yes—
23#1304C2216yes—
24C23×D4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C2×#123—23no—
27C2×#123—23no—
28C2×#123—26no—
29C2×C4×Q8—22no—
30C2×#117—23no—
31C2×#117D416yes—
32C22×C4⋊C4Q812yescomplemented
33C22×C22⋊C4Q812yescomplemented
34C22×C22⋊C4C2311yescharacteristic
35C2×#99—26no—
36C2×#99—26no—
37#400—26no—
38#399—26no—
39#398—212no—
40#396—212no—
41#395—26no—
42#393—26no—
43#381—23no—
44#381D416yes—
45#380—26no—
46#380—26no—
47#379C2313yes—
48#379C2313yes—
49#379C2313yes—
50#379C2313yes—
51C4×C4⋊C4—26no—
52C4×C22⋊C4—26no—
53C4×C22⋊C4—26no—
54#342—26no—
55#342—26no—
56C25C2×Q811yescharacteristic
57C25C2411yescharacteristic
58C22×Q8—22no—
59C22×D4—23no—
60C22×D4—23no—
61C22×D4—82no—
62C23×C4—82no—
63C23×C4C2×D413yes—
64C23×C4C2×D413yes—
65C23×C4C2×Q812yes—
66C23×C4C2×Q811yescharacteristic
67C23×C4C2×Q812yes—
68C23×C4#4116yes—
69#123—43no—
70#123—41no—
71#123—46no—
72#123—43no—
73#123—43no—
74C4×Q8—84no—
75#117—86no—
76#117—86no—
77C2×C4⋊C4—26no—
78C2×C4⋊C4—26no—
79C2×C4⋊C4—26no—
80C2×C4⋊C4—212no—
81C2×C4⋊C4—26no—
82C2×C4⋊C4—43no—
83C2×C22⋊C4—23no—
84C2×C22⋊C4—23no—
85C2×C22⋊C4—26no—
86C2×C22⋊C4—26no—
87C2×C22⋊C4—23no—
88C2×C22⋊C4—23no—
89C2×C22⋊C4—26no—
90C2×C22⋊C4—26no—
91C2×C22⋊C4—26no—
92C2×C22⋊C4—26no—
93C2×C22⋊C4C2×D416yes—
94C2×C22⋊C4C2×D416yes—
95C2×C42—26no—
96C2×C42—43no—
97#99—86no—
98#99—86no—
99#95—26no—
100#95—26no—
101#95—212no—
102#95—46no—
103C24—23no—
104C24—42no—
105C24—82no—
106C24—81no—
107C24—82no—
108C24C22×D413yes—
109C24C22×Q811yescharacteristic
110C24C2×#4113yes—
111C2×Q8—82no—
112C2×Q8—84no—
113C2×D4—46no—
114C2×D4—46no—
115C2×D4—41no—
116C2×D4—43no—
117C2×D4—83no—
118C2×D4—83no—
119C22×C4—26no—
120C22×C4—26no—
121C22×C4—23no—
122C22×C4—26no—
123C22×C4—46no—
124C22×C4—46no—
125C22×C4—43no—
126C22×C4—43no—
127C22×C4—82no—
128C22×C4—86no—
129C22×C4—81no—
130C22×C4—82no—
131C22×C4#12013yes—
132C22×C4#12011yescharacteristic
133C22×C4C22⋊Q816yes—
134C22×C4C22⋊Q816yes—
135C22×C4#12416yes—
136C22×C4#12413yes—
137C22×C4#12413yes—
138C4⋊C4—46no—
139C4⋊C4—42no—
140C4⋊C4—86no—
141C4⋊C4—812no—
142C4⋊C4—86no—
143C4⋊C4—86no—
144C22⋊C4—41no—
145C22⋊C4—43no—
146C22⋊C4—46no—
147C22⋊C4—42no—
148C22⋊C4—46no—
149C22⋊C4—46no—
150C22⋊C4—46no—
151C22⋊C4—46no—
152C22⋊C4—86no—
153C22⋊C4—86no—
154C22⋊C4—86no—
155C22⋊C4—86no—
156C42—412no—
157C42—86no—
158C23—26no—
159C23—41no—
160C23—42no—
161C23—46no—
162C23—46no—
163C23—43no—
164C23—43no—
165C23—43no—
166C23—41no—
167C23—86no—
168C23—88no—
169C23—164no—
170C23C2×#12011yescharacteristic
171C23C2×C22⋊Q813yes—
172C23C2×#12413yes—
173Q8—88no—
174D4—86no—
175D4—82no—
176C2×C4—22no—
177C2×C4—26no—
178C2×C4—43no—
179C2×C4—43no—
180C2×C4—46no—
181C2×C4—46no—
182C2×C4—46no—
183C2×C4—412no—
184C2×C4—412no—
185C2×C4—43no—
186C2×C4—43no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—46no—
190C2×C4—46no—
191C2×C4—86no—
192C2×C4—83no—
193C2×C4—83no—
194C2×C4—86no—
195C2×C4—86no—
196C2×C4#39316yes—
197C2×C4#39312yes—
198C22—22no—
199C22—23no—
200C22—23no—
201C22—26no—
202C22—26no—
203C22—44no—
204C22—86no—
205C22—86no—
206C22—81no—
207C22—82no—
208C22—163no—
209C22—163no—
210C22C2×#39311yescharacteristic
211C22#302612yes—
212C4—22no—
213C4—26no—
214C4—86no—
215C4—86no—
216C4—81no—
217C4—82no—
218C4—163no—
219C4—163no—
220C2—22no—
221C2—43no—
222C2—43no—
223C2—161no—
224C2#2346811yescharacteristic
225C2#2347412yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36674C2semidirect
3. #10393C2semidirect
5. #10391C2semidirect
7. #8102C2semidirect
32. C22×C4⋊C4Q8semidirect
33. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#366742
#217472
#103932
#103922
#103912
#103902
#81022