Finite Group Databasebeta

← #124580

Subgroups — #124580

Order 512.

570 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 102 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#13856C211yesmaximal, characteristic
2#8558C212yesmaximal
3#7123C214yesmaximal
4C2×C4×C4⋊C4—22no—
5C2×C4×C4⋊C4C411yescharacteristic
6C2×C4×C4⋊C4C2211yescharacteristic
7C2×#376C411yescharacteristic
8#1797—22no—
9C2×#324—22no—
10#1444—22no—
11#1444C412yes—
12#1444C2212yes—
13#1280C2214yes—
14C22×C4⋊C4—22no—
15C22×C4⋊C4—22no—
16C22×C4⋊C4C2×C412yes—
17C22×C4⋊C4D412yes—
18C22×C42—21no—
19C22×C42—21no—
20C22×C42C2×C411yescharacteristic
21C22×C42D412yes—
22C2×C4⋊C8—42no—
23C2×C22⋊C8—24no—
24C4×C4⋊C4—41no—
25C4×C4⋊C4—41no—
26C4×C4⋊C4—48no—
27C4×C4⋊C4—41no—
28C4×C4⋊C4—41no—
29#376—22no—
30#376C814yescomplemented
31C2×#95—21no—
32C2×#95—21no—
33C2×#95—22no—
34C2×#95—22no—
35C2×#95C2×C412yes—
36C2×#95C2×C411yescharacteristic
37C2×#95C2311yescharacteristic
38#336—42no—
39#324—44no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4C22⋊C411yescharacteristic
43C23×C4C22⋊C412yes—
44C23×C4C22⋊C411yescharacteristic
45C23×C4C22×C411yescharacteristic
46C23×C4C2×D412yes—
47C22×C8—44no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—44no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4—44no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—44no—
61C2×C42—24no—
62C2×C42—24no—
63C2×C42—24no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—44no—
70C2×C42—44no—
71C2×C42—41no—
72C2×C42—41no—
73C2×C42C2×C812yes—
74C2×C42M1612yes—
75C4⋊C8—84no—
76C22⋊C8—44no—
77C22⋊C8—44no—
78#95—24no—
79#95—22no—
80#95—41no—
81#95—41no—
82#95—44no—
83#95—44no—
84#95—42no—
85#95C2×C814yes—
86C24C2×C22⋊C411yescharacteristic
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C22⋊C812yes—
120C22×C4C22⋊C812yes—
121C22×C4#9911yescharacteristic
122C22×C4#9911yescharacteristic
123C22×C4#10112yes—
124C22×C4C22×C811yescharacteristic
125C22×C4C2×M1611yescharacteristic
126C22×C4AGL(1,ℤ/8)12yes—
127C22×C4#13712yes—
128C2×C8—84no—
129C2×C8—84no—
130C2×C8—84no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—48no—
135C4⋊C4—48no—
136C4⋊C4—416no—
137C4⋊C4—44no—
138C42—216no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—44no—
144C42—82no—
145C42—82no—
146C42—82no—
147C23—22no—
148C23—21no—
149C23—21no—
150C23C2×C22⋊C811yescharacteristic
151C23C2×#9911yescharacteristic
152C23C2×#10111yescharacteristic
153C23#41712yes—
154C23#42112yes—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—28no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#32412yes—
197C2×C4#32412yes—
198C2×C4#35312yes—
199C2×C4#35612yes—
200C8—164no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22C2×#32411yescharacteristic
210C22#147712yes—
211C22#151312yes—
212C22#153711yescharacteristic
213C22#156111yescharacteristic
214C22#212012yes—
215C22C2×#35311yescharacteristic
216C22C2×#35611yescharacteristic
217C4—22no—
218C4—48no—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—82no—
225C4—84no—
226C4#132514yes—
227C2—22no—
228C2—21no—
229C2—21no—
230C2#759811yescharacteristic
231C2C2×#132511yescharacteristic
232C2#1145211yescharacteristic
233C2#1182112yes—
234C2#1183112yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #376C8semidirect

Maximal subgroups

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

subgroupindex
#138562
#85582
#71232