Finite Group Databasebeta

← #124304

Subgroups — #124304

Order 512.

502 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#1984C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4#7128C214yesmaximal, complemented
5C22×C4×Q8C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4⋊C16—21no—
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×C4×C16—21no—
12#1984—21no—
13#1984—21no—
14#1984C412yes—
15#1984C2212yes—
16C2×#336—21no—
17C2×#336—21no—
18#1374C2212yes—
19#1374C2212yes—
20C23×Q8C811yescharacteristic
21C23×C8—21no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8—21no—
26C2×C4×Q8—42no—
27C2×C4×Q8—42no—
28C2×C4×Q8C812yes—
29C2×C4×Q8C2×C412yes—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C811yescharacteristic
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C16—22no—
35C22×C16—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—42no—
38C2×C4⋊C8C2×C412yes—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C4⋊C16—42no—
45C22⋊C16—24no—
46C22⋊C16—42no—
47C4×C16—42no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336—22no—
52C22×Q8—21no—
53C22×Q8—22no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8C1614yescomplemented
57C22×Q8C2×C812yes—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4M1611yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8SD1612yes—
75C22×C8Q1612yes—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4×Q8—44no—
79C4×Q8—44no—
80C4×Q8—42no—
81C4×Q8—42no—
82C4×Q8—42no—
83C4×Q8—42no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C2×C812yes—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—42no—
95C2×C42—42no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C2×C811yescharacteristic
98C2×C42M1611yescharacteristic
99C2×C42C22×C411yescharacteristic
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103C2×C16—42no—
104C2×C16—42no—
105C2×C16—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112C24C22⋊C811yescharacteristic
113C2×Q8—24no—
114C2×Q8—22no—
115C2×Q8—48no—
116C2×Q8—48no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8C2×C1614yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C22⋊C811yescharacteristic
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4#10312yes—
143C22×C4#10412yes—
144C22×C4C2×C1612yes—
145C22×C4M3212yes—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C22×C811yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—84no—
165C2×C8#10312yes—
166C2×C8C2×SD1611yescharacteristic
167C2×C8C2×Q1611yescharacteristic
168C2×C8#13712yes—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—44no—
178C42—42no—
179C42—42no—
180C42#9912yes—
181C42#10012yes—
182C16—44no—
183C16—82no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23Q8⋊C812yes—
189C23C22⋊C1612yes—
190C23#35012yes—
191C23C2×C22⋊C811yescharacteristic
192Q8—44no—
193Q8—44no—
194Q8—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—84no—
219C2×C4#32312yes—
220C2×C4#32312yes—
221C2×C4Q8⋊C812yes—
222C2×C4C22⋊C1612yes—
223C2×C4#34912yes—
224C2×C4C2×#9911yescharacteristic
225C2×C4C2×#10011yescharacteristic
226C2×C4C2×#10311yescharacteristic
227C2×C4#41911yescharacteristic
228C2×C4C2×#10411yescharacteristic
229C2×C4#42111yescharacteristic
230C2×C4C22×C1611yescharacteristic
231C2×C4C2×M3211yescharacteristic
232C8—22no—
233C8—21no—
234C8—22no—
235C8—21no—
236C8—41no—
237C8—82no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22Q8⋊C1614yes—
245C22C2×#32311yescharacteristic
246C22C2×Q8⋊C811yescharacteristic
247C22#147911yescharacteristic
248C22C2×C22⋊C1611yescharacteristic
249C22C2×#34911yescharacteristic
250C22C2×#35011yescharacteristic
251C22#211711yescharacteristic
252C4—22no—
253C4—21no—
254C4—21no—
255C4—22no—
256C4—44no—
257C4—41no—
258C4—82no—
259C4#131414yes—
260C4#150811yescharacteristic
261C4#152011yescharacteristic
262C4#152711yescharacteristic
263C4#154811yescharacteristic
264C2—24no—
265C2#757511yescharacteristic
266C2C2×#131411yescharacteristic
267C2#1142211yescharacteristic
268C2C2×Q8⋊C1611yescharacteristic
269C2#1161111yescharacteristic
270C2#1163711yescharacteristic
271C2#1164211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect
56. C22×Q8C16semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
#127832
C4×C22⋊C162
#71282