Finite Group Databasebeta

← #149574

Subgroups — #149574

Order 512.

526 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 82 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#21200C211yesmaximal, characteristic
2C2×#1325C212yesmaximal, complemented
3#7355C212yesmaximal, complemented
4#7354C212yesmaximal, complemented
5C2×#532C411yescharacteristic
6C2×#417—22no—
7C2×C4×M16C411yescharacteristic
8C2×#384C2211yescharacteristic
9#1959—22no—
10#1957C411yescharacteristic
11#1957C2211yescharacteristic
12#1956C411yescharacteristic
13#1956C2211yescharacteristic
14C2×#324—22no—
15#1325—22no—
16#1325C2214yes—
17C22×#41—21no—
18C22×M16—21no—
19#532—22no—
20#532—42no—
21#532C814yescomplemented
22C2×C4⋊Q8C2×C411yescharacteristic
23C2×#127C2×C411yescharacteristic
24C2×#124—21no—
25C2×#121—41no—
26C2×C4×D4—41no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4D412yes—
29C22×C42C2×C411yescharacteristic
30#417—44no—
31#417—44no—
32C2×#103—22no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#102—22no—
36C2×C22⋊C8—42no—
37C4×M16—42no—
38C2×#97—21no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41#384—41no—
42#384C2×C411yescharacteristic
43#384D412yes—
44#384C2311yescharacteristic
45#324—24no—
46#324—42no—
47C2×#41—24no—
48C2×#41—44no—
49C2×#41—42no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8—21no—
53C22×D4—21no—
54C22×D4—41no—
55C22×D4—41no—
56C23×C4—21no—
57C23×C4—22no—
58C23×C4—41no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—24no—
61C2×M16—42no—
62C2×M16—42no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—82no—
66C4⋊Q8—21no—
67C4⋊Q8—41no—
68C4⋊Q8C2×C812yescomplemented
69#127—21no—
70#127—41no—
71#127C2×C812yescomplemented
72#124—24no—
73#124—42no—
74#121—44no—
75#121—44no—
76C4×D4—44no—
77C4×D4—44no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—84no—
86C2×C4⋊C4C22⋊C412yes—
87C2×C4⋊C4C2×D412yes—
88C2×C22⋊C4—42no—
89C2×C42—21no—
90C2×C42—22no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C2×C812yes—
93C2×C42M1612yes—
94C2×C42C22×C411yescharacteristic
95#103—44no—
96#103—44no—
97#102—44no—
98#102—44no—
99C22⋊C8—44no—
100C22⋊C8—82no—
101#97—42no—
102C4×C8—41no—
103C4×C8—41no—
104C24—41no—
105C24#9911yescharacteristic
106#41—48no—
107#41—48no—
108#41—84no—
109#41—84no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—41no—
114C2×D4—22no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—44no—
120C2×D4—41no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4—82no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—82no—
142C22×C4—84no—
143C22×C4C22⋊C812yes—
144C22×C4#9911yescharacteristic
145C22×C4#10012yes—
146C22×C4C2×C22⋊C411yescharacteristic
147M16—48no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—84no—
155C2×C8—164no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C22⋊C4—48no—
165C42—22no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42C22⋊C812yes—
170C42C22×C811yescharacteristic
171C42C2×M1611yescharacteristic
172C23—41no—
173C23—41no—
174C23—42no—
175C23—82no—
176C23—82no—
177C23—81no—
178C23—81no—
179C23#32312yes—
180C23#35312yes—
181C23#35512yes—
182C23C2×#9911yescharacteristic
183Q8—44no—
184Q8—82no—
185D4—44no—
186D4—82no—
187D4—84no—
188D4—84no—
189D4—88no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—162no—
216C2×C4#32312yes—
217C2×C4C2×C22⋊C811yescharacteristic
218C2×C4C2×#10011yescharacteristic
219C8—42no—
220C8—42no—
221C8—164no—
222C22—21no—
223C22—22no—
224C22—21no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—84no—
229C22—82no—
230C22—162no—
231C22—161no—
232C22—161no—
233C22#131814yes—
234C22C2×#32311yescharacteristic
235C22C2×#35311yescharacteristic
236C22C2×#35511yescharacteristic
237C4—22no—
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—82no—
245C4—82no—
246C2—21no—
247C2—21no—
248C2—24no—
249C2—162no—
250C2C2×#131811yescharacteristic
251C2#1358411yescharacteristic
252C2#1359011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC2×#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
3. #7355C2semidirect
4. #7354C2semidirect
21. #532C8semidirect
68. C4⋊Q8C2×C8semidirect
71. #127C2×C8semidirect

Maximal subgroups

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

subgroupindex
#212002
C2×#13252
#73552
#73542