Finite Group Databasebeta

← #124140

Subgroups — #124140

Order 512.

598 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 106 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#13850C211yesmaximal, characteristic
2C4×#324C212yesmaximal
3#7131C212yesmaximal
4#7104C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×C4×C4⋊C4C2211yescharacteristic
7C2×#376C411yescharacteristic
8C4×C22⋊C8—22no—
9C2×#324—22no—
10#1446C411yescharacteristic
11#1446C2211yescharacteristic
12#1445C411yescharacteristic
13#1445C2211yescharacteristic
14#1441—22no—
15C4×#95—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C2×C4×Q8—22no—
19C2×C4×Q8—41no—
20C2×C4×Q8—41no—
21C2×C4×Q8—42no—
22C2×C4×Q8C814yescomplemented
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C42—21no—
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42D412yes—
30C2×C22⋊C8—24no—
31C2×C4×C8—42no—
32C4×C4⋊C4—41no—
33C4×C4⋊C4—41no—
34#376—42no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—21no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95C2311yescharacteristic
44#336—42no—
45#324—28no—
46C22×Q8—21no—
47C22×Q8—44no—
48C22×Q8—41no—
49C22×Q8—41no—
50C22×Q8C2×C812yes—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D412yes—
60C22×C8—44no—
61C4×Q8—44no—
62C4×Q8—44no—
63C4×Q8—44no—
64C4×Q8—44no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4C2×C812yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—22no—
78C2×C42—24no—
79C2×C42—24no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—42no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42—42no—
86C2×C42—42no—
87C2×C42—44no—
88C2×C42C2×C812yes—
89C2×C42M1612yes—
90C22⋊C8—28no—
91C22⋊C8—44no—
92C4×C8—84no—
93#95—41no—
94#95—41no—
95#95—41no—
96#95—41no—
97#95—44no—
98#95—44no—
99#95—42no—
100#95—44no—
101#95—44no—
102C24C2×C22⋊C411yescharacteristic
103C2×Q8—44no—
104C2×Q8—42no—
105C2×Q8—44no—
106C2×Q8—42no—
107C2×Q8—48no—
108C2×Q8—44no—
109C2×Q8—44no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C22⋊C812yes—
148C22×C4C22⋊C812yes—
149C22×C4#9911yescharacteristic
150C22×C4#9911yescharacteristic
151C22×C4#10012yes—
152C22×C4#10414yes—
153C22×C4C22×C811yescharacteristic
154C22×C4C2×M1611yescharacteristic
155C22×C4#13514yes—
156C2×C8—48no—
157C2×C8—84no—
158C2×C8—84no—
159C4⋊C4—48no—
160C4⋊C4—42no—
161C4⋊C4—48no—
162C4⋊C4—42no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42—22no—
167C42—48no—
168C42—44no—
169C42—44no—
170C42—44no—
171C42—82no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23C2×C22⋊C811yescharacteristic
176C23C2×#9911yescharacteristic
177C23C2×#10011yescharacteristic
178C23#41612yes—
179C23C2×#10412yes—
180Q8—44no—
181Q8—44no—
182Q8—48no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4#32312yes—
229C2×C4#32312yes—
230C2×C4#35412yes—
231C2×C4#35612yes—
232C8—88no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22C2×#32311yescharacteristic
242C22#147612yes—
243C22#150112yes—
244C22#152511yescharacteristic
245C22#154311yescharacteristic
246C22#212012yes—
247C22C2×#35411yescharacteristic
248C22C2×#35611yescharacteristic
249C4—22no—
250C4—42no—
251C4—44no—
252C4—41no—
253C4—41no—
254C4—44no—
255C4—44no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4#131914yes—
260C2—21no—
261C2—21no—
262C2—22no—
263C2#756311yescharacteristic
264C2#1142511yescharacteristic
265C2C2×#131911yescharacteristic
266C2#1172812yes—
267C2#1173612yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
22. C2×C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
#138502
C4×#3242
#71312
#71042