Finite Group Databasebeta

← #124935

Subgroups — #124935

Order 512.

526 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 94 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×#1956C211yesmaximal, characteristic
2#12788C212yesmaximal
3#7108C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C22×#103—22no—
6C22×C4×C8C2211yescharacteristic
7C2×#384C411yescharacteristic
8C2×#364—22no—
9#2112—22no—
10#1956—22no—
11#1956C412yes—
12#1956C2212yes—
13#1376C2214yes—
14C23×Q8—21no—
15C23×C8D412yes—
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8C2×C412yes—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4—41no—
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26C2×#103—24no—
27C2×#103—24no—
28C2×#103—42no—
29C2×#103—42no—
30C2×#103—42no—
31C2×#103—42no—
32C2×C4×C8—42no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#384—21no—
37#384—41no—
38#384C2×C412yes—
39#364—44no—
40C22⋊C16—42no—
41C22⋊C16—42no—
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—42no—
45C22×Q8—44no—
46C22×Q8—41no—
47C22×Q8—42no—
48C22×Q8—42no—
49C23×C4—21no—
50C23×C4—41no—
51C23×C4—41no—
52C23×C4C22⋊C411yescharacteristic
53C22×C8—42no—
54C22×C8—42no—
55C22×C8C22⋊C412yes—
56C22×C8D812yes—
57C22×C8D812yes—
58C22×C8SD1612yes—
59C22×C8SD1612yes—
60C22×C8C2×D412yes—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—42no—
64C4⋊Q8—48no—
65C4⋊Q8—42no—
66C4⋊Q8M1614yes—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—48no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—84no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82M32—44no—
83M32—44no—
84C2×C16—44no—
85#103—44no—
86#103—44no—
87#103—44no—
88#103—44no—
89C4×C8—24no—
90C4×C8—41no—
91C4×C8—41no—
92C24#10011yescharacteristic
93C2×Q8—24no—
94C2×Q8—44no—
95C2×Q8—48no—
96C2×Q8—48no—
97C2×Q8—42no—
98C2×Q8—44no—
99C2×Q8—44no—
100C2×Q8—82no—
101C2×Q8—84no—
102C2×Q8—84no—
103C2×Q8—82no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—81no—
117C22×C4—81no—
118C22×C4—84no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4#10011yescharacteristic
122C22×C4#10212yes—
123C22×C4#10212yes—
124C22×C4C2×C22⋊C411yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8#10212yes—
134C2×C8#10212yes—
135C2×C8C2×D812yes—
136C2×C8C2×SD1612yes—
137C4⋊C4—44no—
138C4⋊C4—48no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—84no—
142C4⋊C4—84no—
143C4⋊C4—82no—
144C42—24no—
145C42—21no—
146C42—21no—
147C42#9912yes—
148C42C2×M1612yes—
149C16—84no—
150C23—22no—
151C23—22no—
152C23—21no—
153C23—21no—
154C23#33111yescharacteristic
155C23#33111yescharacteristic
156C23C2×#10011yescharacteristic
157Q8—48no—
158Q8—88no—
159Q8—84no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—216no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—44no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—84no—
179C2×C4—82no—
180C2×C4—162no—
181C2×C4—162no—
182C2×C4#33112yes—
183C2×C4#40711yescharacteristic
184C2×C4C2×#9911yescharacteristic
185C2×C4C2×#10212yes—
186C2×C4#42112yes—
187C8—22no—
188C8—22no—
189C8—22no—
190C8—22no—
191C22—24no—
192C22—24no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22#135812yes—
200C22#135912yes—
201C22#146211yescharacteristic
202C22#153311yescharacteristic
203C22C2×#33111yescharacteristic
204C4—24no—
205C4—24no—
206C4—22no—
207C4—84no—
208C4—162no—
209C4#148612yes—
210C4#151912yes—
211C2—21no—
212C2—22no—
213C2—22no—
214C2—21no—
215C2#764011yescharacteristic
216C2C2×#135811yescharacteristic
217C2C2×#135911yescharacteristic
subgroupG/H
center, socleC22C2×#331
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×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
3. #7108C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19562
#127882
#71082