Finite Group Databasebeta

← #134152

Subgroups — #134152

Order 512.

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

normal subgroups only

This group has 714 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#35649C211yesmaximal, characteristic
2#11508C214yesmaximal, complemented
3#8554C212yesmaximal, complemented
4C2×#537C411yescharacteristic
5C22×#125C2211yescharacteristic
6#2591—22no—
7#2504—22no—
8#2490—24no—
9#2489—24no—
10#2311C2212yes—
11#2297C412yes—
12#2274C411yescharacteristic
13#1821—22no—
14#1463—22no—
15#1463C2214yes—
16#1319—24no—
17#1318—24no—
18C22×#41C2×C412yes—
19#537—44no—
20C2×#125—24no—
21C2×#125—44no—
22C2×C22⋊Q8—22no—
23C2×#121—22no—
24C2×#121—22no—
25C2×#117C2×C412yes—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31#407—24no—
32#400—24no—
33#397—28no—
34#394—24no—
35#391—22no—
36#391D414yes—
37#390—24no—
38#390—22no—
39#390D414yes—
40#388—28no—
41#386—22no—
42#386—22no—
43#386—24no—
44#382—24no—
45#380—24no—
46C4×C4⋊C4—24no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—24no—
49#343—44no—
50#324—28no—
51#324—44no—
52C2×#41—44no—
53C2×#41—44no—
54C2×#41—44no—
55C22×Q8C22⋊C412yes—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4C22⋊C412yes—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C412yes—
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D412yes—
65C2×M16—44no—
66#125—416no—
67C22⋊Q8—48no—
68#121—48no—
69#121—44no—
70#121—44no—
71#117—44no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4C2×D414yes—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—28no—
89C2×C42—22no—
90C2×C42—24no—
91C2×C42—42no—
92C2×C42C22⋊C414yes—
93C22⋊C8—44no—
94C22⋊C8—44no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—28no—
99C24—22no—
100C24C2×C22⋊C411yescharacteristic
101#41—816no—
102C2×Q8—24no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×D4—24no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—84no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4#9914yes—
133C22×C4C2×C22⋊C412yes—
134C22×C4#12014yes—
135M16—88no—
136C2×C8—84no—
137C4⋊C4—28no—
138C4⋊C4—44no—
139C4⋊C4—48no—
140C4⋊C4—48no—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C22⋊C4—48no—
144C22⋊C4—44no—
145C22⋊C4—88no—
146C22⋊C4—84no—
147C22⋊C4—84no—
148C42—44no—
149C42—44no—
150C42—84no—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—44no—
155C23—44no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23#37911yescharacteristic
160C23C2×#9912yes—
161Q8—84no—
162D4—84no—
163D4—88no—
164D4—164no—
165C2×C4—28no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—88no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4#41314yes—
190C8—164no—
191C22—21no—
192C22—21no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—44no—
197C22—84no—
198C22—84no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22#178111yescharacteristic
203C22#178512yes—
204C4—24no—
205C4—44no—
206C4—44no—
207C4—84no—
208C4—82no—
209C4—81no—
210C4—81no—
211C2—22no—
212C2—42no—
213C2—162no—
214C2#855211yescharacteristic
215C2#1270811yescharacteristic
216C2#1270911yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC2×#537C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11508C2semidirect
3. #8554C2semidirect

Maximal subgroups

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

subgroupindex
#356492
#115082
#85542