Finite Group Databasebeta

← #97025

Subgroups — #97025

Order 512.

764 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2331C211yesmaximal, characteristic
2#8699C212yesmaximal
3#7593C214yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5#2331—28no—
6C2×#391—22no—
7C2×#389C2212yes—
8C2×#387—22no—
9C2×#382C411yescharacteristic
10C2×C4×C4⋊C4C412yes—
11C2×#376C2211yescharacteristic
12#1817C2214yes—
13#1280—22no—
14#1280—22no—
15#1271—28no—
16C23×Q8—21no—
17C2×C4⋊Q8—24no—
18C2×C4⋊Q8—24no—
19C2×C4⋊Q8—22no—
20C2×C4⋊Q8C814yescomplemented
21C22×C4⋊C4—21no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4C2×C412yes—
24C22×C4⋊C4C2×C412yes—
25C22×C42—22no—
26C22×C42C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42C2311yescharacteristic
29C2×C22⋊C8—24no—
30#391—216no—
31#389—24no—
32#389D418yes—
33#387—216no—
34#382—24no—
35C4×C4⋊C4—24no—
36C4×C4⋊C4—24no—
37#376—24no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95D412yes—
43C2×#95D412yes—
44#336—28no—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—24no—
48C22×Q8—84no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C411yescharacteristic
53C23×C4C22⋊C412yes—
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C22×C8—44no—
57C4⋊Q8—216no—
58C4⋊Q8—216no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—28no—
67C2×C4⋊C4—44no—
68C2×C4⋊C4—84no—
69C2×C4⋊C4C22⋊C414yes—
70C2×C4⋊C4C22⋊C414yes—
71C2×C4⋊C4C2×C814yes—
72C2×C4⋊C4M1614yes—
73C2×C42—24no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—28no—
78C2×C42—42no—
79C2×C42—84no—
80C2×C42C2×C812yes—
81C2×C42M1612yes—
82C2×C42C2×D414yes—
83C22⋊C8—48no—
84#95—28no—
85#95—24no—
86#95—24no—
87#95—28no—
88#95—24no—
89#95—24no—
90C24C2×C22⋊C411yescharacteristic
91C2×Q8—216no—
92C2×Q8—88no—
93C2×Q8—88no—
94C2×Q8—88no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4C22⋊C812yes—
120C22×C4C22⋊C812yes—
121C22×C4#10114yes—
122C22×C4C2×C22⋊C412yes—
123C22×C4#12014yes—
124C22×C4C22×C811yescharacteristic
125C22×C4C2×M1612yes—
126C22×C4C2×M1611yescharacteristic
127C2×C8—84no—
128C2×C8—84no—
129C2×C8—84no—
130C4⋊C4—216no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—88no—
137C4⋊C4C22⋊C818yes—
138C42—28no—
139C42—24no—
140C42—24no—
141C42—44no—
142C42—88no—
143C23—22no—
144C23—22no—
145C23#37911yescharacteristic
146C23C2×C22⋊C811yescharacteristic
147C23#40711yescharacteristic
148C23#41012yes—
149C23C2×#10112yes—
150Q8—816no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—48no—
166C2×C4—48no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—84no—
181C2×C4—88no—
182C2×C4C2×C22⋊C812yes—
183C2×C4#40712yes—
184C8—164no—
185C22—24no—
186C22—22no—
187C22—22no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22#146112yes—
193C22#146712yes—
194C22#177911yescharacteristic
195C22#178211yescharacteristic
196C22#178411yescharacteristic
197C22#179512yes—
198C22#179612yes—
199C4—22no—
200C4—22no—
201C4—44no—
202C4—44no—
203C4—48no—
204C4—42no—
205C4—42no—
206C4—84no—
207C2—22no—
208C2—22no—
209C2#830311yescharacteristic
210C2#832211yescharacteristic
211C2#832311yescharacteristic
212C2#851912yes—
213C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
20. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
#86992
#75932