Finite Group Databasebeta

← #97027

Subgroups — #97027

Order 512.

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

normal subgroups only

This group has 872 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×#2327C211yesmaximal, characteristic
2#8699C212yesmaximal
3#7595C214yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×C4×Q8C412yes—
6#2327—28no—
7C2×#391C2212yes—
8C2×#387—24no—
9C2×#382C411yescharacteristic
10C2×#376C2211yescharacteristic
11#1817C2214yes—
12#1280—24no—
13#1272—28no—
14C23×Q8C2×C412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—22no—
18C2×C4⋊Q8C814yescomplemented
19C2×C4×Q8—24no—
20C2×C4×Q8—24no—
21C2×C4×Q8—44no—
22C2×C4×Q8—44no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C412yes—
26C22×C42—22no—
27C22×C42C2×C411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42C2311yescharacteristic
30C2×C22⋊C8—24no—
31#391—24no—
32#391D418yes—
33#387—232no—
34#382—24no—
35#376—24no—
36C2×#95—22no—
37C2×#95—24no—
38C2×#95D414yes—
39#336—28no—
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—22no—
43C22×Q8—42no—
44C22×Q8—42no—
45C22×Q8—42no—
46C22×Q8—42no—
47C22×Q8C22⋊C414yes—
48C22×Q8C2×C814yes—
49C22×Q8M1614yes—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C412yes—
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C22×C8—44no—
57C4⋊Q8—216no—
58C4⋊Q8—216no—
59C4×Q8—416no—
60C4×Q8—416no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—28no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—44no—
69C2×C4⋊C4—44no—
70C2×C4⋊C4C22⋊C414yes—
71C2×C42—24no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—28no—
76C2×C42—42no—
77C2×C42—44no—
78C2×C42—44no—
79C2×C42C2×C812yes—
80C2×C42M1612yes—
81C2×C42C2×D414yes—
82C22⋊C8—48no—
83#95—28no—
84#95—216no—
85#95—28no—
86C24C2×C22⋊C411yescharacteristic
87C2×Q8—24no—
88C2×Q8—24no—
89C2×Q8—24no—
90C2×Q8—44no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—44no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8C22⋊C818yes—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—44no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—48no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C22⋊C814yes—
125C22×C4#10014yes—
126C22×C4C2×C22⋊C412yes—
127C22×C4#12014yes—
128C22×C4C22×C811yescharacteristic
129C22×C4C2×M1612yes—
130C22×C4C2×M1611yescharacteristic
131C2×C8—84no—
132C2×C8—88no—
133C4⋊C4—232no—
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—416no—
137C42—28no—
138C42—24no—
139C42—24no—
140C42—44no—
141C42—416no—
142C23—24no—
143C23#37911yescharacteristic
144C23C2×C22⋊C811yescharacteristic
145C23#40711yescharacteristic
146C23#41012yes—
147C23C2×#10012yes—
148Q8—48no—
149Q8—48no—
150Q8—48no—
151Q8—48no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—28no—
159C2×C4—24no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—48no—
176C2×C4—44no—
177C2×C4—84no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4C2×C22⋊C812yes—
181C2×C4#40712yes—
182C8—164no—
183C22—24no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22#146012yes—
189C22#146612yes—
190C22#177911yescharacteristic
191C22#178211yescharacteristic
192C22#178311yescharacteristic
193C22#179614yes—
194C4—22no—
195C4—22no—
196C4—44no—
197C4—44no—
198C4—48no—
199C4—44no—
200C4—84no—
201C2—22no—
202C2—22no—
203C2#830211yescharacteristic
204C2#832312yes—
205C2#852112yes—
206C2#852412yes—
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
18. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
#86992
#75952