Finite Group Databasebeta

← #133179

Subgroups — #133179

Order 512.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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#37395C211yesmaximal, characteristic
2#11807C212yesmaximal
3#11801C212yesmaximal
4#8745C211yesmaximal, characteristic
5#8740C211yesmaximal, characteristic
6#2868C411yescharacteristic, complemented
7#2864C411yescharacteristic, complemented
8#2793—24no—
9#2791—24no—
10#2621C2212yes—
11#2446—22no—
12#2429—22no—
13#2327C412yescomplemented
14#2298C2211yescharacteristic
15#1822C2211yescharacteristic
16#1820C2211yescharacteristic
17#1464C2212yes—
18#1346—24no—
19#1344—24no—
20#1285—22no—
21#1284—22no—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8C2×C412yes—
24C2×C22⋊Q8—22no—
25C2×C4×Q8—22no—
26C2×#117C2311yescharacteristic
27#432—22no—
28#407—21no—
29#400—24no—
30#398—24no—
31#398—24no—
32#398—22no—
33#398D414yes—
34#397—22no—
35#397D414yes—
36#396—24no—
37#391—22no—
38#391—22no—
39#389—24no—
40#389—22no—
41#388—24no—
42#388—22no—
43#387—24no—
44#387—24no—
45#387—24no—
46#387—22no—
47#387—24no—
48#387—24no—
49#384—24no—
50#382C2×C412yes—
51#381—21no—
52C4×C4⋊C4D412yes—
53C4×C22⋊C4D412yes—
54#376C2×C411yescharacteristic
55C43C2×C411yescharacteristic
56#343—22no—
57#342—22no—
58#324—22no—
59#323—22no—
60C22×Q8—22no—
61C23×C4C2×D411yescharacteristic
62C2×M16—41no—
63C4⋊Q8—44no—
64C4⋊Q8—84no—
65C22⋊Q8—84no—
66C4×Q8—84no—
67#117—42no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4C22⋊C414yes—
77C2×C4⋊C4C2×D412yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4C2×D412yes—
82C2×C42—22no—
83C2×C42—24no—
84C2×C42C22⋊C412yes—
85C2×C42C22⋊C412yes—
86C2×C42C22×C411yescharacteristic
87C2×C42C2×D411yescharacteristic
88C4⋊C8—42no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91#97—42no—
92#95—24no—
93#95—24no—
94#95—24no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95C22⋊C414yes—
103C24#12011yescharacteristic
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—44no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—42no—
113C22×C4—81no—
114C22×C4C2×C22⋊C412yes—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4#12011yescharacteristic
117C22×C4#12012yes—
118M16—82no—
119C2×C8—82no—
120C2×C8—81no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4—48no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—84no—
128C4⋊C4—84no—
129C4⋊C4—84no—
130C4⋊C4—84no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C22⋊C4—84no—
134C22⋊C4—82no—
135C42—22no—
136C42—22no—
137C42—42no—
138C42—42no—
139C42—44no—
140C42—84no—
141C23—21no—
142C23—22no—
143C23—42no—
144C23#37911yescharacteristic
145Q8—88no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—48no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—82no—
165C2×C4—161no—
166C2×C4—162no—
167C2×C4#41012yes—
168C8—162no—
169C8—161no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—82no—
174C22#178211yescharacteristic
175C22#179512yes—
176C4—22no—
177C4—44no—
178C4—41no—
179C4—42no—
180C4—42no—
181C4—82no—
182C4—84no—
183C4—84no—
184C4—321no—
185C2—22no—
186C2—81no—
187C2#851811yescharacteristic
188C2#1272412yes—
subgroupG/H
center, socleC22#1782
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2298 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2868C4semidirect
7. #2864C4semidirect
13. #2327C4semidirect

Maximal subgroups

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

subgroupindex
#373952
#118072
#118012
#87452
#87402