Finite Group Databasebeta

← #94103

Subgroups — #94103

Order 512.

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

normal subgroups only

This group has 322 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
1C2×#1324C212yesmaximal
2#10842C211yesmaximal, characteristic
3C22×C4×C8C411yescharacteristic
4C2×#382C411yescharacteristic
5C2×#382C2211yescharacteristic
6C2×#340—22no—
7C2×#324—22no—
8#1324—22no—
9#1324C414yescomplemented
10C23×C8—21no—
11C22×C4⋊C4—21no—
12C22×C4⋊C4D412yes—
13C22×C42C2×C411yescharacteristic
14C2×C22⋊C8—42no—
15C2×C4×C8—21no—
16C2×C4×C8—41no—
17C2×C4×C8—41no—
18C2×C4×C8C812yescomplemented
19#382—21no—
20#382—41no—
21#382—41no—
22#382C812yescomplemented
23#382C2×C412yes—
24#382D411yescharacteristic
25#382Q811yescharacteristic
26C2×#95—41no—
27C2×#95—41no—
28#340—44no—
29#324—24no—
30#324—42no—
31C23×C4—21no—
32C23×C4—22no—
33C23×C4—41no—
34C23×C4C22⋊C411yescharacteristic
35C22×C8—22no—
36C22×C8—22no—
37C22×C8—22no—
38C22×C8—41no—
39C22×C8—41no—
40C22×C8—41no—
41C22×C8—41no—
42C22×C8—82no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4C22⋊C412yes—
51C2×C4⋊C4C4⋊C412yes—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42C4211yescharacteristic
56C2×C42C4⋊C411yescharacteristic
57C2×C42C2×C811yescharacteristic
58C2×C42C2×C811yescharacteristic
59C2×C42M1611yescharacteristic
60C2×C42M1611yescharacteristic
61C22⋊C8—44no—
62C22⋊C8—82no—
63C4×C8—42no—
64C4×C8—42no—
65#95—42no—
66#95—41no—
67#95—41no—
68#95—42no—
69#95—42no—
70C24#9911yescharacteristic
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—44no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—84no—
88C22×C4—84no—
89C22×C4—81no—
90C22×C4—81no—
91C22×C4#9511yescharacteristic
92C22×C4C22⋊C811yescharacteristic
93C22×C4C22⋊C811yescharacteristic
94C22×C4#9911yescharacteristic
95C22×C4#10011yescharacteristic
96C22×C4#10111yescharacteristic
97C2×C8—24no—
98C2×C8—42no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—84no—
105C2×C8—164no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4—82no—
110C4⋊C4—82no—
111C42—21no—
112C42—21no—
113C42—21no—
114C42—21no—
115C42—21no—
116C42—21no—
117C42C4×C811yescharacteristic
118C42#9711yescharacteristic
119C42C4⋊C811yescharacteristic
120C42C4⋊C811yescharacteristic
121C23—24no—
122C23#32311yescharacteristic
123C23#32411yescharacteristic
124C23#34211yescharacteristic
125C23#35311yescharacteristic
126C23#35411yescharacteristic
127C23#35511yescharacteristic
128C23#35611yescharacteristic
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—84no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—84no—
157C2×C4—82no—
158C2×C4—162no—
159C2×C4—162no—
160C2×C4—162no—
161C2×C4#32311yescharacteristic
162C2×C4#32411yescharacteristic
163C2×C4#33611yescharacteristic
164C2×C4#34311yescharacteristic
165C8—44no—
166C8—164no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22#128011yescharacteristic
174C22#131811yescharacteristic
175C22#131911yescharacteristic
176C22#132411yescharacteristic
177C22#132511yescharacteristic
178C22#139311yescharacteristic
179C22#139411yescharacteristic
180C22#141414yes—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—82no—
190C4—82no—
191C4—162no—
192C2—21no—
193C2—21no—
194C2—21no—
195C2—21no—
196C2#718911yescharacteristic
197C2#735912yes—
198C2#736612yes—
199C2#750212yes—
subgroupG/H
center, socleC23#354
commutatorC42C4×C8
Frattini, cosocleC2×#382C22
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ C2×#382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1324C4semidirect
18. C2×C4×C8C8semidirect
22. #382C8semidirect

Maximal subgroups

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

subgroupindex
C2×#13242
#108422