Finite Group Databasebeta

← #93706

Subgroups — #93706

Order 512.

217 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 32 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#8321C211yesmaximal, characteristic
2#7128C211yesmaximal, characteristic, complemented
3#7057C211yesmaximal, characteristic, complemented
4#3461C411yescharacteristic
5#1984C2211yescharacteristic
6#1477—21no—
7#1476—21no—
8#1461C411yescharacteristic
9#1374—21no—
10#1322—21no—
11#544C811yescharacteristic
12#542—41no—
13#541—41no—
14#539—41no—
15#537C811yescharacteristic
16#536—21no—
17#533—41no—
18#529—41no—
19#529—41no—
20C4×#41—21no—
21C2×C4×Q8C2×C411yescharacteristic
22C22×C16—41no—
23#431—21no—
24C2×C4⋊C8D411yescharacteristic
25C2×C4×C8D411yescharacteristic
26C22⋊C16—41no—
27#336—41no—
28Q8⋊C8—41no—
29Q8⋊C8—41no—
30D4⋊C8—41no—
31D4⋊C8—41no—
32#324—41no—
33C2×#41—21no—
34C22×Q8M1611yescharacteristic
35C22×C8—21no—
36C22×C8—21no—
37C4⋊Q8—41no—
38C4⋊Q8—41no—
39#126—42no—
40#126—42no—
41#125—41no—
42#125—41no—
43#125—42no—
44#124—41no—
45#124—41no—
46#123—42no—
47#123—42no—
48C22⋊Q8—41no—
49C22⋊Q8—41no—
50C22⋊Q8—42no—
51C22⋊Q8—41no—
52C22⋊Q8—41no—
53#121—41no—
54#121—41no—
55C4×Q8—21no—
56C4×Q8—21no—
57C4×Q8—41no—
58C4×Q8—41no—
59C4×Q8—41no—
60C4×D4—21no—
61C4×D4—21no—
62C4×D4—41no—
63C4×D4—41no—
64C4×D4—41no—
65#117—21no—
66#117—41no—
67#117—42no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C42—21no—
71C2×C42—41no—
72C2×C42C22⋊C411yescharacteristic
73C2×C16—42no—
74C2×C16—81no—
75C2×C16—81no—
76C2×C16—81no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C22⋊C8—41no—
80C4×C8—41no—
81C4×C8—41no—
82#41—42no—
83#41—42no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—41no—
87C2×Q8—41no—
88C2×Q8—41no—
89C2×D4—21no—
90C2×D4—41no—
91C2×D4—41no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4C22⋊C811yescharacteristic
98C22×C4#9911yescharacteristic
99C22×C4#10011yescharacteristic
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—81no—
106C2×C8—81no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—41no—
110C4⋊C4—42no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—21no—
125C42—21no—
126C42—41no—
127C42—41no—
128C42—41no—
129C42—41no—
130C42—41no—
131C16—82no—
132C16—161no—
133C23—41no—
134C23#32311yescharacteristic
135Q8—42no—
136Q8—82no—
137Q8—161no—
138D4—42no—
139D4—81no—
140D4—81no—
141D4—161no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—82no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—161no—
160C2×C4#35012yes—
161C2×C4AΣL(1,𝔽16)11yescharacteristic
162C2×C4AΣL(1,𝔽16)11yescharacteristic
163C8—41no—
164C8—41no—
165C8—81no—
166C8—161no—
167C22—21no—
168C22—21no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#131511yescharacteristic
173C22#131611yescharacteristic
174C22#132111yescharacteristic
175C4—22no—
176C4—21no—
177C4—41no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4#140411yescharacteristic
184C4#140511yescharacteristic
185C2—21no—
186C2—21no—
187C2—161no—
188C2#705811yescharacteristic
189C2#734511yescharacteristic
190C2#734611yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×C4⋊C4C2×C8
Frattini, cosocle#1984C22
socleC22#1316
prime radical#537C8
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1984 ◁ G
LECSG ▷ #1984 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7128C2semidirect
3. #7057C2semidirect

Maximal subgroups

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

subgroupindex
#83212
#71282
#70572