Finite Group Databasebeta

← #93678

Subgroups — #93678

Order 512.

279 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 36 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#8319C211yesmaximal, characteristic
2#7128C211yesmaximal, characteristic, complemented
3#7057C211yesmaximal, characteristic, complemented
4C4×2−1+4C411yescharacteristic
5#1984C2211yescharacteristic
6#1477—21no—
7#1476—21no—
8#1461C411yescharacteristic
9#1374—21no—
10#1322—21no—
11C2×2−1+4C811yescharacteristic
12#520—21no—
13#520—41no—
14#520—41no—
15#519—41no—
16#519C811yescharacteristic
17C4×#41—21no—
18C4×#41—41no—
19C4×#41—41no—
20C2×C4×Q8—41no—
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—
332−1+4—21no—
342−1+4—41no—
352−1+4—42no—
362−1+4C1612yescomplemented
37C2×#41—21no—
38C2×#41—41no—
39C2×#41—41no—
40C22×Q8—41no—
41C22×Q8C2×C811yescharacteristic
42C22×C8—21no—
43C22×C8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8—41no—
47C4×Q8—41no—
48C4×Q8—41no—
49C4×Q8—41no—
50C4×Q8—41no—
51C4×Q8—42no—
52C4×Q8—41no—
53C4×Q8—41no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—42no—
57C4×D4—41no—
58C4×D4—42no—
59C4×D4—41no—
60C4×D4—41no—
61C4×D4—41no—
62C4×D4—41no—
63#117—21no—
64#117—41no—
65#117—41no—
66#117—42no—
67#117—42no—
68#117—41no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4M1611yescharacteristic
73C2×C42—21no—
74C2×C42—41no—
75C2×C42—42no—
76C2×C42C22⋊C411yescharacteristic
77C2×C16—42no—
78C2×C16—81no—
79C2×C16—81no—
80C2×C16—81no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C22⋊C8—41no—
84C4×C8—41no—
85C4×C8—41no—
86#41—22no—
87#41—22no—
88#41—41no—
89#41—42no—
90#41—42no—
91#41—42no—
92#41—42no—
93#41—44no—
94#41—42no—
95#41—42no—
96#41—41no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—44no—
104C2×Q8—41no—
105C2×Q8C2×C1611yescharacteristic
106C2×Q8M3211yescharacteristic
107C2×D4—21no—
108C2×D4—41no—
109C2×D4—41no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—41no—
117C22×C4C22⋊C811yescharacteristic
118C22×C4#9911yescharacteristic
119C22×C4#10011yescharacteristic
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—81no—
126C2×C8—81no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—41no—
130C4⋊C4—42no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—42no—
135C4⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—42no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C42—21no—
141C42—21no—
142C42—41no—
143C42—41no—
144C42—42no—
145C42—42no—
146C42—41no—
147C42—41no—
148C42—41no—
149C16—82no—
150C16—161no—
151C23—41no—
152C23#32311yescharacteristic
153Q8—22no—
154Q8—41no—
155Q8—41no—
156Q8—42no—
157Q8—41no—
158Q8—42no—
159Q8—42no—
160D4—22no—
161D4—41no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—42no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—161no—
188C2×C4—162no—
189C2×C4C22⋊C1611yescharacteristic
190C2×C4#34911yescharacteristic
191C2×C4AΣL(1,𝔽16)11yescharacteristic
192C2×C4AΣL(1,𝔽16)11yescharacteristic
193C8—41no—
194C8—41no—
195C8—81no—
196C8—161no—
197C22—21no—
198C22—21no—
199C22—42no—
200C22—81no—
201C22—81no—
202C22#131411yescharacteristic
203C22#131611yescharacteristic
204C22#132011yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—41no—
209C4—41no—
210C4—42no—
211C4—41no—
212C4—81no—
213C4—81no—
214C4—81no—
215C4#140411yescharacteristic
216C4#140511yescharacteristic
217C2—21no—
218C2—21no—
219C2—82no—
220C2#705411yescharacteristic
221C2#734511yescharacteristic
222C2#734711yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×Q8C2×C16
Frattini, cosocle#1984C22
socleC22#1316
prime radicalC2×2−1+4C8
LCSG ▷ C2×Q8 ▷ C2×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
36. 2−1+4C16semidirect

Maximal subgroups

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

subgroupindex
#83192
#71282
#70572