Finite Group Databasebeta

← #94455

Subgroups — #94455

Order 512.

274 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 44 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#12797C211yesmaximal, characteristic
2#8963C211yesmaximal, characteristic
3#7651C211yesmaximal, characteristic
4#2975C411yescharacteristic, complemented
5C4×#127C2211yescharacteristic
6C2×#353C412yes—
7#1831—21no—
8#1799—21no—
9#1478—21no—
10#1460C411yescharacteristic, complemented
11#1393—21no—
12#1276—21no—
13#1270—21no—
14#1270C412yescomplemented
15C2×#127C2×C411yescharacteristic
16C2×C4×D4—41no—
17C2×C4×D4C2×C411yescharacteristic
18C2×C4×D4D411yescharacteristic
19#436—42no—
20#431—21no—
21#431—21no—
22C2×C4⋊C8—41no—
23C2×#99—22no—
24#407—41no—
25#390—41no—
26#390C2×C411yescharacteristic
27#390D411yescharacteristic
28#384Q811yescharacteristic
29#380—41no—
30C43D411yescharacteristic
31#353—42no—
32#342—41no—
33#336—22no—
34#336—41no—
35D4⋊C8—42no—
36#323—41no—
37C22×D4—41no—
38C22×D4C4211yescharacteristic
39C22×D4C22⋊C411yescharacteristic
40C23×C4—21no—
41C23×C4—21no—
42C2×M16—42no—
43C22×C8—42no—
44#127—21no—
45#127—21no—
46#127—41no—
47C4×D4—41no—
48C4×D4—41no—
49C4×D4—41no—
50C4×D4—41no—
51C4×D4—82no—
52C4×D4—82no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—81no—
55C2×C4⋊C4C4⋊C411yescharacteristic
56C2×C4⋊C4C4⋊C411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—41no—
60C2×C22⋊C4—41no—
61C2×C22⋊C4—42no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C22⋊C411yescharacteristic
65C2×C42C22⋊C411yescharacteristic
66C2×C42C4⋊C411yescharacteristic
67C4⋊C8—22no—
68C4⋊C8—41no—
69C4⋊C8—82no—
70#99—42no—
71#99—42no—
72C22⋊C8—42no—
73C22⋊C8—41no—
74C22⋊C8—41no—
75C4×C8—41no—
76C4×C8—41no—
77#95—81no—
78C24—21no—
79C24—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—42no—
88C2×D4—42no—
89C2×D4—82no—
90C2×D4—82no—
91C2×D4#10011yescharacteristic
92C2×D4#10111yescharacteristic
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—82no—
102C22×C4—82no—
103C22×C4—82no—
104C22×C4#9511yescharacteristic
105C22×C4#9911yescharacteristic
106C22×C4#9911yescharacteristic
107M16—162no—
108C2×C8—42no—
109C2×C8—81no—
110C2×C8—82no—
111C2×C8—82no—
112C2×C8—82no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—82no—
116C4⋊C4—162no—
117C22⋊C4—42no—
118C22⋊C4—81no—
119C22⋊C4—81no—
120C22⋊C4—81no—
121C22⋊C4—81no—
122C22⋊C4—81no—
123C22⋊C4—82no—
124C22⋊C4—82no—
125C42—22no—
126C42—22no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—41no—
132C42—81no—
133C23—21no—
134C23—21no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—81no—
141C23—81no—
142C23—81no—
143C23—81no—
144C23#34211yescharacteristic
145D4—41no—
146D4—41no—
147D4—81no—
148D4—81no—
149D4—81no—
150D4—84no—
151D4—84no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—161no—
171C2×C4—161no—
172C2×C4—162no—
173C2×C4—162no—
174C2×C4—162no—
175C2×C4#34311yescharacteristic
176C2×C4#34411yescharacteristic
177C2×C4AΣL(1,𝔽16)11yescharacteristic
178C2×C4#35211yescharacteristic
179C8—82no—
180C8—161no—
181C8—162no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22—81no—
188C22—82no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—161no—
195C22—161no—
196C22#130511yescharacteristic
197C22#139011yescharacteristic
198C22#139111yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—42no—
205C4—41no—
206C4—81no—
207C4—81no—
208C4—322no—
209C4#140212yes—
210C2—21no—
211C2—21no—
212C2—81no—
213C2—81no—
214C2—161no—
215C2#732011yescharacteristic
216C2#748511yescharacteristic
217C2#749411yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC22×D4C42
Frattini, cosocleC4×#127C22
prime radicalC2×#127C2×C4
LCSG ▷ C22×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ C4×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2975C4semidirect
10. #1460C4semidirect
14. #1270C4semidirect

Maximal subgroups

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

subgroupindex
#127972
#89632
#76512