Finite Group Databasebeta

← #94334

Subgroups — #94334

Order 512.

273 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 60 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×#1327C211yesmaximal, characteristic
2#11244C211yesmaximal, characteristic
3#8119C211yesmaximal, characteristic
4#2920—21no—
5C2×#430C411yescharacteristic
6C2×#423C2211yescharacteristic
7C2×#349—21no—
8C2×#344—21no—
9C2×#339—21no—
10C2×H(3,ℤ/4)C411yescharacteristic
11#1327—21no—
12#1327C412yescomplemented
13#1307—21no—
14#1307C412yescomplemented
15C22×M16—21no—
16C22×M16C2×C411yescharacteristic
17C22×M16D411yescharacteristic
18C2×#117—21no—
19C22×C42D411yescharacteristic
20C2×M32—41no—
21#432—22no—
22#432—41no—
23#431—22no—
24#431—41no—
25#430—42no—
26C2×#108—41no—
27#423—41no—
28#423—41no—
29#423C2×C411yescharacteristic
30#423C2×C411yescharacteristic
31#423D411yescharacteristic
32#423Q811yescharacteristic
33C2×C4⋊C8—41no—
34#407—22no—
35C4×M16—41no—
36#349—22no—
37#349—41no—
38#344—42no—
39#339—41no—
40#339—41no—
41H(3,ℤ/4)—41no—
42C23×C4—21no—
43C23×C4C22⋊C411yescharacteristic
44C2×M16—21no—
45C2×M16—21no—
46C2×M16—22no—
47C2×M16—42no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16C4211yescharacteristic
58C2×M16C22⋊C411yescharacteristic
59C2×M16C4⋊C411yescharacteristic
60C2×M16C4⋊C411yescharacteristic
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—41no—
64#117—22no—
65#117—41no—
66#117—41no—
67#117—41no—
68C2×C4⋊C4—41no—
69C2×C22⋊C4—41no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C4⋊C411yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42SD1611yescharacteristic
78C2×C42SD1611yescharacteristic
79C2×C42Q1611yescharacteristic
80M32—42no—
81M32—81no—
82C2×C16—81no—
83#108—42no—
84#108—42no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91C22⋊C8—24no—
92C22⋊C8—42no—
93#97—42no—
94C4×C8—42no—
95C24#10411yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—82no—
107C22×C4—81no—
108C22×C4#9511yescharacteristic
109C22×C4#9911yescharacteristic
110C22×C4#9911yescharacteristic
111C22×C4#10211yescharacteristic
112C22×C4#10311yescharacteristic
113C22×C4#10411yescharacteristic
114M16—42no—
115M16—41no—
116M16—41no—
117M16—41no—
118M16—41no—
119M16—42no—
120M16—42no—
121M16—81no—
122M16—81no—
123M16—84no—
124M16—81no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8—81no—
137C4⋊C4—42no—
138C4⋊C4—81no—
139C22⋊C4—42no—
140C22⋊C4—81no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—42no—
148C42—81no—
149C42#10211yescharacteristic
150C42#10311yescharacteristic
151C42#10611yescharacteristic
152C42#10711yescharacteristic
153C16—82no—
154C23—21no—
155C23—21no—
156C23—81no—
157C23#32711yescharacteristic
158C23#32811yescharacteristic
159C23#33911yescharacteristic
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—162no—
182C2×C4#32711yescharacteristic
183C2×C4#32811yescharacteristic
184C2×C4#34011yescharacteristic
185C2×C4#34211yescharacteristic
186C2×C4AΣL(1,𝔽16)11yescharacteristic
187C2×C4#35211yescharacteristic
188C2×C4#35511yescharacteristic
189C2×C4#35611yescharacteristic
190C8—41no—
191C8—41no—
192C8—41no—
193C8—41no—
194C8—84no—
195C8—82no—
196C22—21no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—81no—
203C22#129411yescharacteristic
204C22#135311yescharacteristic
205C22#135411yescharacteristic
206C4—22no—
207C4—21no—
208C4—21no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—162no—
213C4#134511yescharacteristic
214C4#134611yescharacteristic
215C4#139011yescharacteristic
216C4#139411yescharacteristic
217C2—21no—
218C2—21no—
219C2—21no—
220C2—41no—
221C2—41no—
222C2#725111yescharacteristic
subgroupG/H
centerC4#1345
commutatorC2×M16C42
Frattini, cosocleC2×#423C22
socleC2#7251
prime radicalC24#104
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G
LECSG ▷ C2×#423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1327C4semidirect
14. #1307C4semidirect

Maximal subgroups

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

subgroupindex
C2×#13272
#112442
#81192