Finite Group Databasebeta

← #94331

Subgroups — #94331

Order 512.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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#11243C211yesmaximal, characteristic
3C2×#1309C211yesmaximal, characteristic
4#2911C411yescharacteristic
5C2×#423—21no—
6C2×#423C2211yescharacteristic
7C2×#349—21no—
8#1742—21no—
9C2×#339—21no—
10C2×#338C411yescharacteristic
11#1327—21no—
12#1327C412yescomplemented
13#1309—21no—
14#1309C412yescomplemented
15C22×M16—21no—
16C22×M16—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16D411yescharacteristic
19C2×#117—21no—
20C22×C42D411yescharacteristic
21C2×M32—41no—
22#428—42no—
23#427—41no—
24#423—22no—
25#423—22no—
26#423—41no—
27#423—41no—
28#423—41no—
29#423C2×C411yescharacteristic
30#423C2×C411yescharacteristic
31#423D411yescharacteristic
32#423Q811yescharacteristic
33C2×C4⋊C8—41no—
34C2×C4⋊C8—41no—
35C2×C4⋊C8—41no—
36#349—22no—
37#349—41no—
38#343—42no—
39#339—41no—
40#339—41no—
41#338—41no—
42C23×C4—21no—
43C23×C4C22⋊C411yescharacteristic
44C2×M16—21no—
45C2×M16—21no—
46C2×M16—22no—
47C2×M16—24no—
48C2×M16—42no—
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×M16—41no—
58C2×M16—84no—
59C2×M16C4211yescharacteristic
60C2×M16C22⋊C411yescharacteristic
61C2×M16C4⋊C411yescharacteristic
62C2×M16C4⋊C411yescharacteristic
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8—42no—
67#117—41no—
68#117—41no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—22no—
72C2×C22⋊C4—41no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—41no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C4⋊C411yescharacteristic
79C2×C42D811yescharacteristic
80C2×C42SD1611yescharacteristic
81C2×C42SD1611yescharacteristic
82C2×C42Q1611yescharacteristic
83M32—42no—
84M32—81no—
85C2×C16—81no—
86#107—42no—
87#106—42no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C24#10411yescharacteristic
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—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4#9511yescharacteristic
111C22×C4#9911yescharacteristic
112C22×C4#9911yescharacteristic
113C22×C4#10211yescharacteristic
114C22×C4#10311yescharacteristic
115C22×C4#10411yescharacteristic
116M16—42no—
117M16—41no—
118M16—41no—
119M16—41no—
120M16—41no—
121M16—42no—
122M16—42no—
123M16—81no—
124M16—81no—
125M16—84no—
126M16—81no—
127M16—84no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—44no—
137C2×C8—42no—
138C2×C8—81no—
139C2×C8—81no—
140C2×C8—84no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C22⋊C4—81no—
144C22⋊C4—81no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—81no—
152C42—81no—
153C42#10211yescharacteristic
154C42#10311yescharacteristic
155C42#10611yescharacteristic
156C42#10711yescharacteristic
157C16—82no—
158C23—21no—
159C23—21no—
160C23—42no—
161C23#32711yescharacteristic
162C23#32811yescharacteristic
163C23#33911yescharacteristic
164C2×C4—22no—
165C2×C4—22no—
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—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—82no—
188C2×C4—162no—
189C2×C4#32711yescharacteristic
190C2×C4#32811yescharacteristic
191C2×C4#34011yescharacteristic
192C2×C4#34211yescharacteristic
193C2×C4AΣL(1,𝔽16)11yescharacteristic
194C2×C4#35211yescharacteristic
195C2×C4#35511yescharacteristic
196C2×C4#35611yescharacteristic
197C8—41no—
198C8—41no—
199C8—41no—
200C8—41no—
201C8—82no—
202C8—84no—
203C22—22no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—81no—
210C22#129411yescharacteristic
211C22#134511yescharacteristic
212C22#134611yescharacteristic
213C4—21no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—162no—
221C4#135311yescharacteristic
222C4#135411yescharacteristic
223C4#139011yescharacteristic
224C4#139411yescharacteristic
225C2—21no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2#725111yescharacteristic
230C2#741412yes—
subgroupG/H
center, socleC22#1345
commutatorC2×M16C42
Frattini, cosocleC2×#423C22
prime radicalC24#104
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #1309C4semidirect

Maximal subgroups

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

subgroupindex
C2×#13272
#112432
C2×#13092