Finite Group Databasebeta

← #94367

Subgroups — #94367

Order 512.

305 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C4×#353C211yesmaximal, characteristic
2#8963C211yesmaximal, characteristic
3C4×#331C211yesmaximal, characteristic
4#2975C411yescharacteristic, complemented
5C4×#127C2211yescharacteristic
6C2×#353—21no—
7C4×C4⋊C8—21no—
8C4×#99—21no—
9C2×#331—21no—
10#1478—21no—
11#1460C411yescharacteristic, complemented
12#1393C412yes—
13#1296C412yescomplemented
14#1276—21no—
15C2×#127D411yescharacteristic
16C2×C4×D4—41no—
17C2×C4×D4C2×C411yescharacteristic
18C2×C4×D4D411yescharacteristic
19#436—42no—
20#431—21no—
21#431—21no—
22C2×C4⋊C8—22no—
23C2×#99—22no—
24#407—41no—
25C2×C4×C8—41no—
26#390—41no—
27#390C2×C411yescharacteristic
28#390Q811yescharacteristic
29#384C2×C411yescharacteristic
30C4×C22⋊C4—41no—
31C43D411yescharacteristic
32#353—24no—
33#342—41no—
34#336—41no—
35#331—24no—
36D4⋊C8—42no—
37#323—41no—
38C22×D4—41no—
39C22×D4C22⋊C411yescharacteristic
40C22×D4C4⋊C411yescharacteristic
41C23×C4—21no—
42C23×C4—21no—
43C2×M16—42no—
44C22×C8—42no—
45#127—22no—
46#127—22no—
47C4×D4—41no—
48C4×D4—41no—
49C4×D4—41no—
50C4×D4—41no—
51C4×D4—82no—
52C4×D4—82no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4C4211yescharacteristic
55C2×C4⋊C4C4⋊C411yescharacteristic
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—41no—
60C2×C22⋊C4—41no—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42—81no—
64C2×C42C22⋊C411yescharacteristic
65C2×C42C22⋊C411yescharacteristic
66C2×C42C4⋊C411yescharacteristic
67C4⋊C8—22no—
68C4⋊C8—24no—
69C4⋊C8—41no—
70C4⋊C8—42no—
71#99—24no—
72#99—42no—
73C22⋊C8—42no—
74C22⋊C8—41no—
75C22⋊C8—41no—
76C4×C8—41no—
77C4×C8—41no—
78C4×C8—82no—
79#95—81no—
80C24—21no—
81C24—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—44no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—81no—
91C2×D4—81no—
92C2×D4—82no—
93C2×D4#10412yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—82no—
103C22×C4—82no—
104C22×C4—82no—
105C22×C4#9511yescharacteristic
106C22×C4#9911yescharacteristic
107C22×C4#9911yescharacteristic
108M16—162no—
109C2×C8—44no—
110C2×C8—42no—
111C2×C8—82no—
112C2×C8—82no—
113C2×C8—81no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—81no—
117C4⋊C4—81no—
118C22⋊C4—42no—
119C22⋊C4—44no—
120C22⋊C4—81no—
121C22⋊C4—82no—
122C22⋊C4—82no—
123C22⋊C4—82no—
124C42—22no—
125C42—22no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—41no—
130C42—42no—
131C42—162no—
132C42#10412yes—
133C23—22no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—81no—
140C23—81no—
141C23—81no—
142C23—81no—
143C23#34211yescharacteristic
144D4—42no—
145D4—81no—
146D4—81no—
147D4—81no—
148D4—84no—
149D4—84no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—84no—
169C2×C4—161no—
170C2×C4—161no—
171C2×C4—162no—
172C2×C4—162no—
173C2×C4#33911yescharacteristic
174C2×C4#33911yescharacteristic
175C2×C4AΣL(1,𝔽16)11yescharacteristic
176C2×C4#35211yescharacteristic
177C8—84no—
178C8—82no—
179C8—161no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—82no—
184C22—82no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—161no—
191C22—161no—
192C22#129711yescharacteristic
193C22#139011yescharacteristic
194C22#139511yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—81no—
203C4—81no—
204C4—164no—
205C4#140212yes—
206C2—21no—
207C2—21no—
208C2—82no—
209C2—161no—
210C2#728211yescharacteristic
211C2#748511yescharacteristic
212C2#749011yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×C4⋊C4C42
Frattini, cosocleC4×#127C22
socleC22#1390
prime radicalC2×#127D4
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
11. #1460C4semidirect
13. #1296C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3532
#89632
C4×#3312