Finite Group Databasebeta

← #94143

Subgroups — #94143

Order 512.

308 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 56 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#8959C211yesmaximal, characteristic
3#7569C211yesmaximal, characteristic
4#2972C411yescharacteristic, complemented
5C4×#127C2211yescharacteristic
6C2×#353C412yes—
7C4×C4⋊C8—21no—
8C4×#99—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
19C8×D4—42no—
20#431—21no—
21#431—21no—
22C2×C4⋊C8—41no—
23C2×#99—22no—
24C2×C22⋊C8—41no—
25C2×C4×C8—41no—
26#390—41no—
27#390C2×C411yescharacteristic
28#390D411yescharacteristic
29#384Q811yescharacteristic
30C4×C22⋊C4—41no—
31C43D411yescharacteristic
32#353—22no—
33#353C814yescomplemented
34#342—41no—
35#336—22no—
36D4⋊C8—42no—
37#323—41no—
38C22×D4—41no—
39C22×D4C4211yescharacteristic
40C22×D4C22⋊C411yescharacteristic
41C23×C4—21no—
42C23×C4—21no—
43C22×C8—42no—
44C22×C8—42no—
45#127—41no—
46#127C2×C812yes—
47#127M1612yes—
48C4×D4—41no—
49C4×D4—41no—
50C4×D4—41no—
51C4×D4—41no—
52C4×D4—82no—
53C4×D4—82no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4C4⋊C411yescharacteristic
56C2×C4⋊C4C4⋊C411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—41no—
61C2×C22⋊C4—41no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42—81no—
65C2×C42C22⋊C411yescharacteristic
66C2×C42C22⋊C411yescharacteristic
67C2×C42C4⋊C411yescharacteristic
68C4⋊C8—22no—
69C4⋊C8—41no—
70C4⋊C8—44no—
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—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—42no—
89C2×D4—42no—
90C2×D4—82no—
91C2×D4—82no—
92C2×D4C4×C811yescharacteristic
93C2×D4#9711yescharacteristic
94C2×D4C22⋊C812yes—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—82no—
104C22×C4—82no—
105C22×C4—82no—
106C22×C4#9511yescharacteristic
107C22×C4#9911yescharacteristic
108C22×C4#9911yescharacteristic
109C2×C8—44no—
110C2×C8—42no—
111C2×C8—82no—
112C2×C8—82no—
113C2×C8—81no—
114C2×C8—162no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—82no—
118C22⋊C4—42no—
119C22⋊C4—44no—
120C22⋊C4—81no—
121C22⋊C4—81no—
122C22⋊C4—81no—
123C22⋊C4—81no—
124C22⋊C4—81no—
125C22⋊C4—82no—
126C42—22no—
127C42—22no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—41no—
132C42—42no—
133C42—162no—
134C42C4⋊C812yes—
135C23—22no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—81no—
143C23—81no—
144C23—81no—
145C23—81no—
146C23#34211yescharacteristic
147D4—41no—
148D4—41no—
149D4—42no—
150D4—42no—
151D4—81no—
152D4—84no—
153D4—84no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—84no—
173C2×C4—161no—
174C2×C4—161no—
175C2×C4—162no—
176C2×C4—162no—
177C2×C4#33611yescharacteristic
178C2×C4#34411yescharacteristic
179C2×C4AΣL(1,𝔽16)11yescharacteristic
180C2×C4#35211yescharacteristic
181C8—84no—
182C8—82no—
183C8—161no—
184C22—21no—
185C22—21no—
186C22—42no—
187C22—41no—
188C22—41no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—161no—
197C22—161no—
198C22#128211yescharacteristic
199C22#139011yescharacteristic
200C22#139111yescharacteristic
201C4—24no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—81no—
210C4—164no—
211C4#140212yes—
212C2—21no—
213C2—21no—
214C2—82no—
215C2—81no—
216C2—81no—
217C2#720411yescharacteristic
218C2#748511yescharacteristic
219C2#748611yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×D4C4×C8
Frattini, cosocleC4×#127C22
socleC22#1390
prime radicalC2×#127C2×C4
LCSG ▷ C2×D4 ▷ C2×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. #2972C4semidirect
10. #1460C4semidirect
14. #1270C4semidirect
33. #353C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3532
#89592
#75692