Finite Group Databasebeta

← #94196

Subgroups — #94196

Order 512.

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

normal subgroups only

This group has 312 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
1#11512C211yesmaximal, characteristic
2#10891C211yesmaximal, characteristic
3#8723C211yesmaximal, characteristic
4C2×#384C2211yescharacteristic
5#2274C411yescharacteristic
6#1797—21no—
7C2×#343C411yescharacteristic
8#1736—21no—
9C2×#339—21no—
10#1463—21no—
11#1325—21no—
12#1325C412yescomplemented
13#1296—21no—
14#1296C412yescomplemented
15C22×M16—21no—
16C2×#117—21no—
17C22×C4⋊C4—41no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C4⋊C4D411yescharacteristic
20C22×C42D411yescharacteristic
21C2×C4⋊C8—22no—
22#407—41no—
23C2×C22⋊C8—41no—
24#384—41no—
25#384C2×C411yescharacteristic
26#384C2×C411yescharacteristic
27#384D411yescharacteristic
28#384Q811yescharacteristic
29C4×C4⋊C4—42no—
30C4×C22⋊C4—41no—
31#343—41no—
32#340—42no—
33#339—24no—
34#336—41no—
35#324—22no—
36#324—41no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—22no—
42C2×M16—41no—
43C2×M16—41no—
44C2×M16—41no—
45C2×M16—41no—
46C2×M16—41no—
47C2×M16—81no—
48C22×C8—41no—
49C22×C8—42no—
50#117—22no—
51#117—22no—
52#117—41no—
53#117—41no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—82no—
66C2×C4⋊C4—82no—
67C2×C4⋊C4C4211yescharacteristic
68C2×C4⋊C4C22⋊C411yescharacteristic
69C2×C4⋊C4C4⋊C411yescharacteristic
70C2×C4⋊C4C4⋊C411yescharacteristic
71C2×C22⋊C4—41no—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—41no—
75C2×C42—41no—
76C2×C42—42no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C4⋊C411yescharacteristic
79C4⋊C8—42no—
80C4⋊C8—42no—
81C22⋊C8—42no—
82C22⋊C8—82no—
83C22⋊C8—81no—
84#95—42no—
85C24#9911yescharacteristic
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—82no—
108C22×C4—82no—
109C22×C4#9511yescharacteristic
110C22×C4#9911yescharacteristic
111M16—42no—
112M16—42no—
113M16—81no—
114M16—81no—
115M16—162no—
116C2×C8—42no—
117C2×C8—81no—
118C2×C8—81no—
119C2×C8—82no—
120C2×C8—82no—
121C2×C8—82no—
122C2×C8—82no—
123C4⋊C4—24no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—81no—
131C4⋊C4—81no—
132C4⋊C4—81no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—42no—
143C42—42no—
144C42—42no—
145C42—162no—
146C42#10412yes—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—41no—
151C23#34211yescharacteristic
152C23AΣL(1,𝔽16)11yescharacteristic
153C23#35211yescharacteristic
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—44no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—161no—
189C2×C4—161no—
190C2×C4H(3,ℤ/4)11yescharacteristic
191C2×C4#33911yescharacteristic
192C2×C4#35511yescharacteristic
193C2×C4#35611yescharacteristic
194C8—82no—
195C8—162no—
196C8—162no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22#128411yescharacteristic
207C22#139011yescharacteristic
208C22#139411yescharacteristic
209C22#140212yes—
210C4—22no—
211C4—22no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—42no—
217C4—84no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4—81no—
222C2—21no—
223C2—22no—
224C2—21no—
225C2—41no—
226C2#721511yescharacteristic
227C2#748311yescharacteristic
228C2#748511yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC2×C4⋊C4C42
Frattini, cosocleC2×#384C22
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1325C4semidirect
14. #1296C4semidirect

Maximal subgroups

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

subgroupindex
#115122
#108912
#87232