Finite Group Databasebeta

← #122285

Subgroups — #122285

Order 512.

368 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 76 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#21852C211yesmaximal, characteristic
2#11440C212yesmaximal, complemented
3#7061C212yesmaximal, complemented
4#7060C212yesmaximal, complemented
5C2×#529C411yescharacteristic
6C2×#432C411yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2112—22no—
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1955C411yescharacteristic
12#1955C2211yescharacteristic
13#1869—22no—
14#1798—22no—
15#1322—22no—
16#1322C2214yes—
17C23×C8D412yes—
18#529—42no—
19#529—42no—
20C2×#126—21no—
21C2×#125C811yescharacteristic
22C2×#124C811yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C811yescharacteristic
25C2×#121C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C2×#117—21no—
30C22×C42C2×C411yescharacteristic
31C2×M32—42no—
32#432—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—42no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D412yes—
37C2×C4⋊C8C2311yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×#97C2×C411yescharacteristic
42C2×#97C2×C411yescharacteristic
43C22⋊C16—24no—
44C22⋊C16—42no—
45#336—22no—
46#336—22no—
47C22×Q8C2×C811yescharacteristic
48C22×D4—21no—
49C22×D4C2×C811yescharacteristic
50C23×C4—21no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C2×C811yescharacteristic
53C23×C4M1611yescharacteristic
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—82no—
59C22×C8C22⋊C412yes—
60C22×C8C2×D412yes—
61#126—44no—
62#125—42no—
63#124—42no—
64#123—44no—
65C22⋊Q8—42no—
66#121—42no—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—42no—
72C4×D4—42no—
73#117—42no—
74#117—42no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2×C811yescharacteristic
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×C811yescharacteristic
91C2×C42M1611yescharacteristic
92C2×C42C22×C411yescharacteristic
93M32—84no—
94C2×C16—44no—
95C4⋊C8—41no—
96C4⋊C8—42no—
97C4⋊C8—41no—
98C4⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—82no—
101C22⋊C8—82no—
102#97—41no—
103#97—41no—
104C24—21no—
105C24C22⋊C811yescharacteristic
106C2×Q8—41no—
107C2×Q8—81no—
108C2×D4—41no—
109C2×D4—44no—
110C2×D4—42no—
111C2×D4—81no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—81no—
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4#9912yes—
129C22×C4#10012yes—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C22×C811yescharacteristic
132C22×C4C2×M1611yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—44no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—82no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C42—22no—
157C42—22no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—41no—
163C42—41no—
164C16—84no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—41no—
172C23#32312yes—
173C23C2×C22⋊C811yescharacteristic
174Q8—82no—
175D4—82no—
176D4—84no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4#32312yes—
203C2×C4C2×#9911yescharacteristic
204C2×C4C2×#10011yescharacteristic
205C8—42no—
206C8—42no—
207C8—81no—
208C8—81no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—44no—
215C22—84no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22C2×#32311yescharacteristic
220C22#211712yes—
221C4—21no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4#212711yescharacteristic
231C4#213311yescharacteristic
232C2—22no—
233C2—42no—
234C2—161no—
235C2#1142111yescharacteristic
236C2#1143211yescharacteristic
237C2#1143611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11440C2semidirect
3. #7061C2semidirect
4. #7060C2semidirect

Maximal subgroups

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

subgroupindex
#218522
#114402
#70612
#70602