Finite Group Databasebeta

← #134761

Subgroups — #134761

Order 512.

355 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 88 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#22208C211yesmaximal, characteristic
2#12801C212yesmaximal
3#8617C212yesmaximal, complemented
4#7238C212yesmaximal, complemented
5C2×C8×Q8C2211yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#355—22no—
9#1984C2211yescharacteristic
10#1954C411yescharacteristic
11#1923—22no—
12#1850—22no—
13#1842C411yescharacteristic
14#1801C2214yes—
15#1744C411yescharacteristic
16#1567—22no—
17#1394—22no—
18#1292—22no—
19C23×C8—21no—
20C2×#124D411yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C2×C4×D4C2×C411yescharacteristic
23C8×Q8—42no—
24C8×D4—42no—
25#423—24no—
26C2×C4⋊C8—22no—
27C2×C4⋊C8C2×C411yescharacteristic
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8D412yes—
30C2×#101—24no—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D412yes—
36#391D411yescharacteristic
37#390Q811yescharacteristic
38#388—22no—
39#382Q811yescharacteristic
40C4×C22⋊C4—22no—
41C43C2×C411yescharacteristic
42#355—44no—
43#343—24no—
44#336—21no—
45#336—21no—
46#336—22no—
47#335—44no—
48C22×Q8C2×D411yescharacteristic
49C22×D4C4⋊C411yescharacteristic
50C23×C4—21no—
51C2×M16—44no—
52C22×C8—21no—
53C22×C8—21no—
54C22×C8—42no—
55C22×C8—82no—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C412yes—
58C22×C8C22⋊C411yescharacteristic
59#124—22no—
60#124D812yes—
61#124Q1612yes—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×D4—41no—
65C4×D4—41no—
66C2×C4⋊C4C4⋊C411yescharacteristic
67C2×C4⋊C4C2×Q811yescharacteristic
68C2×C4⋊C4#4112yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C4⋊C411yescharacteristic
75C2×C42C4⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77C2×C42C2×D412yes—
78C4⋊C8—41no—
79C4⋊C8—42no—
80C4⋊C8—41no—
81C4⋊C8—44no—
82C4⋊C8—44no—
83C4⋊C8—82no—
84#101—44no—
85#101—44no—
86C22⋊C8—42no—
87C22⋊C8—82no—
88C4×C8—24no—
89C4×C8—22no—
90C4×C8—41no—
91#95—22no—
92#95—22no—
93C24—21no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8C2×D811yescharacteristic
97C2×Q8C2×Q1611yescharacteristic
98C2×Q8#13512yes—
99C2×D4—21no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4#10712yes—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4C2×C4⋊C411yescharacteristic
112C22×C4#11711yescharacteristic
113C22×C4#12112yes—
114C22×C4C22⋊Q812yes—
115M16—84no—
116M16—84no—
117C2×C8—21no—
118C2×C8—22no—
119C2×C8—41no—
120C2×C8—42no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—42no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—84no—
128C2×C8—84no—
129C2×C8#9912yes—
130C4⋊C4—41no—
131C4⋊C4—42no—
132C4⋊C4—41no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—42no—
136C22⋊C4—82no—
137C42—22no—
138C42—21no—
139C42—21no—
140C42—22no—
141C42—21no—
142C42—41no—
143C42—41no—
144C42—41no—
145C42—81no—
146C42#10712yes—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23#38011yescharacteristic
153Q8—41no—
154Q8—41no—
155Q8—42no—
156D4—42no—
157D4—42no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4C2×#9911yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4C2×#10711yescharacteristic
182C2×C4#42711yescharacteristic
183C2×C4#46612yes—
184C2×C4#46712yes—
185C8—22no—
186C8—42no—
187C8—41no—
188C8—81no—
189C8—81no—
190C8—164no—
191C22—21no—
192C22—21no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—81no—
197C22—81no—
198C22#179911yescharacteristic
199C22#180811yescharacteristic
200C22#181211yescharacteristic
201C4—21no—
202C4—22no—
203C4—21no—
204C4—21no—
205C4—22no—
206C4—42no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—81no—
211C4#213311yescharacteristic
212C4#213411yescharacteristic
213C2—21no—
214C2—21no—
215C2—82no—
216C2#860911yescharacteristic
217C2#1280311yescharacteristic
218C2#1281011yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8617C2semidirect
4. #7238C2semidirect

Maximal subgroups

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

subgroupindex
#222082
#128012
#86172
#72382