Finite Group Databasebeta

← #124202

Subgroups — #124202

Order 512.

356 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C8×#124C211yesmaximal, characteristic
2#12591C212yesmaximal, complemented
3#7128C212yesmaximal, complemented
4#7102C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C411yescharacteristic
8C2×#346—22no—
9#1984C2211yescharacteristic
10#1954C2211yescharacteristic
11#1923—22no—
12#1831C411yescharacteristic
13C8×C22⋊C4—22no—
14C42×C8C2211yescharacteristic
15#1374—22no—
16#1374C2214yes—
17C23×C8—21no—
18C2×#124C811yescharacteristic
19C2×C4×Q8C2×C411yescharacteristic
20C2×C4×D4C2×C411yescharacteristic
21C22×C16—24no—
22C8×Q8—42no—
23C8×D4—42no—
24C2×C4⋊C8—21no—
25C2×C4⋊C8C2×C411yescharacteristic
26C2×C4⋊C8C2×C411yescharacteristic
27C2×C22⋊C8—21no—
28C2×C22⋊C8—22no—
29C2×C4×C8—21no—
30C2×C4×C8C2×C411yescharacteristic
31C2×C4×C8D412yes—
32C2×C4×C8D412yes—
33C2×C4×C8C2311yescharacteristic
34#391C811yescharacteristic
35#390C811yescharacteristic
36#388—22no—
37#382C811yescharacteristic
38C4×C22⋊C4—22no—
39C43C2×C411yescharacteristic
40#346—28no—
41#336—22no—
42#336—22no—
43C22×Q8C2×C811yescharacteristic
44C22×D4C2×C811yescharacteristic
45C23×C4—21no—
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—42no—
49C22×C8—44no—
50C22×C8C22⋊C411yescharacteristic
51C22×C8C22⋊C411yescharacteristic
52C22×C8C2×D412yes—
53#124—22no—
54#124—22no—
55C4×Q8—41no—
56C4×Q8—41no—
57C4×D4—41no—
58C4×D4—41no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2×C811yescharacteristic
61C2×C4⋊C4C2×C811yescharacteristic
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C42—21no—
66C2×C42C22⋊C412yes—
67C2×C42C2×C811yescharacteristic
68C2×C42C2×C811yescharacteristic
69C2×C42M1612yes—
70C2×C42C22×C411yescharacteristic
71C2×C16—28no—
72C2×C16—44no—
73C2×C16—44no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—42no—
77C22⋊C8—42no—
78C22⋊C8—44no—
79C4×C8—22no—
80C4×C8—24no—
81C4×C8—24no—
82C4×C8—41no—
83C4×C8—42no—
84#95—22no—
85#95—22no—
86C24—21no—
87C2×Q8—21no—
88C2×Q8—41no—
89C2×Q8—41no—
90C2×Q8M3212yes—
91C2×D4—21no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4M3212yes—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C22⋊C812yes—
103C22×C4C22⋊C812yes—
104C22×C4C2×C22⋊C411yescharacteristic
105C22×C4C22×C811yescharacteristic
106C22×C4C2×M1611yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—21no—
110C2×C8—22no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—42no—
114C2×C8—42no—
115C2×C8—84no—
116C2×C8#10012yes—
117C2×C8#10414yes—
118C2×C8#13514yes—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C22⋊C4—28no—
123C22⋊C4—44no—
124C22⋊C4—42no—
125C42—21no—
126C42—21no—
127C42—22no—
128C42—22no—
129C42—21no—
130C42—21no—
131C42—41no—
132C42—42no—
133C42—41no—
134C42—41no—
135C16—48no—
136C23—22no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23—42no—
141C23C2×C22⋊C811yescharacteristic
142Q8—42no—
143Q8—42no—
144D4—42no—
145D4—42no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—81no—
164C2×C4—82no—
165C2×C4—81no—
166C2×C4#41011yescharacteristic
167C2×C4C2×#10011yescharacteristic
168C2×C4#41612yes—
169C2×C4C2×#10412yes—
170C2×C4C2×M3211yescharacteristic
171C2×C4C2×M3211yescharacteristic
172C8—24no—
173C8—41no—
174C8—81no—
175C8—81no—
176C8#32914yes—
177C22—44no—
178C22—42no—
179C22—82no—
180C22—81no—
181C22—81no—
182C22#146011yescharacteristic
183C22#147612yes—
184C22#211211yescharacteristic
185C22#211511yescharacteristic
186C22#211712yes—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—42no—
191C4—42no—
192C4—41no—
193C4—81no—
194C4—81no—
195C4#150012yes—
196C4C2×#32911yescharacteristic
197C4#154311yescharacteristic
198C2—82no—
199C2#756711yescharacteristic
200C2#1148811yescharacteristic
201C2#1149111yescharacteristic
202C2#1161512yes—
203C2#1162012yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12591C2semidirect
3. #7128C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#125912
#71282
#71022