Finite Group Databasebeta

← #149695

Subgroups — #149695

Order 512.

402 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 70 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#21692C211yesmaximal, characteristic
2#11475C212yesmaximal
3#7370C212yesmaximal, complemented
4#7355C212yesmaximal, complemented
5C2×#492C411yescharacteristic
6C2×#440—22no—
7C4×C4⋊Q8C411yescharacteristic
8C4×#127C2211yescharacteristic
9#1973—22no—
10#1957C411yescharacteristic
11#1956C2211yescharacteristic
12#1845C2211yescharacteristic
13C4×#97C411yescharacteristic
14#1459—22no—
15#1325C2214yes—
16#1318—22no—
17C22×SD16—22no—
18C2×C4⋊Q8C2×C411yescharacteristic
19C2×#127C2×C411yescharacteristic
20C2×C4×Q8—21no—
21C2×C4×D4—41no—
22C2×C4×D4D412yes—
23#492—22no—
24#492C814yescomplemented
25#440—44no—
26#440—44no—
27C2×#107—22no—
28C2×#103—22no—
29C2×#102—22no—
30C2×C22⋊C8—42no—
31C2×#97—21no—
32C2×#97—21no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35#391—21no—
36#390—41no—
37#390D412yes—
38#389—41no—
39#384C2×C411yescharacteristic
40#384C2311yescharacteristic
41C4×C4⋊C4—41no—
42C43C2×C411yescharacteristic
43#324—24no—
44#323—42no—
45C22×Q8—21no—
46C22×D4—41no—
47C22×D4C22⋊C412yes—
48C23×C4—22no—
49C2×SD16—28no—
50C2×SD16—44no—
51C2×SD16—44no—
52C22×C8—21no—
53C22×C8—21no—
54C22×C8—82no—
55C4⋊Q8—21no—
56C4⋊Q8—41no—
57C4⋊Q8C2×C812yescomplemented
58#127—41no—
59#127C2×C812yes—
60#127M1612yes—
61C4×Q8—42no—
62C4×Q8—42no—
63C4×D4—42no—
64C4×D4—42no—
65C4×D4—84no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C2×D412yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—42no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—41no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22×C411yescharacteristic
80#107—44no—
81#103—44no—
82#102—44no—
83C22⋊C8—44no—
84C22⋊C8—82no—
85#97—24no—
86#97—42no—
87C4×C8—21no—
88C4×C8—41no—
89C4×C8C2×C812yescomplemented
90#95—42no—
91C24—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—41no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—82no—
102C2×D4—82no—
103C2×D4C22⋊C814yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—41no—
111C22×C4—84no—
112C22×C4#9911yescharacteristic
113C22×C4#9911yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115SD16—416no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—84no—
123C2×C8—164no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4—82no—
131C22⋊C4—82no—
132C22⋊C4—84no—
133C42—22no—
134C42—21no—
135C42—41no—
136C42—42no—
137C42—42no—
138C42—81no—
139C42—81no—
140C42C22×C811yescharacteristic
141C42C2×M1611yescharacteristic
142C23—24no—
143C23—42no—
144C23—42no—
145C23—82no—
146C23—82no—
147C23C2×#9911yescharacteristic
148Q8—44no—
149Q8—82no—
150D4—44no—
151D4—82no—
152D4—84no—
153D4—84no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—84no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—162no—
175C2×C4#35312yes—
176C2×C4C2×C22⋊C811yescharacteristic
177C2×C4#41311yescharacteristic
178C8—24no—
179C8—42no—
180C8—164no—
181C22—21no—
182C22—21no—
183C22—44no—
184C22—84no—
185C22—82no—
186C22—82no—
187C22—162no—
188C22#145911yescharacteristic
189C22C2×#35311yescharacteristic
190C22#213211yescharacteristic
191C4—22no—
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—41no—
197C4—81no—
198C4—82no—
199C4—82no—
200C4—161no—
201C2—21no—
202C2—21no—
203C2—84no—
204C2#1147511yescharacteristic
205C2#1358411yescharacteristic
206C2#1359211yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7370C2semidirect
4. #7355C2semidirect
24. #492C8semidirect
57. C4⋊Q8C2×C8semidirect
89. C4×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#216922
#114752
#73702
#73552