Finite Group Databasebeta

← #128967

Subgroups — #128967

Order 512.

369 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 92 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#21635C211yesmaximal, characteristic
2#11734C212yesmaximal
3C8×#101C212yesmaximal, complemented
4#7286C211yesmaximal, characteristic, complemented
5#7284C211yesmaximal, characteristic, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C411yescharacteristic
8#2398C2211yescharacteristic
9#1985C2211yescharacteristic
10#1955C411yescharacteristic
11#1924—22no—
12#1832C411yescharacteristic
13C4×#101C2212yes—
14C8×C22⋊C4—22no—
15#1726C411yescharacteristic
16#1461C2212yes—
17C8×M16—22no—
18#1353C414yes—
19#1345C414yes—
20#1283—22no—
21C23×C8—21no—
22C2×C4×Q8C2311yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C8×Q8—41no—
25C8×Q8—41no—
26C8×D4—41no—
27C8×D4—41no—
28#431—22no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8D412yes—
32C2×#101—22no—
33C2×C22⋊C8—21no—
34C2×C22⋊C8—22no—
35C4×M16—22no—
36C2×#97—21no—
37C2×#97C2×C411yescharacteristic
38C2×C4×C8—21no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D412yes—
41#400C2×C412yes—
42#397—21no—
43#394C2×C412yes—
44#393—21no—
45C4×C22⋊C4—22no—
46#376C2×C411yescharacteristic
47#343C2×C414yes—
48#336—22no—
49#336—22no—
50#324—22no—
51#324C2×C414yes—
52#322—42no—
53C82—42no—
54C22×Q8#4111yescharacteristic
55C22×D4C4211yescharacteristic
56C23×C4—21no—
57C2×M16—24no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—44no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8C22⋊C412yes—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4C4211yescharacteristic
71C2×C4⋊C4C22×C412yes—
72C2×C4⋊C4#4111yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42C22×C411yescharacteristic
78C2×C42C2×D412yes—
79C4⋊C8—41no—
80C4⋊C8—42no—
81C4⋊C8—41no—
82C4⋊C8—42no—
83#101—28no—
84C22⋊C8—24no—
85C22⋊C8—44no—
86C22⋊C8—42no—
87#97—41no—
88#97—42no—
89#97—42no—
90C4×C8—22no—
91C4×C8—21no—
92C4×C8—22no—
93C4×C8—21no—
94C4×C8—42no—
95C4×C8—42no—
96C4×C8—42no—
97#95—21no—
98#95—22no—
99#95—22no—
100#95C4212yes—
101C24—21no—
102C2×Q8—24no—
103C2×Q8—41no—
104C2×D4—41no—
105C2×D4—42no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×C4211yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4#11711yescharacteristic
115C22×C4C4×D412yes—
116C22×C4C4×D412yes—
117M16—28no—
118M16—44no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—44no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8#9911yescharacteristic
134C2×C8#9911yescharacteristic
135C4⋊C4—42no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C22⋊C4—42no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C42—22no—
142C42—21no—
143C42—22no—
144C42—21no—
145C42—41no—
146C42—41no—
147C23—22no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23C4×C22⋊C411yescharacteristic
153Q8—44no—
154D4—42no—
155D4—42no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×#9911yescharacteristic
175C2×C4#41011yescharacteristic
176C2×C4#44314yes—
177C8—21no—
178C8—42no—
179C8—48no—
180C8—41no—
181C8—42no—
182C8—84no—
183C8—81no—
184C8#35512yes—
185C22—21no—
186C22—21no—
187C22—44no—
188C22—42no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22C4×#9911yescharacteristic
193C22#176412yes—
194C4—24no—
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—82no—
200C4—81no—
201C4C2×#35511yescharacteristic
202C4#213211yescharacteristic
203C2—22no—
204C2—82no—
205C2#823811yescharacteristic
206C2#1262311yescharacteristic
207C2C4×#35511yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC22×D4C42
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ 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. C8×#101C2semidirect
4. #7286C2semidirect
5. #7284C2semidirect

Maximal subgroups

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

subgroupindex
#216352
#117342
C8×#1012
#72862
#72842