Finite Group Databasebeta

← #102865

Subgroups — #102865

Order 512.

515 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 96 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#23837C211yesmaximal, characteristic, complemented
2#23815C212yesmaximal
3#9644C211yesmaximal, characteristic
4#9643C212yesmaximal, complemented
5#7014C211yesmaximal, characteristic, complemented
6C4×C4⋊Q8C2212yes—
7C4×#127C2211yescharacteristic
8#2099—22no—
9#2084—22no—
10#2071—22no—
11#2058—22no—
12#2041—22no—
13#2023—22no—
14#1842C2211yescharacteristic
15C4×#106—22no—
16C4×#102—24no—
17#1744C2212yes—
18#1744C2211yescharacteristic
19C2×Q8⋊C8—24no—
20C2×D4⋊C8—22no—
21C2×C4⋊Q8D412yes—
22C2×#127D411yescharacteristic
23C2×C4×Q8—22no—
24C2×C4×D4—21no—
25C2×C4×D4—22no—
26C2×#106—24no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—22no—
29C2×#102—24no—
30C2×#102—24no—
31C2×C4×C8D414yes—
32C2×C4×C8D412yes—
33#391—22no—
34#390—22no—
35#390—21no—
36#389—22no—
37#389—22no—
38#384D411yescharacteristic
39#384D412yes—
40C4×C4⋊C4—22no—
41C4×C4⋊C4—22no—
42C43C2311yescharacteristic
43#340—22no—
44#340—24no—
45#340—22no—
46#340—24no—
47#336—22no—
48#336—22no—
49#336—22no—
50Q8⋊C8—48no—
51D4⋊C8—44no—
52C22×Q8—22no—
53C22×D4—22no—
54C22×D4—21no—
55C23×C4—41no—
56C22×C8—22no—
57C22×C8—24no—
58C22×C8—41no—
59C4⋊Q8—44no—
60#127—42no—
61C4×Q8—44no—
62C4×Q8—44no—
63C4×D4—42no—
64C4×D4—44no—
65C4×D4—44no—
66C4×D4—42no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C42—42no—
78C2×C42C2×D411yescharacteristic
79C2×C42C2×D412yes—
80C2×C42C2×D411yescharacteristic
81C2×C42C2×D412yes—
82C2×C42#4111yescharacteristic
83#106—48no—
84C4⋊C8—42no—
85C4⋊C8—44no—
86#102—48no—
87#102—48no—
88C4×C8—24no—
89C4×C8—24no—
90C4×C8—22no—
91C4×C8SD1614yes—
92#95—42no—
93#95—42no—
94C24—41no—
95C2×Q8—24no—
96C2×Q8—42no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—44no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—41no—
105C2×D4—44no—
106C2×D4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4—81no—
111C22×C4—82no—
112C22×C4—82no—
113C22×C4—81no—
114C22×C4#12011yescharacteristic
115C22×C4#12012yes—
116C22×C4#12111yescharacteristic
117C22×C4#12112yes—
118C22×C4#12711yescharacteristic
119C2×C8—24no—
120C2×C8—44no—
121C2×C8—44no—
122C2×C8—44no—
123C2×C8—44no—
124C2×C8—81no—
125C2×C8—82no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C4⋊C4—84no—
136C22⋊C4—82no—
137C22⋊C4—84no—
138C42—41no—
139C42—42no—
140C42—41no—
141C42—82no—
142C42—82no—
143C42C2×SD1612yes—
144C42#13514yes—
145C42AGL(1,ℤ/8)14yes—
146C42#13712yes—
147C23—42no—
148C23—82no—
149C23—82no—
150C23—81no—
151C23—81no—
152C23#39211yescharacteristic
153Q8—48no—
154Q8—84no—
155D4—48no—
156D4—44no—
157D4—84no—
158D4—82no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—161no—
175C2×C4—161no—
176C2×C4—162no—
177C2×C4#44812yes—
178C2×C4#44914yes—
179C2×C4#46112yes—
180C2×C4#46314yes—
181C2×C4#49211yescharacteristic
182C2×C4#49512yes—
183C2×C4#49612yes—
184C2×C4#49711yescharacteristic
185C8—44no—
186C8—84no—
187C8—161no—
188C22—84no—
189C22—82no—
190C22—161no—
191C22—162no—
192C22—161no—
193C22#201011yescharacteristic
194C22#201012yes—
195C22#201212yes—
196C22#201311yescharacteristic
197C22#201711yescharacteristic
198C4—24no—
199C4—22no—
200C4—24no—
201C4—22no—
202C4—84no—
203C4—162no—
204C4—161no—
205C4#170811yescharacteristic
206C4#171511yescharacteristic
207C4#172012yes—
208C2—162no—
209C2#1001312yes—
210C2#1001411yescharacteristic
211C2#1003312yes—
212C2#1003611yescharacteristic
213C2#1031211yescharacteristic
subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC43C23
socleC23#392
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23837C2semidirect
4. #9643C2semidirect
5. #7014C2semidirect

Maximal subgroups

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

subgroupindex
#238372
#238152
#96442
#96432
#70142