Finite Group Databasebeta

← #16854944

Subgroups — #16854944

Order 3968.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic62C211yesmaximal, characteristic, complemented
2C2×#11758528C212yesmaximal, complemented
3#14521204C214yesmaximal, complemented
4#14521187C214yesmaximal, complemented
5#14521057C212yesmaximal, complemented
6#14521013C211yesmaximal, characteristic, complemented
7C31×#525C211yesmaximal, characteristic
8C2×#41×C31C2211yescharacteristic
9C23×C4×C31C2211yescharacteristic
10C23×Dic31—21no—
11C22×Dic62—22no—
12C22×Dic62—21no—
13C22×Dic62—28no—
14C22×Dic62C2214yes—
15C22×Dic62C2212yes—
16C31×#123—21no—
17C31×C22⋊Q8C2212yes—
18C31×#121C2212yes—
19C31×#120—21no—
20C4×D4×C31—22no—
21C31×#117C2211yescharacteristic
22#11758528—28no—
23C2×#92482C2214yescomplemented
24C2×#92480C2214yescomplemented
25C2×#92474C2212yes—
26C22⋊(C31⋊C8)C2214yes—
27#11758398C2218yescomplemented
28C24×C31D411yescharacteristic
29#41×C31—24no—
30C2×Q8×C31C2311yescharacteristic
31C2×D4×C31—22no—
32C2×D4×C31—21no—
33C2×D4×C31C2311yescharacteristic
34C22×C4×C31—24no—
35C22×C4×C31—21no—
36C22×C4×C31D412yes—
37C22×C4×C31D411yescharacteristic
38C22×C4×C31C2312yes—
39C22×C4×C31C2311yescharacteristic
40C22×Dic31—28no—
41C22×Dic31—24no—
42C22×Dic31—22no—
43C2×Dic62—28no—
44C2×Dic62—216no—
45C2×Dic62—24no—
46C2×Dic62—22no—
47C2×Dic62—24no—
48C2×Dic62—216no—
49C2×Dic62D418yescomplemented
50C2×Dic62C2314yes—
51C4⋊C4×C31—22no—
52C4⋊C4×C31C2312yes—
53C22⋊C4×C31—21no—
54C22⋊C4×C31—22no—
55C22⋊C4×C31—22no—
56C42×C31—21no—
57#92482—28no—
58#92480—28no—
59#92474—24no—
60C2×C31⋊C8C2314yes—
61C23×C31—22no—
62C23×C31—24no—
63C23×C31—21no—
64C23×C31C2×D411yescharacteristic
65C23×C31C2×D412yes—
66Q8×C31—22no—
67D4×C31—22no—
68D4×C31—42no—
69D4×C31—44no—
70C2×C4×C31—22no—
71C2×C4×C31—24no—
72C2×C4×C31—24no—
73C2×C4×C31—22no—
74C2×C4×C31—21no—
75C2×C4×C31—42no—
76C2×C4×C31C2×D414yes—
77C2×C4×C31C2×D412yes—
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2411yescharacteristic
80C2×Dic31—216no—
81C2×Dic31—28no—
82C2×Dic31—24no—
83Dic62—216no—
84Dic62—28no—
85Dic62—24no—
86Dic62C2×D418yes—
87C31⋊C8—24no—
88#3004—311nomaximal, Sylow
89C22×C31—28no—
90C22×C31—22no—
91C22×C31—22no—
92C22×C31—22no—
93C22×C31—41no—
94C22×C31—42no—
95C22×C31#12012yes—
96C22×C31#13714yescomplemented
97C22×C31C22×D411yescharacteristic
98C4×C31—22no—
99C4×C31—41no—
100C4×C31—42no—
101C4×C31#12012yes—
102C4×C31C22×D412yes—
103Dic31—28no—
104C23×Q8—311no—
105C2×#137—312no—
106#525D3111yescharacteristic
107#451—314no—
108#448—314no—
109#419—312no—
110#407—311no—
111C2×C31—22no—
112C2×C31—24no—
113C2×C31—81no—
114C2×C31C2×#12011yescharacteristic
115C2×C31C2×#13712yes—
116C2×#41C2×D3111yescharacteristic
117C22×Q8—312no—
118C22×Q8—314no—
119C22×Q8—628no—
120C22×Q8—622no—
121C22×Q8—621no—
122C23×C4—621no—
123C23×C4C2×D3111yescharacteristic
124#137—628no—
125C2×Q16—314no—
126C2×SD16—314no—
127C2×M16—312no—
128#123—21no—
129C22⋊Q8C2×D3112yes—
130#121C2×D3112yes—
131#120—21no—
132C4×D4—22no—
133#117C2×D3111yescharacteristic
134#103—318no—
135C22⋊C8—314no—
136C31#300411yescharacteristic, complemented, Sylow
137C24C22⋊D3111yescharacteristic
138#41—24no—
139C2×Q8—318no—
140C2×Q8—314no—
141C2×Q8—6216no—
142C2×Q8—628no—
143C2×Q8—6216no—
144C2×Q8—624no—
145C2×Q8—624no—
146C2×Q8—622no—
147C2×Q8C22×D3111yescharacteristic
148C2×D4—22no—
149C2×D4—21no—
150C2×D4C22×D3111yescharacteristic
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—628no—
154C22×C4—624no—
155C22×C4—622no—
156C22×C4C22⋊D3112yes—
157C22×C4C22⋊D3111yescharacteristic
158C22×C4C22×D3112yes—
159C22×C4C22×D3111yescharacteristic
160Q16—628no—
161SD16—628no—
162M16—624no—
163C2×C8—314no—
164C4⋊C4—22no—
165C4⋊C4C22×D3112yes—
166C22⋊C4—21no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—21no—
170C23—22no—
171C23—24no—
172C23—21no—
173C23C2×C22⋊D3111yescharacteristic
174C23C2×C22⋊D3112yes—
175Q8—22no—
176Q8—318no—
177Q8—6216no—
178Q8—628no—
179Q8—624no—
180D4—22no—
181D4—42no—
182D4—44no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—6216no—
190C2×C4—628no—
191C2×C4—624no—
192C2×C4D4×D3114yes—
193C2×C4C2×C22⋊D3112yes—
194C2×C4C2×C22⋊D3111yescharacteristic
195C2×C4C23×D3111yescharacteristic
196C8—624no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—41no—
202C22—42no—
203C22#1175851412yes—
204C22#1175852814yescomplemented
205C22C22×C22⋊D3111yescharacteristic
206C4—22no—
207C4—41no—
208C4—42no—
209C4—628no—
210C4#1175851412yes—
211C4C2×D4×D3112yes—
212C2—22no—
213C2—24no—
214C2—81no—
215C2C2×#1175851411yescharacteristic
216C2C2×#1175852812yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC31×#525C2
socleC22×C31C22×D4
hypercenter#525D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×Dic62C2semidirect
2. C2×#11758528C2semidirect
3. #14521204C2semidirect
4. #14521187C2semidirect
5. #14521057C2semidirect
6. #14521013C2semidirect
23. C2×#92482C22semidirect
24. C2×#92480C22semidirect
27. #11758398C22semidirect
49. C2×Dic62D4semidirect
96. C22×C31#137semidirect
136. C31#3004semidirect
204. C22#11758528semidirect

Maximal subgroups

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

subgroupindex
C23×Dic622
C2×#117585282
#145212042
#145211872
#145210572
#145210132
C31×#5252
#300431