Finite Group Databasebeta

← #16856175

Subgroups — #16856175

Order 3968.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11758519C212yesmaximal, complemented
2C2×#11758509C212yesmaximal, complemented
3#14521599C211yesmaximal, characteristic, complemented
4#14521444C212yesmaximal, complemented
5#14521439C212yesmaximal, complemented
6#14521431C214yesmaximal, complemented
7C4×#92474C211yesmaximal, characteristic, complemented
8C2×C31×#124C211yesmaximal, characteristic, complemented
9C22×Q8×C31C2211yescharacteristic
10C22×D4×C31C2211yescharacteristic
11C2×#92495C2212yes—
12C31×#124—22no—
13C31×#124C2212yes—
14C31×#124C2212yes—
15C2×C22⋊C4×C31—22no—
16C2×C42×C31C2211yescharacteristic
17C2×#92482C2214yescomplemented
18#11758519—28no—
19C2×#92481C2214yescomplemented
20C2×#92480C2214yes—
21#11758509—28no—
22C2×#92479C2214yes—
23#11758507—22no—
24C2×#92474C2212yes—
25#11758452C2212yescomplemented
26#11758451C2211yescharacteristic, complemented
27C4×D124—22no—
28#11758446C2211yescharacteristic, complemented
29#11758391C2212yescomplemented
30C4×C31⋊C8C2212yescomplemented
31C24×C31—21no—
32C2×Q8×C31—21no—
33C2×Q8×C31—24no—
34C2×Q8×C31C2312yes—
35C2×D4×C31—24no—
36C2×D4×C31—21no—
37C2×D4×C31C2312yes—
38C22×C4×C31—21no—
39C22×C4×C31D412yes—
40C22×C4×C31C2311yescharacteristic
41C2×C22⋊D31—22no—
42#92495—24no—
43#92495—24no—
44C2×D124—22no—
45C2×D124C2312yes—
46C2×C4×D31—22no—
47C2×Dic62C2312yes—
48C22⋊C4×C31—24no—
49C22⋊C4×C31—24no—
50C42×C31D412yes—
51C42×C31C2311yescharacteristic
52C42×C31C2311yescharacteristic
53#92482—28no—
54#92481—28no—
55#92480—28no—
56#92479—28no—
57(C2×D31)⋊C4—24no—
58C4⋊Dic31—22no—
59#92474D418yescomplemented
60C2×C31⋊C8C2314yes—
61C23×C31—22no—
62C23×C31—44no—
63C23×C31—42no—
64C23×C31C2×D411yescharacteristic
65C22×D31—22no—
66Q8×C31—24no—
67Q8×C31—42no—
68D4×C31—24no—
69D4×C31—42no—
70C2×C4×C31—22no—
71C2×C4×C31—42no—
72C2×C4×C31—42no—
73C2×C4×C31C2×D414yes—
74C2×C4×C31C2×D412yes—
75C2×C4×C31C2×D412yes—
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2411yescharacteristic
78C22⋊D31—44no—
79C2×Dic31—22no—
80D124—22no—
81D124—22no—
82D124—44no—
83C4×D31—44no—
84Dic62—22no—
85Dic62—22no—
86C31⋊C8C2×D418yes—
87#3153—311nomaximal, Sylow
88C22×C31—22no—
89C22×C31—42no—
90C22×C31—44no—
91C22×C31—84no—
92C22×C31#12712yes—
93C22×C31C22×D411yescharacteristic
94C2×D31—42no—
95C2×D31—44no—
96C4×C31—22no—
97C4×C31—22no—
98C4×C31—42no—
99C4×C31#12712yes—
100C4×C31C22×D412yes—
101Dic31—42no—
102C2×#137—312no—
103C2×AGL(1,ℤ/8)—312no—
104#532—311no—
105C2×#124D3111yescharacteristic, complemented
106#497—312no—
107#496—312no—
108#495—314no—
109C4×M16—311no—
110C2×C31—22no—
111C2×C31—82no—
112C2×C31C2×#12711yescharacteristic
113C2×C31#57512yes—
114D31—82no—
115C2×#41—312no—
116C22×Q8C2×D3111yescharacteristic
117C22×D4C2×D3111yescharacteristic
118#137—628no—
119AGL(1,ℤ/8)—628no—
120C2×Q16—314no—
121C2×SD16—314no—
122C2×SD16—314no—
123C2×D8—314no—
124C2×M16—312no—
125C4⋊Q8—311no—
126#127—311no—
127#124—22no—
128#124—312no—
129#124C2×D3112yes—
130#124C2×D3112yes—
131#121—622no—
132C4×D4—622no—
133C2×C22⋊C4—22no—
134C2×C42C2×D3111yescharacteristic
135#97—312no—
136C4×C8—312no—
137C31#315311yescharacteristic, complemented, Sylow
138C24—21no—
139#41—624no—
140#41—624no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—312no—
144C2×Q8C22×D3112yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—312no—
148C2×D4—622no—
149C2×D4—622no—
150C2×D4C22×D3112yes—
151C22×C4—21no—
152C22×C4—622no—
153C22×C4C22⋊D3112yes—
154C22×C4C22×D3111yescharacteristic
155Q16—628no—
156SD16—628no—
157SD16—628no—
158D8—628no—
159M16—318no—
160C2×C8—314no—
161C4⋊C4—622no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—624no—
165C42C22⋊D3112yes—
166C42C22×D3111yescharacteristic
167C42C22×D3111yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—622no—
172C23C2×C22⋊D3111yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—622no—
176Q8—622no—
177D4—24no—
178D4—42no—
179D4—622no—
180D4—622no—
181D4—1244no—
182D4—1244no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—622no—
187C2×C4—1244no—
188C2×C4D4×D3114yes—
189C2×C4C2×C22⋊D3112yes—
190C2×C4C2×C22⋊D3112yes—
191C2×C4C2×C22⋊D3111yescharacteristic
192C2×C4C23×D3111yescharacteristic
193C8—318no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—1242no—
199C22—1244no—
200C22#1175851712yes—
201C22C22×C22⋊D3111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—1242no—
206C4#1175851712yes—
207C4C2×D4×D3112yes—
208C2—22no—
209C2—82no—
210C2—2482no—
211C2C2×#1175851711yescharacteristic
212C2#1452184712yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#124C2
socleC22×C31C22×D4
hypercenterC2×#124D31
hypocenterC31#3153
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#11758519C2semidirect
2. C2×#11758509C2semidirect
3. #14521599C2semidirect
4. #14521444C2semidirect
5. #14521439C2semidirect
6. #14521431C2semidirect
7. C4×#92474C2semidirect
8. C2×C31×#124C2semidirect
17. C2×#92482C22semidirect
19. C2×#92481C22semidirect
25. #11758452C22semidirect
26. #11758451C22semidirect
28. #11758446C22semidirect
29. #11758391C22semidirect
30. C4×C31⋊C8C22semidirect
59. #92474D4semidirect
105. C2×#124D31semidirect
137. C31#3153semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
C2×#117585092
#145215992
#145214442
#145214392
#145214312
C4×#924742
C2×C31×#1242
#315331