Finite Group Databasebeta

← #16856179

Subgroups — #16856179

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×#11758528C212yesmaximal, complemented
2C2×#11758526C212yesmaximal, complemented
3C2×#11758452C211yesmaximal, characteristic, complemented
4#14521441C212yesmaximal, complemented
5#14521436C212yesmaximal, complemented
6#14521431C214yesmaximal, complemented
7C4×#92474C211yesmaximal, characteristic, complemented
8C31×#532C211yesmaximal, characteristic, complemented
9C2×#41×C31C2212yes—
10C22×D124C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C31×C4⋊Q8C2211yescharacteristic, complemented
13C31×#127C2211yescharacteristic, complemented
14C31×#124C2212yescomplemented
15C31×#121—22no—
16C4×D4×C31—22no—
17C2×C42×C31C2211yescharacteristic
18#11758528—28no—
19#11758526—28no—
20C2×#92482C2214yescomplemented
21C2×#92481C2214yes—
22C2×#92480C2214yescomplemented
23C2×#92479C2214yes—
24C2×(C2×D31)⋊C4—22no—
25C2×#92474C2212yes—
26#11758452—22no—
27#11758452C2212yes—
28#11758452C2212yes—
29#11758391C2212yescomplemented
30C4×C31⋊C8C2212yescomplemented
31C23×D31—21no—
32#41×C31—24no—
33#41×C31—24no—
34C2×Q8×C31C2312yes—
35C2×D4×C31—22no—
36C2×D4×C31—22no—
37C2×D4×C31C2312yes—
38C22×C4×C31—22no—
39C22×C4×C31D412yes—
40C22×C4×C31C2311yescharacteristic
41C22×Dic31—21no—
42C2×D124—24no—
43C2×D124—21no—
44C2×D124C2312yes—
45C2×Dic62—24no—
46C2×Dic62—21no—
47C2×Dic62C2312yes—
48C4⋊C4×C31—22no—
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—
58(C2×D31)⋊C4—24no—
59#92474D418yescomplemented
60C2×C31⋊C8C2314yes—
61C23×C31—22no—
62C23×C31C2×D411yescharacteristic
63C22×D31—22no—
64C22×D31—44no—
65C22×D31—42no—
66Q8×C31—22no—
67Q8×C31—22no—
68D4×C31—22no—
69D4×C31—22no—
70D4×C31—44no—
71D4×C31—44no—
72C2×C4×C31—22no—
73C2×C4×C31—42no—
74C2×C4×C31—44no—
75C2×C4×C31C2×D414yes—
76C2×C4×C31C2×D412yes—
77C2×C4×C31C2×D412yes—
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2411yescharacteristic
80C2×Dic31—22no—
81C2×Dic31—42no—
82D124—24no—
83D124—42no—
84Dic62—24no—
85Dic62—42no—
86C31⋊C8C2×D418yes—
87#3153—311nomaximal, Sylow
88C22×C31—22no—
89C22×C31—42no—
90C22×C31—44no—
91C22×C31#12712yes—
92C22×C31C22×D411yescharacteristic
93C2×D31—42no—
94C2×D31—44no—
95C2×D31—84no—
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#532D3111yescharacteristic, complemented
105C2×#124—311no—
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×#41C2×D3112yes—
116C22×Q8—311no—
117C22×D4—311no—
118#137—628no—
119AGL(1,ℤ/8)—628no—
120C2×Q16—314no—
121C2×SD16—314no—
122C2×SD16—314no—
123C2×D8—314no—
124C2×M16—312no—
125C4⋊Q8C2×D3111yescharacteristic, complemented
126#127C2×D3111yescharacteristic, complemented
127#124—312no—
128#124—312no—
129#124—622no—
130#124C2×D3112yescomplemented
131#121—22no—
132C4×D4—22no—
133C2×C22⋊C4—622no—
134C2×C42C2×D3111yescharacteristic
135#97—312no—
136C4×C8—312no—
137C31#315311yescharacteristic, complemented, Sylow
138C24—621no—
139#41—24no—
140#41—24no—
141C2×Q8—312no—
142C2×Q8—621no—
143C2×Q8—624no—
144C2×Q8C22×D3112yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—312no—
148C2×D4—624no—
149C2×D4—621no—
150C2×D4C22×D3112yes—
151C22×C4—22no—
152C22×C4—621no—
153C22×C4C22⋊D3112yes—
154C22×C4C22×D3111yescharacteristic
155Q16—628no—
156SD16—628no—
157SD16—628no—
158D8—628no—
159M16—318no—
160C2×C8—314no—
161C4⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—624no—
164C22⋊C4—624no—
165C42C22⋊D3112yes—
166C42C22×D3111yescharacteristic
167C42C22×D3111yescharacteristic
168C23—22no—
169C23—622no—
170C23—1244no—
171C23—1242no—
172C23C2×C22⋊D3111yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—624no—
176Q8—1242no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—44no—
181D4—624no—
182D4—1242no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—622no—
187C2×C4—1242no—
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—1242no—
198C22—1244no—
199C22—2484no—
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
FittingC31×#532C2
socleC22×C31C22×D4
hypercenter#532D31
hypocenterC31#3153
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C2×#11758526C2semidirect
3. C2×#11758452C2semidirect
4. #14521441C2semidirect
5. #14521436C2semidirect
6. #14521431C2semidirect
7. C4×#92474C2semidirect
8. C31×#532C2semidirect
12. C31×C4⋊Q8C22semidirect
13. C31×#127C22semidirect
14. C31×#124C22semidirect
20. C2×#92482C22semidirect
22. C2×#92480C22semidirect
29. #11758391C22semidirect
30. C4×C31⋊C8C22semidirect
59. #92474D4semidirect
104. #532D31semidirect
125. C4⋊Q8C2×D31semidirect
126. #127C2×D31semidirect
130. #124C2×D31semidirect
137. C31#3153semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C2×#117585262
C2×#117584522
#145214412
#145214362
#145214312
C4×#924742
C31×#5322
#315331