Finite Group Databasebeta

← #16856198

Subgroups — #16856198

Order 3968.

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

normal subgroups only

This group has 596 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×#11758526C212yesmaximal, complemented
2C22×D248C211yesmaximal, characteristic, complemented
3#14521613C212yesmaximal, complemented
4#14521435C214yesmaximal, complemented
5#14521434C212yesmaximal, complemented
6#14521424C212yesmaximal, complemented
7#14521004C211yesmaximal, characteristic, complemented
8C2×C31×#135C211yesmaximal, characteristic, complemented
9C2×#41×C31C2212yes—
10C22×D124C2212yes—
11C31×#135—24no—
12C2×Q16×C31C2211yescharacteristic, complemented
13C2×SD16×C31C2212yescomplemented
14C2×D8×C31C2211yescharacteristic, complemented
15C22×C8×C31C2211yescharacteristic
16#11758526—28no—
17#11758524C2214yescomplemented
18C2×#92481C2214yescomplemented
19#11758517C2214yescomplemented
20#11758514—24no—
21C2×#92479C2214yescomplemented
22#11758507—24no—
23#11758503C2211yescharacteristic
24C2×#92474C2211yescharacteristic
25C2×D248—24no—
26C2×D248C2212yes—
27C2×D248C2212yes—
28#11758403C2212yescomplemented
29C8×Dic31C2212yescomplemented
30C23×D31—22no—
31#41×C31—24no—
32#41×C31—24no—
33C2×Q8×C31C2312yes—
34C2×D4×C31—22no—
35C2×D4×C31C2312yes—
36C22×C4×C31—22no—
37C22×C4×C31C2311yescharacteristic
38C2×C22⋊D31—28no—
39C2×D124—24no—
40C2×D124—24no—
41C2×D124—22no—
42C2×D124C2314yes—
43Q16×C31—22no—
44SD16×C31—24no—
45D8×C31—22no—
46C2×C8×C31D414yescomplemented
47C2×C8×C31C2311yescharacteristic
48C2×C8×C31C2311yescharacteristic
49C22⋊Dic31D412yes—
50#92481—28no—
51#92479—28no—
52(C2×D31)⋊C4—28no—
53C4⋊Dic31D412yes—
54C4×Dic31C2312yes—
55#92474D414yescomplemented
56C2×C31⋊C8C2312yes—
57D248—24no—
58D248—24no—
59C23×C31—22no—
60C23×C31C2×D411yescharacteristic
61C22×D31—24no—
62C22×D31—44no—
63C22×D31—84no—
64Q8×C31—22no—
65Q8×C31—22no—
66D4×C31—22no—
67D4×C31—22no—
68D4×C31—44no—
69C2×C4×C31—22no—
70C2×C4×C31—44no—
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31C2×D412yes—
73C2×C4×C31C2×D412yes—
74C2×C4×C31C2411yescharacteristic
75C22⋊D31—88no—
76C2×Dic31C2×D414yes—
77D124—24no—
78D124—24no—
79D124—44no—
80C8×C31C2×D412yes—
81C8×C31C2×D412yes—
82C31⋊C8C2×D414yes—
83#3155—311nomaximal, Sylow
84C22×C31—22no—
85C22×C31—42no—
86C22×C31—44no—
87C22×C31#12712yes—
88C22×C31C22×D411yescharacteristic
89C2×D31—44no—
90C2×D31—48no—
91C2×D31—88no—
92C4×C31—42no—
93C4×C31#12712yes—
94C4×C31C22×D411yescharacteristic
95C4×C31C22×D411yescharacteristic
96Dic31—24no—
97C2×AGL(1,ℤ/8)—312no—
98C2×#135D3111yescharacteristic, complemented
99C22×D8—311no—
100#535—312no—
101#496—314no—
102#495—312no—
103#493—312no—
104#405—311no—
105C2×C31—22no—
106C2×C31—82no—
107C2×C31C2×#12711yescharacteristic
108C2×C31#57612yes—
109D31—84no—
110C2×#41C2×D3112yes—
111C22×D4—312no—
112AGL(1,ℤ/8)—628no—
113#135—24no—
114C2×Q16C2×D3111yescharacteristic, complemented
115C2×SD16—314no—
116C2×SD16C2×D3112yescomplemented
117C2×D8—314no—
118C2×D8—312no—
119C2×D8—312no—
120C2×D8—624no—
121C2×D8C2×D3111yescharacteristic, complemented
122C2×M16—311no—
123C22×C8C2×D3111yescharacteristic
124#127—314no—
125#124—314no—
126#121—624no—
127#120—624no—
128#117—311no—
129#97—312no—
130C4×C8—312no—
131C31#315511yescharacteristic, complemented, Sylow
132C24—622no—
133#41—24no—
134#41—24no—
135C2×Q8C22×D3112yes—
136C2×D4—22no—
137C2×D4—314no—
138C2×D4—628no—
139C2×D4—624no—
140C2×D4—622no—
141C2×D4—624no—
142C2×D4C22×D3112yes—
143C22×C4—22no—
144C22×C4C22×D3111yescharacteristic
145Q16—22no—
146SD16—24no—
147SD16—628no—
148D8—22no—
149D8—628no—
150D8—624no—
151D8—624no—
152M16—314no—
153C2×C8—312no—
154C2×C8C22⋊D3114yescomplemented
155C2×C8C22×D3111yescharacteristic
156C2×C8C22×D3111yescharacteristic
157C4⋊C4—312no—
158C22⋊C4—312no—
159C22⋊C4—628no—
160C42—312no—
161C23—22no—
162C23—624no—
163C23—1244no—
164C23—2484no—
165C23D4×D3111yescharacteristic
166Q8—22no—
167Q8—22no—
168D4—22no—
169D4—22no—
170D4—44no—
171D4—624no—
172D4—624no—
173D4—1244no—
174D4—2488no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—314no—
178C2×C4D4×D3111yescharacteristic
179C2×C4D4×D3112yes—
180C2×C4C2×C22⋊D3112yes—
181C2×C4C23×D3111yescharacteristic
182C8—314no—
183C8C2×C22⋊D3112yes—
184C8C2×C22⋊D3112yes—
185C22—22no—
186C22—42no—
187C22—44no—
188C22—1244no—
189C22—1248no—
190C22—2488no—
191C22#1175851712yes—
192C22C2×D4×D3111yescharacteristic
193C4—42no—
194C4—624no—
195C4#1175851712yes—
196C4C2×D4×D3111yescharacteristic
197C4C22×C22⋊D3111yescharacteristic
198C2—22no—
199C2—82no—
200C2—2484no—
201C2C2×#1175851711yescharacteristic
202C2#1452185112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#135C2
socleC22×C31C22×D4
hypercenterC2×#135D31
hypocenterC31#3155
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C22×D248C2semidirect
3. #14521613C2semidirect
4. #14521435C2semidirect
5. #14521434C2semidirect
6. #14521424C2semidirect
7. #14521004C2semidirect
8. C2×C31×#135C2semidirect
12. C2×Q16×C31C22semidirect
13. C2×SD16×C31C22semidirect
14. C2×D8×C31C22semidirect
17. #11758524C22semidirect
18. C2×#92481C22semidirect
19. #11758517C22semidirect
21. C2×#92479C22semidirect
28. #11758403C22semidirect
29. C8×Dic31C22semidirect
46. C2×C8×C31D4semidirect
55. #92474D4semidirect
98. C2×#135D31semidirect
114. C2×Q16C2×D31semidirect
116. C2×SD16C2×D31semidirect
121. C2×D8C2×D31semidirect
131. C31#3155semidirect
154. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
C22×D2482
#145216132
#145214352
#145214342
#145214242
#145210042
C2×C31×#1352
#315531