Finite Group Databasebeta

← #16847379

Subgroups — #16847379

Order 3968.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#14521617C211yesmaximal, characteristic
2#99×D31C212yesmaximal, complemented
3#14520844C211yesmaximal, characteristic, complemented
4#14520841C212yesmaximal, complemented
5C31×#354C211yesmaximal, characteristic, complemented
6C2×#92499C411yescharacteristic, complemented
7C2×D4×D31C2211yescharacteristic
8C31×#124C2211yescharacteristic
9#11758524C411yescharacteristic, complemented
10#11758517C2211yescharacteristic
11#11758461—22no—
12#11758459—22no—
13C22⋊C4×D31—22no—
14#11758451C411yescharacteristic, complemented
15#11758449C411yescharacteristic, complemented
16C31×#99C2212yes—
17#11758422C2212yes—
18#11758394—22no—
19C23×D31D412yes—
20C2×Q8×C31C2×C411yescharacteristic, complemented
21C2×D4×C31C2311yescharacteristic
22C2×C22⋊D31—22no—
23C2×C22⋊D31D412yes—
24#92499—42no—
25D4×D31—42no—
26C2×D124—21no—
27C2×D124—22no—
28C2×D124C2×C411yescharacteristic
29C2×D124C2×C411yescharacteristic, complemented
30C2×C4×D31—21no—
31C2×C4×D31—42no—
32C2×C4×D31C2×C411yescharacteristic
33C22⋊C4×C31—22no—
34C22⋊C4×C31—22no—
35C42×C31C2×C411yescharacteristic, complemented
36C22⋊Dic31—22no—
37(C2×D31)⋊C4—21no—
38(C2×D31)⋊C4—22no—
39(C2×D31)⋊C4—42no—
40Dic31⋊C4—21no—
41C4×Dic31C2×C411yescharacteristic, complemented
42C23×C31C2×D412yes—
43C22×D31—22no—
44C22×D31—22no—
45C22×D31—21no—
46C22×D31—82no—
47C22×D31C22⋊C412yes—
48C22×D31C22⋊C411yescharacteristic
49Q8×C31—41no—
50D4×C31—41no—
51C2×C4×C31—21no—
52C2×C4×C31—21no—
53C2×C4×C31—42no—
54C2×C4×C31C22×C411yescharacteristic
55C22⋊D31—42no—
56C22⋊D31—82no—
57C2×Dic31—21no—
58C2×Dic31—42no—
59C2×Dic31C22⋊C411yescharacteristic
60D124—41no—
61D124—41no—
62D124—42no—
63D124—82no—
64C4×D31—22no—
65C4×D31—42no—
66C4×D31—84no—
67#2127—311nomaximal, Sylow
68C22×C31—22no—
69C22×C31—42no—
70C22×C31C2×C22⋊C411yescharacteristic
71C2×D31—22no—
72C2×D31—41no—
73C2×D31—44no—
74C2×D31—42no—
75C2×D31—42no—
76C2×D31—41no—
77C2×D31—82no—
78C2×D31#9912yes—
79C4×C31—21no—
80C4×C31—41no—
81C4×C31—41no—
82C4×C31—82no—
83Dic31—21no—
84Dic31—41no—
85Dic31—82no—
86#535—311no—
87C2×#99—312no—
88#354—311no—
89#354D3111yescharacteristic, complemented
90#353—312no—
91C2×C31—21no—
92C2×C31—42no—
93C2×C31C2×#9911yescharacteristic
94D31—22no—
95D31—42no—
96D31—81no—
97C2×#41—311no—
98C22×D4—311no—
99#127—311no—
100#127—311no—
101#124—311no—
102#124C2×D3111yescharacteristic
103#121—622no—
104#120—622no—
105#117—311no—
106C2×C22⋊C4—622no—
107#99—312no—
108#99—622no—
109#99C2×D3112yes—
110C31#212711yescharacteristic, complemented, Sylow
111C24—312no—
112#41—1242no—
113C2×Q8C4×D3111yescharacteristic, complemented
114C2×D4—312no—
115C2×D4—311no—
116C2×D4—311no—
117C2×D4—621no—
118C2×D4—622no—
119C2×D4—622no—
120C2×D4—1242no—
121C2×D4C22×D3111yescharacteristic
122C22×C4—311no—
123C22×C4—621no—
124C22×C4—1242no—
125C4⋊C4—621no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—621no—
129C22⋊C4—622no—
130C22⋊C4—622no—
131C22⋊C4—1242no—
132C42—311no—
133C42C4×D3111yescharacteristic, complemented
134C23—311no—
135C23—312no—
136C23—622no—
137C23—622no—
138C23—621no—
139C23—2482no—
140C23D4×D3112yes—
141Q8—41no—
142D4—41no—
143D4—1241no—
144D4—1241no—
145D4—1242no—
146D4—1242no—
147D4—2482no—
148D4—2482no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—311no—
153C2×C4—622no—
154C2×C4—621no—
155C2×C4—1242no—
156C2×C4—1242no—
157C2×C4—2484no—
158C2×C4C2×C4×D3111yescharacteristic
159C22—22no—
160C22—42no—
161C22—312no—
162C22—622no—
163C22—1241no—
164C22—1242no—
165C22—1242no—
166C22—1244no—
167C22—1241no—
168C22—2482no—
169C22C22⋊C4×D3111yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—621no—
175C4—1241no—
176C4—2482no—
177C2—21no—
178C2—42no—
179C2—622no—
180C2—1242no—
181C2—2481no—
182C2#99×D3111yescharacteristic
subgroupG/H
centerC2#99×D31
commutatorC2×C4×C31C22×C4
FrattiniC2×D4C22×D31
FittingC31×#354C2
socleC2×C31C2×#99
prime radical#14521617C2
cosocleC2×D4×C31C23
hypercenter#354D31
hypocenterC31#2127
LCSG ▷ C2×C4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D31C2semidirect
3. #14520844C2semidirect
4. #14520841C2semidirect
5. C31×#354C2semidirect
6. C2×#92499C4semidirect
9. #11758524C4semidirect
14. #11758451C4semidirect
15. #11758449C4semidirect
20. C2×Q8×C31C2×C4semidirect
29. C2×D124C2×C4semidirect
35. C42×C31C2×C4semidirect
41. C4×Dic31C2×C4semidirect
89. #354D31semidirect
110. C31#2127semidirect
113. C2×Q8C4×D31semidirect
133. C42C4×D31semidirect

Maximal subgroups

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

subgroupindex
#145216172
#99×D312
#145208442
#145208412
C31×#3542
#212731