Finite Group Databasebeta

← #16844878

Subgroups — #16844878

Order 3968.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D124C211yesmaximal, characteristic
2#14521400C211yesmaximal, characteristic, complemented
3#14521344C211yesmaximal, characteristic, complemented
4#14521340C211yesmaximal, characteristic
5#14520794C211yesmaximal, characteristic, complemented
6#14520778C211yesmaximal, characteristic
7C31×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D31—21no—
9C4×Q8×C31C2211yescharacteristic
10#11758474—21no—
11#11758474—22no—
12C4×D124—21no—
13C4×D124C2211yescharacteristic
14#11758446—21no—
15#11758446C2211yescharacteristic
16#11758446C2211yescharacteristic
17C31×C4⋊C8C2211yescharacteristic
18C4×C8×C31C2211yescharacteristic
19#11758423—21no—
20#11758409—21no—
21#11758405—21no—
22#11758404—22no—
23#11758395—21no—
24C4⋊(C31⋊C8)C2211yescharacteristic
25C2×Q8×C31D411yescharacteristic
26Q8×D31—42no—
27Q8×D31—42no—
28C2×D124D411yescharacteristic
29C2×C4×D31—22no—
30C2×C4×D31—21no—
31C2×Dic62—22no—
32C2×Dic62—21no—
33C2×Dic62—21no—
34C2×C8×C31—21no—
35C2×C8×C31—21no—
36C4⋊C4×C31—21no—
37C4⋊C4×C31D411yescharacteristic
38C42×C31—21no—
39C42×C31C2311yescharacteristic
40(C2×D31)⋊C4—22no—
41(C2×D31)⋊C4—21no—
42C4⋊Dic31—22no—
43C4⋊Dic31—21no—
44C4⋊Dic31—21no—
45C4⋊Dic31—21no—
46C4⋊Dic31D412yes—
47C4⋊Dic31D411yescharacteristic
48C2×C31⋊C8—21no—
49C22×D31—21no—
50Q8×C31—21no—
51Q8×C31SD1612yes—
52C2×C4×C31—21no—
53C2×C4×C31—21no—
54C2×C4×C31C2×D411yescharacteristic
55C2×C4×C31C2×D411yescharacteristic
56C2×C4×C31C2×D411yescharacteristic
57C2×Dic31—21no—
58C2×Dic31—22no—
59C2×Dic31—41no—
60D124—21no—
61D124Q1612yes—
62C4×D31—44no—
63C4×D31—41no—
64C4×D31—41no—
65Dic62—44no—
66Dic62—41no—
67Dic62—41no—
68Dic62—42no—
69C8×C31—41no—
70C8×C31—41no—
71C31⋊C8—41no—
72#1637—311nomaximal, Sylow
73C22×C31#12011yescharacteristic
74C2×D31—22no—
75C2×D31—41no—
76C2×D31—41no—
77C4×C31—22no—
78C4×C31—21no—
79C4×C31—41no—
80C4×C31C2×SD1611yescharacteristic
81C4×C31C2×Q1611yescharacteristic
82C4×C31AGL(1,ℤ/8)11yescharacteristic
83C4×C31#13711yescharacteristic
84Dic31—41no—
85Dic31—42no—
86Dic31—81no—
87D4×Q8—311no—
88#487—311no—
89#477—311no—
90#476—311no—
91#332—311no—
92Q8⋊C8D3111yescharacteristic, complemented
93D4⋊C8—311no—
94C2×C31#45011yescharacteristic
95C2×C31#45111yescharacteristic
96C2×C31#45411yescharacteristic
97D31—42no—
98C22×Q8—621no—
99C4⋊Q8—311no—
100C4⋊Q8—311no—
101C4⋊Q8—621no—
102C22⋊Q8—622no—
103C22⋊Q8—621no—
104C4×Q8C2×D3111yescharacteristic
105C4×D4—311no—
106C4×D4—621no—
107#107—621no—
108#106—621no—
109C4⋊C8—311no—
110C4⋊C8C2×D3111yescharacteristic
111#103—622no—
112#103—621no—
113#102—621no—
114C4×C8C2×D3111yescharacteristic
115C31#163711yescharacteristic, complemented, Sylow
116C2×Q8—622no—
117C2×Q8—621no—
118C2×Q8—621no—
119C2×Q8—1242no—
120C2×Q8—1242no—
121C2×Q8D12411yescharacteristic
122C2×D4—311no—
123C22×C4—622no—
124C22×C4—621no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—621no—
128C4⋊C4—21no—
129C4⋊C4—312no—
130C4⋊C4—311no—
131C4⋊C4—622no—
132C4⋊C4—621no—
133C4⋊C4—621no—
134C4⋊C4—621no—
135C4⋊C4D12411yescharacteristic
136C22⋊C4—622no—
137C22⋊C4—621no—
138C42—21no—
139C42C22×D3111yescharacteristic
140C23—621no—
141Q8—21no—
142Q8—1244no—
143Q8—1242no—
144Q8—1241no—
145Q8—1241no—
146Q8#9247012yes—
147D4—312no—
148D4—621no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—622no—
152C2×C4—621no—
153C2×C4—1244no—
154C2×C4—1241no—
155C2×C4—1241no—
156C2×C4—1241no—
157C2×C4C2×D12411yescharacteristic
158C2×C4D4×D3111yescharacteristic
159C2×C4D4×D3111yescharacteristic
160C8—41no—
161C8—41no—
162C8—1241no—
163C22—622no—
164C22—1241no—
165C22—1241no—
166C22#1175845911yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—1242no—
171C4—1241no—
172C4—2481no—
173C4C2×#9247011yescharacteristic
174C4#1175848511yescharacteristic
175C4#1175849211yescharacteristic
176C4Q16×D3111yescharacteristic
177C2—1242no—
178C2#1452120211yescharacteristic
179C2#1452120611yescharacteristic
180C2#1452122311yescharacteristic
subgroupG/H
centerC22#11758459
commutator, cosocleC42×C31C23
FrattiniC42C22×D31
FittingC31×Q8⋊C8C2
socleC22×C31#120
prime radicalC2×D124D4
hypercenterQ8⋊C8D31
hypocenterC31#1637
LCSG ▷ C42×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14521400C2semidirect
3. #14521344C2semidirect
5. #14520794C2semidirect
7. C31×Q8⋊C8C2semidirect
92. Q8⋊C8D31semidirect
115. C31#1637semidirect

Maximal subgroups

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

subgroupindex
Q8×D1242
#145214002
#145213442
#145213402
#145207942
#145207782
C31×Q8⋊C82
#163731