Finite Group Databasebeta

← #28335713

Subgroups — #28335713

Order 13696.

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

normal subgroups only

This group has 234 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
1D4×Dic214C211yesmaximal, characteristic
2#20293032C211yesmaximal, characteristic, complemented
3#20292907C211yesmaximal, characteristic
4#20292838C211yesmaximal, characteristic, complemented
5#20292374C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9C2×Dic428—21no—
10#16812109C2211yescharacteristic
11#16812109C2211yescharacteristic
12#16812108—21no—
13#16812088—21no—
14#16812086—22no—
15C4×Dic214C2211yescharacteristic
16D4×Dic107—21no—
17#16812043—22no—
18#16812042—21no—
19C4⋊(C107⋊C8)C2211yescharacteristic
20#16812034—21no—
21#16812030—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—42no—
29C2×Dic214—42no—
30C2×Dic214D411yescharacteristic
31C22×Dic107—22no—
32Dic428—42no—
33C2×C107⋊C8—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107D411yescharacteristic
38C4⋊Dic107D412yes—
39Dic107⋊C4—22no—
40Dic107⋊C4—21no—
41C22⋊Dic107—22no—
42C4×Dic107—21no—
43C2×D4×C107D411yescharacteristic
44C22×C4×C107—21no—
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C4⋊C4×C107D411yescharacteristic
48C22⋊C4×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—21no—
51D4×C107—21no—
52D4×C107Q1612yescomplemented
53C2×C4×C107—21no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—21no—
60C2×Dic107—21no—
61C2×Dic107—22no—
62C2×Dic107—44no—
63C2×Dic107—41no—
64Dic214—21no—
65Dic214—44no—
66Dic214—41no—
67Dic214—41no—
68Dic214—41no—
69Dic214—41no—
70Dic214D812yes—
71C8×C107—22no—
72C8×C107—41no—
73C107⋊C8—41no—
74C22×C107—22no—
75C22×C107—41no—
76C22×C107—41no—
77C22×C107#12011yescharacteristic
78C4×C107—21no—
79C4×C107—41no—
80C4×C107C2×D811yescharacteristic
81C4×C107C2×Q1611yescharacteristic
82C4×C107AGL(1,ℤ/8)11yescharacteristic
83C4×C107#13711yescharacteristic
84Dic107—22no—
85Dic107—42no—
86Dic107—41no—
87Dic107—81no—
88C2×C107—42no—
89C2×C107#44711yescharacteristic
90C2×C107#45111yescharacteristic
91C2×C107#45611yescharacteristic
92#1649—1071nomaximal, Sylow
93C107#164911yescharacteristic, complemented, Sylow
94D4×Q8—1071no—
95#500—1071no—
96#474—1071no—
97#462—1071no—
98#332—1071no—
99Q8⋊C8—1071no—
100D4⋊C8D10711yescharacteristic
101C22×Q8—2141no—
102C2×Q16—2141no—
103C4⋊Q8—1071no—
104C4⋊Q8—1071no—
105C4⋊Q8—2141no—
106C22⋊Q8—2142no—
107C22⋊Q8—2141no—
108C4×Q8—1071no—
109C4×D4—2141no—
110C4×D4C2×D10711yescharacteristic
111#107—2142no—
112#107—2141no—
113C4⋊C8—1071no—
114C4⋊C8C2×D10711yescharacteristic
115#103—2141no—
116#102—2141no—
117C4×C8C2×D10711yescharacteristic
118C2×Q8—1071no—
119C2×Q8—4282no—
120C2×Q8—4282no—
121C2×Q8—2142no—
122C2×Q8—2141no—
123C2×Q8—2141no—
124C2×D4D42811yescharacteristic
125C22×C4—21no—
126C22×C4—2142no—
127Q16—4282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2141no—
131C4⋊C4—1071no—
132C4⋊C4—1072no—
133C4⋊C4—2142no—
134C4⋊C4—2141no—
135C4⋊C4—2141no—
136C4⋊C4—2141no—
137C4⋊C4—2141no—
138C4⋊C4D42811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2142no—
141C42—2141no—
142C42C22×D10711yescharacteristic
143C23—21no—
144Q8—1072no—
145Q8—4284no—
146Q8—4281no—
147Q8—4281no—
148Q8—4281no—
149Q8—4281no—
150Q8—2141no—
151D4—21no—
152D4Dic42812yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—4281no—
157C2×C4—4284no—
158C2×C4—2141no—
159C2×C4—2142no—
160C2×C4—2141no—
161C2×C4D4×D10711yescharacteristic
162C2×C4D4×D10711yescharacteristic
163C2×C4C2×D42811yescharacteristic
164C8—22no—
165C8—41no—
166C8—4281no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1681207911yescharacteristic
171C4—21no—
172C4—41no—
173C4—8561no—
174C4—4282no—
175C4—4281no—
176C4—2142no—
177C4D8×D10711yescharacteristic
178C4C2×Dic42811yescharacteristic
179C4#1681213611yescharacteristic
180C4#1681214211yescharacteristic
181C2—42no—
182C2#2029276211yescharacteristic
183C2#2029278511yescharacteristic
184C2#2029281111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenterD4⋊C8D107
hypocenterC107#1649
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20293032C2semidirect
4. #20292838C2semidirect
5. #20292374C2semidirect
6. #20292359C2semidirect
52. D4×C107Q16semidirect
93. C107#1649semidirect
152. D4Dic428semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202930322
#202929072
#202928382
#202923742
#202923592
D4⋊C8×C1072
#1649107