Finite Group Databasebeta

← #28335629

Subgroups — #28335629

Order 13696.

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
1D4×Dic214C211yesmaximal, characteristic
2#20292978C211yesmaximal, characteristic, complemented
3#20292923C211yesmaximal, characteristic, complemented
4#20292917C211yesmaximal, characteristic
5#20292374C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9#16812109C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812108—21no—
12#16812088—21no—
13#16812086—22no—
14C4×Dic214C2211yescharacteristic
15D4×Dic107—21no—
16#16812043—21no—
17#16812040—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812034—21no—
20#16812034—22no—
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—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107D412yes—
37C4⋊Dic107D411yescharacteristic
38Dic107⋊C4—22no—
39Dic107⋊C4—21no—
40C22⋊Dic107—22no—
41C4×Dic107—21no—
42C2×D4×C107D411yescharacteristic
43C22×C4×C107—21no—
44C2×C8×C107—21no—
45C2×C8×C107—21no—
46C4⋊C4×C107D411yescharacteristic
47C22⋊C4×C107—21no—
48C42×C107C2311yescharacteristic
49C23×C107—21no—
50D4×C107—21no—
51D4×C107Q1612yes—
52C2×C4×C107—21no—
53C2×C4×C107—41no—
54C2×C4×C107—41no—
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×Dic107—21no—
59C2×Dic107—21no—
60C2×Dic107—22no—
61C2×Dic107—44no—
62C2×Dic107—41no—
63Dic214—21no—
64Dic214—44no—
65Dic214—41no—
66Dic214—41no—
67Dic214—42no—
68Dic214SD1612yes—
69C8×C107—41no—
70C8×C107—41no—
71C107⋊C8—41no—
72C22×C107—22no—
73C22×C107—41no—
74C22×C107—41no—
75C22×C107#12011yescharacteristic
76C4×C107—21no—
77C4×C107—41no—
78C4×C107C2×SD1611yescharacteristic
79C4×C107C2×Q1611yescharacteristic
80C4×C107AGL(1,ℤ/8)11yescharacteristic
81C4×C107#13711yescharacteristic
82Dic107—22no—
83Dic107—42no—
84Dic107—41no—
85Dic107—81no—
86C2×C107—42no—
87C2×C107#45011yescharacteristic
88C2×C107#45111yescharacteristic
89C2×C107#45411yescharacteristic
90#1637—1071nomaximal, Sylow
91C107#163711yescharacteristic, complemented, Sylow
92D4×Q8—1071no—
93#487—1071no—
94#477—1071no—
95#476—1071no—
96#332—1071no—
97Q8⋊C8—1071no—
98D4⋊C8D10711yescharacteristic
99C22×Q8—2141no—
100C4⋊Q8—1071no—
101C4⋊Q8—1071no—
102C4⋊Q8—2141no—
103C22⋊Q8—2142no—
104C22⋊Q8—2141no—
105C4×Q8—1071no—
106C4×D4—2141no—
107C4×D4C2×D10711yescharacteristic
108#107—2141no—
109#106—2141no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—2142no—
113#103—2141no—
114#102—2141no—
115C4×C8C2×D10711yescharacteristic
116C2×Q8—1071no—
117C2×Q8—4282no—
118C2×Q8—4282no—
119C2×Q8—2142no—
120C2×Q8—2141no—
121C2×Q8—2141no—
122C2×D4D42811yescharacteristic
123C22×C4—21no—
124C22×C4—2142no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2141no—
128C4⋊C4—1072no—
129C4⋊C4—1071no—
130C4⋊C4—2142no—
131C4⋊C4—2141no—
132C4⋊C4—2141no—
133C4⋊C4—2141no—
134C4⋊C4—2141no—
135C4⋊C4D42811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2142no—
138C42—2141no—
139C42C22×D10711yescharacteristic
140C23—21no—
141Q8—1072no—
142Q8—4282no—
143Q8—4284no—
144Q8—4281no—
145Q8—4281no—
146Q8—2141no—
147D4—21no—
148D4Dic42812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—4281no—
153C2×C4—4284no—
154C2×C4—2141no—
155C2×C4—2142no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×D42811yescharacteristic
160C8—41no—
161C8—41no—
162C8—4281no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1681207911yescharacteristic
167C4—21no—
168C4—41no—
169C4—8561no—
170C4—4282no—
171C4—4281no—
172C4—2142no—
173C4SD16×D10711yescharacteristic
174C4C2×Dic42811yescharacteristic
175C4#1681213311yescharacteristic
176C4#1681214211yescharacteristic
177C2—42no—
178C2#2029277911yescharacteristic
179C2#2029278511yescharacteristic
180C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenterD4⋊C8D107
hypocenterC107#1637
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. #20292978C2semidirect
3. #20292923C2semidirect
5. #20292374C2semidirect
6. #20292359C2semidirect
91. C107#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202929782
#202929232
#202929172
#202923742
#202923592
D4⋊C8×C1072
#1637107