Finite Group Databasebeta

← #36323306

Subgroups — #36323306

Order 21376.

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×Dic334C211yesmaximal, characteristic
2#24923620C211yesmaximal, characteristic, complemented
3#24923565C211yesmaximal, characteristic, complemented
4#24923559C211yesmaximal, characteristic
5#24923016C211yesmaximal, characteristic, complemented
6#24923001C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic
8C22×Dic334—21no—
9#19128832C2211yescharacteristic
10#19128832C2211yescharacteristic
11#19128831—21no—
12#19128811—21no—
13#19128809—22no—
14C4×Dic334C2211yescharacteristic
15D4×Dic167—21no—
16#19128766—21no—
17#19128763—21no—
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128757—21no—
20#19128757—22no—
21#19128753—21no—
22C4×D4×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24C4×C8×C167C2211yescharacteristic
25C2×Dic334—22no—
26C2×Dic334—21no—
27C2×Dic334—21no—
28C2×Dic334—42no—
29C2×Dic334—42no—
30C2×Dic334D411yescharacteristic
31C22×Dic167—22no—
32C2×C167⋊C8—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167—21no—
36C4⋊Dic167D412yes—
37C4⋊Dic167D411yescharacteristic
38Dic167⋊C4—22no—
39Dic167⋊C4—21no—
40C22⋊Dic167—22no—
41C4×Dic167—21no—
42C2×D4×C167D411yescharacteristic
43C22×C4×C167—21no—
44C2×C8×C167—21no—
45C2×C8×C167—21no—
46C4⋊C4×C167D411yescharacteristic
47C22⋊C4×C167—21no—
48C42×C167C2311yescharacteristic
49Dic334—21no—
50Dic334—44no—
51Dic334—41no—
52Dic334—41no—
53Dic334—42no—
54Dic334SD1612yes—
55C2×Dic167—21no—
56C2×Dic167—21no—
57C2×Dic167—22no—
58C2×Dic167—44no—
59C2×Dic167—41no—
60C167⋊C8—41no—
61C23×C167—21no—
62D4×C167—21no—
63D4×C167Q1612yes—
64C2×C4×C167—21no—
65C2×C4×C167—41no—
66C2×C4×C167—41no—
67C2×C4×C167C2×D411yescharacteristic
68C2×C4×C167C2×D411yescharacteristic
69C2×C4×C167C2×D411yescharacteristic
70C8×C167—41no—
71C8×C167—41no—
72C22×C167—22no—
73C22×C167—41no—
74C22×C167—41no—
75C22×C167#12011yescharacteristic
76C4×C167—21no—
77C4×C167—41no—
78C4×C167C2×SD1611yescharacteristic
79C4×C167C2×Q1611yescharacteristic
80C4×C167AGL(1,ℤ/8)11yescharacteristic
81C4×C167#13711yescharacteristic
82Dic167—22no—
83Dic167—42no—
84Dic167—41no—
85Dic167—81no—
86C2×C167—42no—
87C2×C167#45011yescharacteristic
88C2×C167#45111yescharacteristic
89C2×C167#45411yescharacteristic
90C167#163711yescharacteristic, complemented, Sylow
91#1637—1671nomaximal, Sylow
92D4×Q8—1671no—
93#487—1671no—
94#477—1671no—
95#476—1671no—
96#332—1671no—
97Q8⋊C8—1671no—
98D4⋊C8D16711yescharacteristic
99C22×Q8—3341no—
100C4⋊Q8—3341no—
101C4⋊Q8—1671no—
102C4⋊Q8—1671no—
103C22⋊Q8—3342no—
104C22⋊Q8—3341no—
105C4×Q8—1671no—
106C4×D4—3341no—
107C4×D4C2×D16711yescharacteristic
108#107—3341no—
109#106—3341no—
110C4⋊C8—1671no—
111C4⋊C8C2×D16711yescharacteristic
112#103—3341no—
113#103—3342no—
114#102—3341no—
115C4×C8C2×D16711yescharacteristic
116C2×Q8—3342no—
117C2×Q8—3341no—
118C2×Q8—3341no—
119C2×Q8—6682no—
120C2×Q8—6682no—
121C2×Q8—1671no—
122C2×D4D66811yescharacteristic
123C22×C4—21no—
124C22×C4—3342no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3341no—
128C4⋊C4—3342no—
129C4⋊C4—3341no—
130C4⋊C4—3341no—
131C4⋊C4—3341no—
132C4⋊C4—3341no—
133C4⋊C4—1672no—
134C4⋊C4—1671no—
135C4⋊C4D66811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3342no—
138C42—3341no—
139C42C22×D16711yescharacteristic
140C23—21no—
141Q8—3341no—
142Q8—6682no—
143Q8—6684no—
144Q8—6681no—
145Q8—6681no—
146Q8—1672no—
147D4—21no—
148D4Dic66812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3341no—
153C2×C4—3341no—
154C2×C4—3342no—
155C2×C4—6681no—
156C2×C4—6684no—
157C2×C4D4×D16711yescharacteristic
158C2×C4D4×D16711yescharacteristic
159C2×C4C2×D66811yescharacteristic
160C8—41no—
161C8—41no—
162C8—6681no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1912880211yescharacteristic
167C4—21no—
168C4—41no—
169C4—13361no—
170C4—3342no—
171C4—6682no—
172C4—6681no—
173C4SD16×D16711yescharacteristic
174C4C2×Dic66811yescharacteristic
175C4#1912885611yescharacteristic
176C4#1912886511yescharacteristic
177C2—42no—
178C2#2492342111yescharacteristic
179C2#2492342711yescharacteristic
180C2#2492344311yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
FittingD4⋊C8×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenterD4⋊C8D167
hypocenterC167#1637
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #24923620C2semidirect
3. #24923565C2semidirect
5. #24923016C2semidirect
6. #24923001C2semidirect
90. C167#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3342
#249236202
#249235652
#249235592
#249230162
#249230012
D4⋊C8×C1672
#1637167