Finite Group Databasebeta

← #36323390

Subgroups — #36323390

Order 21376.

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×Dic334C211yesmaximal, characteristic
2#24923674C211yesmaximal, characteristic, complemented
3#24923549C211yesmaximal, characteristic
4#24923480C211yesmaximal, characteristic, complemented
5#24923016C211yesmaximal, characteristic, complemented
6#24923001C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic
8C22×Dic334—21no—
9C2×Dic668—21no—
10#19128832C2211yescharacteristic
11#19128832C2211yescharacteristic
12#19128831—21no—
13#19128811—21no—
14#19128809—22no—
15C4×Dic334C2211yescharacteristic
16D4×Dic167—21no—
17#19128766—22no—
18#19128765—21no—
19C4⋊(C167⋊C8)C2211yescharacteristic
20#19128757—21no—
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—
32Dic668—42no—
33C2×C167⋊C8—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167—21no—
36C4⋊Dic167—21no—
37C4⋊Dic167D412yes—
38C4⋊Dic167D411yescharacteristic
39Dic167⋊C4—22no—
40Dic167⋊C4—21no—
41C22⋊Dic167—22no—
42C4×Dic167—21no—
43C2×D4×C167D411yescharacteristic
44C22×C4×C167—21no—
45C2×C8×C167—21no—
46C2×C8×C167—21no—
47C4⋊C4×C167D411yescharacteristic
48C22⋊C4×C167—21no—
49C42×C167C2311yescharacteristic
50Dic334—21no—
51Dic334—44no—
52Dic334—41no—
53Dic334—41no—
54Dic334—41no—
55Dic334—41no—
56Dic334D812yes—
57C2×Dic167—21no—
58C2×Dic167—21no—
59C2×Dic167—22no—
60C2×Dic167—44no—
61C2×Dic167—41no—
62C167⋊C8—41no—
63C23×C167—21no—
64D4×C167—21no—
65D4×C167Q1612yescomplemented
66C2×C4×C167—21no—
67C2×C4×C167—41no—
68C2×C4×C167—41no—
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167C2×D411yescharacteristic
72C8×C167—22no—
73C8×C167—41no—
74C22×C167—22no—
75C22×C167—41no—
76C22×C167—41no—
77C22×C167#12011yescharacteristic
78C4×C167—21no—
79C4×C167—41no—
80C4×C167C2×D811yescharacteristic
81C4×C167C2×Q1611yescharacteristic
82C4×C167AGL(1,ℤ/8)11yescharacteristic
83C4×C167#13711yescharacteristic
84Dic167—22no—
85Dic167—42no—
86Dic167—41no—
87Dic167—81no—
88C2×C167—42no—
89C2×C167#44711yescharacteristic
90C2×C167#45111yescharacteristic
91C2×C167#45611yescharacteristic
92C167#164911yescharacteristic, complemented, Sylow
93#1649—1671nomaximal, Sylow
94D4×Q8—1671no—
95#500—1671no—
96#474—1671no—
97#462—1671no—
98#332—1671no—
99Q8⋊C8—1671no—
100D4⋊C8D16711yescharacteristic
101C22×Q8—3341no—
102C2×Q16—3341no—
103C4⋊Q8—3341no—
104C4⋊Q8—1671no—
105C4⋊Q8—1671no—
106C22⋊Q8—3342no—
107C22⋊Q8—3341no—
108C4×Q8—1671no—
109C4×D4—3341no—
110C4×D4C2×D16711yescharacteristic
111#107—3342no—
112#107—3341no—
113C4⋊C8—1671no—
114C4⋊C8C2×D16711yescharacteristic
115#103—3341no—
116#102—3341no—
117C4×C8C2×D16711yescharacteristic
118C2×Q8—3342no—
119C2×Q8—3341no—
120C2×Q8—3341no—
121C2×Q8—6682no—
122C2×Q8—6682no—
123C2×Q8—1671no—
124C2×D4D66811yescharacteristic
125C22×C4—21no—
126C22×C4—3342no—
127Q16—6682no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3341no—
131C4⋊C4—3342no—
132C4⋊C4—3341no—
133C4⋊C4—3341no—
134C4⋊C4—3341no—
135C4⋊C4—3341no—
136C4⋊C4—1671no—
137C4⋊C4—1672no—
138C4⋊C4D66811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3342no—
141C42—3341no—
142C42C22×D16711yescharacteristic
143C23—21no—
144Q8—3341no—
145Q8—6684no—
146Q8—6681no—
147Q8—6681no—
148Q8—6681no—
149Q8—6681no—
150Q8—1672no—
151D4—21no—
152D4Dic66812yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3341no—
157C2×C4—3341no—
158C2×C4—3342no—
159C2×C4—6681no—
160C2×C4—6684no—
161C2×C4D4×D16711yescharacteristic
162C2×C4D4×D16711yescharacteristic
163C2×C4C2×D66811yescharacteristic
164C8—22no—
165C8—41no—
166C8—6681no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1912880211yescharacteristic
171C4—21no—
172C4—41no—
173C4—13361no—
174C4—3342no—
175C4—6682no—
176C4—6681no—
177C4D8×D16711yescharacteristic
178C4C2×Dic66811yescharacteristic
179C4#1912885911yescharacteristic
180C4#1912886511yescharacteristic
181C2—42no—
182C2#2492340411yescharacteristic
183C2#2492342711yescharacteristic
184C2#2492345311yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
FittingD4⋊C8×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenterD4⋊C8D167
hypocenterC167#1649
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. #24923674C2semidirect
4. #24923480C2semidirect
5. #24923016C2semidirect
6. #24923001C2semidirect
65. D4×C167Q16semidirect
92. C167#1649semidirect
152. D4Dic668semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3342
#249236742
#249235492
#249234802
#249230162
#249230012
D4⋊C8×C1672
#1649167