Finite Group Databasebeta

← #36323274

Subgroups — #36323274

Order 21376.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#24923976C211yesmaximal, characteristic
2#24923649C211yesmaximal, characteristic, complemented
3#24923565C211yesmaximal, characteristic, complemented
4#24923559C211yesmaximal, characteristic
5#24923017C211yesmaximal, characteristic, complemented
6#24923001C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic
8C2×Dic668—22no—
9#19128832C2211yescharacteristic
10#19128832C2211yescharacteristic
11#19128823—21no—
12#19128811—21no—
13#19128809—22no—
14C4×Dic334C2211yescharacteristic
15D4×Dic167—21no—
16C2×C4⋊Dic167—21no—
17#19128766—21no—
18#19128763—21no—
19C4⋊(C167⋊C8)C2211yescharacteristic
20#19128757—21no—
21#19128753—21no—
22C4×D4×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24C4×C8×C167C2211yescharacteristic
25C2×Dic334—21no—
26C2×Dic334D411yescharacteristic
27C2×Dic334D412yes—
28C22×Dic167—22no—
29Dic668—44no—
30C2×C167⋊C8—21no—
31C4⋊Dic167—22no—
32C4⋊Dic167—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167—21no—
36C4⋊Dic167D411yescharacteristic
37Dic167⋊C4—22no—
38Dic167⋊C4—21no—
39C22⋊Dic167—22no—
40C4×Dic167—21no—
41C2×D4×C167D411yescharacteristic
42C22×C4×C167—21no—
43C2×C8×C167—21no—
44C2×C8×C167—21no—
45C4⋊C4×C167D411yescharacteristic
46C22⋊C4×C167—21no—
47C42×C167C2311yescharacteristic
48Dic334—21no—
49Dic334—21no—
50Dic334—42no—
51Dic334—42no—
52C2×Dic167—21no—
53C2×Dic167—21no—
54C2×Dic167—22no—
55C2×Dic167—41no—
56C2×Dic167—82no—
57C167⋊C8—41no—
58C23×C167—21no—
59D4×C167—21no—
60D4×C167Q1612yescomplemented
61C2×C4×C167—21no—
62C2×C4×C167—41no—
63C2×C4×C167—41no—
64C2×C4×C167C2×D411yescharacteristic
65C2×C4×C167C2×D411yescharacteristic
66C2×C4×C167C2×D411yescharacteristic
67C8×C167—22no—
68C8×C167—41no—
69C22×C167—22no—
70C22×C167—41no—
71C22×C167—41no—
72C22×C167#12011yescharacteristic
73C4×C167—21no—
74C4×C167—41no—
75C4×C167C2×Q1611yescharacteristic
76C4×C167#13511yescharacteristic
77C4×C167#13711yescharacteristic
78C4×C167#13711yescharacteristic
79Dic167—41no—
80Dic167—41no—
81Dic167—42no—
82Dic167—81no—
83C2×C167—42no—
84C2×C167#45111yescharacteristic
85C2×C167#45211yescharacteristic
86C2×C167#45311yescharacteristic
87C167#163211yescharacteristic, complemented, Sylow
88#1632—1671nomaximal, Sylow
89#554—1671no—
90#494—1671no—
91#477—1671no—
92#476—1671no—
93#333—1671no—
94Q8⋊C8—1671no—
95D4⋊C8D16711yescharacteristic
96C2×Q16—3342no—
97C4⋊Q8—1671no—
98C4⋊Q8—1671no—
99#125—3341no—
100C22⋊Q8—3342no—
101C22⋊Q8—3341no—
102C4×Q8—1671no—
103C4×D4—3341no—
104C4×D4C2×D16711yescharacteristic
105C2×C4⋊C4—3341no—
106#107—3341no—
107#106—3341no—
108C4⋊C8—1671no—
109C4⋊C8C2×D16711yescharacteristic
110#103—3341no—
111#102—3341no—
112C4×C8C2×D16711yescharacteristic
113C2×Q8—3341no—
114C2×Q8—1671no—
115C2×Q8—1672no—
116C2×D4D66811yescharacteristic
117C22×C4—21no—
118C22×C4—3342no—
119Q16—6684no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3341no—
123C4⋊C4—3342no—
124C4⋊C4—3342no—
125C4⋊C4—3341no—
126C4⋊C4—3341no—
127C4⋊C4—3341no—
128C4⋊C4—3341no—
129C4⋊C4—3341no—
130C4⋊C4—1671no—
131C4⋊C4D66811yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3342no—
134C42—3341no—
135C42C22×D16711yescharacteristic
136C23—21no—
137Q8—3341no—
138Q8—3341no—
139Q8—6682no—
140Q8—6682no—
141D4—21no—
142D4Dic66812yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—13362no—
147C2×C4—3341no—
148C2×C4—3341no—
149C2×C4—3342no—
150C2×C4—6681no—
151C2×C4D4×D16711yescharacteristic
152C2×C4D4×D16711yescharacteristic
153C2×C4C2×D66811yescharacteristic
154C8—22no—
155C8—41no—
156C8—6681no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1912880211yescharacteristic
161C4—21no—
162C4—41no—
163C4—13361no—
164C4—6681no—
165C4—6681no—
166C4—6682no—
167C4C2×Dic66811yescharacteristic
168C4#1912885111yescharacteristic
169C4#1912886111yescharacteristic
170C4#1912886511yescharacteristic
171C2—42no—
172C2#2492342711yescharacteristic
173C2#2492343211yescharacteristic
174C2#2492343811yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
FittingD4⋊C8×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenterD4⋊C8D167
hypocenterC167#1632
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. #24923649C2semidirect
3. #24923565C2semidirect
5. #24923017C2semidirect
6. #24923001C2semidirect
60. D4×C167Q16semidirect
87. C167#1632semidirect
142. D4Dic668semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249236492
#249235652
#249235592
#249230172
#249230012
D4⋊C8×C1672
#1632167