Finite Group Databasebeta

← #36323355

Subgroups — #36323355

Order 21376.

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

normal subgroups only

This group has 208 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#24923620C211yesmaximal, characteristic, complemented
3#24923549C211yesmaximal, characteristic
4#24923480C211yesmaximal, characteristic, complemented
5#24923017C211yesmaximal, characteristic, complemented
6#24923001C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic
8C2×Dic668—21no—
9#19128832C2211yescharacteristic
10#19128832C2211yescharacteristic
11#19128823—21no—
12#19128811—21no—
13#19128809—22no—
14C4×Dic334C2211yescharacteristic
15D4×Dic167—21no—
16C2×C4⋊Dic167—21no—
17#19128765—21no—
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128757—22no—
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—42no—
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—41no—
52Dic334—41no—
53C2×Dic167—21no—
54C2×Dic167—21no—
55C2×Dic167—22no—
56C2×Dic167—41no—
57C2×Dic167—82no—
58C167⋊C8—41no—
59C23×C167—21no—
60D4×C167—21no—
61D4×C167Q1612yescomplemented
62C2×C4×C167—21no—
63C2×C4×C167—41no—
64C2×C4×C167—41no—
65C2×C4×C167C2×D411yescharacteristic
66C2×C4×C167C2×D411yescharacteristic
67C2×C4×C167C2×D411yescharacteristic
68C8×C167—41no—
69C8×C167—41no—
70C22×C167—22no—
71C22×C167—41no—
72C22×C167—41no—
73C22×C167#12011yescharacteristic
74C4×C167—21no—
75C4×C167—41no—
76C4×C167C2×Q1611yescharacteristic
77C4×C167#13511yescharacteristic
78C4×C167AGL(1,ℤ/8)11yescharacteristic
79C4×C167#13711yescharacteristic
80Dic167—41no—
81Dic167—41no—
82Dic167—42no—
83Dic167—81no—
84C2×C167—42no—
85C2×C167#44911yescharacteristic
86C2×C167#45111yescharacteristic
87C2×C167#45511yescharacteristic
88C167#164411yescharacteristic, complemented, Sylow
89#1644—1671nomaximal, Sylow
90#554—1671no—
91#487—1671no—
92#474—1671no—
93#462—1671no—
94#333—1671no—
95Q8⋊C8—1671no—
96D4⋊C8D16711yescharacteristic
97C2×Q16—3341no—
98C4⋊Q8—1671no—
99C4⋊Q8—1671no—
100#125—3341no—
101C22⋊Q8—3342no—
102C22⋊Q8—3341no—
103C4×Q8—1671no—
104C4×D4—3341no—
105C4×D4C2×D16711yescharacteristic
106C2×C4⋊C4—3341no—
107#107—3341no—
108C4⋊C8—1671no—
109C4⋊C8C2×D16711yescharacteristic
110#103—3342no—
111#103—3341no—
112#102—3341no—
113C4×C8C2×D16711yescharacteristic
114C2×Q8—3341no—
115C2×Q8—1671no—
116C2×Q8—1672no—
117C2×D4D66811yescharacteristic
118C22×C4—21no—
119C22×C4—3342no—
120Q16—6682no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3341no—
124C4⋊C4—3342no—
125C4⋊C4—3342no—
126C4⋊C4—3341no—
127C4⋊C4—3341no—
128C4⋊C4—3341no—
129C4⋊C4—3341no—
130C4⋊C4—3341no—
131C4⋊C4—1671no—
132C4⋊C4D66811yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—3342no—
135C42—3341no—
136C42C22×D16711yescharacteristic
137C23—21no—
138Q8—3341no—
139Q8—3341no—
140Q8—6682no—
141Q8—6681no—
142Q8—6681no—
143D4—21no—
144D4Dic66812yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—13362no—
149C2×C4—3341no—
150C2×C4—3341no—
151C2×C4—3342no—
152C2×C4—6681no—
153C2×C4D4×D16711yescharacteristic
154C2×C4D4×D16711yescharacteristic
155C2×C4C2×D66811yescharacteristic
156C8—41no—
157C8—41no—
158C8—6681no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1912880211yescharacteristic
163C4—21no—
164C4—41no—
165C4—13361no—
166C4—6681no—
167C4—6681no—
168C4—6682no—
169C4C2×Dic66811yescharacteristic
170C4#1912885311yescharacteristic
171C4#1912885611yescharacteristic
172C4#1912886511yescharacteristic
173C2—42no—
174C2#2492341511yescharacteristic
175C2#2492342711yescharacteristic
176C2#2492344811yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
FittingD4⋊C8×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenterD4⋊C8D167
hypocenterC167#1644
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
4. #24923480C2semidirect
5. #24923017C2semidirect
6. #24923001C2semidirect
61. D4×C167Q16semidirect
88. C167#1644semidirect
144. D4Dic668semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249236202
#249235492
#249234802
#249230172
#249230012
D4⋊C8×C1672
#1644167