Finite Group Databasebeta

← #37532973

Subgroups — #37532973

Order 23168.

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×Dic362C211yesmaximal, characteristic
2#26041684C211yesmaximal, characteristic, complemented
3#26041559C211yesmaximal, characteristic
4#26041490C211yesmaximal, characteristic, complemented
5#26041026C211yesmaximal, characteristic, complemented
6#26041011C211yesmaximal, characteristic, complemented
7D4⋊C8×C181C211yesmaximal, characteristic
8C22×Dic362—21no—
9C2×Dic724—21no—
10#19153816C2211yescharacteristic
11#19153816C2211yescharacteristic
12#19153815—21no—
13#19153795—21no—
14#19153793—22no—
15C4×Dic362C2211yescharacteristic
16D4×Dic181—21no—
17#19153750—22no—
18#19153749—21no—
19C4⋊#13089678C2211yescharacteristic
20#19153741—21no—
21#19153737—21no—
22C4×D4×C181C2211yescharacteristic
23C4⋊C8×C181C2211yescharacteristic
24C4×C8×C181C2211yescharacteristic
25C2×Dic362—22no—
26C2×Dic362—21no—
27C2×Dic362—21no—
28C2×Dic362—42no—
29C2×Dic362—42no—
30C2×Dic362D411yescharacteristic
31C22×Dic181—22no—
32Dic724—42no—
33C2×#13089678—21no—
34C4⋊Dic181—21no—
35C4⋊Dic181—21no—
36C4⋊Dic181—21no—
37C4⋊Dic181D411yescharacteristic
38C4⋊Dic181D412yes—
39#15668428—22no—
40#15668428—21no—
41C22⋊Dic181—22no—
42C4×Dic181—21no—
43C2×D4×C181D411yescharacteristic
44C22×C4×C181—21no—
45C2×C8×C181—21no—
46C2×C8×C181—21no—
47C4⋊C4×C181D411yescharacteristic
48C22⋊C4×C181—21no—
49C42×C181C2311yescharacteristic
50Dic362—21no—
51Dic362—44no—
52Dic362—41no—
53Dic362—41no—
54Dic362—41no—
55Dic362—41no—
56Dic362D812yes—
57C2×Dic181—21no—
58C2×Dic181—21no—
59C2×Dic181—22no—
60C2×Dic181—44no—
61C2×Dic181—41no—
62#13089678—41no—
63C23×C181—21no—
64D4×C181—21no—
65D4×C181Q1612yescomplemented
66C2×C4×C181—21no—
67C2×C4×C181—41no—
68C2×C4×C181—41no—
69C2×C4×C181C2×D411yescharacteristic
70C2×C4×C181C2×D411yescharacteristic
71C2×C4×C181C2×D411yescharacteristic
72C8×C181—22no—
73C8×C181—41no—
74C22×C181—22no—
75C22×C181—41no—
76C22×C181—41no—
77C22×C181#12011yescharacteristic
78C4×C181—21no—
79C4×C181—41no—
80C4×C181C2×D811yescharacteristic
81C4×C181C2×Q1611yescharacteristic
82C4×C181AGL(1,ℤ/8)11yescharacteristic
83C4×C181#13711yescharacteristic
84Dic181—22no—
85Dic181—42no—
86Dic181—41no—
87Dic181—81no—
88C2×C181—42no—
89C2×C181#44711yescharacteristic
90C2×C181#45111yescharacteristic
91C2×C181#45611yescharacteristic
92C181#164911yescharacteristic, complemented, Sylow
93#1649—1811nomaximal, Sylow
94D4×Q8—1811no—
95#500—1811no—
96#474—1811no—
97#462—1811no—
98#332—1811no—
99Q8⋊C8—1811no—
100D4⋊C8D18111yescharacteristic
101C22×Q8—3621no—
102C2×Q16—3621no—
103C4⋊Q8—3621no—
104C4⋊Q8—1811no—
105C4⋊Q8—1811no—
106C22⋊Q8—3622no—
107C22⋊Q8—3621no—
108C4×Q8—1811no—
109C4×D4—3621no—
110C4×D4C2×D18111yescharacteristic
111#107—3622no—
112#107—3621no—
113C4⋊C8—1811no—
114C4⋊C8C2×D18111yescharacteristic
115#103—3621no—
116#102—3621no—
117C4×C8C2×D18111yescharacteristic
118C2×Q8—3622no—
119C2×Q8—3621no—
120C2×Q8—3621no—
121C2×Q8—1811no—
122C2×Q8—7242no—
123C2×Q8—7242no—
124C2×D4D72411yescharacteristic
125C22×C4—21no—
126C22×C4—3622no—
127Q16—7242no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3621no—
131C4⋊C4—3622no—
132C4⋊C4—3621no—
133C4⋊C4—3621no—
134C4⋊C4—3621no—
135C4⋊C4—3621no—
136C4⋊C4—1811no—
137C4⋊C4—1812no—
138C4⋊C4D72411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3622no—
141C42—3621no—
142C42C22×D18111yescharacteristic
143C23—21no—
144Q8—3621no—
145Q8—1812no—
146Q8—7244no—
147Q8—7241no—
148Q8—7241no—
149Q8—7241no—
150Q8—7241no—
151D4—21no—
152D4Dic72412yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3621no—
157C2×C4—3621no—
158C2×C4—3622no—
159C2×C4—7244no—
160C2×C4—7241no—
161C2×C4D4×D18111yescharacteristic
162C2×C4D4×D18111yescharacteristic
163C2×C4C2×D72411yescharacteristic
164C8—22no—
165C8—41no—
166C8—7241no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1915378611yescharacteristic
171C4—21no—
172C4—41no—
173C4—3622no—
174C4—14481no—
175C4—7242no—
176C4—7241no—
177C4D8×D18111yescharacteristic
178C4C2×Dic72411yescharacteristic
179C4#1915384311yescharacteristic
180C4#1915384911yescharacteristic
181C2—42no—
182C2#2604141411yescharacteristic
183C2#2604143711yescharacteristic
184C2#2604146311yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
FittingD4⋊C8×C181C2
socleC22×C181#120
prime radicalC2×D4×C181D4
hypercenterD4⋊C8D181
hypocenterC181#1649
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #26041684C2semidirect
4. #26041490C2semidirect
5. #26041026C2semidirect
6. #26041011C2semidirect
65. D4×C181Q16semidirect
92. C181#1649semidirect
152. D4Dic724semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3622
#260416842
#260415592
#260414902
#260410262
#260410112
D4⋊C8×C1812
#1649181