Finite Group Databasebeta

← #26373919

Subgroups — #26373919

Order 12032.

562 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×D376C211yesmaximal, characteristic, complemented
2#20245611C213yesmaximal, complemented
3#20245525C213yesmaximal, complemented
4#20245420C213yesmaximal, complemented
5Q8×C47⋊C16C211yesmaximal, characteristic, complemented
6C4×#13093965C213yesmaximal, complemented
7C47×#3391C211yesmaximal, characteristic, complemented
8Q8×D188—21no—
9C2×#13093965C2213yes—
10#15689528C2213yes—
11#15689404—23no—
12C4×D376C2213yes—
13#15688948C2213yes—
14C4⋊(C47⋊C16)C2213yes—
15#15688924C2216yes—
16#15688924C2213yes—
17#15688923C2211yescharacteristic
18C4×C47⋊C16C2213yes—
19C47×#550—21no—
20C47×#493C2213yes—
21C47×#459—23no—
22C8×Q8×C47C2211yescharacteristic
23C4×D8×C47C2213yes—
24C2×Q8×D47—41no—
25C2×D376C2311yescharacteristic
26#13094024—23no—
27#13094004—43no—
28C4×D188—23no—
29#13093965—26no—
30C2×C47⋊C16C2313yes—
31C2×C47⋊C16C2311yescharacteristic
32#13093958—23no—
33#13093958C2313yes—
34#13093946—23no—
35C2×#41×C47—41no—
36C2×D8×C47—23no—
37C2×D8×C47C2313yes—
38#127×C47—23no—
39#121×C47—43no—
40C4×Q8×C47D411yescharacteristic
41C4×D4×C47—23no—
42#107×C47C2311yescharacteristic
43C4⋊C8×C47D413yes—
44#102×C47—23no—
45C4×C8×C47C2313yes—
46#41×C47—84no—
47C2×Q8×C47D811yescharacteristic
48C2×D4×C47—23no—
49C2×D4×C47—43no—
50C2×D4×C47—43no—
51C22×C4×C47—43no—
52Q8×D47—84no—
53C2×D188—21no—
54C2×C4×D47—43no—
55C2×Dic94—43no—
56D8×C47—23no—
57D8×C47—23no—
58D8×C47—83no—
59C2×C8×C47C2×D413yes—
60C2×C8×C47C2411yescharacteristic
61C4⋊C4×C47—21no—
62C4⋊C4×C47D813yes—
63C22⋊C4×C47—43no—
64C42×C47C2×D413yes—
65(C2×D47)⋊C4—43no—
66C4⋊Dic47—23no—
67C4⋊Dic47—43no—
68D376—21no—
69D376#4112yescomplemented
70C47⋊C16—23no—
71C47⋊C16—21no—
72C23×C47—43no—
73C22×D47—41no—
74Q8×C47D1614yescomplemented
75D4×C47—43no—
76D4×C47—43no—
77D4×C47—86no—
78D4×C47—163no—
79C2×C4×C47—41no—
80C2×C4×C47—86no—
81C2×C4×C47C2×D813yes—
82C2×C4×C47C2×D813yes—
83C2×C4×C47C22×D411yescharacteristic
84C2×Dic47—43no—
85D188—22no—
86D188—41no—
87C4×D47—86no—
88Dic94—86no—
89C8×C47—23no—
90C8×C47C2×#4111yescharacteristic
91C8×C472−1+411yescharacteristic
92#33925—471nomaximal, Sylow
93C22×C47—83no—
94C22×C47—83no—
95C22×C47—83no—
96C22×C47C22×D811yescharacteristic
97C2×D47—42no—
98C2×D47—81no—
99C2×D47—81no—
100C4×C47—23no—
101C4×C47—81no—
102C4×C47C2×D1616yes—
103C4×C47#54811yescharacteristic
104C4×C47#57711yescharacteristic
105Dic47—83no—
106#3391D4711yescharacteristic, complemented
107Q8×D8—471no—
108#2240—473no—
109#2224—473no—
110#2206—473no—
111Q8×C16—471no—
112C4×D16—473no—
113C2×C47—163no—
114C2×C47#335811yescharacteristic
115C2×C47C22×D1611yescharacteristic
116C2×C47#341611yescharacteristic
117D47—82no—
118#550—21no—
119D4×Q8—941no—
120C2×D16—473no—
121#500—473no—
122#493C2×D4713yes—
123#474—943no—
124#459—23no—
125C8×Q8C2×D4711yescharacteristic
126C4×D8—473no—
127C4×D8C2×D4713yes—
128#366—473no—
129C4⋊C16—473no—
130#357—473no—
131#357—476no—
132#357—471no—
133C4×C16—473no—
134C47#3392511yescharacteristic, complemented, Sylow
135C2×#41—41no—
136C22×Q8—1881no—
137C2×D8—23no—
138C2×D8—471no—
139C2×D8C22×D4713yes—
140C4⋊Q8—943no—
141#127—23no—
142C22⋊Q8—1883no—
143#121—43no—
144C4×Q8C22⋊D4711yescharacteristic
145C4×D4—23no—
146C4×D4—943no—
147D16—946no—
148C2×C16—471no—
149C2×C16—473no—
150#107—473no—
151#107—943no—
152#107C22×D4711yescharacteristic
153C4⋊C8C22⋊D4713yes—
154#102—23no—
155#102—943no—
156C4×C8C22×D4713yes—
157#41—84no—
158C2×Q8—3764no—
159C2×Q8—1883no—
160C2×Q8#1062533411yescharacteristic
161C2×D4—23no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—941no—
165C22×C4—43no—
166C22×C4—1883no—
167D8—23no—
168D8—23no—
169D8—83no—
170D8—472no—
171D8—941no—
172C2×C8C2×C22⋊D4713yes—
173C2×C8C23×D4711yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—943no—
176C4⋊C4—1883no—
177C4⋊C4#1062533413yes—
178C22⋊C4—43no—
179C22⋊C4—1883no—
180C42C2×C22⋊D4713yes—
181C16—943no—
182C16—941no—
183C23—43no—
184C23—1881no—
185Q8—3766no—
186Q8#1309396514yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—163no—
191D4—942no—
192D4—1881no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—3766no—
196C2×C4—1883no—
197C2×C4C2×#1062533413yes—
198C2×C4C2×#1062533413yes—
199C2×C4C22×C22⋊D4711yescharacteristic
200C8—23no—
201C8#41×D4711yescharacteristic
202C8#1309407111yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—3761no—
207C22—3761no—
208C22—1882no—
209C22C22×#1062533411yescharacteristic
210C4—23no—
211C4—81no—
212C4—3763no—
213C4C2×#1309396516yes—
214C4#1568978011yescharacteristic
215C4#1568993411yescharacteristic
216C2—163no—
217C2—3762no—
218C2#2025601911yescharacteristic
219C2C22×#1309396511yescharacteristic
220C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3391C2
socleC22×C47C22×D8
hypercenter#3391D47
hypocenterC47#33925
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×D376C2semidirect
2. #20245611C2semidirect
3. #20245525C2semidirect
4. #20245420C2semidirect
5. Q8×C47⋊C16C2semidirect
6. C4×#13093965C2semidirect
7. C47×#3391C2semidirect
69. D376#41semidirect
74. Q8×C47D16semidirect
106. #3391D47semidirect
134. C47#33925semidirect
186. Q8#13093965semidirect

Maximal subgroups

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

subgroupindex
Q8×D3762
#202456112
#202455252
#202454202
Q8×C47⋊C162
C4×#130939652
C47×#33912
#3392547