Finite Group Databasebeta

← #26373924

Subgroups — #26373924

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
1#20256363C211yesmaximal, characteristic, complemented
2Q8×D376C211yesmaximal, characteristic, complemented
3#20245612C213yesmaximal, complemented
4#20245526C213yesmaximal, complemented
5#20245424C213yesmaximal, complemented
6C4×D752C213yesmaximal, complemented
7Q8×C16×C47C211yesmaximal, characteristic, complemented
8#15689795—21no—
9Q8×D188—21no—
10C2×D752C2213yes—
11#15689528C2213yes—
12#15689500C2213yes—
13#15689404—23no—
14#15689319—23no—
15C4×D376C2213yes—
16C4×D376C2213yes—
17#15688949C2213yes—
18#15688925C2216yes—
19#15688925C2213yes—
20#15688925C2211yescharacteristic
21C8×Q8×C47C2211yescharacteristic
22C47×C4⋊C16C2213yes—
23C4×C16×C47C2213yes—
24C2×#10625354—41no—
25C2×Q8×D47—41no—
26C2×D376—23no—
27C2×D376C2313yes—
28C2×D376C2311yescharacteristic
29#13094024—23no—
30#13094021—23no—
31#13094004—43no—
32#13093999—43no—
33C4×D188—23no—
34C4×D188—23no—
35D752—26no—
36#13093958—23no—
37#13093958C2313yes—
38#13093958C2311yescharacteristic
39#13093946—23no—
40#13093946—23no—
41C4×Q8×C47D411yescharacteristic
42C2×C16×C47C2313yes—
43C2×C16×C47C2311yescharacteristic
44C4⋊C8×C47D413yes—
45C4×C8×C47C2313yes—
46C2×Q8×C47D811yescharacteristic
47#10625354—84no—
48Q8×D47—84no—
49C2×D188—21no—
50C2×D188—23no—
51C2×D188—43no—
52C2×D188—43no—
53C2×C4×D47—43no—
54C2×C4×D47—43no—
55C2×Dic94—43no—
56C2×C8×C47C2×D413yes—
57C2×C8×C47C2411yescharacteristic
58C4⋊C4×C47D813yes—
59C42×C47C2×D413yes—
60(C2×D47)⋊C4—43no—
61(C2×D47)⋊C4—43no—
62C4⋊Dic47—21no—
63C4⋊Dic47—23no—
64C4⋊Dic47—43no—
65D376—23no—
66D376—21no—
67D376—23no—
68D376—83no—
69D376#4112yescomplemented
70C16×C47—23no—
71C16×C47—21no—
72C22×D47—41no—
73C22×D47—43no—
74Q8×C47D1614yescomplemented
75C2×C4×C47C2×D813yes—
76C2×C4×C47C2×D813yes—
77C2×C4×C47C22×D411yescharacteristic
78C2×Dic47—43no—
79C2×Dic47—41no—
80D188—22no—
81D188—43no—
82D188—41no—
83D188—43no—
84D188—86no—
85D188—163no—
86C4×D47—86no—
87C4×D47—86no—
88Dic94—86no—
89C8×C47—23no—
90C8×C47C2×#4111yescharacteristic
91C8×C472−1+411yescharacteristic
92#33925—471nomaximal, Sylow
93C22×C47C22×D811yescharacteristic
94C2×D47—42no—
95C2×D47—81no—
96C2×D47—81no—
97C2×D47—83no—
98C2×D47—83no—
99C2×D47—83no—
100C4×C47—23no—
101C4×C47C2×D1616yes—
102C4×C47#54811yescharacteristic
103C4×C47#57711yescharacteristic
104Dic47—83no—
105Dic47—81no—
106#3391—471no—
107Q8×D8—471no—
108#2240—473no—
109#2224—473no—
110#2206—473no—
111Q8×C16D4711yescharacteristic, complemented
112C4×D16—473no—
113C2×C47#335811yescharacteristic
114C2×C47C22×D1611yescharacteristic
115C2×C47#341611yescharacteristic
116D47—82no—
117D47—163no—
118#550—941no—
119D4×Q8—941no—
120C2×D16—473no—
121#500—473no—
122#493—473no—
123#474—943no—
124#459—943no—
125C8×Q8C2×D4711yescharacteristic
126C4×D8—473no—
127C4×D8—473no—
128#366—473no—
129C4⋊C16C2×D4713yes—
130#357—471no—
131#357—473no—
132#357—476no—
133C4×C16C2×D4713yes—
134C47#3392511yescharacteristic, complemented, Sylow
135C2×#41—1881no—
136C22×Q8—1881no—
137C2×D8—471no—
138C2×D8—473no—
139C2×D8—943no—
140C4⋊Q8—943no—
141#127—943no—
142C22⋊Q8—1883no—
143#121—1883no—
144C4×Q8D18811yescharacteristic
145C4×D4—943no—
146C4×D4—943no—
147D16—946no—
148C2×C16C22×D4711yescharacteristic
149C2×C16C22×D4713yes—
150#107—473no—
151#107—471no—
152#107—943no—
153C4⋊C8D18813yes—
154#102—943no—
155#102—943no—
156C4×C8C22×D4713yes—
157#41—3764no—
158C2×Q8—3764no—
159C2×Q8—1883no—
160C2×Q8D37611yescharacteristic
161C2×D4—941no—
162C2×D4—943no—
163C2×D4—1883no—
164C2×D4—1883no—
165C22×C4—1883no—
166C22×C4—1883no—
167D8—472no—
168D8—943no—
169D8—941no—
170D8—943no—
171D8—3763no—
172C2×C8C2×D18813yes—
173C2×C8C23×D4711yescharacteristic
174C4⋊C4—941no—
175C4⋊C4—943no—
176C4⋊C4—1883no—
177C4⋊C4D37613yes—
178C22⋊C4—1883no—
179C22⋊C4—1883no—
180C42C2×D18813yes—
181C16—23no—
182C16—21no—
183C23—1883no—
184C23—1881no—
185Q8—3766no—
186Q8D75214yescomplemented
187D4—942no—
188D4—3766no—
189D4—1883no—
190D4—1883no—
191D4—1881no—
192D4—7523no—
193C2×C4—3766no—
194C2×C4—3766no—
195C2×C4—1881no—
196C2×C4—1883no—
197C2×C4C2×D37613yes—
198C2×C4C2×D37613yes—
199C2×C4C22×D18811yescharacteristic
200C8—23no—
201C8#41×D4711yescharacteristic
202C8#1309407211yescharacteristic
203C22—3761no—
204C22—3761no—
205C22—3763no—
206C22—3763no—
207C22—3763no—
208C22—1882no—
209C22C22×D37611yescharacteristic
210C4—23no—
211C4—3761no—
212C4—3763no—
213C4C2×D75216yes—
214C4#1568978511yescharacteristic
215C4#1568993511yescharacteristic
216C2—3762no—
217C2—7523no—
218C2#2025602411yescharacteristic
219C2C22×D75211yescharacteristic
220C2#2025653611yescharacteristic
subgroupG/H
centerC22C22×D376
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingQ8×C16×C47C2
socleC22×C47C22×D8
hypercenterQ8×C16D47
hypocenterC47#33925
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×C16

Complements

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

NH ≅ G/Ntype
1. #20256363C2semidirect
2. Q8×D376C2semidirect
3. #20245612C2semidirect
4. #20245526C2semidirect
5. #20245424C2semidirect
6. C4×D752C2semidirect
7. Q8×C16×C47C2semidirect
69. D376#41semidirect
74. Q8×C47D16semidirect
111. Q8×C16D47semidirect
134. C47#33925semidirect
186. Q8D752semidirect

Maximal subgroups

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

subgroupindex
#202563632
Q8×D3762
#202456122
#202455262
#202454242
C4×D7522
Q8×C16×C472
#3392547