Finite Group Databasebeta

← #37571827

Subgroups — #37571827

Order 24448.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#27133405C211yesmaximal, characteristic
2#27133058C211yesmaximal, characteristic, complemented
3#27133023C212yesmaximal, complemented
4#27132402C212yesmaximal, complemented
5#329×C191C211yesmaximal, characteristic, complemented
6C2×#15690277—21no—
7C2×#15690275—21no—
8#20259322C2211yescharacteristic, complemented
9#20259311—21no—
10#20259310C2212yes—
11#20259308—22no—
12#20259292—22no—
13#20259290—21no—
14#20259247—22no—
15#20259244—22no—
16#20259233C2211yescharacteristic, complemented
17#124×C191C2211yescharacteristic
18#97×C191C2212yescomplemented
19C23×D191—21no—
20C2×Dic382—21no—
21C2×Dic382D412yes—
22C2×D764—21no—
23C2×D764D412yes—
24C2×C22⋊D191—22no—
25#15690277—42no—
26#15690275—42no—
27C2×C191⋊C8—21no—
28C4⋊Dic191—22no—
29C22⋊Dic191—21no—
30(C2×D191)⋊C4—22no—
31(C2×D191)⋊C4—22no—
32(C2×D191)⋊C4—22no—
33C4×Dic191—21no—
34C2×Q8×C191D411yescharacteristic
35C2×D4×C191D411yescharacteristic
36C2×C8×C191—22no—
37C22⋊C4×C191—22no—
38C42×C191C2311yescharacteristic
39C22×D191—22no—
40C22×D191—81no—
41C22×D191—82no—
42Dic382—42no—
43Dic382—41no—
44Dic382—41no—
45D764—42no—
46D764—81no—
47C22⋊D191—82no—
48C2×Dic191—22no—
49C2×Dic191—41no—
50C191⋊C8—22no—
51C23×C191—21no—
52Q8×C191—41no—
53D4×C191—41no—
54C2×C4×C191—21no—
55C2×C4×C191C2×D412yes—
56C2×C4×C191C2×D411yescharacteristic
57C8×C191—42no—
58C22×C191—41no—
59C22×C191—42no—
60C22×C191#12011yescharacteristic
61C2×D191—42no—
62C2×D191—42no—
63C2×D191—42no—
64C2×D191—84no—
65C4×C191—21no—
66C4×C191—22no—
67C4×C191—41no—
68Dic191—42no—
69Dic191—81no—
70C2×C191—81no—
71C2×C191#45311yescharacteristic
72C2×C191#45512yes—
73D191—82no—
74C191#165911yescharacteristic, complemented, Sylow
75#1659—1911nomaximal, Sylow
76H(3,𝔽4)—1911no—
77#497—1911no—
78#489—1912no—
79#329—1912no—
80#329D19111yescharacteristic, complemented
81C2×Q16—3821no—
82C2×SD16—3821no—
83C4⋊Q8—1911no—
84#124—3822no—
85#124—3821no—
86#124—1912no—
87#124C2×D19111yescharacteristic
88#120—3822no—
89#120—3821no—
90#103—3822no—
91#102—3822no—
92#97—1911no—
93#97C2×D19112yescomplemented
94C24—3821no—
95C2×Q8—3821no—
96C2×Q8—1912no—
97C2×Q8D76411yescharacteristic
98C2×D4—3821no—
99C2×D4—3822no—
100C2×D4—1912no—
101C2×D4D76411yescharacteristic
102Q16—7642no—
103SD16—7642no—
104C2×C8—22no—
105C2×C8—3821no—
106C4⋊C4—3822no—
107C22⋊C4—22no—
108C22⋊C4—3822no—
109C22⋊C4—3821no—
110C22⋊C4—3822no—
111C22⋊C4—3822no—
112C42—3821no—
113C42C22×D19111yescharacteristic
114C23—21no—
115C23—3822no—
116C23—15281no—
117C23—15282no—
118Q8—41no—
119Q8—7641no—
120Q8—7641no—
121Q8—7642no—
122D4—41no—
123D4—15281no—
124D4—15282no—
125D4—7642no—
126C2×C4—21no—
127C2×C4—3822no—
128C2×C4—7641no—
129C2×C4D4×D19112yes—
130C2×C4C2×D76411yescharacteristic
131C8—42no—
132C8—3822no—
133C22—41no—
134C22—42no—
135C22—15284no—
136C22—7642no—
137C22—7642no—
138C22—7642no—
139C22#2025929211yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—15281no—
144C4—7642no—
145C2—81no—
146C2—15282no—
147C2#2713283311yescharacteristic
148C2#2713284212yes—
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#329×C191C2
socleC22×C191#120
prime radical#27133405C2
hypercenter#329D191
hypocenterC191#1659
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27133058C2semidirect
3. #27133023C2semidirect
4. #27132402C2semidirect
5. #329×C191C2semidirect
8. #20259322C22semidirect
16. #20259233C22semidirect
18. #97×C191C22semidirect
74. C191#1659semidirect
80. #329D191semidirect
93. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334052
#271330582
#271330232
#271324022
#329×C1912
#1659191