Finite Group Databasebeta

← #27143757

Subgroups — #27143757

Order 12416.

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

normal subgroups only

This group has 216 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#20260917C212yesmaximal, complemented
2#20260907C211yesmaximal, characteristic
3#20260534C212yesmaximal, complemented
4#20260532C211yesmaximal, characteristic, complemented
5#563×C97C211yesmaximal, characteristic, complemented
6#15690858C2211yescharacteristic
7#15690841—22no—
8#15690836—21no—
9#15690738—21no—
10#15690738—22no—
11#15690729C2211yescharacteristic
12#15690729C2212yes—
13C4⋊Q8×C97C2212yes—
14#126×C97—22no—
15#125×C97—21no—
16#124×C97C2211yescharacteristic
17#123×C97—21no—
18C22⋊Q8×C97—22no—
19#13095512—41no—
20#13095511—21no—
21#13095511—22no—
22Q8×D97—41no—
23#13095501—42no—
24#13095499—41no—
25#13095498—42no—
26#13095496—41no—
27#13095495—21no—
28#13095495—22no—
29C2×Q8×C97D412yes—
30C2×Q8×C97C2311yescharacteristic
31C2×D4×C97D411yescharacteristic
32C22×C4×C97—21no—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97—21no—
36C4⋊C4×C97—22no—
37C4⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—21no—
40C22⋊C4×C97—22no—
41C42×C97D411yescharacteristic
42C42×C97D412yes—
43C23×C97—21no—
44Q8×C97—41no—
45Q8×C97—42no—
46D4×C97—41no—
47C2×C4×C97—21no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—81no—
51C2×C4×C97C2×D412yes—
52C2×C4×C97C2×D411yescharacteristic
53C22⋊D97—41no—
54D388—21no—
55D388—22no—
56C4×D97—41no—
57C4×D97—42no—
58Dic194—21no—
59Dic194—22no—
60#11715895—41no—
61#11715895—42no—
62C22×C97—42no—
63C22×C97—41no—
64C22×C97#12011yescharacteristic
65C2×D97—41no—
66C4×C97—21no—
67C4×C97—22no—
68C4×C97—41no—
69C4×C97—42no—
70C4×C97—42no—
71Dic97—41no—
72C2×C97—21no—
73C2×C97—81no—
74C2×C97B(4,𝔽2)11yescharacteristic
75D97—81no—
76#2203—971nomaximal, Sylow
77C97#220311yescharacteristic, complemented, Sylow
78#563D9711yescharacteristic, complemented
79#456—972no—
80#454—971no—
81#356—972no—
82#355—971no—
832−1+4—971no—
84#137—1942no—
85AGL(1,ℤ/8)—1941no—
86C4⋊Q8C2×D9712yes—
87#126—22no—
88#125—21no—
89#124C2×D9711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—1941no—
93#104—1942no—
94#101—972no—
95#101—971no—
96#41—3881no—
97#41—1941no—
98#41—1942no—
99C2×Q8—3881no—
100C2×Q8C22⋊D9712yes—
101C2×Q8C22×D9711yescharacteristic
102C2×D4C22⋊D9711yescharacteristic
103C22×C4—21no—
104Q16—3882no—
105SD16—3881no—
106SD16—3882no—
107D8—3881no—
108M16—1941no—
109M16—1942no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D9711yescharacteristic
119C42C22⋊D9712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—1941no—
124Q8—1942no—
125D4—41no—
126D4—3881no—
127D4—1941no—
128D4—1942no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—3881no—
134C2×C4—3882no—
135C2×C4C2×C22⋊D9712yes—
136C2×C4C2×C22⋊D9711yescharacteristic
137C8—3881no—
138C8—3882no—
139C22—42no—
140C22—41no—
141C22—3881no—
142C22#1569077911yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—3881no—
149C2—21no—
150C2—81no—
151C2—7761no—
152C2#2026092211yescharacteristic
subgroupG/H
centerC2#20260922
commutator, cosocleC2×Q8×C97C23
FrattiniC2×Q8C22×D97
Fitting#563×C97C2
socleC2×C97B(4,𝔽2)
prime radical#20260907C2
hypercenter#563D97
hypocenterC97#2203
LCSG ▷ C2×Q8×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20260917C2semidirect
3. #20260534C2semidirect
4. #20260532C2semidirect
5. #563×C97C2semidirect
77. C97#2203semidirect
78. #563D97semidirect

Maximal subgroups

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

subgroupindex
#202609172
#202609072
#202605342
#202605322
#563×C972
#220397