Finite Group Databasebeta

← #27142893

Subgroups — #27142893

Order 12416.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261366C211yesmaximal, characteristic, complemented
2#20260741C212yesmaximal
3#20260636C211yesmaximal, characteristic
4#20260496C212yesmaximal, complemented
5#343×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7#15690809—22no—
8#15690801—22no—
9#15690789—22no—
10#15690769C2212yes—
11C2×#13095488—22no—
12C2×C4×Dic97C2211yescharacteristic
13#15690732—22no—
14#15690732—22no—
15C2×M16×C97C2212yes—
16C2×C4⋊C4×C97C2211yescharacteristic
17C23×D97—21no—
18C2×Dic194—22no—
19C2×D388—22no—
20C2×D388—22no—
21C2×D388—21no—
22C2×D388Q812yescomplemented
23C22×Dic97—21no—
24C4⋊Dic97—22no—
25C4⋊Dic97D412yes—
26C4⋊Dic97D412yes—
27#13095490—24no—
28C22⋊Dic97—22no—
29#13095488—44no—
30C4×Dic97—22no—
31C4×Dic97—22no—
32C22×C4×C97D412yes—
33C22×C4×C97C2311yescharacteristic
34M16×C97—22no—
35M16×C97—22no—
36C2×C8×C97—22no—
37C4⋊C4×C97—22no—
38C23×C97C2×D411yescharacteristic
39C22×D97—22no—
40C22×D97—42no—
41C22×D97—42no—
42C22×D97—41no—
43C22×D97—41no—
44C2×C4×C97—22no—
45C2×C4×C97—42no—
46C2×C4×C97C2×D412yes—
47C2×C4×C97C2×Q811yescharacteristic
48C2×C4×C97#4111yescharacteristic
49C2×C4×C97#4112yes—
50C2×Dic97—22no—
51C2×Dic97—22no—
52C2×Dic97—22no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55D388—24no—
56D388—42no—
57Dic194—82no—
58C8×C97—42no—
59C22×C97—21no—
60C22×C97—21no—
61C22×C97#12011yescharacteristic
62C22×C97C22⋊Q811yescharacteristic
63C22×C97#12411yescharacteristic
64C2×D97—44no—
65C2×D97—42no—
66C2×D97—82no—
67C2×D97—81no—
68C2×D97—81no—
69C4×C97—42no—
70C4×C97C22⋊Q812yes—
71C4×C97#12412yes—
72Dic97—42no—
73Dic97—42no—
74Dic97—42no—
75C2×C97—21no—
76C2×C97—21no—
77C2×C97#39311yescharacteristic
78C2×C97#45311yescharacteristic
79C2×C97#45411yescharacteristic
80D97—82no—
81#2027—971nomaximal, Sylow
82C97#202711yescharacteristic, complemented, Sylow
83#544—971no—
84#418—972no—
85#390—971no—
86#343D9711yescharacteristic, complemented
87#339—972no—
88C22×D4—971no—
89C2×M16C2×D9712yes—
90C4⋊Q8—1942no—
91#125—1942no—
92C22⋊Q8—1942no—
93#117—972no—
94C2×C4⋊C4C2×D9711yescharacteristic
95C2×C22⋊C4—1942no—
96C2×C42—971no—
97#102—1942no—
98#102—1942no—
99C24—1941no—
100C2×Q8—1942no—
101C2×D4—972no—
102C2×D4—1942no—
103C2×D4—1941no—
104C2×D4—1942no—
105C22×C4—1941no—
106C22×C4D38812yes—
107C22×C4C22×D9711yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—972no—
113C4⋊C4—972no—
114C4⋊C4—1944no—
115C4⋊C4—1942no—
116C22⋊C4—3884no—
117C22⋊C4—1942no—
118C42—1942no—
119C42—1942no—
120C23—3882no—
121C23—3882no—
122C23—3881no—
123C23—3881no—
124C23—1942no—
125C23C2×D38811yescharacteristic
126Q8—7762no—
127D4—3882no—
128D4—1944no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—3881no—
132C2×C4—3881no—
133C2×C4—1942no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4D4×D9712yes—
137C2×C4Q8×D9711yescharacteristic
138C2×C4#1309551011yescharacteristic
139C2×C4#1309551112yes—
140C8—42no—
141C22—21no—
142C22—21no—
143C22—7762no—
144C22—7761no—
145C22—7761no—
146C22—3884no—
147C22—3882no—
148C22#1569078011yescharacteristic
149C22#1569079011yescharacteristic
150C22#1569079811yescharacteristic
151C4—42no—
152C4—3882no—
153C4—3882no—
154C4—3882no—
155C4#1569078812yes—
156C4#1569079612yes—
157C2—21no—
158C2—21no—
159C2—7762no—
160C2#2026064811yescharacteristic
161C2#2026090511yescharacteristic
162C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#343×C97C2
socleC22×C97#120
prime radicalC22×D388C22
hypercenter#343D97
hypocenterC97#2027
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. #20261366C2semidirect
4. #20260496C2semidirect
5. #343×C97C2semidirect
22. C2×D388Q8semidirect
82. C97#2027semidirect
86. #343D97semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202607412
#202606362
#202604962
#343×C972
#202797