Finite Group Databasebeta

← #27142898

Subgroups — #27142898

Order 12416.

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

normal subgroups only

This group has 246 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
2#20260745C212yesmaximal
3#20260640C211yesmaximal, characteristic
4#20260496C212yesmaximal
5#343×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7#15690809—22no—
8#15690801—22no—
9#15690789—22no—
10#15690769C2212yes—
11C2×C4×Dic97C2211yescharacteristic
12#15690735—22no—
13#15690735—22no—
14#15690717—22no—
15C2×M16×C97C2212yes—
16C2×C4⋊C4×C97C2211yescharacteristic
17C2×Dic194—22no—
18C2×Dic194—22no—
19C2×Dic194—21no—
20C2×Dic194—22no—
21C2×Dic194Q812yescomplemented
22C22×Dic97—21no—
23C22×Dic97—21no—
24C4⋊Dic97—22no—
25C4⋊Dic97D412yes—
26C4⋊Dic97D412yes—
27#13095490—24no—
28C22⋊Dic97—22no—
29C4×Dic97—22no—
30C4×Dic97—22no—
31C22×C4×C97D412yes—
32C22×C4×C97C2311yescharacteristic
33M16×C97—22no—
34M16×C97—22no—
35C2×C8×C97—22no—
36C4⋊C4×C97—22no—
37C23×C97C2×D411yescharacteristic
38C2×C4×C97—22no—
39C2×C4×C97—42no—
40C2×C4×C97C2×D412yes—
41C2×C4×C97C2×Q811yescharacteristic
42C2×C4×C97#4111yescharacteristic
43C2×C4×C97#4112yes—
44C2×Dic97—22no—
45C2×Dic97—22no—
46C2×Dic97—22no—
47C2×Dic97—22no—
48C2×Dic97—41no—
49C2×Dic97—41no—
50C2×Dic97—41no—
51C2×Dic97—41no—
52Dic194—24no—
53Dic194—42no—
54Dic194—82no—
55C8×C97—42no—
56C22×C97—21no—
57C22×C97—21no—
58C22×C97#12011yescharacteristic
59C22×C97C22⋊Q811yescharacteristic
60C22×C97#12411yescharacteristic
61C4×C97—42no—
62C4×C97C22⋊Q812yes—
63C4×C97#12412yes—
64Dic97—42no—
65Dic97—42no—
66Dic97—42no—
67Dic97—42no—
68C2×C97—21no—
69C2×C97—21no—
70C2×C97#39311yescharacteristic
71C2×C97#45511yescharacteristic
72C2×C97#45611yescharacteristic
73#2028—971nomaximal, Sylow
74C97#202811yescharacteristic, complemented, Sylow
75#544—971no—
76#419—972no—
77#391—971no—
78#343D9711yescharacteristic
79#339—972no—
80C22×Q8—971no—
81C2×M16C2×D9712yes—
82C4⋊Q8—1942no—
83#125—1942no—
84C22⋊Q8—1942no—
85#117—972no—
86C2×C4⋊C4C2×D9711yescharacteristic
87C2×C42—971no—
88#103—1942no—
89#103—1942no—
90#95—1942no—
91C2×Q8—972no—
92C2×Q8—1942no—
93C2×Q8—1942no—
94C2×Q8—1942no—
95C2×Q8—1941no—
96C22×C4—1941no—
97C22×C4—1941no—
98C22×C4D38812yes—
99C22×C4C22×D9711yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—972no—
105C4⋊C4—972no—
106C4⋊C4—1944no—
107C4⋊C4—1942no—
108C22⋊C4—1942no—
109C42—1942no—
110C42—1942no—
111C23C2×D38811yescharacteristic
112Q8—7762no—
113Q8—3882no—
114Q8—1944no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—3881no—
118C2×C4—3881no—
119C2×C4—3881no—
120C2×C4—3881no—
121C2×C4—1942no—
122C2×C4—1942no—
123C2×C4—1942no—
124C2×C4—1942no—
125C2×C4D4×D9712yes—
126C2×C4Q8×D9711yescharacteristic
127C2×C4#1309551011yescharacteristic
128C2×C4#1309551112yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#1569078011yescharacteristic
133C22#1569079011yescharacteristic
134C22#1569079811yescharacteristic
135C4—42no—
136C4—3882no—
137C4—3882no—
138C4—3882no—
139C4—3882no—
140C4#1569078812yes—
141C4#1569079612yes—
142C2—21no—
143C2—21no—
144C2#2026064811yescharacteristic
145C2#2026091511yescharacteristic
146C2#2026092011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#343×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#343D97
hypocenterC97#2028
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
21. C2×Dic194Q8semidirect
74. C97#2028semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202607452
#202606402
#202604962
#343×C972
#202897