Finite Group Databasebeta

← #27142258

Subgroups — #27142258

Order 12416.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261306C211yesmaximal, characteristic
2#20260745C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic
4#20260492C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7#15690808C412yescomplemented
8#15690797C412yescomplemented
9#15690791—22no—
10#15690789—22no—
11#15690769C2212yes—
12#15690769C2211yescharacteristic
13#15690735—24no—
14#15690725—22no—
15C2×#41×C97C2211yescharacteristic
16#131×C97—24no—
17C2×M16×C97—22no—
18C2×M16×C97C2212yes—
19C2×Dic194—22no—
20C2×Dic194—22no—
21C2×Dic194—21no—
22C2×Dic194C2×C412yes—
23C22×Dic97—21no—
24C4⋊Dic97—21no—
25C4⋊Dic97—22no—
26#13095490—24no—
27C22⋊Dic97—24no—
28C22⋊Dic97—21no—
29C22⋊Dic97—22no—
30C4×Dic97—22no—
31C4×Dic97C2×C412yes—
32#41×C97—22no—
33#41×C97—22no—
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97C2×C411yescharacteristic
36C2×D4×C97D412yes—
37C22×C4×C97D412yes—
38C22×C4×C97C2311yescharacteristic
39M16×C97—24no—
40M16×C97—22no—
41M16×C97—22no—
42C2×C8×C97—24no—
43C2×C8×C97D414yes—
44C23×C97C22⋊C412yes—
45C23×C97C2×D411yescharacteristic
46Q8×C97—22no—
47D4×C97—24no—
48D4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97C22⋊C412yes—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D412yes—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97#4112yes—
56C2×Dic97—22no—
57C2×Dic97—22no—
58C2×Dic97—42no—
59C2×Dic97—42no—
60Dic194—24no—
61Dic194—41no—
62Dic194—41no—
63C8×C97—24no—
64C22×C97—22no—
65C22×C97—41no—
66C22×C97—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#12112yes—
69C4×C97—22no—
70C4×C97C4×D412yes—
71C4×C97#12011yescharacteristic
72C4×C97#12311yescharacteristic
73Dic97—42no—
74Dic97—42no—
75Dic97—82no—
76C2×C97—21no—
77C2×C97—22no—
78C2×C97—42no—
79C2×C97#38611yescharacteristic
80#1909—971nomaximal, Sylow
81C97#190911yescharacteristic, complemented, Sylow
82#568D9711yescharacteristic
83#536—971no—
84#419—972no—
85#410—971no—
86H(3,ℤ/4)—972no—
87C2×#41C2×D9711yescharacteristic
88C22×Q8—971no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D9712yes—
92C4⋊Q8—972no—
93#124—972no—
94#123—1942no—
95C22⋊Q8—1942no—
96#117—971no—
97#117—972no—
98#103—1944no—
99#99—1942no—
100#41—22no—
101#41—22no—
102C2×Q8—972no—
103C2×Q8—1942no—
104C2×Q8—1942no—
105C2×Q8—1941no—
106C2×Q8C4×D9711yescharacteristic
107C2×D4C4×D9711yescharacteristic
108C2×D4D38812yes—
109C22×C4—1941no—
110C22×C4C22⋊D9712yes—
111C22×C4C22×D9711yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D9714yes—
117C4⋊C4—1942no—
118C4⋊C4—1944no—
119C4⋊C4—1941no—
120C22⋊C4—1942no—
121C22⋊C4—1944no—
122C22⋊C4—1941no—
123C42—972no—
124C42—1942no—
125C23#1309548812yes—
126C23C2×D38811yescharacteristic
127Q8—22no—
128Q8—3881no—
129Q8—3881no—
130Q8—1944no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—3882no—
136C2×C4—3882no—
137C2×C4—1942no—
138C2×C4—1942no—
139C2×C4#1309548812yes—
140C2×C4C2×C4×D9711yescharacteristic
141C2×C4C2×C22⋊D9712yes—
142C2×C4C2×C22⋊D9711yescharacteristic
143C2×C4#1309551112yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1309548811yescharacteristic
149C22#1569078412yes—
150C4—22no—
151C4—7762no—
152C4—3882no—
153C4—3882no—
154C4C4×C22⋊D9712yes—
155C4#1569077911yescharacteristic
156C4#1569079311yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1909
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20260745C2semidirect
4. #20260492C2semidirect
7. #15690808C4semidirect
8. #15690797C4semidirect
81. C97#1909semidirect

Maximal subgroups

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

subgroupindex
#202613062
#202607452
#202607092
#202604922
#568×C972
#190997