Finite Group Databasebeta

← #27142256

Subgroups — #27142256

Order 12416.

312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 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#20261502C211yesmaximal, characteristic
2#20261306C211yesmaximal, characteristic
3#20260743C212yesmaximal, complemented
4#20260491C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7#15690817—24no—
8C2×#13095495—22no—
9C2×#13095495C2212yes—
10#15690808C412yescomplemented
11#15690797C412yescomplemented
12#15690791—22no—
13#15690789—22no—
14#15690769C2211yescharacteristic
15#15690733—24no—
16C2×#41×C97C2211yescharacteristic
17#117×C97C2212yes—
18#99×C97—22no—
19C2×Dic194—22no—
20C2×Dic194—22no—
21C2×Dic194—21no—
22C2×Dic194C2×C412yes—
23C22×Dic97—21no—
24#13095495—24no—
25#13095495—22no—
26#13095495—22no—
27C2×#11715895—24no—
28C2×#11715895D414yes—
29C4⋊Dic97—21no—
30#13095490—24no—
31C22⋊Dic97—24no—
32C22⋊Dic97—21no—
33C4×Dic97C2×C412yes—
34#41×C97—22no—
35#41×C97—22no—
36C2×Q8×C97C2×C411yescharacteristic
37C2×D4×C97C2×C411yescharacteristic
38C2×D4×C97D412yes—
39C22×C4×C97D412yes—
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C42×C97—22no—
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×C97—42no—
52C2×C4×C97C22⋊C412yes—
53C2×C4×C97C22×C411yescharacteristic
54C2×C4×C97C2×D412yes—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97#4112yes—
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—42no—
60Dic194—24no—
61Dic194—41no—
62Dic194—41no—
63#11715895—24no—
64C22×C97—22no—
65C22×C97—41no—
66C22×C97—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#12112yes—
69C4×C97—22no—
70C4×C97—82no—
71C4×C97C4×D412yes—
72C4×C97#12011yescharacteristic
73C4×C97#12311yescharacteristic
74Dic97—42no—
75Dic97—42no—
76C2×C97—21no—
77C2×C97—22no—
78C2×C97—42no—
79C2×C97#38611yescharacteristic
80#1909—971nomaximal, Sylow
81C97#190911yescharacteristic, complemented, Sylow
82#568—971no—
83#536—971no—
84#419—972no—
85#410D9711yescharacteristic
86H(3,ℤ/4)—972no—
87C2×#41C2×D9711yescharacteristic
88C22×Q8—971no—
89#131—1944no—
90C2×M16—972no—
91C2×M16—1942no—
92C4⋊Q8—972no—
93#124—972no—
94#123—1942no—
95C22⋊Q8—1942no—
96#117—971no—
97#117C2×D9712yes—
98#103—1944no—
99#99—22no—
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×D4C22⋊D9712yes—
109C22×C4—1941no—
110C22×C4D38812yes—
111C22×C4C22×D9711yescharacteristic
112M16—1944no—
113M16—1942no—
114M16—1942no—
115C2×C8—974no—
116C2×C8—1944no—
117C4⋊C4—22no—
118C4⋊C4—1944no—
119C4⋊C4—1941no—
120C22⋊C4—22no—
121C22⋊C4—1944no—
122C22⋊C4—1941no—
123C42—22no—
124C42—972no—
125C23#1309548812yes—
126C23C2×C22⋊D9711yescharacteristic
127Q8—22no—
128Q8—3881no—
129Q8—3881no—
130Q8—1944no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3882no—
137C2×C4—1942no—
138C2×C4—1942no—
139C2×C4#1309548812yes—
140C2×C4C2×C4×D9711yescharacteristic
141C2×C4D4×D9712yes—
142C2×C4C2×D38811yescharacteristic
143C2×C4#1309551012yes—
144C8—1944no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1309548811yescharacteristic
149C22#1569078212yes—
150C4—22no—
151C4—82no—
152C4—3882no—
153C4—3882no—
154C4#1569077012yes—
155C4#1569078011yescharacteristic
156C4#1569079411yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2026061211yescharacteristic
subgroupG/H
centerC2#20260612
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1909
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #20260743C2semidirect
4. #20260491C2semidirect
10. #15690808C4semidirect
11. #15690797C4semidirect
81. C97#1909semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202613062
#202607432
#202604912
#410×C972
#190997