Finite Group Databasebeta

← #27141837

Subgroups — #27141837

Order 12416.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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#41×Dic97C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic
3#20260494C212yesmaximal, complemented
4#20260491C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7Q8×Dic97—21no—
8D4×Dic97—21no—
9D4×Dic97—22no—
10#15690769C412yescomplemented
11#15690769C2211yescharacteristic
12C2×C4×Dic97C412yescomplemented
13C2×C4×Dic97C2211yescharacteristic
14#15690729—21no—
15#15690727—21no—
16C2×#41×C97C2211yescharacteristic
17#117×C97C2212yes—
18#99×C97—22no—
19C22×Dic97—22no—
20C22×Dic97—21no—
21#13095495—22no—
22#13095495—22no—
23C2×#11715895—22no—
24C4⋊Dic97—22no—
25C4⋊Dic97—21no—
26C22⋊Dic97—22no—
27C22⋊Dic97—21no—
28C4×Dic97—22no—
29C4×Dic97—21no—
30C4×Dic97C2×C411yescharacteristic
31C4×Dic97C2×C412yes—
32C4×Dic97C2×C411yescharacteristic
33#41×C97—22no—
34#41×C97—22no—
35C2×Q8×C97D411yescharacteristic
36C2×D4×C97D411yescharacteristic
37C2×D4×C97D412yes—
38C22×C4×C97C2×C412yes—
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—22no—
41C22⋊C4×C97—22no—
42C42×C97—22no—
43C23×C97C22⋊C412yes—
44C23×C97C2×D411yescharacteristic
45Q8×C97—22no—
46D4×C97—24no—
47D4×C97—22no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—42no—
51C2×C4×C97C22⋊C412yes—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97#4112yes—
55C2×C4×C97#4112yes—
56C2×Dic97—21no—
57C2×Dic97—22no—
58C2×Dic97—21no—
59C2×Dic97—42no—
60C2×Dic97—44no—
61#11715895—42no—
62C22×C97—22no—
63C22×C97—41no—
64C22×C97—42no—
65C22×C97C2×C22⋊C411yescharacteristic
66C22×C97#12312yes—
67C4×C97—22no—
68C4×C97—82no—
69C4×C97#11711yescharacteristic
70C4×C97#11711yescharacteristic
71C4×C97#12312yes—
72Dic97—22no—
73Dic97—42no—
74Dic97—41no—
75C2×C97—21no—
76C2×C97—22no—
77C2×C97—42no—
78C2×C97#38111yescharacteristic
79#1827—971nomaximal, Sylow
80C97#182711yescharacteristic, complemented, Sylow
81C4×#41—971no—
82#413—971no—
83#410D9711yescharacteristic
84#339—972no—
85H(3,ℤ/4)—972no—
86C2×#41C2×D9711yescharacteristic
87C2×M16—972no—
88C4×Q8—1941no—
89C4×D4—1941no—
90C4×D4—1942no—
91#117—972no—
92#117—971no—
93#117C2×D9712yes—
94C2×C42—972no—
95C2×C42—971no—
96#101—1941no—
97#100—1941no—
98#99—22no—
99#41—22no—
100#41—22no—
101C2×Q8D38811yescharacteristic
102C2×D4D38811yescharacteristic
103C2×D4C22⋊D9712yes—
104C22×C4—1942no—
105C22×C4—1941no—
106C22×C4C4×D9712yes—
107C22×C4C22×D9711yescharacteristic
108M16—1942no—
109M16—1942no—
110C2×C8—1942no—
111C4⋊C4—22no—
112C4⋊C4—1942no—
113C4⋊C4—1941no—
114C22⋊C4—22no—
115C22⋊C4—1942no—
116C22⋊C4—1941no—
117C42—22no—
118C42—971no—
119C42—972no—
120C42—971no—
121C42—1942no—
122C42—1941no—
123C23#1309548812yes—
124C23C2×C22⋊D9711yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—3882no—
132C2×C4—3884no—
133C2×C4—1942no—
134C2×C4—1941no—
135C2×C4—1941no—
136C2×C4#1309548812yes—
137C2×C4C2×C4×D9711yescharacteristic
138C2×C4C2×D38811yescharacteristic
139C2×C4#1309551012yes—
140C2×C4#1309551012yes—
141C8—3882no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×#1309548811yescharacteristic
146C22#1569079112yes—
147C4—22no—
148C4—82no—
149C4—3881no—
150C4—3882no—
151C4—1942no—
152C4#1569076611yescharacteristic
153C4#1569076611yescharacteristic
154C4#1569079412yes—
155C2—22no—
156C2—21no—
157C2—42no—
158C2#2026059611yescharacteristic
subgroupG/H
centerC4#15690766
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#381
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1827
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #20260494C2semidirect
4. #20260491C2semidirect
10. #15690769C4semidirect
12. C2×C4×Dic97C4semidirect
80. C97#1827semidirect

Maximal subgroups

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

subgroupindex
#41×Dic972
#202607192
#202604942
#202604912
#410×C972
#182797