Finite Group Databasebeta

← #27142271

Subgroups — #27142271

Order 12416.

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

normal subgroups only

This group has 314 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#20261309C211yesmaximal, characteristic
2#20261213C211yesmaximal, characteristic, complemented
3#20260751C212yesmaximal
4#20260492C212yesmaximal, complemented
5C2×#101×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7#15690808C412yescomplemented
8#15690799C412yescomplemented
9#15690793—22no—
10#15690786—22no—
11Q8×Dic97—24no—
12#15690769—22no—
13#15690769C2211yescharacteristic
14#15690769C2212yes—
15#15690736—24no—
16C22×Q8×C97C2211yescharacteristic
17C2×M16×C97C2212yes—
18#101×C97—22no—
19#13095511—22no—
20#13095511—22no—
21C2×Dic194C2×C411yescharacteristic
22C2×D388C2×C411yescharacteristic
23C2×C22⋊D97—21no—
24C2×C4×D97—21no—
25C4⋊Dic97—24no—
26C4⋊Dic97—22no—
27C4⋊Dic97—21no—
28#13095490—24no—
29C22⋊Dic97—22no—
30C22⋊Dic97—21no—
31#13095488—24no—
32C4×Dic97—24no—
33C4×Dic97C2×C412yes—
34C4×Dic97D414yes—
35C2×Q8×C97—22no—
36C2×Q8×C97—22no—
37C2×Q8×C97C2×C412yes—
38C2×Q8×C97D412yes—
39C22×C4×C97D412yes—
40C22×C4×C97C2311yescharacteristic
41M16×C97—22no—
42M16×C97—22no—
43C2×C8×C97—22no—
44C23×C97C2×D411yescharacteristic
45C22×D97—21no—
46Q8×C97—24no—
47Q8×C97—24no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97C22⋊C414yes—
51C2×C4×C97C22×C411yescharacteristic
52C2×C4×C97C2×D412yes—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97#4112yes—
55C22⋊D97—41no—
56C22⋊D97—41no—
57C2×Dic97—22no—
58C2×Dic97—21no—
59C2×Dic97—24no—
60D388—22no—
61C4×D97—42no—
62Dic194—22no—
63C8×C97—42no—
64C22×C97—41no—
65C22×C97C2×C22⋊C411yescharacteristic
66C22×C97#12011yescharacteristic
67C22×C97#12311yescharacteristic
68C2×D97—42no—
69C2×D97—41no—
70C4×C97—24no—
71C4×C97C4×D412yes—
72C4×C97#12112yes—
73Dic97—44no—
74Dic97—42no—
75Dic97—41no—
76C2×C97—21no—
77C2×C97—21no—
78C2×C97—21no—
79C2×C97#38611yescharacteristic
80D97—81no—
81#1912—971nomaximal, Sylow
82C97#191211yescharacteristic, complemented, Sylow
83#536—971no—
84#519—971no—
85#421—972no—
86C2×#101D9711yescharacteristic, complemented
87H(3,ℤ/4)—972no—
88C2×#41—971no—
89C22×Q8C2×D9711yescharacteristic
90C2×M16C2×D9712yes—
91C4⋊Q8—972no—
92#124—972no—
93#123—1942no—
94C22⋊Q8—1942no—
95C4×Q8—1944no—
96#117—972no—
97#117—971no—
98#117—1942no—
99#104—1944no—
100#101—22no—
101#41—1942no—
102#41—1942no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—971no—
106C2×Q8C4×D9712yes—
107C2×Q8C22⋊D9712yes—
108C2×D4—971no—
109C2×D4—1941no—
110C22×C4—1941no—
111C22×C4D38812yes—
112C22×C4C22×D9711yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—1944no—
117C4⋊C4—1944no—
118C4⋊C4—1942no—
119C4⋊C4—1941no—
120C22⋊C4—1942no—
121C22⋊C4—1941no—
122C22⋊C4—1944no—
123C42—972no—
124C42—974no—
125C42—1944no—
126C23—1941no—
127C23C2×D38811yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—1942no—
131D4—3881no—
132D4—3881no—
133D4—1942no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—3882no—
137C2×C4—1942no—
138C2×C4—1941no—
139C2×C4—1944no—
140C2×C4#1309548814yes—
141C2×C4C2×C4×D9711yescharacteristic
142C2×C4D4×D9712yes—
143C2×C4C2×C22⋊D9711yescharacteristic
144C2×C4#1309551012yes—
145C8—42no—
146C22—41no—
147C22—3882no—
148C22—3881no—
149C22C2×#1309548811yescharacteristic
150C22#1569078011yescharacteristic
151C22#1569079411yescharacteristic
152C4—24no—
153C4—3884no—
154C4—3882no—
155C4—3881no—
156C4#1569077012yes—
157C4#1569078212yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—7761no—
162C2#2026061211yescharacteristic
subgroupG/H
centerC2#20260612
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#101×C97C2
socleC2×C97#386
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenterC2×#101D97
hypocenterC97#1912
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #20261213C2semidirect
4. #20260492C2semidirect
5. C2×#101×C97C2semidirect
7. #15690808C4semidirect
8. #15690799C4semidirect
82. C97#1912semidirect
86. C2×#101D97semidirect

Maximal subgroups

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

subgroupindex
#202613092
#202612132
#202607512
#202604922
C2×#101×C972
#191297