Finite Group Databasebeta

← #27142274

Subgroups — #27142274

Order 12416.

314 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261213C211yesmaximal, characteristic, complemented
2#20260752C212yesmaximal
3C2×#15690729C211yesmaximal, characteristic, complemented
4#20260491C212yesmaximal, complemented
5#536×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7Q8×Dic97—24no—
8#15690769—22no—
9#15690769C2212yes—
10#15690737—24no—
11#15690729—22no—
12C2×#41×C97C2211yescharacteristic
13C22×Q8×C97C2211yescharacteristic
14C4⋊Q8×C97C412yescomplemented
15#124×C97C412yescomplemented
16#123×C97—22no—
17C22⋊Q8×C97—22no—
18#117×C97C2211yescharacteristic
19#13095495—22no—
20#13095495—22no—
21C2×#11715895—22no—
22C4⋊Dic97—24no—
23C4⋊Dic97—22no—
24C22⋊Dic97—22no—
25C4×Dic97—24no—
26C4×Dic97D414yes—
27#41×C97—22no—
28#41×C97—22no—
29C2×Q8×C97—22no—
30C2×Q8×C97—22no—
31C2×Q8×C97C2×C411yescharacteristic
32C2×Q8×C97C2×C412yes—
33C2×Q8×C97D412yes—
34C2×D4×C97—21no—
35C2×D4×C97C2×C411yescharacteristic
36C22×C4×C97—21no—
37C22×C4×C97D412yes—
38C22×C4×C97C2311yescharacteristic
39C4⋊C4×C97—24no—
40C4⋊C4×C97—21no—
41C22⋊C4×C97—21no—
42C22⋊C4×C97—24no—
43C42×C97C2×C412yes—
44C23×C97—21no—
45C23×C97C2×D411yescharacteristic
46Q8×C97—24no—
47Q8×C97—24no—
48Q8×C97—22no—
49D4×C97—22no—
50D4×C97—41no—
51D4×C97—41no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97—42no—
57C2×C4×C97C22⋊C414yes—
58C2×C4×C97C22×C411yescharacteristic
59C2×C4×C97C2×D412yes—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97#4112yes—
62C2×Dic97—24no—
63#11715895—42no—
64C22×C97—41no—
65C22×C97—42no—
66C22×C97—41no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#12011yescharacteristic
69C22×C97#12311yescharacteristic
70C4×C97—24no—
71C4×C97—42no—
72C4×C97—41no—
73C4×C97C4×D412yes—
74C4×C97#12112yes—
75Dic97—44no—
76C2×C97—21no—
77C2×C97—21no—
78C2×C97—21no—
79C2×C97—81no—
80C2×C97#38611yescharacteristic
81#1912—971nomaximal, Sylow
82C97#191211yescharacteristic, complemented, Sylow
83#536D9711yescharacteristic
84#519—971no—
85#421—972no—
86C2×#101—971no—
87H(3,ℤ/4)—972no—
88C2×#41C2×D9711yescharacteristic
89C22×Q8C2×D9711yescharacteristic
90C2×M16—972no—
91C4⋊Q8Dic9712yescomplemented
92#124Dic9712yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—1944no—
96#117—972no—
97#117—1942no—
98#117C2×D9711yescharacteristic
99#104—1944no—
100#101—1942no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic9711yescharacteristic
106C2×Q8C2×Dic9712yes—
107C2×Q8C22⋊D9712yes—
108C2×D4—21no—
109C2×D4C2×Dic9711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D9712yes—
112C22×C4C22×D9711yescharacteristic
113M16—1942no—
114M16—1942no—
115C2×C8—1942no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—1944no—
119C4⋊C4—1942no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—1942no—
123C42—974no—
124C42—1944no—
125C42C2×Dic9712yes—
126C23—21no—
127C23C2×C22⋊D9711yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—1944no—
140C2×C4C22⋊Dic9714yes—
141C2×C4C22×Dic9711yescharacteristic
142C2×C4D4×D9712yes—
143C2×C4C2×C22⋊D9711yescharacteristic
144C2×C4#1309551012yes—
145C8—3882no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic9711yescharacteristic
150C22#1569077811yescharacteristic
151C22#1569079111yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—3884no—
156C4D4×Dic9712yes—
157C4#1569078212yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#536×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#536D97
hypocenterC97#1912
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #20261213C2semidirect
3. C2×#15690729C2semidirect
4. #20260491C2semidirect
14. C4⋊Q8×C97C4semidirect
15. #124×C97C4semidirect
82. C97#1912semidirect
91. C4⋊Q8Dic97semidirect
92. #124Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612132
#202607522
C2×#156907292
#202604912
#536×C972
#191297