Finite Group Databasebeta

← #27171670

Subgroups — #27171670

Order 12928.

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#20265655C211yesmaximal, characteristic, complemented
2#20265194C212yesmaximal
3C2×#15692174C211yesmaximal, characteristic, complemented
4#20264933C212yesmaximal, complemented
5#536×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7Q8×Dic101—24no—
8#15692214—22no—
9#15692214C2212yes—
10#15692182—24no—
11#15692174—22no—
12C2×#41×C101C2211yescharacteristic
13C22×Q8×C101C2211yescharacteristic
14C4⋊Q8×C101C412yescomplemented
15#124×C101C412yescomplemented
16#123×C101—22no—
17C22⋊Q8×C101—22no—
18#117×C101C2211yescharacteristic
19#13108058—22no—
20#13108058—22no—
21C2×#11717579—22no—
22C4⋊Dic101—24no—
23C4⋊Dic101—22no—
24C22⋊Dic101—22no—
25C4×Dic101—24no—
26C4×Dic101D414yes—
27#41×C101—22no—
28#41×C101—22no—
29C2×Q8×C101—22no—
30C2×Q8×C101—22no—
31C2×Q8×C101C2×C411yescharacteristic
32C2×Q8×C101C2×C412yes—
33C2×Q8×C101D412yes—
34C2×D4×C101—21no—
35C2×D4×C101C2×C411yescharacteristic
36C22×C4×C101—21no—
37C22×C4×C101D412yes—
38C22×C4×C101C2311yescharacteristic
39C4⋊C4×C101—24no—
40C4⋊C4×C101—21no—
41C22⋊C4×C101—21no—
42C22⋊C4×C101—24no—
43C42×C101C2×C412yes—
44C23×C101—21no—
45C23×C101C2×D411yescharacteristic
46Q8×C101—24no—
47Q8×C101—24no—
48Q8×C101—22no—
49D4×C101—22no—
50D4×C101—41no—
51D4×C101—41no—
52C2×C4×C101—22no—
53C2×C4×C101—22no—
54C2×C4×C101—22no—
55C2×C4×C101—21no—
56C2×C4×C101—42no—
57C2×C4×C101C22⋊C414yes—
58C2×C4×C101C22×C411yescharacteristic
59C2×C4×C101C2×D412yes—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101#4112yes—
62C2×Dic101—24no—
63#11717579—42no—
64C22×C101—41no—
65C22×C101—42no—
66C22×C101—41no—
67C22×C101C2×C22⋊C411yescharacteristic
68C22×C101#12011yescharacteristic
69C22×C101#12311yescharacteristic
70C4×C101—24no—
71C4×C101—42no—
72C4×C101—41no—
73C4×C101C4×D412yes—
74C4×C101#12112yes—
75Dic101—44no—
76C2×C101—21no—
77C2×C101—21no—
78C2×C101—21no—
79C2×C101—81no—
80C2×C101#38611yescharacteristic
81#1912—1011nomaximal, Sylow
82C101#191211yescharacteristic, complemented, Sylow
83#536D10111yescharacteristic
84#519—1011no—
85#421—1012no—
86C2×#101—1011no—
87H(3,ℤ/4)—1012no—
88C2×#41C2×D10111yescharacteristic
89C22×Q8C2×D10111yescharacteristic
90C2×M16—1012no—
91C4⋊Q8Dic10112yescomplemented
92#124Dic10112yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—2024no—
96#117—1012no—
97#117—2022no—
98#117C2×D10111yescharacteristic
99#104—2024no—
100#101—2022no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic10111yescharacteristic
106C2×Q8C2×Dic10112yes—
107C2×Q8C22⋊D10112yes—
108C2×D4—21no—
109C2×D4C2×Dic10111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D10112yes—
112C22×C4C22×D10111yescharacteristic
113M16—2022no—
114M16—2022no—
115C2×C8—2022no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—2024no—
119C4⋊C4—2022no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—2022no—
123C42—1014no—
124C42—2024no—
125C42C2×Dic10112yes—
126C23—21no—
127C23C2×C22⋊D10111yescharacteristic
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—2024no—
140C2×C4C22⋊Dic10114yes—
141C2×C4C22×Dic10111yescharacteristic
142C2×C4D4×D10112yes—
143C2×C4C2×C22⋊D10111yescharacteristic
144C2×C4#1310807312yes—
145C8—4042no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic10111yescharacteristic
150C22#1569222311yescharacteristic
151C22#1569223611yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—4044no—
156C4D4×Dic10112yes—
157C4#1569222712yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2026505711yescharacteristic
subgroupG/H
centerC2#20265057
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#536×C101C2
socleC2×C101#386
prime radicalC2×#41×C101C22
cosocleC22×C4×C101C23
hypercenter#536D101
hypocenterC101#1912
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #20265655C2semidirect
3. C2×#15692174C2semidirect
4. #20264933C2semidirect
14. C4⋊Q8×C101C4semidirect
15. #124×C101C4semidirect
82. C101#1912semidirect
91. C4⋊Q8Dic101semidirect
92. #124Dic101semidirect

Maximal subgroups

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

subgroupindex
#202656552
#202651942
C2×#156921742
#202649332
#536×C1012
#1912101