Finite Group Databasebeta

← #27171341

Subgroups — #27171341

Order 12928.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265652C211yesmaximal, characteristic
2#20265217C212yesmaximal, complemented
3C2×#15692172C211yesmaximal, characteristic
4#20264937C212yesmaximal, complemented
5#390×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7D4×Dic101—22no—
8D4×Dic101—22no—
9#15692214C2212yes—
10C2×C22⋊Dic101—22no—
11#15692188—22no—
12#15692186—22no—
13#15692172C414yescomplemented
14C22×D4×C101C2211yescharacteristic
15C2×C4⋊C4×C101C2211yescharacteristic
16C2×C22⋊C4×C101—22no—
17C2×C42×C101C2211yescharacteristic
18C22×Dic101—22no—
19C22×Dic101—22no—
20#13108058—22no—
21#13108058C2×C414yescomplemented
22C2×#11717579—22no—
23C4⋊Dic101D412yes—
24C4⋊Dic101D412yes—
25C22⋊Dic101—24no—
26C22⋊Dic101—22no—
27C4×Dic101—22no—
28C24×C101D412yes—
29C2×D4×C101—22no—
30C2×D4×C101—22no—
31C2×D4×C101C2×C412yes—
32C2×D4×C101Q812yes—
33C22×C4×C101—21no—
34C22×C4×C101—21no—
35C22×C4×C101C2311yescharacteristic
36C4⋊C4×C101—22no—
37C22⋊C4×C101—44no—
38C42×C101—22no—
39C23×C101—22no—
40C23×C101—22no—
41C23×C101—42no—
42C23×C101—42no—
43C23×C101C4⋊C414yes—
44C23×C101C2×D411yescharacteristic
45D4×C101—24no—
46D4×C101—24no—
47C2×C4×C101—22no—
48C2×C4×C101—41no—
49C2×C4×C101—41no—
50C2×C4×C101—42no—
51C2×C4×C101—41no—
52C2×C4×C101C22×C411yescharacteristic
53C2×C4×C101C2×D412yes—
54C2×C4×C101C2×Q811yescharacteristic
55C2×C4×C101#4112yes—
56C2×Dic101—22no—
57C2×Dic101—22no—
58C2×Dic101—82no—
59C2×Dic101—82no—
60#11717579—24no—
61C22×C101—21no—
62C22×C101—21no—
63C22×C101—24no—
64C22×C101—44no—
65C22×C101—42no—
66C22×C101—42no—
67C22×C101—82no—
68C22×C101C2×C4⋊C411yescharacteristic
69C22×C101#12011yescharacteristic
70C22×C101#12311yescharacteristic
71C4×C101—42no—
72C4×C101—81no—
73C4×C101C4×D412yes—
74C4×C101C22⋊Q812yes—
75Dic101—42no—
76Dic101—42no—
77C2×C101—21no—
78C2×C101—21no—
79C2×C101—44no—
80C2×C101#38511yescharacteristic
81C2×C101#45311yescharacteristic
82C2×C101#45511yescharacteristic
83#1856—1011nomaximal, Sylow
84C101#185611yescharacteristic, complemented, Sylow
85#518—1011no—
86#428—1012no—
87C2×#100—1011no—
88#390D10111yescharacteristic
89#339—1012no—
90C22×D4C2×D10111yescharacteristic
91C2×M16—1012no—
92C4×D4—2022no—
93C4×D4—2022no—
94#117—1012no—
95C2×C4⋊C4C2×D10111yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—2022no—
98C2×C42C2×D10111yescharacteristic
99#107—2022no—
100#106—2022no—
101#100—1014no—
102C24C22⋊D10112yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4Dic20212yes—
106C2×D4C4×D10112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—2022no—
110C22×C4—2022no—
111C22×C4C22×D10111yescharacteristic
112M16—1014no—
113M16—2022no—
114C2×C8—2022no—
115C4⋊C4—22no—
116C4⋊C4—1012no—
117C4⋊C4—1012no—
118C22⋊C4—44no—
119C22⋊C4—2024no—
120C22⋊C4—2022no—
121C42—22no—
122C42—2022no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23#1310805314yes—
128C23C2×C22⋊D10111yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—8082no—
137C2×C4—8082no—
138C2×C4—2022no—
139C2×C4—2022no—
140C2×C4C2×C4×D10111yescharacteristic
141C2×C4D4×D10112yes—
142C2×C4C2×Dic20211yescharacteristic
143C2×C4#1310807312yes—
144C8—2024no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×#1310805311yescharacteristic
153C22#1569222311yescharacteristic
154C22#1569223611yescharacteristic
155C4—42no—
156C4—81no—
157C4—4042no—
158C4—4042no—
159C4#1569221512yes—
160C4#1569223212yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#2026504911yescharacteristic
165C2#2026534411yescharacteristic
166C2#2026535411yescharacteristic
subgroupG/H
centerC22#15692223
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#390×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
cosocleC22×C4×C101C23
hypercenter#390D101
hypocenterC101#1856
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20265217C2semidirect
4. #20264937C2semidirect
13. #15692172C4semidirect
21. #13108058C2×C4semidirect
84. C101#1856semidirect

Maximal subgroups

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

subgroupindex
#202656522
#202652172
C2×#156921722
#202649372
#390×C1012
#1856101