Finite Group Databasebeta

← #27170438

Subgroups — #27170438

Order 12928.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265572C211yesmaximal, characteristic, complemented
2#20265536C212yesmaximal, complemented
3#20264917C212yesmaximal, complemented
4#20264917C211yesmaximal, characteristic, complemented
5H(3,𝔽4)×C101C211yesmaximal, characteristic
6C2×#13108064—21no—
7C2×#13108062—21no—
8#15692255C2211yescharacteristic, complemented
9#15692179—22no—
10#15692176—22no—
11#15692166C2212yescomplemented
12#15692166C2211yescharacteristic, complemented
13#124×C101—22no—
14#124×C101—21no—
15#124×C101C2212yes—
16#124×C101C2211yescharacteristic
17#120×C101—22no—
18#120×C101—21no—
19C2×Dic202—21no—
20#13108064—42no—
21#13108062—42no—
22C2×#11717579—21no—
23C2×#11717579—22no—
24C4⋊Dic101—22no—
25C24×C101—21no—
26C2×Q8×C101D412yes—
27C2×Q8×C101D411yescharacteristic
28C2×D4×C101—21no—
29C2×D4×C101—22no—
30C2×D4×C101D412yes—
31C2×D4×C101D411yescharacteristic
32C22⋊C4×C101—22no—
33C22⋊C4×C101—21no—
34C22⋊C4×C101—22no—
35C22⋊C4×C101—22no—
36C22⋊C4×C101—22no—
37C42×C101—21no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40C23×C101—22no—
41C23×C101—81no—
42C23×C101—82no—
43Q8×C101—42no—
44Q8×C101—41no—
45D4×C101—42no—
46D4×C101—41no—
47D4×C101—81no—
48D4×C101—82no—
49C2×C4×C101—21no—
50C2×C4×C101—22no—
51C2×C4×C101C2×D412yes—
52C2×C4×C101C2×D411yescharacteristic
53C2×Dic101—41no—
54Dic202—41no—
55Dic202—41no—
56#11717579—22no—
57#11717579—42no—
58C22×C101—41no—
59C22×C101—42no—
60C22×C101—42no—
61C22×C101—42no—
62C22×C101—42no—
63C22×C101—84no—
64C22×C101#12011yescharacteristic
65C4×C101—21no—
66C4×C101—22no—
67C4×C101—41no—
68C4×C101—42no—
69Dic101—81no—
70C2×C101—81no—
71C2×C101—82no—
72C2×C101#45311yescharacteristic
73C2×C101#45512yes—
74#1659—1011nomaximal, Sylow
75C101#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)D10111yescharacteristic
77#497—1011no—
78#489—1012no—
79#329—1011no—
80#329—1012no—
81C2×Q16—2021no—
82C2×SD16—2021no—
83C4⋊Q8—1011no—
84#124—22no—
85#124—21no—
86#124C2×D10112yes—
87#124C2×D10111yescharacteristic
88#120—22no—
89#120—21no—
90#103—2022no—
91#102—2022no—
92#97—1012no—
93#97—1011no—
94C24—21no—
95C2×Q8—2021no—
96C2×Q8C22⋊D10112yes—
97C2×Q8C22⋊D10111yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D10112yes—
101C2×D4C22⋊D10111yescharacteristic
102Q16—4042no—
103SD16—4042no—
104C2×C8—2022no—
105C2×C8—2021no—
106C4⋊C4—2022no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D10111yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—4041no—
121Q8—4041no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—4041no—
129C2×C4C2×C22⋊D10112yes—
130C2×C4C2×C22⋊D10111yescharacteristic
131C8—4042no—
132C8—2022no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1569222411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—8081no—
145C2—81no—
146C2—82no—
147C2#2026534511yescharacteristic
148C2#2026535512yes—
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radicalH(3,𝔽4)×C101C2
socleC22×C101#120
hypercenterH(3,𝔽4)D101
hypocenterC101#1659
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20265572C2semidirect
2. #20265536C2semidirect
3. #20264917C2semidirect
4. #20264917C2semidirect
8. #15692255C22semidirect
11. #15692166C22semidirect
12. #15692166C22semidirect
75. C101#1659semidirect

Maximal subgroups

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

subgroupindex
#202655722
#202655362
#202649172
H(3,𝔽4)×C1012
#1659101