Finite Group Databasebeta

← #27170439

Subgroups — #27170439

Order 12928.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265919C211yesmaximal, characteristic
2#20265572C211yesmaximal, characteristic, complemented
3#20265537C212yesmaximal, complemented
4#20264916C212yesmaximal, complemented
5#329×C101C211yesmaximal, characteristic, complemented
6C2×#13108064—21no—
7C2×#13108062—21no—
8#15692255C2211yescharacteristic, complemented
9#15692244—21no—
10#15692243C2212yes—
11#15692241—22no—
12#15692225—22no—
13#15692223—21no—
14#15692180—22no—
15#15692177—22no—
16#15692166C2211yescharacteristic, complemented
17#124×C101C2211yescharacteristic
18#97×C101C2212yescomplemented
19C23×D101—21no—
20C2×Dic202—21no—
21C2×Dic202D412yes—
22C2×D404—21no—
23C2×D404D412yes—
24C2×C22⋊D101—22no—
25#13108064—42no—
26#13108062—42no—
27C2×#11717579—21no—
28C4⋊Dic101—22no—
29C22⋊Dic101—21no—
30#13108051—22no—
31#13108051—22no—
32#13108051—22no—
33C4×Dic101—21no—
34C2×Q8×C101D411yescharacteristic
35C2×D4×C101D411yescharacteristic
36C2×C8×C101—22no—
37C22⋊C4×C101—22no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40C22×D101—22no—
41C22×D101—81no—
42C22×D101—82no—
43Q8×C101—41no—
44D4×C101—41no—
45C2×C4×C101—21no—
46C2×C4×C101C2×D412yes—
47C2×C4×C101C2×D411yescharacteristic
48C22⋊D101—82no—
49C2×Dic101—22no—
50C2×Dic101—41no—
51D404—42no—
52D404—81no—
53Dic202—42no—
54Dic202—41no—
55Dic202—41no—
56C8×C101—42no—
57#11717579—22no—
58C22×C101—41no—
59C22×C101—42no—
60C22×C101#12011yescharacteristic
61C2×D101—42no—
62C2×D101—42no—
63C2×D101—42no—
64C2×D101—84no—
65C4×C101—21no—
66C4×C101—22no—
67C4×C101—41no—
68Dic101—42no—
69Dic101—81no—
70C2×C101—81no—
71C2×C101#45311yescharacteristic
72C2×C101#45512yes—
73D101—82no—
74#1659—1011nomaximal, Sylow
75C101#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—1011no—
77#497—1011no—
78#489—1012no—
79#329—1012no—
80#329D10111yescharacteristic, complemented
81C2×Q16—2021no—
82C2×SD16—2021no—
83C4⋊Q8—1011no—
84#124—1012no—
85#124—2022no—
86#124—2021no—
87#124C2×D10111yescharacteristic
88#120—2022no—
89#120—2021no—
90#103—2022no—
91#102—2022no—
92#97—1011no—
93#97C2×D10112yescomplemented
94C24—2021no—
95C2×Q8—1012no—
96C2×Q8—2021no—
97C2×Q8D40411yescharacteristic
98C2×D4—1012no—
99C2×D4—2021no—
100C2×D4—2022no—
101C2×D4D40411yescharacteristic
102Q16—4042no—
103SD16—4042no—
104C2×C8—22no—
105C2×C8—2021no—
106C4⋊C4—2022no—
107C22⋊C4—22no—
108C22⋊C4—2022no—
109C22⋊C4—2021no—
110C22⋊C4—2022no—
111C22⋊C4—2022no—
112C42—2021no—
113C42C22×D10111yescharacteristic
114C23—21no—
115C23—8081no—
116C23—8082no—
117C23—2022no—
118Q8—41no—
119Q8—4041no—
120Q8—4041no—
121Q8—4042no—
122D4—41no—
123D4—8081no—
124D4—8082no—
125D4—4042no—
126C2×C4—21no—
127C2×C4—4041no—
128C2×C4—2022no—
129C2×C4D4×D10112yes—
130C2×C4C2×D40411yescharacteristic
131C8—42no—
132C8—2022no—
133C22—41no—
134C22—42no—
135C22—8084no—
136C22—4042no—
137C22—4042no—
138C22—4042no—
139C22#1569222511yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—8081no—
144C4—4042no—
145C2—81no—
146C2—8082no—
147C2#2026534711yescharacteristic
148C2#2026535612yes—
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101#120
prime radical#20265919C2
hypercenter#329D101
hypocenterC101#1659
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20265572C2semidirect
3. #20265537C2semidirect
4. #20264916C2semidirect
5. #329×C101C2semidirect
8. #15692255C22semidirect
16. #15692166C22semidirect
18. #97×C101C22semidirect
75. C101#1659semidirect
80. #329D101semidirect
93. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659192
#202655722
#202655372
#202649162
#329×C1012
#1659101