Finite Group Databasebeta

← #27170431

Subgroups — #27170431

Order 12928.

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

normal subgroups only

This group has 206 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#20265927C211yesmaximal, characteristic, complemented
2#20265567C211yesmaximal, characteristic
3#20265542C212yesmaximal, complemented
4#20264918C212yesmaximal, complemented
5#329×C101C211yesmaximal, characteristic, complemented
6C2×#13108061—21no—
7C2×#13108059—21no—
8#15692253—21no—
9#15692252C2211yescharacteristic
10#15692248—22no—
11#15692247C2212yes—
12#15692236—21no—
13#15692232—22no—
14#15692177—22no—
15#15692177—22no—
16#15692166C2211yescharacteristic
17#124×C101C2211yescharacteristic
18#97×C101C2212yes—
19C2×Dic202—21no—
20C2×D404—22no—
21C2×D404—21no—
22C22×Dic101—21no—
23#13108061—42no—
24#13108059—42no—
25C2×#11717579—21no—
26C4⋊Dic101D412yes—
27C4⋊Dic101D412yes—
28#13108053—22no—
29#13108053—22no—
30#13108053—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—21no—
33C4×Dic101—21no—
34C2×Q8×C101D411yescharacteristic
35C2×D4×C101D411yescharacteristic
36C2×C8×C101—22no—
37C22⋊C4×C101—22no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40C22×D101—41no—
41Q8×C101—41no—
42D4×C101—41no—
43C2×C4×C101—21no—
44C2×C4×C101C2×D412yes—
45C2×C4×C101C2×D411yescharacteristic
46C2×Dic101—22no—
47C2×Dic101—22no—
48C2×Dic101—81no—
49D404—41no—
50D404—41no—
51D404—44no—
52Dic202—81no—
53C8×C101—42no—
54#11717579—22no—
55C22×C101—41no—
56C22×C101—42no—
57C22×C101#12011yescharacteristic
58C2×D101—81no—
59C2×D101—82no—
60C4×C101—21no—
61C4×C101—22no—
62C4×C101—41no—
63Dic101—42no—
64Dic101—42no—
65C2×C101—81no—
66C2×C101#45311yescharacteristic
67C2×C101#45412yes—
68D101—161no—
69#1657—1011nomaximal, Sylow
70C101#165711yescharacteristic, complemented, Sylow
71#563—1011no—
72#496—1011no—
73#490—1012no—
74#330—1012no—
75#329D10111yescharacteristic, complemented
76C2×SD16—2021no—
77C2×D8—2021no—
78C4⋊Q8—2021no—
79#127—1011no—
80#126—2022no—
81#125—1012no—
82#124C2×D10111yescharacteristic
83#123—2021no—
84C22⋊Q8—2022no—
85#102—2022no—
86#102—2022no—
87#97—1011no—
88#97C2×D10112yes—
89C2×Q8—2021no—
90C2×Q8D40411yescharacteristic
91C2×D4—2022no—
92C2×D4—2021no—
93C2×D4D40411yescharacteristic
94C22×C4—2021no—
95SD16—4042no—
96D8—4042no—
97C2×C8—22no—
98C2×C8—2021no—
99C4⋊C4—1012no—
100C4⋊C4—1012no—
101C4⋊C4—2022no—
102C4⋊C4—2022no—
103C4⋊C4—2022no—
104C22⋊C4—22no—
105C22⋊C4—2022no—
106C22⋊C4—2021no—
107C42—2021no—
108C42C22×D10111yescharacteristic
109C23—21no—
110C23—4041no—
111Q8—41no—
112Q8—8081no—
113D4—41no—
114D4—4041no—
115D4—4041no—
116D4—4044no—
117C2×C4—21no—
118C2×C4—8081no—
119C2×C4—2022no—
120C2×C4—2022no—
121C2×C4D4×D10112yes—
122C2×C4C2×D40411yescharacteristic
123C8—42no—
124C8—2022no—
125C22—41no—
126C22—42no—
127C22—8081no—
128C22—8082no—
129C22#1569222511yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—4042no—
134C4—4042no—
135C2—81no—
136C2—16161no—
137C2#2026534711yescharacteristic
138C2#2026535112yes—
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101#120
prime radical#20265567C2
hypercenter#329D101
hypocenterC101#1657
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #20265927C2semidirect
3. #20265542C2semidirect
4. #20264918C2semidirect
5. #329×C101C2semidirect
70. C101#1657semidirect
75. #329D101semidirect

Maximal subgroups

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

subgroupindex
#202659272
#202655672
#202655422
#202649182
#329×C1012
#1657101