Finite Group Databasebeta

← #27172300

Subgroups — #27172300

Order 12928.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#20265803C211yesmaximal, characteristic
2#20265038C212yesmaximal
3#20265038C211yesmaximal, characteristic
4#20264943C212yesmaximal
5#342×C101C211yesmaximal, characteristic
6C23×Dic101C2211yescharacteristic
7#15692234—22no—
8#15692232—24no—
9C2×C22⋊Dic101C2211yescharacteristic
10C2×C22⋊Dic101C2212yes—
11#15692162—24no—
12#15692162—22no—
13C2×C22⋊C4×C101C2212yes—
14C2×C22⋊C4×C101C2211yescharacteristic
15C2×Dic202—22no—
16C22×Dic101—22no—
17C22×Dic101—21no—
18C22×Dic101—21no—
19C22×Dic101—22no—
20C22×Dic101—21no—
21C22×Dic101D414yes—
22C22×Dic101Q812yescomplemented
23C4⋊Dic101—22no—
24#13108053—24no—
25C22⋊Dic101—22no—
26C22⋊Dic101—22no—
27C22⋊Dic101—22no—
28C24×C101C2311yescharacteristic
29C22×C4×C101—22no—
30C22×C4×C101D412yes—
31C22⋊C4×C101—22no—
32C22⋊C4×C101—22no—
33C22⋊C4×C101—22no—
34C23×C101—42no—
35C23×C101C2×D412yes—
36C23×C101C2×D411yescharacteristic
37C23×C101C2×Q811yescharacteristic
38C23×C101#4111yescharacteristic
39C23×C101#4112yes—
40C2×C4×C101—22no—
41C2×C4×C101—42no—
42C2×C4×C101—44no—
43C2×C4×C101—42no—
44C2×Dic101—24no—
45C2×Dic101—24no—
46C2×Dic101—22no—
47C2×Dic101—41no—
48C2×Dic101—42no—
49C2×Dic101—41no—
50C2×Dic101—44no—
51C2×Dic101—42no—
52C2×Dic101—41no—
53Dic202—82no—
54C22×C101—24no—
55C22×C101—22no—
56C22×C101—41no—
57C22×C101—41no—
58C22×C101—42no—
59C22×C101#12011yescharacteristic
60C22×C101C22⋊Q812yes—
61C22×C101C22⋊Q811yescharacteristic
62C22×C101#12412yes—
63C22×C101#12411yescharacteristic
64C4×C101—42no—
65C4×C101—82no—
66Dic101—42no—
67Dic101—44no—
68Dic101—81no—
69C2×C101—21no—
70C2×C101—22no—
71C2×C101—22no—
72C2×C101—21no—
73C2×C101#39311yescharacteristic
74C2×C101#45812yes—
75#2030—1011nomaximal, Sylow
76C101#203011yescharacteristic, complemented, Sylow
77#543—1011no—
78#381—1012no—
79#381—1011no—
80#342—1012no—
81#342D10111yescharacteristic
82C23×C4—1011no—
83C22⋊Q8—2024no—
84C22⋊Q8—2022no—
85C2×C22⋊C4—1012no—
86C2×C22⋊C4—1011no—
87C2×C22⋊C4C2×D10112yes—
88C2×C22⋊C4C2×D10111yescharacteristic
89#95—2024no—
90#95—2022no—
91C24C22×D10111yescharacteristic
92C2×Q8—2022no—
93C22×C4—22no—
94C22×C4—1014no—
95C22×C4—1012no—
96C22×C4—2021no—
97C22×C4—2022no—
98C22×C4—2021no—
99C22×C4—2021no—
100C22×C4—2022no—
101C22×C4D40412yes—
102C4⋊C4—2024no—
103C4⋊C4—2022no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—2022no—
108C22⋊C4—2022no—
109C22⋊C4—2022no—
110C23—42no—
111C23D4×D10112yes—
112C23C2×D40411yescharacteristic
113C23Q8×D10111yescharacteristic
114C23#1310807311yescharacteristic
115C23#1310807412yes—
116Q8—8082no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—4042no—
122C2×C4—4041no—
123C2×C4—4041no—
124C2×C4—4042no—
125C2×C4—4041no—
126C2×C4—4044no—
127C2×C4—2024no—
128C2×C4—2024no—
129C2×C4—2022no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1569222511yescharacteristic
136C22#1569223312yes—
137C22#1569223511yescharacteristic
138C22#1569224112yes—
139C22#1569224311yescharacteristic
140C4—42no—
141C4—82no—
142C4—8081no—
143C4—4042no—
144C4—4044no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2026509011yescharacteristic
150C2#2026536812yes—
subgroupG/H
centerC22#15692225
commutator, prime radical, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101#120
hypercenter#342D101
hypocenterC101#2030
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic101Q8semidirect
76. C101#2030semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202650382
#202649432
#342×C1012
#2030101