Finite Group Databasebeta

← #27170108

Subgroups — #27170108

Order 12928.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265808C211yesmaximal, characteristic
2#20265335C212yesmaximal
3#20265097C211yesmaximal, characteristic
4#20264915C212yesmaximal
5#324×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7C2×Dic404—22no—
8#15692254—22no—
9#15692246—22no—
10#15692234—21no—
11#15692234C2212yes—
12#15692214—21no—
13C2×#13108053—21no—
14#15692180—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C2×Dic202—22no—
20C2×Dic202—21no—
21C2×Dic202—42no—
22C2×Dic202D412yes—
23C22×Dic101—21no—
24C22×Dic101—21no—
25Dic404—44no—
26C4⋊Dic101—21no—
27C4⋊Dic101—21no—
28C4⋊Dic101—21no—
29C4⋊Dic101D412yes—
30#13108053—22no—
31#13108053—22no—
32#13108053—42no—
33C22⋊Dic101—22no—
34C4×Dic101—22no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C4⋊C4×C101—22no—
40C23×C101C2×D411yescharacteristic
41C2×C4×C101—21no—
42C2×C4×C101—21no—
43C2×C4×C101—21no—
44C2×C4×C101—41no—
45C2×C4×C101—41no—
46C2×C4×C101Q1612yescomplemented
47C2×C4×C101C2×D412yes—
48C2×Dic101—22no—
49C2×Dic101—22no—
50C2×Dic101—22no—
51C2×Dic101—22no—
52C2×Dic101—41no—
53C2×Dic101—41no—
54C2×Dic101—41no—
55C2×Dic101—41no—
56Dic202—24no—
57Dic202—42no—
58Dic202—42no—
59Dic202—81no—
60C8×C101—42no—
61C22×C101—21no—
62C22×C101—21no—
63C22×C101#12011yescharacteristic
64C22×C101C2×Q1611yescharacteristic
65C22×C101#13711yescharacteristic
66C4×C101—22no—
67C4×C101—22no—
68C4×C101—41no—
69Dic101—42no—
70Dic101—42no—
71Dic101—42no—
72Dic101—42no—
73C2×C101—21no—
74C2×C101—21no—
75C2×C101#45111yescharacteristic
76C2×C101#45611yescharacteristic
77C2×C101B(4,𝔽2)11yescharacteristic
78#1605—1011nomaximal, Sylow
79C101#160511yescharacteristic, complemented, Sylow
80#544—1011no—
81#451—1012no—
82#395—1011no—
83#328—1012no—
84#324D10111yescharacteristic
85C22×Q8—1011no—
86C2×Q16—2022no—
87C4⋊Q8—2022no—
88#125—2022no—
89C22⋊Q8—1012no—
90C22⋊Q8—2021no—
91#117—2021no—
92C2×C4⋊C4—2021no—
93C2×C4⋊C4C2×D10111yescharacteristic
94#103—2022no—
95C22⋊C8C2×D10112yes—
96#95—1011no—
97#95—2021no—
98C2×Q8—1012no—
99C2×Q8—4042no—
100C2×Q8—2021no—
101C2×Q8—2022no—
102C22×C4—2021no—
103C22×C4—2021no—
104C22×C4D40411yescharacteristic
105C22×C4D40411yescharacteristic
106C22×C4C22×D10111yescharacteristic
107Q16—4044no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—1012no—
111C4⋊C4—4042no—
112C4⋊C4—2022no—
113C4⋊C4—2021no—
114C4⋊C4—2021no—
115C4⋊C4—2021no—
116C4⋊C4—2022no—
117C22⋊C4—2022no—
118C42—2022no—
119C23C2×D40411yescharacteristic
120Q8—8081no—
121Q8—4042no—
122Q8—4042no—
123Q8—2024no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—4041no—
130C2×C4—4041no—
131C2×C4—4041no—
132C2×C4—4041no—
133C2×C4—2022no—
134C2×C4—2022no—
135C2×C4—2022no—
136C2×C4—2022no—
137C2×C4Dic40412yescomplemented
138C2×C4D4×D10112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1569222511yescharacteristic
143C22C2×Dic40411yescharacteristic
144C22#1569228811yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—4042no—
149C4—4042no—
150C4—4042no—
151C4—4042no—
152C2—21no—
153C2—21no—
154C2#2026533511yescharacteristic
155C2#2026536211yescharacteristic
156C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
hypercenter#324D101
hypocenterC101#1605
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
46. C2×C4×C101Q16semidirect
79. C101#1605semidirect
137. C2×C4Dic404semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202653352
#202650972
#202649152
#324×C1012
#1605101