Finite Group Databasebeta

← #27173152

Subgroups — #27173152

Order 12928.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#20265360C212yesmaximal, complemented
3#20264975C212yesmaximal, complemented
4#20264974C211yesmaximal, characteristic, complemented
5#454×C101C211yesmaximal, characteristic
6#15692285—22no—
7#15692253C2212yes—
8#15692248—22no—
9#15692247—21no—
10#15692236—21no—
11#15692232—22no—
12#15692182—22no—
13#15692174C2212yes—
142−1+4×C101C2211yescharacteristic
15AGL(1,ℤ/8)×C101—21no—
16#124×C101C2211yescharacteristic
17#104×C101—21no—
18#101×C101C2211yescharacteristic
19C2×Dic202D412yes—
20C22×Dic101—21no—
21#13108064—42no—
22#13108062—42no—
23#13108058—22no—
24C4⋊Dic101—21no—
25C4⋊Dic101—21no—
26#13108053—22no—
27#13108053—22no—
28#13108053—22no—
29C22⋊Dic101—22no—
30C22⋊Dic101—21no—
31C4×Dic101D412yes—
32#41×C101—21no—
33#41×C101—22no—
34#41×C101—41no—
35C2×Q8×C101—41no—
36C2×Q8×C101C2311yescharacteristic
37C2×D4×C101D411yescharacteristic
38SD16×C101—41no—
39D8×C101—41no—
40M16×C101—21no—
41C22⋊C4×C101—22no—
42C42×C101D411yescharacteristic
43C23×C101—21no—
44Q8×C101—21no—
45Q8×C101—22no—
46Q8×C101—41no—
47D4×C101—21no—
48D4×C101—22no—
49D4×C101—41no—
50D4×C101—41no—
51C2×C4×C101—21no—
52C2×C4×C101—42no—
53C2×C4×C101—41no—
54C2×C4×C101C2×D412yes—
55C2×C4×C101C2×D411yescharacteristic
56C2×Dic101—22no—
57C2×Dic101—22no—
58C2×Dic101—81no—
59Dic202—42no—
60C8×C101—41no—
61#11717579—42no—
62C22×C101—42no—
63C22×C101—41no—
64C22×C101—41no—
65C22×C101#12011yescharacteristic
66C4×C101—22no—
67C4×C101—21no—
68C4×C101—41no—
69C4×C101—41no—
70Dic101—42no—
71Dic101—42no—
72C2×C101—21no—
73C2×C101—81no—
74C2×C101—81no—
75C2×C101B(4,𝔽2)11yescharacteristic
76#2203—1011nomaximal, Sylow
77C101#220311yescharacteristic, complemented, Sylow
78#563—1011no—
79#456—1012no—
80#454D10111yescharacteristic
81#356—1012no—
82#355—1011no—
832−1+4C2×D10111yescharacteristic
84#137—2022no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1012no—
87#126—2022no—
88#125—2021no—
89#124C2×D10111yescharacteristic
90#123—2021no—
91C22⋊Q8—2022no—
92#104—21no—
93#104—2022no—
94#101—1012no—
95#101C2×D10111yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1012no—
101C2×Q8C22×D10111yescharacteristic
102C2×D4C22⋊D10111yescharacteristic
103C22×C4—2021no—
104Q16—4042no—
105SD16—41no—
106SD16—4042no—
107D8—41no—
108M16—21no—
109M16—2022no—
110C4⋊C4—2022no—
111C4⋊C4—2021no—
112C4⋊C4—2021no—
113C4⋊C4—2022no—
114C4⋊C4—2022no—
115C22⋊C4—22no—
116C22⋊C4—2022no—
117C22⋊C4—2021no—
118C42—1012no—
119C42C22⋊D10111yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—4042no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—8081no—
133C2×C4—2022no—
134C2×C4—2022no—
135C2×C4D4×D10112yes—
136C2×C4C2×C22⋊D10111yescharacteristic
137C8—41no—
138C8—4042no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1569222311yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—4042no—
148C4—4042no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting, prime radical#454×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#454D101
hypocenterC101#2203
LCSG ▷ C2×Q8×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20265927C2semidirect
2. #20265360C2semidirect
3. #20264975C2semidirect
4. #20264974C2semidirect
77. C101#2203semidirect

Maximal subgroups

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

subgroupindex
#202659272
#202653602
#202649752
#202649742
#454×C1012
#2203101