Finite Group Databasebeta

← #27173161

Subgroups — #27173161

Order 12928.

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

normal subgroups only

This group has 224 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#20265907C211yesmaximal, characteristic, complemented
2#20265360C212yesmaximal
3#20264976C211yesmaximal, characteristic, complemented
4#20264975C212yesmaximal, complemented
5#456×C101C211yesmaximal, characteristic
6#15692285—22no—
7#15692254—22no—
8#15692253—21no—
9#15692253C2212yes—
10C4×Dic202—21no—
11#15692220—22no—
12#15692182—22no—
13#15692174C2212yes—
142−1+4×C101C2211yescharacteristic
15#137×C101—21no—
16C4⋊Q8×C101C2211yescharacteristic
17#104×C101—21no—
18#101×C101C2211yescharacteristic
19C2×Dic202—21no—
20C2×Dic202—21no—
21C2×Dic202D412yes—
22#13108064—42no—
23#13108062—42no—
24#13108058—22no—
25C4⋊Dic101—21no—
26#13108053—22no—
27#13108053—22no—
28#13108053—22no—
29C4×Dic101—21no—
30C4×Dic101—22no—
31C4×Dic101D412yes—
32#41×C101—21no—
33#41×C101—22no—
34#41×C101—41no—
35C2×Q8×C101—41no—
36C2×Q8×C101D411yescharacteristic
37C2×Q8×C101C2311yescharacteristic
38Q16×C101—41no—
39SD16×C101—41no—
40M16×C101—21no—
41C4⋊C4×C101—22no—
42C42×C101D411yescharacteristic
43Q8×C101—21no—
44Q8×C101—22no—
45Q8×C101—41no—
46Q8×C101—41no—
47D4×C101—21no—
48D4×C101—22no—
49D4×C101—41no—
50C2×C4×C101—21no—
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—
58Dic202—24no—
59Dic202—42no—
60Dic202—81no—
61C8×C101—41no—
62#11717579—42no—
63C22×C101—41no—
64C22×C101#12011yescharacteristic
65C4×C101—22no—
66C4×C101—22no—
67C4×C101—21no—
68C4×C101—41no—
69C4×C101—41no—
70Dic101—24no—
71Dic101—42no—
72C2×C101—21no—
73C2×C101—81no—
74C2×C101B(4,𝔽2)11yescharacteristic
75#2205—1011nomaximal, Sylow
76C101#220511yescharacteristic, complemented, Sylow
77Q82—1011no—
78#456—1012no—
79#456D10111yescharacteristic
80#356—1011no—
81#356—1012no—
822−1+4C2×D10111yescharacteristic
83#137—21no—
84#137—2022no—
85C4⋊Q8—1012no—
86C4⋊Q8—2022no—
87C4⋊Q8—2021no—
88C4⋊Q8C2×D10111yescharacteristic
89C4×Q8—2022no—
90C4×Q8—2021no—
91#104—21no—
92#104—2022no—
93#101—1012no—
94#101C2×D10111yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—1012no—
100C2×Q8—2021no—
101C2×Q8—2021no—
102C2×Q8C22⋊D10111yescharacteristic
103C2×Q8C22×D10111yescharacteristic
104Q16—41no—
105Q16—4042no—
106SD16—41no—
107SD16—4042no—
108M16—21no—
109M16—2022no—
110C4⋊C4—22no—
111C4⋊C4—2022no—
112C4⋊C4—2021no—
113C4⋊C4—2022no—
114C4⋊C4—2022no—
115C42—1012no—
116C42—2022no—
117C42—2021no—
118C42C22⋊D10111yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—8081no—
124Q8—4042no—
125Q8—2024no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—2022no—
134C2×C4—2022no—
135C2×C4D4×D10112yes—
136C2×C4C2×C22⋊D10111yescharacteristic
137C8—41no—
138C8—4042no—
139C22—41no—
140C22#1569222311yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—4042no—
147C4—2024no—
148C2—21no—
149C2—81no—
150C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting#456×C101C2
socleC2×C101B(4,𝔽2)
prime radical2−1+4×C101C22
hypercenter#456D101
hypocenterC101#2205
LCSG ▷ C2×Q8×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20265907C2semidirect
3. #20264976C2semidirect
4. #20264975C2semidirect
76. C101#2205semidirect

Maximal subgroups

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

subgroupindex
#202659072
#202653602
#202649762
#202649752
#456×C1012
#2205101