Finite Group Databasebeta

← #27173157

Subgroups — #27173157

Order 12928.

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

normal subgroups only

This group has 236 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#20265893C211yesmaximal, characteristic, complemented
2#20265350C212yesmaximal, complemented
3#20264976C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#456×C101C211yesmaximal, characteristic, complemented
6#15692280—22no—
7#15692244—21no—
8#15692244C2212yes—
9#15692230—22no—
10C4×D404—21no—
11#15692213—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×D404—21no—
20C2×D404—21no—
21C2×D404D412yes—
22C2×C4×D101—22no—
23#13108061—42no—
24#13108059—42no—
25#13108058—22no—
26C4⋊Dic101—21no—
27#13108051—22no—
28#13108051—22no—
29#13108051—22no—
30C4×Dic101—21no—
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
43C22×D101—22no—
44Q8×C101—21no—
45Q8×C101—22no—
46Q8×C101—41no—
47Q8×C101—41no—
48D4×C101—21no—
49D4×C101—22no—
50D4×C101—41no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—42no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D412yes—
56C2×C4×C101C2×D411yescharacteristic
57C2×Dic101—22no—
58D404—42no—
59D404—42no—
60D404—81no—
61C4×D101—44no—
62C8×C101—41no—
63#11717579—42no—
64C22×C101—41no—
65C22×C101#12011yescharacteristic
66C2×D101—42no—
67C2×D101—42no—
68C2×D101—42no—
69C4×C101—22no—
70C4×C101—22no—
71C4×C101—21no—
72C4×C101—41no—
73C4×C101—41no—
74Dic101—42no—
75C2×C101—21no—
76C2×C101—81no—
77C2×C101B(4,𝔽2)11yescharacteristic
78D101—82no—
79#2204—1011nomaximal, Sylow
80C101#220411yescharacteristic, complemented, Sylow
81#555—1011no—
82#456D10111yescharacteristic, complemented
83#454—1012no—
84#356—1011no—
85#355—1012no—
862−1+4C2×D10111yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—2022no—
89C4⋊Q8C2×D10111yescharacteristic
90#124—1012no—
91#124—2021no—
92#121—2022no—
93C4×D4—2021no—
94#117—2022no—
95#104—21no—
96#104—2022no—
97#101—1012no—
98#101C2×D10111yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D10111yescharacteristic
104C2×Q8C22×D10111yescharacteristic
105C2×D4—1012no—
106C2×D4—2021no—
107C2×D4—2021no—
108C22×C4—2022no—
109Q16—41no—
110SD16—41no—
111SD16—4042no—
112D8—4042no—
113M16—21no—
114M16—2022no—
115C4⋊C4—22no—
116C4⋊C4—2021no—
117C22⋊C4—2022no—
118C22⋊C4—2022no—
119C22⋊C4—2022no—
120C42—1012no—
121C42—2021no—
122C42C22⋊D10111yescharacteristic
123C23—2022no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—8081no—
132D4—4042no—
133D4—4042no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—4044no—
139C2×C4—2022no—
140C2×C4D4×D10112yes—
141C2×C4C2×C22⋊D10111yescharacteristic
142C8—41no—
143C8—4042no—
144C22—41no—
145C22—4042no—
146C22—4042no—
147C22—4042no—
148C22#1569222311yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—4042no—
155C2—21no—
156C2—81no—
157C2—8082no—
158C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting#456×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#456D101
hypocenterC101#2204
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. #20265893C2semidirect
2. #20265350C2semidirect
3. #20264976C2semidirect
4. #20264973C2semidirect
5. #456×C101C2semidirect
80. C101#2204semidirect
82. #456D101semidirect

Maximal subgroups

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

subgroupindex
#202658932
#202653502
#202649762
#202649732
#456×C1012
#2204101