Finite Group Databasebeta

← #27173148

Subgroups — #27173148

Order 12928.

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

normal subgroups only

This group has 248 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#20265919C211yesmaximal, characteristic, complemented
2#20265350C212yesmaximal, complemented
3#20264974C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#454×C101C211yesmaximal, characteristic, complemented
6#15692280—22no—
7#15692244C2212yes—
8#15692243—21no—
9#15692241—22no—
10#15692225—22no—
11#15692223—21no—
12#15692182—22no—
13#15692174C2212yescomplemented
142−1+4×C101C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C101—21no—
16#124×C101C2211yescharacteristic
17#104×C101—21no—
18#101×C101C2211yescharacteristic, complemented
19C23×D101—21no—
20C2×Dic202—21no—
21C2×D404—21no—
22C2×D404D412yes—
23C2×C22⋊D101—22no—
24#13108061—42no—
25#13108059—42no—
26#13108058—22no—
27C22⋊Dic101—21no—
28#13108051—22no—
29#13108051—22no—
30#13108051—22no—
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—
44C22×D101—22no—
45C22×D101—82no—
46C22×D101—81no—
47Q8×C101—21no—
48Q8×C101—22no—
49Q8×C101—41no—
50D4×C101—21no—
51D4×C101—22no—
52D4×C101—41no—
53D4×C101—41no—
54C2×C4×C101—21no—
55C2×C4×C101—42no—
56C2×C4×C101—41no—
57C2×C4×C101C2×D412yes—
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—82no—
60C2×Dic101—22no—
61D404—42no—
62D404—81no—
63Dic202—81no—
64C8×C101—41no—
65#11717579—42no—
66C22×C101—42no—
67C22×C101—41no—
68C22×C101—41no—
69C22×C101#12011yescharacteristic
70C2×D101—42no—
71C2×D101—42no—
72C2×D101—42no—
73C2×D101—84no—
74C4×C101—22no—
75C4×C101—21no—
76C4×C101—41no—
77C4×C101—41no—
78Dic101—42no—
79C2×C101—21no—
80C2×C101—81no—
81C2×C101—81no—
82C2×C101B(4,𝔽2)11yescharacteristic
83D101—82no—
84#2202—1011nomaximal, Sylow
85C101#220211yescharacteristic, complemented, Sylow
86H(3,𝔽4)—1011no—
87#454—1012no—
88#454D10111yescharacteristic, complemented
89#355—1011no—
90#355—1012no—
912−1+4C2×D10111yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—2022no—
94#124—1012no—
95#124—2022no—
96#124—2021no—
97#124C2×D10111yescharacteristic
98#120—2022no—
99#120—2021no—
100#104—21no—
101#104—2022no—
102#101—1012no—
103#101C2×D10111yescharacteristic, complemented
104C24—2021no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—2021no—
110C2×Q8C22×D10111yescharacteristic
111C2×D4—1012no—
112C2×D4—2021no—
113C2×D4—2022no—
114C2×D4C22⋊D10111yescharacteristic
115SD16—41no—
116SD16—4042no—
117D8—41no—
118D8—4042no—
119M16—21no—
120M16—2022no—
121C22⋊C4—22no—
122C22⋊C4—2022no—
123C22⋊C4—2021no—
124C22⋊C4—2022no—
125C22⋊C4—2022no—
126C42—1012no—
127C42C22⋊D10111yescharacteristic
128C23—21no—
129C23—8082no—
130C23—8081no—
131C23—2022no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—8081no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—8081no—
141D4—8082no—
142D4—4042no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—2022no—
147C2×C4D4×D10112yes—
148C2×C4C2×C22⋊D10111yescharacteristic
149C8—41no—
150C8—4042no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—8084no—
155C22—4042no—
156C22—4042no—
157C22—4042no—
158C22#1569222311yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—4042no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—8082no—
168C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting#454×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#454D101
hypocenterC101#2202
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. #20265919C2semidirect
2. #20265350C2semidirect
3. #20264974C2semidirect
4. #20264973C2semidirect
5. #454×C101C2semidirect
13. #15692174C22semidirect
14. 2−1+4×C101C22semidirect
18. #101×C101C22semidirect
85. C101#2202semidirect
88. #454D101semidirect
91. 2−1+4C2×D101semidirect
103. #101C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659192
#202653502
#202649742
#202649732
#454×C1012
#2202101