Finite Group Databasebeta

← #27173080

Subgroups — #27173080

Order 12928.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265977C211yesmaximal, characteristic, complemented
2#20265615C212yesmaximal, complemented
3#20265430C211yesmaximal, characteristic, complemented
4#20264988C212yesmaximal, complemented
5#349×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7#15692288—22no—
8#15692283—22no—
9#15692277—21no—
10#15692277C2212yes—
11C2×D808C2211yescharacteristic, complemented
12#15692229—21no—
13#15692200—22no—
14D1616—22no—
15#15692187C2211yescharacteristic, complemented
16#15692177—21no—
17C2×M16×C101C2211yescharacteristic
18M32×C101C2212yescomplemented
19#13108074—22no—
20#13108074—22no—
21#13108074—41no—
22#13108074—41no—
23C2×Dic202—21no—
24C2×D404—21no—
25C2×D404—21no—
26C2×C22⋊D101—41no—
27C2×C22⋊D101—41no—
28C2×C4×D101—41no—
29Dic404—21no—
30Dic404D412yescomplemented
31#13108063—22no—
32#13108063—22no—
33D808—21no—
34D808—22no—
35D808D412yescomplemented
36C4⋊Dic101—21no—
37#13108051—41no—
38C22×C4×C101D411yescharacteristic
39M16×C101—22no—
40C2×C8×C101D411yescharacteristic
41C2×C8×C101C2311yescharacteristic
42C16×C101—22no—
43C23×C101D811yescharacteristic
44C22×D101—41no—
45C22×D101—41no—
46C2×C4×C101—21no—
47C2×C4×C101—21no—
48C2×C4×C101D811yescharacteristic
49C2×C4×C101C2×D411yescharacteristic
50C22⋊D101—42no—
51C22⋊D101—42no—
52C22⋊D101—82no—
53C2×Dic101—41no—
54C2×Dic101—41no—
55D404—22no—
56D404—41no—
57D404—41no—
58D404—41no—
59C4×D101—42no—
60C4×D101—42no—
61Dic202—22no—
62Dic202—41no—
63C8×C101—21no—
64C8×C101C2×D412yes—
65C22×C101—21no—
66C22×C101—41no—
67C22×C101C2×D811yescharacteristic
68C2×D101—42no—
69C2×D101—82no—
70C2×D101—81no—
71C2×D101—81no—
72C2×D101—81no—
73C4×C101—21no—
74C4×C101#12011yescharacteristic
75C4×C101AGL(1,ℤ/8)11yescharacteristic
76Dic101—42no—
77Dic101—81no—
78C2×C101—21no—
79C2×C101—41no—
80C2×C101#44711yescharacteristic
81D101—82no—
82D101—161no—
83#2190—1011nomaximal, Sylow
84C101#219011yescharacteristic, complemented, Sylow
85#575—1011no—
86#509—1012no—
87#469—1011no—
88#360—1012no—
89#349D10111yescharacteristic, complemented
90C2×#41—2021no—
91#137—2022no—
92AGL(1,ℤ/8)—2022no—
93#135—1012no—
94#135—2021no—
95C2×D8—1011no—
96C2×M16C2×D10111yescharacteristic
97#121—2021no—
98SD32—2022no—
99D16—2022no—
100M32C2×D10112yescomplemented
101#106—1011no—
102#102—2021no—
103#41—4041no—
104#41—4041no—
105#41—2022no—
106#41—2022no—
107C2×Q8—2021no—
108C2×D4—4041no—
109C2×D4—4041no—
110C2×D4—2021no—
111C2×D4—2021no—
112C22×C4—4041no—
113C22×C4D40411yescharacteristic
114Q16—1012no—
115Q16—2021no—
116SD16—2022no—
117SD16—2022no—
118D8—1012no—
119D8—2021no—
120D8—2022no—
121M16—22no—
122C2×C8D40411yescharacteristic
123C2×C8C22×D10111yescharacteristic
124C4⋊C4—2021no—
125C22⋊C4—4041no—
126C16—22no—
127C23—4041no—
128C23—4041no—
129C23D80811yescharacteristic
130Q8—4041no—
131Q8—2022no—
132D4—8082no—
133D4—4041no—
134D4—4042no—
135D4—4042no—
136D4—4041no—
137D4—4041no—
138D4—2022no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—4042no—
142C2×C4—4041no—
143C2×C4—4041no—
144C2×C4—4042no—
145C2×C4D80811yescharacteristic
146C2×C4C2×D40411yescharacteristic
147C8—21no—
148C8D4×D10112yes—
149C22—21no—
150C22—41no—
151C22—8081no—
152C22—8081no—
153C22—8082no—
154C22—8081no—
155C22—4042no—
156C22C2×D80811yescharacteristic
157C4—21no—
158C4—8081no—
159C4—4042no—
160C4#1569222511yescharacteristic
161C4#1569228311yescharacteristic
162C2—21no—
163C2—41no—
164C2—8082no—
165C2—16161no—
166C2#2026531311yescharacteristic
subgroupG/H
centerC2#20265313
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#349×C101C2
socleC2×C101#447
hypercenter#349D101
hypocenterC101#2190
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #20265977C2semidirect
2. #20265615C2semidirect
3. #20265430C2semidirect
4. #20264988C2semidirect
5. #349×C101C2semidirect
11. C2×D808C22semidirect
15. #15692187C22semidirect
18. M32×C101C22semidirect
30. Dic404D4semidirect
35. D808D4semidirect
84. C101#2190semidirect
89. #349D101semidirect
100. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659772
#202656152
#202654302
#202649882
#349×C1012
#2190101