Finite Group Databasebeta

← #27170218

Subgroups — #27170218

Order 12928.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265830C211yesmaximal, characteristic, complemented
2#20265369C211yesmaximal, characteristic, complemented
3#20265369C211yesmaximal, characteristic, complemented
4#20264923C211yesmaximal, characteristic, complemented
5#20264906C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7#486×C101C211yesmaximal, characteristic, complemented
8C2×#13108075—21no—
9C2×#13108059—21no—
10C2×#13108059—21no—
11#15692240—22no—
12#15692231—21no—
13#15692230—21no—
14C4×D404C2211yescharacteristic
15C4×D404C2211yescharacteristic
16C4⋊C4×D101—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692176—21no—
20#15692176—21no—
21C4⋊Q8×C101C2211yescharacteristic
22#127×C101C2211yescharacteristic
23#102×C101—22no—
24C4×C8×C101C2211yescharacteristic
25#13108075—42no—
26#13108075—42no—
27C2×D404—22no—
28C2×D404D411yescharacteristic
29C2×D404D411yescharacteristic
30C2×C4×D101—22no—
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33#13108059—42no—
34#13108059—42no—
35C2×#11717579—21no—
36C2×#11717579—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D411yescharacteristic
39#13108053—22no—
40#13108051—22no—
41#13108051—21no—
42#13108051—21no—
43C2×Q8×C101—21no—
44C2×D4×C101—21no—
45C2×D4×C101—21no—
46C2×D4×C101—21no—
47C2×C8×C101—21no—
48C4⋊C4×C101—21no—
49C4⋊C4×C101D412yes—
50C42×C101C2311yescharacteristic
51C23×C101—41no—
52C22×D101—21no—
53C22×D101—21no—
54Q8×C101—41no—
55Q8×C101—41no—
56D4×C101—42no—
57D4×C101—41no—
58D4×C101—41no—
59D4×C101—41no—
60D4×C101—41no—
61C2×C4×C101—22no—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×Dic101—21no—
66C2×Dic101—21no—
67D404—21no—
68D404—21no—
69D404—21no—
70D404—44no—
71D404D812yescomplemented
72C4×D101—44no—
73C4×D101—41no—
74C4×D101—41no—
75C4×D101—41no—
76C4×D101—41no—
77C8×C101—41no—
78#11717579—41no—
79#11717579—41no—
80C22×C101—81no—
81C22×C101—81no—
82C22×C101—81no—
83C22×C101#12011yescharacteristic
84C2×D101—22no—
85C2×D101—41no—
86C2×D101—41no—
87C2×D101—41no—
88C2×D101—41no—
89C2×D101—41no—
90C4×C101—21no—
91C4×C101—42no—
92C4×C101C2×D811yescharacteristic
93C4×C101#13511yescharacteristic
94C4×C101AGL(1,ℤ/8)11yescharacteristic
95C4×C101AGL(1,ℤ/8)11yescharacteristic
96Dic101—41no—
97Dic101—41no—
98C2×C101—161no—
99C2×C101#44711yescharacteristic
100C2×C101#44911yescharacteristic
101C2×C101#45411yescharacteristic
102D101—42no—
103D101—81no—
104#1625—1011nomaximal, Sylow
105C101#162511yescharacteristic, complemented, Sylow
106#547—1011no—
107#486D10111yescharacteristic, complemented
108#459—1011no—
109#459—1011no—
110#332—1011no—
111D4⋊C8—1011no—
112D4⋊C8—1011no—
113C2×#41—2021no—
114C2×D8—2021no—
115C2×D8—2021no—
116C4⋊Q8C2×D10111yescharacteristic
117#127C2×D10111yescharacteristic
118#123—2022no—
119C22⋊Q8—2021no—
120#121—2021no—
121C4×D4—1011no—
122C4×D4—1011no—
123C2×C4⋊C4—2021no—
124C4⋊C8—1011no—
125C4⋊C8—1011no—
126#102—22no—
127#102—2021no—
128#102—2021no—
129C4×C8C2×D10111yescharacteristic
130#41—4042no—
131#41—4042no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—1011no—
137C2×D4—1011no—
138C2×D4—2022no—
139C22×C4—2022no—
140C22×C4—2021no—
141C22×C4—2021no—
142D8—4042no—
143D8—4042no—
144C2×C8—21no—
145C2×C8—2021no—
146C2×C8—2021no—
147C4⋊C4—21no—
148C4⋊C4—1011no—
149C4⋊C4—1011no—
150C4⋊C4—2022no—
151C4⋊C4C22⋊D10112yes—
152C22⋊C4—2022no—
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C42C22×D10111yescharacteristic
156C23—41no—
157C23—2021no—
158C23—2021no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—1012no—
167D4—4044no—
168D4—2021no—
169D4—2021no—
170D4—2021no—
171C2×C4—22no—
172C2×C4—4044no—
173C2×C4—4041no—
174C2×C4—4041no—
175C2×C4—4041no—
176C2×C4—4041no—
177C2×C4—2021no—
178C2×C4—2021no—
179C2×C4D4×D10111yescharacteristic
180C2×C4D4×D10111yescharacteristic
181C2×C4C2×C22⋊D10111yescharacteristic
182C8—41no—
183C8—4041no—
184C8—4041no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—4041no—
189C22—4041no—
190C22—4041no—
191C22—4041no—
192C22—4041no—
193C22—2022no—
194C22#1569222311yescharacteristic
195C4—21no—
196C4—42no—
197C4—4041no—
198C4—4041no—
199C4D8×D10111yescharacteristic
200C4#1569227611yescharacteristic
201C4#1569227911yescharacteristic
202C4#1569228211yescharacteristic
203C2—161no—
204C2—8081no—
205C2—4042no—
206C2#2026530911yescharacteristic
207C2#2026532111yescharacteristic
208C2#2026534911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#486×C101C2
socleC22×C101#120
hypercenter#486D101
hypocenterC101#1625
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20265830C2semidirect
2. #20265369C2semidirect
3. #20265369C2semidirect
4. #20264923C2semidirect
5. #20264906C2semidirect
6. #20264906C2semidirect
7. #486×C101C2semidirect
71. D404D8semidirect
105. C101#1625semidirect
107. #486D101semidirect

Maximal subgroups

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

subgroupindex
#202658302
#202653692
#202649232
#202649062
#486×C1012
#1625101