Finite Group Databasebeta

← #27170221

Subgroups — #27170221

Order 12928.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#20265525C211yesmaximal, characteristic, complemented
2#20265372C211yesmaximal, characteristic, complemented
3#20265372C211yesmaximal, characteristic, complemented
4#20264923C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#547×C101C211yesmaximal, characteristic, complemented
8C2×#13108059—21no—
9C2×#13108059—21no—
10#15692252C2211yescharacteristic
11C4⋊#11717579C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13#15692175—22no—
14#15692175—21no—
15#15692175—21no—
16C4×#11717579C2211yescharacteristic
17C2×#41×C101—21no—
18C4⋊Q8×C101C2211yescharacteristic
19#123×C101—22no—
20C22⋊Q8×C101—21no—
21#121×C101—21no—
22C4×D4×C101C2211yescharacteristic
23C4×D4×C101C2211yescharacteristic
24C2×C4⋊C4×C101—21no—
25C2×D404—21no—
26C2×D404—21no—
27C2×D404—21no—
28#13108059—42no—
29#13108059—42no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33#41×C101—42no—
34#41×C101—42no—
35C2×Q8×C101—21no—
36C2×D4×C101—22no—
37C2×D4×C101D411yescharacteristic
38C2×D4×C101D411yescharacteristic
39C22×C4×C101—22no—
40C22×C4×C101—21no—
41C22×C4×C101—21no—
42C4⋊C4×C101—22no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101D411yescharacteristic
45C4⋊C4×C101D411yescharacteristic
46C4⋊C4×C101D412yes—
47C22⋊C4×C101—22no—
48C22⋊C4×C101—21no—
49C22⋊C4×C101—21no—
50C42×C101C2311yescharacteristic
51C23×C101—21no—
52C23×C101—21no—
53C22×D101—41no—
54Q8×C101—41no—
55Q8×C101—41no—
56D4×C101—21no—
57D4×C101—21no—
58D4×C101—21no—
59D4×C101—44no—
60D4×C101D812yescomplemented
61C2×C4×C101—21no—
62C2×C4×C101—21no—
63C2×C4×C101—22no—
64C2×C4×C101—44no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72D404—42no—
73D404—41no—
74D404—41no—
75D404—41no—
76D404—41no—
77#11717579—41no—
78#11717579—41no—
79#11717579—41no—
80C22×C101—22no—
81C22×C101—41no—
82C22×C101—41no—
83C22×C101—41no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101#12011yescharacteristic
87C2×D101—81no—
88C2×D101—81no—
89C2×D101—81no—
90C4×C101—21no—
91C4×C101—41no—
92C4×C101—42no—
93C4×C101—41no—
94C4×C101C2×D811yescharacteristic
95C4×C101#13511yescharacteristic
96C4×C101AGL(1,ℤ/8)11yescharacteristic
97C4×C101AGL(1,ℤ/8)11yescharacteristic
98C2×C101—42no—
99C2×C101—81no—
100C2×C101#44711yescharacteristic
101C2×C101#44911yescharacteristic
102C2×C101#45411yescharacteristic
103D101—161no—
104#1625—1011nomaximal, Sylow
105C101#162511yescharacteristic, complemented, Sylow
106#547D10111yescharacteristic, complemented
107#486—1011no—
108#459—1011no—
109#459—1011no—
110#332—1011no—
111D4⋊C8—1011no—
112D4⋊C8—1011no—
113C2×#41—21no—
114C2×D8—2021no—
115C2×D8—2021no—
116C4⋊Q8C2×D10111yescharacteristic
117#127—1011no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D10111yescharacteristic
122C4×D4C2×D10111yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—1011no—
125C4⋊C8—1011no—
126#102—2021no—
127#102—2021no—
128#102—2022no—
129C4×C8—1011no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—2021no—
135C2×D4—2021no—
136C2×D4—2021no—
137C2×D4C22⋊D10111yescharacteristic
138C2×D4C22⋊D10111yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—4042no—
143D8—4042no—
144C2×C8—2021no—
145C2×C8—2021no—
146C2×C8—2021no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D10111yescharacteristic
150C4⋊C4C22⋊D10111yescharacteristic
151C4⋊C4C22⋊D10112yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D10111yescharacteristic
156C23—21no—
157C23—21no—
158C23—4041no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—4042no—
166D4—4041no—
167D4—4041no—
168D4—4041no—
169D4—4041no—
170D4#1310805912yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D10111yescharacteristic
180C2×C4C2×C22⋊D10111yescharacteristic
181C2×C4C2×C22⋊D10111yescharacteristic
182C8—4041no—
183C8—4041no—
184C8—4041no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—8081no—
192C22—8081no—
193C22—8081no—
194C22#1569222411yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1310805911yescharacteristic
200C4#1569227511yescharacteristic
201C4#1569228011yescharacteristic
202C4#1569228111yescharacteristic
203C2—42no—
204C2—81no—
205C2—16161no—
206C2#2026531111yescharacteristic
207C2#2026532211yescharacteristic
208C2#2026535011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#547×C101C2
socleC22×C101#120
hypercenter#547D101
hypocenterC101#1625
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20265525C2semidirect
2. #20265372C2semidirect
3. #20265372C2semidirect
4. #20264923C2semidirect
5. D4⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
7. #547×C101C2semidirect
60. D4×C101D8semidirect
105. C101#1625semidirect
106. #547D101semidirect
170. D4#13108059semidirect

Maximal subgroups

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

subgroupindex
#202655252
#202653722
#202649232
D4⋊#117175792
#547×C1012
#1625101