Finite Group Databasebeta

← #27170330

Subgroups — #27170330

Order 12928.

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

normal subgroups only

This group has 250 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#20265848C211yesmaximal, characteristic
2#20265525C211yesmaximal, characteristic, complemented
3#20265459C211yesmaximal, characteristic, complemented
4#20264920C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#461×C101C211yesmaximal, characteristic, complemented
8C2×#13108074—21no—
9#15692252C2211yescharacteristic
10#15692251—21no—
11#15692229—21no—
12#15692228—22no—
13C4×Dic202C2211yescharacteristic
14#15692217—21no—
15#15692186—21no—
16C4⋊#11717579C2211yescharacteristic
17#15692178—21no—
18#15692175—22no—
19C4×#11717579C2211yescharacteristic
20C2×SD16×C101—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25#13108074—42no—
26#13108074—42no—
27C2×Dic202D411yescharacteristic
28C2×D404—21no—
29C2×D404—21no—
30C2×D404D412yes—
31C2×C22⋊D101—22no—
32C2×C22⋊D101—22no—
33C2×C4×D101—22no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C4⋊Dic101D411yescharacteristic
37#13108053—21no—
38#13108051—22no—
39C4×Dic101—21no—
40C2×Q8×C101—21no—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101—21no—
43SD16×C101—42no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51C22×D101—22no—
52Q8×C101—41no—
53Q8×C101—41no—
54D4×C101—21no—
55D4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C22⋊D101—44no—
64C22⋊D101—82no—
65C2×Dic101—21no—
66C2×Dic101—21no—
67D404—42no—
68D404—41no—
69D404—41no—
70D404—41no—
71D404—41no—
72C4×D101—44no—
73Dic202—21no—
74Dic202SD1612yescomplemented
75C8×C101—41no—
76#11717579—41no—
77#11717579—41no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101—41no—
81C22×C101#12011yescharacteristic
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—42no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101—81no—
88C4×C101C2×SD1611yescharacteristic
89C4×C101#13511yescharacteristic
90C4×C101AGL(1,ℤ/8)11yescharacteristic
91C4×C101AGL(1,ℤ/8)11yescharacteristic
92Dic101—22no—
93Dic101—41no—
94C2×C101—81no—
95C2×C101#44911yescharacteristic
96C2×C101#45011yescharacteristic
97C2×C101#45511yescharacteristic
98D101—82no—
99#1642—1011nomaximal, Sylow
100C101#164211yescharacteristic, complemented, Sylow
101#550—1011no—
102#486—1011no—
103#475—1011no—
104#461D10111yescharacteristic, complemented
105#331—1011no—
106Q8⋊C8—1011no—
107D4⋊C8—1011no—
108C2×#41—2021no—
109C2×SD16—21no—
110C4⋊Q8C2×D10111yescharacteristic
111#127—1011no—
112#127—2021no—
113#121—2022no—
114#121—2021no—
115C4×Q8—1011no—
116C4×D4—2021no—
117C4×D4C2×D10111yescharacteristic
118#106—2021no—
119C4⋊C8—1011no—
120C4⋊C8C2×D10111yescharacteristic
121#103—21no—
122#103—2021no—
123#102—2022no—
124C4×C8—1011no—
125#41—4042no—
126#41—4042no—
127C2×Q8—21no—
128C2×Q8—1011no—
129C2×D4—1012no—
130C2×D4—2022no—
131C2×D4—2022no—
132C2×D4—2021no—
133C2×D4—2021no—
134C2×D4C22⋊D10111yescharacteristic
135C22×C4—21no—
136C22×C4—2022no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—2021no—
140C2×C8—2021no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1011no—
144C4⋊C4—2021no—
145C4⋊C4C22⋊D10111yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—2022no—
148C42—2021no—
149C42C22×D10111yescharacteristic
150C23—21no—
151C23—2022no—
152Q8—41no—
153Q8—41no—
154Q8—1012no—
155Q8—2021no—
156D4—21no—
157D4—21no—
158D4—8082no—
159D4—4042no—
160D4—4044no—
161D4—4041no—
162D4—4041no—
163D4—4041no—
164D4—4041no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—4044no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4D4×D10111yescharacteristic
173C2×C4D4×D10111yescharacteristic
174C2×C4C2×C22⋊D10111yescharacteristic
175C8—41no—
176C8—4041no—
177C8—4041no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—4042no—
182C22—4042no—
183C22—4042no—
184C22#1569222311yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—4041no—
189C4—2022no—
190C4SD16×D10111yescharacteristic
191C4#1569227511yescharacteristic
192C4#1569228111yescharacteristic
193C4#1569228211yescharacteristic
194C2—81no—
195C2—8082no—
196C2#2026532011yescharacteristic
197C2#2026532611yescharacteristic
198C2#2026535311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#461×C101C2
socleC22×C101#120
prime radical#20265848C2
hypercenter#461D101
hypocenterC101#1642
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20265525C2semidirect
3. #20265459C2semidirect
4. #20264920C2semidirect
5. #20264907C2semidirect
6. D4⋊#11717579C2semidirect
7. #461×C101C2semidirect
74. Dic202SD16semidirect
100. C101#1642semidirect
104. #461D101semidirect

Maximal subgroups

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

subgroupindex
#202658482
#202655252
#202654592
#202649202
#202649072
D4⋊#117175792
#461×C1012
#1642101