Finite Group Databasebeta

← #27170448

Subgroups — #27170448

Order 12928.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265921C211yesmaximal, characteristic
2#20265583C211yesmaximal, characteristic, complemented
3#20265569C211yesmaximal, characteristic, complemented
4#20264918C211yesmaximal, characteristic, complemented
5#20264917C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic, complemented
7#489×C101C211yesmaximal, characteristic, complemented
8C2×#13108064—21no—
9C2×#13108061—21no—
10#15692247C2211yescharacteristic
11#15692243C2211yescharacteristic
12#15692239—21no—
13#15692237—21no—
14#15692232—21no—
15#15692231—21no—
16#15692228—21no—
17#15692227—21no—
18#15692188—21no—
19#15692186—21no—
20#15692166C2211yescharacteristic
21#15692166C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#124×C101C2211yescharacteristic
24#103×C101—21no—
25#102×C101—21no—
26#97×C101C2211yescharacteristic
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C22×Dic101—21no—
32C2×C4×D101—21no—
33#13108064—42no—
34#13108061—42no—
35C2×#11717579—21no—
36C2×#11717579—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101D411yescharacteristic
39C4⋊Dic101D411yescharacteristic
40#13108053—21no—
41#13108053—21no—
42#13108053—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47C2×Q8×C101—21no—
48C2×Q8×C101D411yescharacteristic
49C2×D4×C101D411yescharacteristic
50C2×C8×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101—21no—
53C22⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C42×C101C2311yescharacteristic
56C23×C101—21no—
57C22×D101—21no—
58Q8×C101—41no—
59Q8×C101—41no—
60Q8×C101—41no—
61D4×C101—41no—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C22⋊D101—81no—
68C22⋊D101—81no—
69C2×Dic101—21no—
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—81no—
73D404—41no—
74C4×D101—81no—
75Dic202—41no—
76C8×C101—41no—
77#11717579—22no—
78#11717579—22no—
79C22×C101—41no—
80C22×C101—41no—
81C22×C101—41no—
82C22×C101#12011yescharacteristic
83C2×D101—41no—
84C2×D101—41no—
85C2×D101—41no—
86C4×C101—21no—
87C4×C101—21no—
88C4×C101—21no—
89C4×C101—41no—
90C4×C101—81no—
91Dic101—41no—
92Dic101—41no—
93Dic101—41no—
94C2×C101—81no—
95C2×C101#45311yescharacteristic
96C2×C101#45511yescharacteristic
97C2×C101#45611yescharacteristic
98D101—81no—
99#1661—1011nomaximal, Sylow
100C101#166111yescharacteristic, complemented, Sylow
101#562—1011no—
102#501—1011no—
103#497—1011no—
104#489D10111yescharacteristic, complemented
105#330—1011no—
106#329—1011no—
107#329—1011no—
108C2×Q16—2021no—
109C2×SD16—2021no—
110C4⋊Q8C2×D10111yescharacteristic
111#125—1011no—
112#124—1011no—
113#124C2×D10111yescharacteristic
114#123—2021no—
115#123—2021no—
116C22⋊Q8—2021no—
117C22⋊Q8—2021no—
118#121—2021no—
119#121—2021no—
120#107—2021no—
121#106—2021no—
122#103—21no—
123#102—21no—
124#97—1011no—
125#97—1011no—
126#97C2×D10111yescharacteristic
127C2×Q8—21no—
128C2×Q8—1011no—
129C2×Q8C22⋊D10111yescharacteristic
130C2×D4—1011no—
131C2×D4—2021no—
132C2×D4—2021no—
133C2×D4C22⋊D10111yescharacteristic
134C22×C4—2021no—
135C22×C4—2021no—
136Q16—4042no—
137SD16—4042no—
138C2×C8—21no—
139C2×C8—2021no—
140C2×C8—2021no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1011no—
144C4⋊C4—1011no—
145C4⋊C4—2021no—
146C4⋊C4—2021no—
147C4⋊C4—2021no—
148C4⋊C4—2021no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—2021no—
152C22⋊C4—2021no—
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C42C22×D10111yescharacteristic
156C23—21no—
157C23—2021no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—4041no—
162D4—41no—
163D4—8081no—
164D4—8081no—
165D4—4041no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—8081no—
169C2×C4—8081no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4—2021no—
173C2×C4D4×D10111yescharacteristic
174C2×C4D4×D10111yescharacteristic
175C2×C4C2×C22⋊D10111yescharacteristic
176C8—41no—
177C8—2022no—
178C8—2022no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—4041no—
183C22—4041no—
184C22—4041no—
185C22#1569222311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—4041no—
192C4—4041no—
193C4—4041no—
194C2—81no—
195C2—8081no—
196C2#2026534311yescharacteristic
197C2#2026535411yescharacteristic
198C2#2026535811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#489×C101C2
socleC22×C101#120
prime radical#20265921C2
hypercenter#489D101
hypocenterC101#1661
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #20265583C2semidirect
3. #20265569C2semidirect
4. #20264918C2semidirect
5. #20264917C2semidirect
6. #20264916C2semidirect
7. #489×C101C2semidirect
100. C101#1661semidirect
104. #489D101semidirect

Maximal subgroups

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

subgroupindex
#202659212
#202655832
#202655692
#202649182
#202649172
#202649162
#489×C1012
#1661101