Finite Group Databasebeta

← #27170442

Subgroups — #27170442

Order 12928.

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

normal subgroups only

This group has 196 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#20265931C211yesmaximal, characteristic
2#20265583C211yesmaximal, characteristic, complemented
3#20265569C211yesmaximal, characteristic
4#20264919C211yesmaximal, characteristic, complemented
5#20264918C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic
7#501×C101C211yesmaximal, characteristic, complemented
8C2×#13108064—21no—
9C2×#13108061—21no—
10#15692254—21no—
11#15692250—21no—
12#15692250—21no—
13#15692247C2211yescharacteristic
14#15692243C2211yescharacteristic
15#15692240—21no—
16#15692235—21no—
17#15692233—21no—
18#15692188—21no—
19#15692186—21no—
20#15692166C2211yescharacteristic
21#15692166C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#125×C101C2211yescharacteristic
24#107×C101—21no—
25#106×C101—21no—
26#97×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202D411yescharacteristic
29C2×D404D411yescharacteristic
30C2×C4×D101—21no—
31#13108064—42no—
32#13108061—42no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37C4⋊Dic101D411yescharacteristic
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41#13108051—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101—21no—
47C2×Q8×C101—21no—
48C2×C8×C101—21no—
49C4⋊C4×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101D411yescharacteristic
54C4⋊C4×C101D411yescharacteristic
55C42×C101C2311yescharacteristic
56C22×D101—21no—
57Q8×C101—41no—
58Q8×C101—41no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—41no—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101—21no—
68D404—41no—
69C4×D101—81no—
70Dic202—41no—
71Dic202—81no—
72C8×C101—22no—
73#11717579—22no—
74#11717579—22no—
75C22×C101#12011yescharacteristic
76C2×D101—41no—
77C2×D101—41no—
78C2×D101—41no—
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101—41no—
84C4×C101—81no—
85Dic101—41no—
86Dic101—41no—
87Dic101—41no—
88C2×C101#45311yescharacteristic
89C2×C101#45611yescharacteristic
90C2×C101#45611yescharacteristic
91D101—81no—
92#1660—1011nomaximal, Sylow
93C101#166011yescharacteristic, complemented, Sylow
94#563—1011no—
95#501—1011no—
96#501D10111yescharacteristic, complemented
97#497—1011no—
98#330—1011no—
99#330—1011no—
100#329—1011no—
101C2×Q16—2021no—
102C2×SD16—2021no—
103C4⋊Q8—2021no—
104C4⋊Q8C2×D10111yescharacteristic
105#126—2021no—
106#126—2021no—
107#125—1011no—
108#125C2×D10111yescharacteristic
109#124—1011no—
110#123—2021no—
111C22⋊Q8—2021no—
112C22⋊Q8—2021no—
113#107—21no—
114#107—2021no—
115#106—21no—
116#106—2021no—
117#97—1011no—
118#97—1011no—
119#97C2×D10111yescharacteristic
120C2×Q8—21no—
121C2×Q8—1011no—
122C2×Q8—2021no—
123C2×D4—1011no—
124C22×C4—2021no—
125Q16—4042no—
126SD16—4042no—
127C2×C8—21no—
128C2×C8—2021no—
129C2×C8—2021no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1011no—
135C4⋊C4—1011no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4—2021no—
139C4⋊C4—2021no—
140C4⋊C4C22⋊D10111yescharacteristic
141C4⋊C4C22⋊D10111yescharacteristic
142C22⋊C4—2021no—
143C22⋊C4—2021no—
144C22⋊C4—2021no—
145C22⋊C4—2021no—
146C22⋊C4—2021no—
147C42—2021no—
148C42C22×D10111yescharacteristic
149C23—2021no—
150Q8—41no—
151Q8—41no—
152Q8—8081no—
153Q8—4041no—
154D4—4041no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—8081no—
159C2×C4—2021no—
160C2×C4—2021no—
161C2×C4—2021no—
162C2×C4D4×D10111yescharacteristic
163C2×C4D4×D10111yescharacteristic
164C2×C4C2×C22⋊D10111yescharacteristic
165C8—22no—
166C8—2022no—
167C8—2022no—
168C22—4041no—
169C22—4041no—
170C22—4041no—
171C22#1569222311yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—4041no—
179C4—4041no—
180C4—4041no—
181C2—8081no—
182C2#2026534311yescharacteristic
183C2#2026535811yescharacteristic
184C2#2026535911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#501×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#501D101
hypocenterC101#1660
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #20265583C2semidirect
4. #20264919C2semidirect
5. #20264918C2semidirect
7. #501×C101C2semidirect
93. C101#1660semidirect
96. #501D101semidirect

Maximal subgroups

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

subgroupindex
#202659312
#202655832
#202655692
#202649192
#202649182
#202649162
#501×C1012
#1660101