Finite Group Databasebeta

← #27170305

Subgroups — #27170305

Order 12928.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265901C211yesmaximal, characteristic
2#20265529C211yesmaximal, characteristic, complemented
3#20265473C211yesmaximal, characteristic, complemented
4#20265468C211yesmaximal, characteristic
5#20264907C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#332×C101C211yesmaximal, characteristic, complemented
8#15692255C2211yescharacteristic
9#15692250—22no—
10#15692244—21no—
11#15692235—21no—
12Q8×Dic101—21no—
13C4×Dic202C2211yescharacteristic
14C4×D404C2211yescharacteristic
15#15692213—21no—
16#15692189—21no—
17#15692187—21no—
18#15692180—21no—
19#15692178—22no—
20#15692177—21no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D411yescharacteristic
27C2×D404D411yescharacteristic
28C2×C4×D101—21no—
29C2×#11717579—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101—21no—
32C4⋊Dic101—21no—
33C4⋊Dic101D411yescharacteristic
34C4⋊Dic101D411yescharacteristic
35#13108053—21no—
36#13108051—22no—
37#13108051—22no—
38#13108051—21no—
39C4×Dic101—22no—
40C4×Dic101—21no—
41C2×Q8×C101—21no—
42C2×C8×C101—21no—
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D412yes—
46C42×C101C2311yescharacteristic
47C22×D101—21no—
48Q8×C101—41no—
49Q8×C101—41no—
50C2×C4×C101—22no—
51C2×C4×C101C2×D411yescharacteristic
52C2×C4×C101C2×D411yescharacteristic
53C2×C4×C101C2×D411yescharacteristic
54C2×Dic101—21no—
55C2×Dic101—21no—
56C2×Dic101—21no—
57C2×Dic101—41no—
58D404—21no—
59D404—21no—
60C4×D101—41no—
61C4×D101—41no—
62Dic202—21no—
63Dic202—21no—
64Dic202—42no—
65C8×C101—41no—
66C8×C101—41no—
67#11717579—41no—
68C22×C101#12011yescharacteristic
69C2×D101—41no—
70C2×D101—41no—
71C2×D101—41no—
72C4×C101—21no—
73C4×C101—42no—
74C4×C101#13511yescharacteristic
75C4×C101#13511yescharacteristic
76C4×C101#13711yescharacteristic
77C4×C101#13711yescharacteristic
78Dic101—41no—
79Dic101—41no—
80Dic101—41no—
81Dic101—81no—
82C2×C101#45211yescharacteristic
83C2×C101#45211yescharacteristic
84C2×C101#45411yescharacteristic
85D101—81no—
86#1638—1011nomaximal, Sylow
87C101#163811yescharacteristic, complemented, Sylow
88#556—1011no—
89#487—1011no—
90#477—1011no—
91#476—1011no—
92#332D10111yescharacteristic, complemented
93Q8⋊C8—1011no—
94D4⋊C8—1011no—
95C4⋊Q8—1011no—
96C4⋊Q8C2×D10111yescharacteristic
97#126—2022no—
98#124—2021no—
99C22⋊Q8—2021no—
100C4×Q8—1011no—
101C4×Q8—2021no—
102C4×D4—1011no—
103#117—2021no—
104#107—2021no—
105#106—2021no—
106C4⋊C8C2×D10111yescharacteristic
107C4⋊C8C2×D10111yescharacteristic
108#103—2021no—
109#103—2022no—
110#102—2021no—
111C4×C8—1011no—
112C2×Q8—21no—
113C2×Q8—1011no—
114C2×Q8—2021no—
115C2×D4—1011no—
116C22×C4—2021no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—2021no—
120C4⋊C4—21no—
121C4⋊C4—1011no—
122C4⋊C4—1011no—
123C4⋊C4—2022no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4D40412yes—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C22⋊C4—2021no—
131C42—2022no—
132C42—2021no—
133C42C22×D10111yescharacteristic
134C23—2021no—
135Q8—41no—
136Q8—41no—
137Q8—4042no—
138Q8—2021no—
139Q8—2021no—
140D4—2021no—
141D4—2021no—
142C2×C4—22no—
143C2×C4—4041no—
144C2×C4—4041no—
145C2×C4—4041no—
146C2×C4—2021no—
147C2×C4—2021no—
148C2×C4—2021no—
149C2×C4D4×D10111yescharacteristic
150C2×C4D4×D10111yescharacteristic
151C2×C4C2×D40411yescharacteristic
152C8—41no—
153C8—41no—
154C8—4041no—
155C22—4041no—
156C22—4041no—
157C22—4041no—
158C22#1569222511yescharacteristic
159C4—21no—
160C4—42no—
161C4—8081no—
162C4—4041no—
163C4—4041no—
164C4—4041no—
165C4#1569227411yescharacteristic
166C4#1569227611yescharacteristic
167C4#1569228411yescharacteristic
168C4#1569228711yescharacteristic
169C2—8081no—
170C2#2026534011yescharacteristic
171C2#2026534211yescharacteristic
172C2#2026535211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#332D101
hypocenterC101#1638
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20265529C2semidirect
3. #20265473C2semidirect
5. #20264907C2semidirect
7. #332×C101C2semidirect
87. C101#1638semidirect
92. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202659012
#202655292
#202654732
#202654682
#202649072
#202649042
#332×C1012
#1638101