Finite Group Databasebeta

← #27170340

Subgroups — #27170340

Order 12928.

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

normal subgroups only

This group has 208 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#20265888C211yesmaximal, characteristic
2#20265527C211yesmaximal, characteristic, complemented
3#20265463C211yesmaximal, characteristic, complemented
4#20265383C211yesmaximal, characteristic
5#20264907C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#333×C101C211yesmaximal, characteristic, complemented
8C2×#13108063—21no—
9#15692255C2211yescharacteristic
10#15692246—21no—
11#15692235—21no—
12#15692231—22no—
13C4×Dic202C2211yescharacteristic
14C4×D404C2211yescharacteristic
15#15692217—21no—
16C4⋊C4×D101—21no—
17#15692187—21no—
18#15692180—21no—
19#15692180—21no—
20#15692179—22no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D411yescharacteristic
27C2×D404D411yescharacteristic
28C2×C4×D101—21no—
29C2×C4×D101—22no—
30#13108063—42no—
31C2×#11717579—21no—
32C4⋊Dic101—22no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D411yescharacteristic
37#13108053—22no—
38#13108053—21no—
39#13108051—22no—
40#13108051—21no—
41C4×Dic101—21no—
42C2×Q8×C101D412yes—
43C2×C8×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—21no—
49Q8×C101—42no—
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—
59D404SD1612yes—
60C4×D101—41no—
61C4×D101—41no—
62C4×D101—82no—
63Dic202—21no—
64Dic202—21no—
65Dic202—41no—
66Dic202—41no—
67C8×C101—41no—
68C8×C101—41no—
69#11717579—41no—
70C22×C101#12011yescharacteristic
71C2×D101—22no—
72C2×D101—41no—
73C2×D101—41no—
74C4×C101—21no—
75C4×C101—42no—
76C4×C101C2×SD1611yescharacteristic
77C4×C101#13511yescharacteristic
78C4×C101#13711yescharacteristic
79C4×C101#13711yescharacteristic
80Dic101—41no—
81Dic101—41no—
82Dic101—41no—
83Dic101—81no—
84C2×C101#44811yescharacteristic
85C2×C101#45211yescharacteristic
86C2×C101#45511yescharacteristic
87D101—42no—
88#1643—1011nomaximal, Sylow
89C101#164311yescharacteristic, complemented, Sylow
90#554—1011no—
91#487—1011no—
92#475—1011no—
93#461—1011no—
94#333D10111yescharacteristic, complemented
95Q8⋊C8—1011no—
96D4⋊C8—1011no—
97C2×SD16—2021no—
98C4⋊Q8—1011no—
99C4⋊Q8C2×D10111yescharacteristic
100#125—2021no—
101C22⋊Q8—2021no—
102C22⋊Q8—2022no—
103C4×Q8—1011no—
104C4×D4—1011no—
105C4×D4—2021no—
106C2×C4⋊C4—2021no—
107#106—2021no—
108C4⋊C8C2×D10111yescharacteristic
109C4⋊C8C2×D10111yescharacteristic
110#103—2021no—
111#103—2021no—
112#103—2022no—
113C4×C8—1011no—
114C2×Q8—1011no—
115C2×Q8—2021no—
116C2×Q8D40412yes—
117C2×D4—1011no—
118C22×C4—2022no—
119C22×C4—2021no—
120SD16—4042no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—2021no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1011no—
127C4⋊C4—1011no—
128C4⋊C4—2022no—
129C4⋊C4—2022no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C22⋊C4—2022no—
134C22⋊C4—2021no—
135C42—2021no—
136C42C22×D10111yescharacteristic
137C23—2021no—
138Q8—42no—
139Q8—4041no—
140Q8—4041no—
141Q8—2021no—
142Q8—2021no—
143D4—1012no—
144D4—2021no—
145C2×C4—22no—
146C2×C4—8082no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—4041no—
150C2×C4—2021no—
151C2×C4—2021no—
152C2×C4—2021no—
153C2×C4D4×D10111yescharacteristic
154C2×C4D4×D10111yescharacteristic
155C2×C4C2×D40411yescharacteristic
156C8—41no—
157C8—41no—
158C8—4041no—
159C22—4041no—
160C22—4041no—
161C22—2022no—
162C22#1569222511yescharacteristic
163C4—21no—
164C4—42no—
165C4—8081no—
166C4—4041no—
167C4—4041no—
168C4—4041no—
169C4SD16×D10111yescharacteristic
170C4#1569227611yescharacteristic
171C4#1569228711yescharacteristic
172C4#1569228711yescharacteristic
173C2—4042no—
174C2#2026531711yescharacteristic
175C2#2026534211yescharacteristic
176C2#2026535711yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#333×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#333D101
hypocenterC101#1643
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20265527C2semidirect
3. #20265463C2semidirect
5. #20264907C2semidirect
7. #333×C101C2semidirect
89. C101#1643semidirect
94. #333D101semidirect

Maximal subgroups

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

subgroupindex
#202658882
#202655272
#202654632
#202653832
#202649072
#202649042
#333×C1012
#1643101