Finite Group Databasebeta

← #49120316

Subgroups — #49120316

Order 36224.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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#31827100C211yesmaximal, characteristic, complemented
2#31826841C211yesmaximal, characteristic, complemented
3#31826661C212yesmaximal, complemented
4#31826192C212yesmaximal, complemented
5#331×C283C211yesmaximal, characteristic, complemented
6C2×D4×D283—22no—
7C2×D2264—22no—
8C2×#17985377—22no—
9#22583607C2211yescharacteristic, complemented
10#22583581—22no—
11#22583578—22no—
12C4×D1132C2212yes—
13#22583532—22no—
14C4×C283⋊C8C2211yescharacteristic, complemented
15#127×C283C2211yescharacteristic
16C4⋊C8×C283C2212yescomplemented
17C23×D283—22no—
18C2×D1132—21no—
19C2×D1132—22no—
20C2×D1132D412yes—
21C2×C22⋊D283—24no—
22D4×D283—44no—
23D4×D283—44no—
24C2×C4×D283—22no—
25D2264—44no—
26#17985377—44no—
27C2×C283⋊C8—21no—
28C4⋊Dic283D412yes—
29C22⋊Dic283—22no—
30(C2×D283)⋊C4—22no—
31C2×D4×C283—22no—
32C2×D4×C283D412yes—
33C2×C8×C283—22no—
34C42×C283C2311yescharacteristic
35C22×D283—22no—
36C22×D283—48no—
37C22×D283—41no—
38C22×D283—82no—
39D1132—22no—
40D1132—42no—
41D1132—42no—
42D1132—42no—
43D1132D814yescomplemented
44C22⋊D283—48no—
45C2×Dic283—22no—
46C4×D283—42no—
47C4×D283—42no—
48C283⋊C8—22no—
49C23×C283—22no—
50D4×C283—42no—
51D4×C283—42no—
52D4×C283—42no—
53C2×C4×C283C2×D412yes—
54C2×C4×C283C2×D411yescharacteristic
55C8×C283—42no—
56C22×C283—44no—
57C22×C283—42no—
58C22×C283#12011yescharacteristic
59C2×D283—24no—
60C2×D283—42no—
61C2×D283—42no—
62C2×D283—88no—
63C2×D283—82no—
64C2×D283—81no—
65C4×C283—21no—
66C4×C283C2×D812yes—
67C4×C283AGL(1,ℤ/8)12yes—
68Dic283—42no—
69C2×C283—82no—
70C2×C283#44712yes—
71C2×C283#45311yescharacteristic
72D283—44no—
73D283—161no—
74C283#161911yescharacteristic, complemented, Sylow
75#1619—2831nomaximal, Sylow
76D42—2831no—
77#493—2831no—
78#459—2832no—
79#331D28311yescharacteristic, complemented
80D4⋊C8—2832no—
81C22×D4—5662no—
82C2×D8—5662no—
83C2×D8—5662no—
84#127—2831no—
85#127C2×D28311yescharacteristic
86#121—5662no—
87#120—5662no—
88C4×D4—2832no—
89C4⋊C8C2×D28312yescomplemented
90#102—5662no—
91C4×C8—2831no—
92C24—5662no—
93C2×D4—22no—
94C2×D4—2832no—
95C2×D4—5664no—
96C2×D4—5662no—
97C2×D4—5661no—
98C2×D4—11324no—
99C2×D4—11324no—
100C2×D4D113212yes—
101C22×C4—5662no—
102D8—11324no—
103D8—11324no—
104C2×C8—22no—
105C2×C8—5661no—
106C4⋊C4—2832no—
107C22⋊C4—5662no—
108C22⋊C4—5662no—
109C42C22×D28311yescharacteristic
110C23—22no—
111C23—5662no—
112C23—11328no—
113C23—11321no—
114C23—22642no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—2834no—
119D4—5662no—
120D4—11322no—
121D4—11328no—
122D4—11322no—
123D4—11322no—
124C2×C4—5662no—
125C2×C4—11322no—
126C2×C4—11322no—
127C2×C4D4×D28312yes—
128C2×C4C2×D113211yescharacteristic
129C8—42no—
130C8—5662no—
131C22—44no—
132C22—42no—
133C22—5664no—
134C22—11322no—
135C22—11322no—
136C22—22648no—
137C22—22642no—
138C22—22641no—
139C22#2258358011yescharacteristic
140C4—21no—
141C4—11322no—
142C4D8×D28312yes—
143C4#2258363712yes—
144C2—82no—
145C2—11324no—
146C2—45281no—
147C2#3182660012yes—
148C2#3182663511yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#331×C283C2
socleC22×C283#120
hypercenter#331D283
hypocenterC283#1619
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #31827100C2semidirect
2. #31826841C2semidirect
3. #31826661C2semidirect
4. #31826192C2semidirect
5. #331×C283C2semidirect
9. #22583607C22semidirect
14. C4×C283⋊C8C22semidirect
16. C4⋊C8×C283C22semidirect
43. D1132D8semidirect
74. C283#1619semidirect
79. #331D283semidirect
89. C4⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271002
#318268412
#318266612
#318261922
#331×C2832
#1619283