Finite Group Databasebeta

← #49122419

Subgroups — #49122419

Order 36224.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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#31826473C212yesmaximal
2#31826370C211yesmaximal, characteristic
3#31826234C211yesmaximal, characteristic
4#31826224C212yesmaximal
5#544×C283C211yesmaximal, characteristic
6C22×Dic566C2211yescharacteristic
7C2×#17985376C2212yes—
8C2×C4×Dic283C2211yescharacteristic
9#22583533—22no—
10#22583533—22no—
11C283⋊#95—22no—
12C4⋊Q8×C283—22no—
13#125×C283—22no—
14C22⋊Q8×C283—22no—
15#117×C283C2212yes—
16C2×C4⋊C4×C283C2211yescharacteristic
17C2×Dic566—22no—
18C2×Dic566—21no—
19C2×Dic566—22no—
20C2×Dic566Q812yescomplemented
21C22×Dic283—21no—
22C22×Dic283—21no—
23#17985376—22no—
24#17985376—22no—
25C2×C283⋊C8—22no—
26C4×Dic283—22no—
27C2×Q8×C283—22no—
28C22×C4×C283D412yes—
29C22×C4×C283C2311yescharacteristic
30C4⋊C4×C283—22no—
31C4⋊C4×C283—24no—
32C4⋊C4×C283—22no—
33C4⋊C4×C283D412yes—
34C4⋊C4×C283D412yes—
35C22⋊C4×C283—22no—
36C42×C283—22no—
37Dic566—24no—
38Dic566—42no—
39C2×Dic283—22no—
40C2×Dic283—22no—
41C2×Dic283—41no—
42C2×Dic283—41no—
43C2×Dic283—41no—
44C2×Dic283—41no—
45C283⋊C8—42no—
46C23×C283C2×D411yescharacteristic
47Q8×C283—82no—
48C2×C4×C283—22no—
49C2×C4×C283—22no—
50C2×C4×C283—22no—
51C2×C4×C283—42no—
52C2×C4×C283C2×D412yes—
53C2×C4×C283C2×Q811yescharacteristic
54C2×C4×C283#4112yes—
55C2×C4×C283#4111yescharacteristic
56C22×C283—21no—
57C22×C283—21no—
58C22×C283#12011yescharacteristic
59C22×C283C22⋊Q811yescharacteristic
60C22×C283#12411yescharacteristic
61C4×C283—42no—
62C4×C283—42no—
63C4×C283—42no—
64C4×C283C22⋊Q812yes—
65C4×C283#12412yes—
66Dic283—42no—
67Dic283—42no—
68C2×C283—21no—
69C2×C283—21no—
70C2×C283#39311yescharacteristic
71C2×C283#45511yescharacteristic
72C2×C283#45611yescharacteristic
73C283#202811yescharacteristic, complemented, Sylow
74#2028—2831nomaximal, Sylow
75#544D28311yescharacteristic
76#419—2832no—
77#391—2831no—
78#343—2831no—
79#339—2832no—
80C22×Q8—2831no—
81C2×M16—2832no—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C2×D28312yes—
86C2×C4⋊C4C2×D28311yescharacteristic
87C2×C42—2831no—
88#103—5662no—
89#103—5662no—
90#95—5662no—
91C2×Q8—22no—
92C2×Q8—2832no—
93C2×Q8—5661no—
94C2×Q8—5662no—
95C2×Q8—5662no—
96C22×C4—5661no—
97C22×C4—5661no—
98C22×C4C22⋊D28312yes—
99C22×C4C22×D28311yescharacteristic
100M16—5662no—
101M16—5662no—
102C2×C8—5662no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4C22⋊D28312yes—
107C4⋊C4C22⋊D28312yes—
108C22⋊C4—22no—
109C42—22no—
110C42—5662no—
111C23C2×C22⋊D28311yescharacteristic
112Q8—82no—
113Q8—5664no—
114Q8—11322no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—5662no—
120C2×C4—5662no—
121C2×C4—11321no—
122C2×C4—11321no—
123C2×C4—11321no—
124C2×C4—11321no—
125C2×C4C2×C22⋊D28312yes—
126C2×C4Q8×D28311yescharacteristic
127C2×C4#1798539312yes—
128C2×C4#1798539311yescharacteristic
129C8—11322no—
130C22—21no—
131C22—21no—
132C22#2258357911yescharacteristic
133C22#2258358611yescharacteristic
134C22#2258359911yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—11322no—
139C4—11322no—
140C4#2258358612yes—
141C4#2258359912yes—
142C2—21no—
143C2—21no—
144C2#3182637611yescharacteristic
145C2#3182664311yescharacteristic
146C2#3182664811yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#544×C283C2
socleC22×C283#120
prime radicalC23×C283C2×D4
hypercenter#544D283
hypocenterC283#2028
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. C2×Dic566Q8semidirect
73. C283#2028semidirect

Maximal subgroups

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

subgroupindex
#318264732
#318263702
#318262342
#318262242
#544×C2832
#2028283