Finite Group Databasebeta

← #49122429

Subgroups — #49122429

Order 36224.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#31827091C211yesmaximal, characteristic
2#31826326C211yesmaximal, characteristic
3#31826326C212yesmaximal
4#31826231C212yesmaximal
5#342×C283C211yesmaximal, characteristic
6C23×Dic283C2211yescharacteristic
7#22583589—22no—
8#22583587—24no—
9C2×C22⋊Dic283C2211yescharacteristic
10C2×C22⋊Dic283C2212yes—
11C283⋊#95—24no—
12C283⋊#95—22no—
13C2×C22⋊C4×C283C2212yes—
14C2×C22⋊C4×C283C2211yescharacteristic
15C2×Dic566—22no—
16C22×Dic283—22no—
17C22×Dic283—21no—
18C22×Dic283—21no—
19C22×Dic283—21no—
20C22×Dic283—22no—
21C22×Dic283D414yes—
22C22×Dic283Q812yescomplemented
23C4⋊Dic283—22no—
24Dic283⋊C4—24no—
25C22⋊Dic283—22no—
26C22⋊Dic283—22no—
27C22⋊Dic283—22no—
28C24×C283C2311yescharacteristic
29C22×C4×C283—22no—
30C22×C4×C283D412yes—
31C22⋊C4×C283—22no—
32C22⋊C4×C283—22no—
33C22⋊C4×C283—22no—
34Dic566—82no—
35C2×Dic283—24no—
36C2×Dic283—24no—
37C2×Dic283—22no—
38C2×Dic283—41no—
39C2×Dic283—42no—
40C2×Dic283—41no—
41C2×Dic283—44no—
42C2×Dic283—42no—
43C2×Dic283—41no—
44C23×C283—42no—
45C23×C283C2×D412yes—
46C23×C283C2×D411yescharacteristic
47C23×C283C2×Q811yescharacteristic
48C23×C283#4111yescharacteristic
49C23×C283#4112yes—
50C2×C4×C283—22no—
51C2×C4×C283—42no—
52C2×C4×C283—44no—
53C2×C4×C283—42no—
54C22×C283—24no—
55C22×C283—22no—
56C22×C283—41no—
57C22×C283—41no—
58C22×C283—42no—
59C22×C283#12011yescharacteristic
60C22×C283C22⋊Q812yes—
61C22×C283C22⋊Q811yescharacteristic
62C22×C283#12412yes—
63C22×C283#12411yescharacteristic
64C4×C283—42no—
65C4×C283—82no—
66Dic283—42no—
67Dic283—44no—
68Dic283—81no—
69C2×C283—21no—
70C2×C283—22no—
71C2×C283—22no—
72C2×C283—21no—
73C2×C283#39311yescharacteristic
74C2×C283#45812yes—
75C283#203011yescharacteristic, complemented, Sylow
76#2030—2831nomaximal, Sylow
77#543—2831no—
78#381—2832no—
79#381—2831no—
80#342—2832no—
81#342D28311yescharacteristic
82C23×C4—2831no—
83C22⋊Q8—5664no—
84C22⋊Q8—5662no—
85C2×C22⋊C4—2832no—
86C2×C22⋊C4—2831no—
87C2×C22⋊C4C2×D28312yes—
88C2×C22⋊C4C2×D28311yescharacteristic
89#95—5664no—
90#95—5662no—
91C24C22×D28311yescharacteristic
92C2×Q8—5662no—
93C22×C4—22no—
94C22×C4—2832no—
95C22×C4—2834no—
96C22×C4—5662no—
97C22×C4—5661no—
98C22×C4—5661no—
99C22×C4—5662no—
100C22×C4—5661no—
101C22×C4D113212yes—
102C4⋊C4—5664no—
103C4⋊C4—5662no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—5662no—
108C22⋊C4—5662no—
109C22⋊C4—5662no—
110C23—42no—
111C23D4×D28312yes—
112C23C2×D113211yescharacteristic
113C23Q8×D28311yescharacteristic
114C23#1798539111yescharacteristic
115C23#1798539212yes—
116Q8—22642no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—5664no—
122C2×C4—5664no—
123C2×C4—5662no—
124C2×C4—11321no—
125C2×C4—11322no—
126C2×C4—11321no—
127C2×C4—11324no—
128C2×C4—11322no—
129C2×C4—11321no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#2258358011yescharacteristic
136C22#2258358812yes—
137C22#2258359011yescharacteristic
138C22#2258359612yes—
139C22#2258359811yescharacteristic
140C4—42no—
141C4—82no—
142C4—11322no—
143C4—11324no—
144C4—22641no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#3182637811yescharacteristic
150C2#3182665612yes—
subgroupG/H
centerC22#22583580
commutator, prime radical, cosocleC24×C283C23
FrattiniC24C22×D283
Fitting#342×C283C2
socleC22×C283#120
hypercenter#342D283
hypocenterC283#2030
LCSG ▷ C24×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic283Q8semidirect
75. C283#2030semidirect

Maximal subgroups

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

subgroupindex
#318270912
#318263262
#318262312
#342×C2832
#2030283