Finite Group Databasebeta

← #49123282

Subgroups — #49123282

Order 36224.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#31826647C212yesmaximal, complemented
2#31826637C211yesmaximal, characteristic
3#31826264C212yesmaximal, complemented
4#31826262C211yesmaximal, characteristic, complemented
5#563×C283C211yesmaximal, characteristic, complemented
6#22583658C2211yescharacteristic
7#22583641—22no—
8#22583636—21no—
9#22583538—21no—
10#22583538—22no—
11#22583529C2212yes—
12#22583529C2211yescharacteristic
13C4⋊Q8×C283C2212yes—
14#126×C283—22no—
15#125×C283—21no—
16#124×C283C2211yescharacteristic
17#123×C283—21no—
18C22⋊Q8×C283—22no—
19#17985393—41no—
20#17985392—21no—
21#17985392—22no—
22Q8×D283—41no—
23#17985382—42no—
24#17985380—41no—
25#17985379—42no—
26#17985377—41no—
27#17985376—21no—
28#17985376—22no—
29C2×Q8×C283D412yes—
30C2×Q8×C283C2311yescharacteristic
31C2×D4×C283D411yescharacteristic
32C22×C4×C283—21no—
33C4⋊C4×C283—21no—
34C4⋊C4×C283—22no—
35C4⋊C4×C283—21no—
36C4⋊C4×C283—22no—
37C4⋊C4×C283—22no—
38C22⋊C4×C283—22no—
39C22⋊C4×C283—21no—
40C22⋊C4×C283—22no—
41C42×C283D411yescharacteristic
42C42×C283D412yes—
43Dic566—21no—
44Dic566—22no—
45D1132—21no—
46D1132—22no—
47C22⋊D283—41no—
48C4×D283—41no—
49C4×D283—42no—
50C283⋊C8—41no—
51C283⋊C8—42no—
52C23×C283—21no—
53Q8×C283—41no—
54Q8×C283—42no—
55D4×C283—41no—
56C2×C4×C283—21no—
57C2×C4×C283—22no—
58C2×C4×C283—22no—
59C2×C4×C283—81no—
60C2×C4×C283C2×D412yes—
61C2×C4×C283C2×D411yescharacteristic
62C22×C283—42no—
63C22×C283—41no—
64C22×C283#12011yescharacteristic
65C2×D283—41no—
66C4×C283—21no—
67C4×C283—22no—
68C4×C283—41no—
69C4×C283—42no—
70C4×C283—42no—
71Dic283—41no—
72C2×C283—21no—
73C2×C283—81no—
74C2×C283B(4,𝔽2)11yescharacteristic
75D283—81no—
76C283#220311yescharacteristic, complemented, Sylow
77#2203—2831nomaximal, Sylow
78#563D28311yescharacteristic, complemented
79#456—2832no—
80#454—2831no—
81#356—2832no—
82#355—2831no—
832−1+4—2831no—
84#137—5662no—
85AGL(1,ℤ/8)—5661no—
86C4⋊Q8C2×D28312yes—
87#126—22no—
88#125—21no—
89#124C2×D28311yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—5661no—
93#104—5662no—
94#101—2832no—
95#101—2831no—
96#41—5661no—
97#41—5662no—
98#41—11321no—
99C2×Q8—11321no—
100C2×Q8C22⋊D28312yes—
101C2×Q8C22×D28311yescharacteristic
102C2×D4C22⋊D28311yescharacteristic
103C22×C4—21no—
104Q16—11322no—
105SD16—11321no—
106SD16—11322no—
107D8—11321no—
108M16—5661no—
109M16—5662no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D28311yescharacteristic
119C42C22⋊D28312yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—5661no—
124Q8—5662no—
125D4—41no—
126D4—5661no—
127D4—5662no—
128D4—11321no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—11321no—
134C2×C4—11322no—
135C2×C4C2×C22⋊D28312yes—
136C2×C4C2×C22⋊D28311yescharacteristic
137C8—11321no—
138C8—11322no—
139C22—42no—
140C22—41no—
141C22—11321no—
142C22#2258357911yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—11321no—
149C2—21no—
150C2—81no—
151C2—22641no—
152C2#3182665211yescharacteristic
subgroupG/H
centerC2#31826652
commutator, cosocleC2×Q8×C283C23
FrattiniC2×Q8C22×D283
Fitting#563×C283C2
socleC2×C283B(4,𝔽2)
prime radical#31826637C2
hypercenter#563D283
hypocenterC283#2203
LCSG ▷ C2×Q8×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #31826647C2semidirect
3. #31826264C2semidirect
4. #31826262C2semidirect
5. #563×C283C2semidirect
76. C283#2203semidirect
78. #563D283semidirect

Maximal subgroups

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

subgroupindex
#318266472
#318266372
#318262642
#318262622
#563×C2832
#2203283