Finite Group Databasebeta

← #49123281

Subgroups — #49123281

Order 36224.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#31827215C211yesmaximal, characteristic, complemented
2#31826648C212yesmaximal, complemented
3#31826263C212yesmaximal, complemented
4#31826262C211yesmaximal, characteristic, complemented
5#454×C283C211yesmaximal, characteristic
6#22583640—22no—
7#22583608C2212yes—
8#22583603—22no—
9#22583602—21no—
10#22583591—21no—
11#22583587—22no—
12#22583537—22no—
13#22583529C2212yes—
142−1+4×C283C2211yescharacteristic
15AGL(1,ℤ/8)×C283—21no—
16#124×C283C2211yescharacteristic
17#104×C283—21no—
18#101×C283C2211yescharacteristic
19C2×Dic566D412yes—
20C22×Dic283—21no—
21#17985382—42no—
22#17985380—42no—
23#17985376—22no—
24C4⋊Dic283—21no—
25C4⋊Dic283—21no—
26Dic283⋊C4—22no—
27Dic283⋊C4—22no—
28Dic283⋊C4—22no—
29C22⋊Dic283—22no—
30C22⋊Dic283—21no—
31C4×Dic283D412yes—
32#41×C283—22no—
33#41×C283—21no—
34#41×C283—41no—
35C2×Q8×C283—41no—
36C2×Q8×C283C2311yescharacteristic
37C2×D4×C283D411yescharacteristic
38SD16×C283—41no—
39D8×C283—41no—
40M16×C283—21no—
41C22⋊C4×C283—22no—
42C42×C283D411yescharacteristic
43Dic566—42no—
44C2×Dic283—22no—
45C2×Dic283—22no—
46C2×Dic283—81no—
47C283⋊C8—42no—
48C23×C283—21no—
49Q8×C283—21no—
50Q8×C283—22no—
51Q8×C283—41no—
52D4×C283—21no—
53D4×C283—22no—
54D4×C283—41no—
55D4×C283—41no—
56C2×C4×C283—21no—
57C2×C4×C283—41no—
58C2×C4×C283—42no—
59C2×C4×C283C2×D412yes—
60C2×C4×C283C2×D411yescharacteristic
61C8×C283—41no—
62C22×C283—42no—
63C22×C283—41no—
64C22×C283—41no—
65C22×C283#12011yescharacteristic
66C4×C283—21no—
67C4×C283—22no—
68C4×C283—41no—
69C4×C283—41no—
70Dic283—42no—
71Dic283—42no—
72C2×C283—21no—
73C2×C283—81no—
74C2×C283—81no—
75C2×C283B(4,𝔽2)11yescharacteristic
76C283#220311yescharacteristic, complemented, Sylow
77#2203—2831nomaximal, Sylow
78#563—2831no—
79#456—2832no—
80#454D28311yescharacteristic
81#356—2832no—
82#355—2831no—
832−1+4C2×D28311yescharacteristic
84#137—5662no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—2832no—
87#126—5662no—
88#125—5661no—
89#124C2×D28311yescharacteristic
90#123—5661no—
91C22⋊Q8—5662no—
92#104—21no—
93#104—5662no—
94#101—2832no—
95#101C2×D28311yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—2832no—
101C2×Q8C22×D28311yescharacteristic
102C2×D4C22⋊D28311yescharacteristic
103C22×C4—5661no—
104Q16—11322no—
105SD16—41no—
106SD16—11322no—
107D8—41no—
108M16—21no—
109M16—5662no—
110C4⋊C4—5662no—
111C4⋊C4—5661no—
112C4⋊C4—5661no—
113C4⋊C4—5662no—
114C4⋊C4—5662no—
115C22⋊C4—22no—
116C22⋊C4—5662no—
117C22⋊C4—5661no—
118C42—2832no—
119C42C22⋊D28311yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—11322no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—5662no—
133C2×C4—5662no—
134C2×C4—22641no—
135C2×C4D4×D28312yes—
136C2×C4C2×C22⋊D28311yescharacteristic
137C8—41no—
138C8—11322no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#2258357811yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—11322no—
148C4—11322no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#3182665111yescharacteristic
subgroupG/H
centerC2#31826651
commutator, cosocleC2×Q8×C283C23
FrattiniC2×Q8C22×D283
Fitting, prime radical#454×C283C2
socleC2×C283B(4,𝔽2)
hypercenter#454D283
hypocenterC283#2203
LCSG ▷ C2×Q8×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #31827215C2semidirect
2. #31826648C2semidirect
3. #31826263C2semidirect
4. #31826262C2semidirect
76. C283#2203semidirect

Maximal subgroups

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

subgroupindex
#318272152
#318266482
#318262632
#318262622
#454×C2832
#2203283