Finite Group Databasebeta

← #44506402

Subgroups — #44506402

Order 30848.

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

normal subgroups only

This group has 256 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#29507872C212yesmaximal, complemented
2#29507629C211yesmaximal, characteristic
3#29507454C212yesmaximal, complemented
4#29507441C211yesmaximal, characteristic
5#543×C241C211yesmaximal, characteristic
6C22×Dic482C2211yescharacteristic
7C2×#16838760—22no—
8C2×C22⋊Dic241—21no—
9#21440565—22no—
10C22⋊#14515349C2212yes—
11#21440549—21no—
12#21440549C2211yescharacteristic
13C22⋊Q8×C241—22no—
14C22⋊Q8×C241—21no—
15C22⋊Q8×C241—22no—
16C22⋊Q8×C241C2212yes—
17C2×C22⋊C4×C241—21no—
18C2×C22⋊C4×C241C2211yescharacteristic
19C2×Dic482—22no—
20C2×Dic482—42no—
21C22×Dic241—21no—
22C22×Dic241—21no—
23#16838760—44no—
24C2×#14515349—22no—
25C22⋊Dic241—42no—
26C24×C241D411yescharacteristic
27C2×Q8×C241—21no—
28C2×Q8×C241D412yes—
29C22×C4×C241—21no—
30C22×C4×C241—21no—
31C22×C4×C241D411yescharacteristic
32C22×C4×C241C2311yescharacteristic
33C4⋊C4×C241—21no—
34C4⋊C4×C241—22no—
35C4⋊C4×C241—22no—
36C4⋊C4×C241D412yes—
37C22⋊C4×C241—22no—
38C22⋊C4×C241—22no—
39C22⋊C4×C241—22no—
40Dic482—24no—
41Dic482—42no—
42C2×Dic241—22no—
43C2×Dic241—41no—
44C2×Dic241—41no—
45C2×Dic241—41no—
46C2×Dic241—41no—
47C2×Dic241—41no—
48#14515349—42no—
49C23×C241—21no—
50C23×C241—21no—
51C23×C241—41no—
52C23×C241—41no—
53C23×C241Q1612yescomplemented
54C23×C241C2×D411yescharacteristic
55Q8×C241—42no—
56Q8×C241—81no—
57C2×C4×C241—22no—
58C2×C4×C241—21no—
59C2×C4×C241—22no—
60C2×C4×C241—41no—
61C2×C4×C241—41no—
62C2×C4×C241—81no—
63C2×C4×C241—81no—
64C2×C4×C241C2×D412yes—
65C22×C241—21no—
66C22×C241—21no—
67C22×C241—22no—
68C22×C241—42no—
69C22×C241—41no—
70C22×C241—41no—
71C22×C241—81no—
72C22×C241#12011yescharacteristic
73C22×C241C2×Q1611yescharacteristic
74C22×C241#13711yescharacteristic
75C4×C241—22no—
76C4×C241—41no—
77C4×C241—42no—
78C4×C241—42no—
79Dic241—42no—
80Dic241—81no—
81C2×C241—21no—
82C2×C241—21no—
83C2×C241—42no—
84C2×C241#45111yescharacteristic
85C2×C241#45511yescharacteristic
86C2×C241B(4,𝔽2)11yescharacteristic
87C241#160211yescharacteristic, complemented, Sylow
88#1602—2411nomaximal, Sylow
89#543D24111yescharacteristic
90#451—2412no—
91#393—2411no—
92#328—2412no—
93#323—2411no—
94C22×Q8—2411no—
95C2×Q16—4822no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D24112yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—4821no—
102C2×C22⋊C4C2×D24111yescharacteristic
103#103—4822no—
104C22⋊C8—2412no—
105#95—4821no—
106#95—2411no—
107C24C22⋊D24111yescharacteristic
108C2×Q8—21no—
109C2×Q8—9642no—
110C2×Q8—4822no—
111C2×Q8C22⋊D24112yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—4821no—
115C22×C4—4821no—
116C22×C4C22⋊D24111yescharacteristic
117C22×C4C22×D24111yescharacteristic
118Q16—9644no—
119C2×C8—4822no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D24112yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—9642no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1683876012yescomplemented
133C23C2×C22⋊D24111yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—9642no—
137Q8—4824no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—9641no—
146C2×C4—9641no—
147C2×C4—9641no—
148C2×C4—9641no—
149C2×C4—9641no—
150C2×C4—4822no—
151C2×C4C2×C22⋊D24112yes—
152C8—9642no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2144061111yescharacteristic
161C22C2×#1683876011yescharacteristic
162C22#2144067311yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—19281no—
168C4—9642no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2950787311yescharacteristic
173C2#2950789511yescharacteristic
174C2#2950790411yescharacteristic
subgroupG/H
centerC22#21440611
commutator, cosocleC22×C4×C241C23
FrattiniC22×C4C22×D241
Fitting#543×C241C2
socleC22×C241#120
prime radicalC24×C241D4
hypercenter#543D241
hypocenterC241#1602
LCSG ▷ C22×C4×C241 ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #29507872C2semidirect
3. #29507454C2semidirect
53. C23×C241Q16semidirect
87. C241#1602semidirect
132. C23#16838760semidirect

Maximal subgroups

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

subgroupindex
#295078722
#295076292
#295074542
#295074412
#543×C2412
#1602241