Finite Group Databasebeta

← #44506396

Subgroups — #44506396

Order 30848.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#29508279C211yesmaximal, characteristic
2#29507856C212yesmaximal, complemented
3#29507450C212yesmaximal, complemented
4#29507441C211yesmaximal, characteristic, complemented
5#393×C241C211yesmaximal, characteristic, complemented
6C22×C22⋊D241—21no—
7C2×#16838757—22no—
8#21440626—22no—
9#21440616—21no—
10#21440616C2212yes—
11#21440612—22no—
12#21440566—22no—
13C22⋊#14515349C2212yescomplemented
14C22×Q8×C241C2211yescharacteristic, complemented
15C2×C22⋊C4×C241—21no—
16C2×C22⋊C4×C241C2211yescharacteristic
17#95×C241—21no—
18#95×C241C2211yescharacteristic, complemented
19C23×D241—21no—
20C2×D964—21no—
21C2×D964D412yes—
22C2×C22⋊D241—22no—
23C2×C22⋊D241—22no—
24C2×C22⋊D241—42no—
25C2×C22⋊D241—42no—
26C22×Dic241—21no—
27#16838757—44no—
28C2×#14515349—22no—
29C4⋊Dic241—21no—
30C4⋊Dic241D412yes—
31#16838747—22no—
32#16838747—22no—
33C24×C241D411yescharacteristic
34C2×Q8×C241—22no—
35C2×Q8×C241—42no—
36C22×C4×C241—21no—
37C22×C4×C241—21no—
38C22×C4×C241D411yescharacteristic
39C22×C4×C241C2311yescharacteristic
40C22⋊C4×C241—22no—
41C22⋊C4×C241—42no—
42C22×D241—22no—
43C22×D241—42no—
44C22×D241—81no—
45C22×D241—81no—
46D964—42no—
47D964—81no—
48C22⋊D241—44no—
49C22⋊D241—42no—
50C22⋊D241—42no—
51C2×Dic241—22no—
52C2×Dic241—42no—
53#14515349—42no—
54C23×C241—21no—
55C23×C241—21no—
56C23×C241—41no—
57C23×C241—41no—
58C23×C241SD1612yescomplemented
59C23×C241C2×D411yescharacteristic
60Q8×C241—24no—
61Q8×C241—42no—
62C2×C4×C241—22no—
63C2×C4×C241—21no—
64C2×C4×C241—41no—
65C2×C4×C241—41no—
66C2×C4×C241—41no—
67C2×C4×C241—41no—
68C2×C4×C241—41no—
69C2×C4×C241—41no—
70C2×C4×C241—41no—
71C2×C4×C241C2×D412yes—
72C22×C241—21no—
73C22×C241—21no—
74C22×C241—22no—
75C22×C241—42no—
76C22×C241—41no—
77C22×C241—41no—
78C22×C241—81no—
79C22×C241#12011yescharacteristic
80C22×C241C2×SD1611yescharacteristic
81C22×C241AGL(1,ℤ/8)11yescharacteristic
82C2×D241—42no—
83C2×D241—42no—
84C2×D241—42no—
85C2×D241—84no—
86C4×C241—22no—
87C4×C241—41no—
88C4×C241—42no—
89C4×C241—81no—
90Dic241—42no—
91C2×C241—21no—
92C2×C241—21no—
93C2×C241—42no—
94C2×C241#45011yescharacteristic
95C2×C241#45511yescharacteristic
96C2×C241B(4,𝔽2)11yescharacteristic
97D241—82no—
98C241#160111yescharacteristic, complemented, Sylow
99#1601—2411nomaximal, Sylow
100#534—2411no—
101#448—2412no—
102#393D24111yescharacteristic, complemented
103#327—2412no—
104#323—2411no—
105C22×Q8C2×D24111yescharacteristic, complemented
106C22×D4—4821no—
107C2×SD16—4822no—
108#123—4822no—
109#121—4821no—
110#121—2412no—
111#120—4822no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D24111yescharacteristic
114#103—4822no—
115C22⋊C8—2412no—
116#95—21no—
117#95C2×D24111yescharacteristic, complemented
118C24—4821no—
119C24C22⋊D24111yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—9642no—
123C2×D4—9642no—
124C2×D4—4821no—
125C2×D4—4822no—
126C2×D4—4822no—
127C2×D4—2412no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—4821no—
131C22×C4C22⋊D24111yescharacteristic
132C22×C4C22×D24111yescharacteristic
133SD16—9644no—
134C2×C8—4822no—
135C4⋊C4—4821no—
136C4⋊C4—2412no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—4822no—
140C22⋊C4—4822no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—19281no—
146C23—19281no—
147C23—9642no—
148C23—4822no—
149C23#1683875712yescomplemented
150C23C2×C22⋊D24111yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—19281no—
154D4—9642no—
155D4—9644no—
156D4—9642no—
157D4—9642no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—9642no—
168C2×C4—4822no—
169C2×C4D4×D24112yes—
170C8—9642no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—19284no—
179C22—9642no—
180C22—9642no—
181C22—9642no—
182C22#2144061011yescharacteristic
183C22C2×#1683875711yescharacteristic
184C22#2144066711yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—9642no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—19282no—
194C2#2950786711yescharacteristic
195C2#2950789411yescharacteristic
196C2#2950790311yescharacteristic
subgroupG/H
centerC22#21440610
commutator, cosocleC22×C4×C241C23
FrattiniC22×C4C22×D241
Fitting#393×C241C2
socleC22×C241#120
prime radical#29508279C2
hypercenter#393D241
hypocenterC241#1601
LCSG ▷ C22×C4×C241 ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #29507856C2semidirect
3. #29507450C2semidirect
4. #29507441C2semidirect
5. #393×C241C2semidirect
13. C22⋊#14515349C22semidirect
14. C22×Q8×C241C22semidirect
18. #95×C241C22semidirect
58. C23×C241SD16semidirect
98. C241#1601semidirect
102. #393D241semidirect
105. C22×Q8C2×D241semidirect
117. #95C2×D241semidirect
149. C23#16838757semidirect

Maximal subgroups

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

subgroupindex
#295082792
#295078562
#295074502
#295074412
#393×C2412
#1601241