Finite Group Databasebeta

← #44506403

Subgroups — #44506403

Order 30848.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#29508342C211yesmaximal, characteristic
2#29507875C211yesmaximal, characteristic, complemented
3#29507872C211yesmaximal, characteristic, complemented
4#29507630C211yesmaximal, characteristic
5#29507453C211yesmaximal, characteristic, complemented
6#29507440C211yesmaximal, characteristic
7#328×C241C211yesmaximal, characteristic, complemented
8C22×Dic482C2211yescharacteristic
9C2×Dic964—21no—
10C2×#16838760—21no—
11#21440622—21no—
12#21440621C2211yescharacteristic
13#21440620—21no—
14#21440620—21no—
15#21440619—21no—
16#21440618—21no—
17C2×#16838747—21no—
18C2×#16838747C2211yescharacteristic
19C22⋊C4×D241—21no—
20#21440567—21no—
21#21440565—21no—
22C22⋊#14515349C2211yescharacteristic
23#21440549—21no—
24C22⋊Q8×C241C2211yescharacteristic
25C22⋊C8×C241C2211yescharacteristic
26#95×C241C2211yescharacteristic
27C23×D241D411yescharacteristic
28C2×Dic482—21no—
29C2×Dic482—21no—
30C2×Dic482—21no—
31C2×Dic482—42no—
32C2×Dic482D411yescharacteristic
33C22×Dic241—21no—
34C22×Dic241D411yescharacteristic
35C2×C4×D241—21no—
36C2×C4×D241—21no—
37Dic964—42no—
38#16838760—42no—
39C2×#14515349—21no—
40C4⋊Dic241—21no—
41C4⋊Dic241D411yescharacteristic
42#16838749—21no—
43#16838749—21no—
44#16838749—21no—
45C22⋊Dic241—21no—
46#16838747—21no—
47#16838747—21no—
48#16838747—21no—
49#16838747—21no—
50#16838747—42no—
51C2×Q8×C241D411yescharacteristic
52C22×C4×C241—21no—
53C22×C4×C241C2311yescharacteristic
54C2×C8×C241—21no—
55C4⋊C4×C241—21no—
56C4⋊C4×C241D411yescharacteristic
57C22⋊C4×C241—21no—
58C22×D241—21no—
59C22×D241—21no—
60C22×D241—41no—
61C22×D241—41no—
62C22×D241Q1612yescomplemented
63Dic482—22no—
64Dic482—22no—
65Dic482—41no—
66Dic482—41no—
67Dic482—41no—
68Dic482—81no—
69C2×Dic241—22no—
70C2×Dic241—21no—
71C2×Dic241—21no—
72C2×Dic241—21no—
73C2×Dic241—41no—
74C2×Dic241—41no—
75C2×Dic241—41no—
76C4×D241—81no—
77C4×D241—81no—
78#14515349—41no—
79C23×C241C2×D411yescharacteristic
80Q8×C241—41no—
81C2×C4×C241—21no—
82C2×C4×C241—21no—
83C2×C4×C241—41no—
84C2×C4×C241—41no—
85C2×C4×C241—41no—
86C2×C4×C241—41no—
87C2×C4×C241C2×D411yescharacteristic
88C2×C4×C241C2×D411yescharacteristic
89C8×C241—41no—
90C22×C241—21no—
91C22×C241—21no—
92C22×C241#12011yescharacteristic
93C22×C241C2×Q1611yescharacteristic
94C22×C241#13711yescharacteristic
95C2×D241—22no—
96C2×D241—41no—
97C2×D241—41no—
98C2×D241—42no—
99C2×D241—81no—
100C4×C241—21no—
101C4×C241—21no—
102C4×C241—41no—
103C4×C241—41no—
104C4×C241—81no—
105Dic241—41no—
106Dic241—41no—
107Dic241—41no—
108Dic241—42no—
109C2×C241—21no—
110C2×C241—21no—
111C2×C241#45111yescharacteristic
112C2×C241#45511yescharacteristic
113C2×C241B(4,𝔽2)11yescharacteristic
114D241—42no—
115C241#160211yescharacteristic, complemented, Sylow
116#1602—2411nomaximal, Sylow
117#543—2411no—
118#451—2411no—
119#451—2411no—
120#393—2411no—
121#328—2411no—
122#328D24111yescharacteristic, complemented
123#323—2411no—
124C22×Q8—2411no—
125C2×Q16—4821no—
126C2×Q16—4821no—
127C22⋊Q8—4821no—
128C22⋊Q8—4821no—
129C22⋊Q8—4821no—
130C22⋊Q8—4821no—
131C22⋊Q8—4821no—
132C22⋊Q8—2411no—
133C22⋊Q8C2×D24111yescharacteristic
134C2×C22⋊C4—4821no—
135C2×C22⋊C4—4821no—
136C2×C22⋊C4—2411no—
137#103—4821no—
138#103—4821no—
139C22⋊C8—2411no—
140C22⋊C8C2×D24111yescharacteristic
141#95—4821no—
142#95C2×D24111yescharacteristic
143C24—2411no—
144C2×Q8—9642no—
145C2×Q8—4821no—
146C2×Q8—4821no—
147C2×Q8—4821no—
148C2×Q8—2411no—
149C2×Q8D96411yescharacteristic
150C22×C4—21no—
151C22×C4—4821no—
152C22×C4—4821no—
153C22×C4—4821no—
154C22×C4—2411no—
155C22×C4C22×D24111yescharacteristic
156Q16—9642no—
157Q16—9642no—
158C2×C8—21no—
159C2×C8—4821no—
160C4⋊C4—21no—
161C4⋊C4—4821no—
162C4⋊C4—4821no—
163C4⋊C4—4821no—
164C4⋊C4—4821no—
165C4⋊C4—2411no—
166C4⋊C4D96411yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—9642no—
169C22⋊C4—4821no—
170C22⋊C4—4821no—
171C22⋊C4—4821no—
172C22⋊C4—4821no—
173C22⋊C4—4821no—
174C23—9641no—
175C23—9641no—
176C23—4821no—
177C23—4821no—
178C23—2412no—
179C23D4×D24111yescharacteristic
180Q8—41no—
181Q8—19281no—
182Q8—9641no—
183Q8—9641no—
184Q8—9641no—
185Q8—4822no—
186Q8—4822no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—19281no—
194C2×C4—19281no—
195C2×C4—9641no—
196C2×C4—9641no—
197C2×C4—9641no—
198C2×C4—4822no—
199C2×C4—4821no—
200C2×C4—4821no—
201C2×C4—4821no—
202C2×C4D4×D24111yescharacteristic
203C2×C4C2×D96411yescharacteristic
204C8—41no—
205C8—9641no—
206C22—21no—
207C22—21no—
208C22—19281no—
209C22—9642no—
210C22—9641no—
211C22—9641no—
212C22—4822no—
213C22#2144061211yescharacteristic
214C22Q16×D24111yescharacteristic
215C22#2144067111yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—9641no—
222C4—9641no—
223C4—9641no—
224C4—9642no—
225C2—21no—
226C2—21no—
227C2—9642no—
228C2#2950787411yescharacteristic
229C2#2950789611yescharacteristic
230C2#2950790511yescharacteristic
subgroupG/H
centerC22#21440612
commutator, cosocleC22×C4×C241C23
FrattiniC22×C4C22×D241
Fitting#328×C241C2
socleC22×C241#120
prime radicalC23×D241D4
hypercenter#328D241
hypocenterC241#1602
LCSG ▷ C22×C4×C241 ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #29507875C2semidirect
3. #29507872C2semidirect
5. #29507453C2semidirect
7. #328×C241C2semidirect
62. C22×D241Q16semidirect
115. C241#1602semidirect
122. #328D241semidirect

Maximal subgroups

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

subgroupindex
#295083422
#295078752
#295078722
#295076302
#295074532
#295074402
#328×C2412
#1602241