Finite Group Databasebeta

← #44509444

Subgroups — #44509444

Order 30848.

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

normal subgroups only

This group has 270 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#29508460C211yesmaximal, characteristic, complemented
2#29507904C211yesmaximal, characteristic, complemented
3#29507895C211yesmaximal, characteristic, complemented
4#29507511C211yesmaximal, characteristic, complemented
5#29507504C211yesmaximal, characteristic, complemented
6#29507503C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C241C211yesmaximal, characteristic, complemented
8#21440672—21no—
9#21440629—21no—
10#21440629C2211yescharacteristic
11#21440628—21no—
12#21440628—21no—
13#21440628—21no—
14#21440611—21no—
15#21440610—21no—
16#21440610—21no—
17#21440610C2211yescharacteristic
18#21440569—21no—
19#21440559C2211yescharacteristic
20#21440557—21no—
21#21440557C2211yescharacteristic
22H(5,𝔽2)×C241C2211yescharacteristic
23#120×C241—21no—
24#120×C241C2211yescharacteristic
25#99×C241—21no—
26#99×C241C2211yescharacteristic
27C23×D241D411yescharacteristic, complemented
28C2×Dic482—21no—
29C2×Dic482D411yescharacteristic, complemented
30C2×C22⋊D241—21no—
31C2×C22⋊D241—21no—
32C2×C22⋊D241—21no—
33C2×C22⋊D241—41no—
34#16838760—41no—
35#16838758—41no—
36#16838754—21no—
37C22⋊Dic241—21no—
38C22⋊Dic241—21no—
39C22⋊Dic241—21no—
40C22⋊Dic241—21no—
41C22⋊Dic241—21no—
42C22⋊Dic241D411yescharacteristic, complemented
43#16838747—21no—
44#16838747—21no—
45#16838747—21no—
46#16838747—41no—
47C4×Dic241—21no—
48C4×Dic241D411yescharacteristic, complemented
49C24×C241D411yescharacteristic, complemented
50#41×C241—21no—
51#41×C241—41no—
52C2×D4×C241—21no—
53C2×D4×C241—21no—
54C2×D4×C241—21no—
55C2×D4×C241—41no—
56C2×D4×C241—41no—
57C2×D4×C241C2311yescharacteristic
58C22⋊C4×C241—21no—
59C22⋊C4×C241—21no—
60C22⋊C4×C241—41no—
61C22⋊C4×C241D411yescharacteristic, complemented
62C22×D241—21no—
63C22×D241—41no—
64C22×D241—81no—
65Dic482—41no—
66Dic482—81no—
67C22⋊D241—81no—
68C22⋊D241—81no—
69C22⋊D241—81no—
70C22⋊D241—82no—
71C2×Dic241—21no—
72C2×Dic241—21no—
73C2×Dic241—21no—
74C2×Dic241—41no—
75#14515349—41no—
76C23×C241—21no—
77C23×C241—21no—
78C23×C241—21no—
79C23×C241—41no—
80C23×C241—81no—
81C23×C241C2×D411yescharacteristic
82C23×C241C2×D411yescharacteristic
83Q8×C241—21no—
84D4×C241—21no—
85D4×C241—41no—
86D4×C241—41no—
87D4×C241—41no—
88D4×C241—41no—
89D4×C241—81no—
90D4×C241—82no—
91C2×C4×C241—21no—
92C2×C4×C241—21no—
93C2×C4×C241—21no—
94C2×C4×C241—41no—
95C2×C4×C241—41no—
96C2×C4×C241C2×D411yescharacteristic
97C22×C241—21no—
98C22×C241—21no—
99C22×C241—41no—
100C22×C241—41no—
101C22×C241—41no—
102C22×C241—41no—
103C22×C241—42no—
104C22×C241—41no—
105C22×C241—41no—
106C22×C241—81no—
107C22×C241—81no—
108C22×C241—81no—
109C22×C241#12011yescharacteristic
110C2×D241—41no—
111C2×D241—41no—
112C2×D241—42no—
113C2×D241—41no—
114C2×D241—81no—
115C4×C241—21no—
116C4×C241—41no—
117C4×C241—41no—
118C4×C241—81no—
119Dic241—41no—
120Dic241—41no—
121Dic241—41no—
122Dic241—81no—
123C2×C241—21no—
124C2×C241—41no—
125C2×C241—41no—
126C2×C241—81no—
127C2×C241—81no—
128C2×C241B(4,𝔽2)11yescharacteristic
129D241—81no—
130C241#219911yescharacteristic, complemented, Sylow
131#2199—2411nomaximal, Sylow
132H(3,𝔽4)—2411no—
133B(4,𝔽2)—2411no—
134B(4,𝔽2)D24111yescharacteristic, complemented
135#455—2411no—
136#354—2411no—
137AΣL(1,𝔽16)—2411no—
138AΣL(1,𝔽16)—2411no—
139H(5,𝔽2)C2×D24111yescharacteristic
140#137—4821no—
141#124—4821no—
142#124—4821no—
143#124—4821no—
144#124—4821no—
145#124—2411no—
146#120—21no—
147#120—4821no—
148#120—4821no—
149#120—4821no—
150#120—2411no—
151#120C2×D24111yescharacteristic
152#104—4821no—
153#100—2411no—
154#99—21no—
155#99—4821no—
156#99—2411no—
157#99C2×D24111yescharacteristic
158C24—2411no—
159C24C22⋊D24111yescharacteristic, complemented
160#41—21no—
161#41—41no—
162C2×Q8—4821no—
163C2×Q8—2411no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—9641no—
170C2×D4—4821no—
171C2×D4—4821no—
172C2×D4—4821no—
173C2×D4C22×D24111yescharacteristic
174Q16—9641no—
175SD16—9641no—
176M16—4821no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—9641no—
181C22⋊C4—4821no—
182C22⋊C4—4821no—
183C22⋊C4—4821no—
184C22⋊C4—4821no—
185C22⋊C4—4821no—
186C22⋊C4—4821no—
187C22⋊C4—4821no—
188C22⋊C4—4821no—
189C22⋊C4—2411no—
190C22⋊C4C22⋊D24111yescharacteristic, complemented
191C42—4821no—
192C42—2411no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—19281no—
199C23—9641no—
200C23—4821no—
201C23D4×D24111yescharacteristic
202C23C2×C22⋊D24111yescharacteristic
203Q8—21no—
204Q8—19281no—
205Q8—9641no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—19282no—
214D4—19281no—
215D4—19281no—
216D4—19281no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—9641no—
223C2×C4—4821no—
224C2×C4—4821no—
225C2×C4—4821no—
226C2×C4D4×D24111yescharacteristic
227C8—9641no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—19281no—
241C22—9641no—
242C22—9641no—
243C22—9642no—
244C22—9641no—
245C22#2144061011yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—19281no—
251C4—9641no—
252C4—9641no—
253C4—9641no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—19281no—
260C2#2950790311yescharacteristic
subgroupG/H
centerC2#29507903
commutator, cosocleC2×D4×C241C23
FrattiniC2×D4C22×D241
FittingB(4,𝔽2)×C241C2
socleC2×C241B(4,𝔽2)
hypercenterB(4,𝔽2)D241
hypocenterC241#2199
LCSG ▷ C2×D4×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #29508460C2semidirect
2. #29507904C2semidirect
3. #29507895C2semidirect
4. #29507511C2semidirect
5. #29507504C2semidirect
6. #29507503C2semidirect
7. B(4,𝔽2)×C241C2semidirect
27. C23×D241D4semidirect
29. C2×Dic482D4semidirect
42. C22⋊Dic241D4semidirect
48. C4×Dic241D4semidirect
49. C24×C241D4semidirect
61. C22⋊C4×C241D4semidirect
130. C241#2199semidirect
134. B(4,𝔽2)D241semidirect
159. C24C22⋊D241semidirect
190. C22⋊C4C22⋊D241semidirect

Maximal subgroups

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

subgroupindex
#295084602
#295079042
#295078952
#295075112
#295075042
#295075032
B(4,𝔽2)×C2412
#2199241