Finite Group Databasebeta

← #44508471

Subgroups — #44508471

Order 30848.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#29508281C211yesmaximal, characteristic
2C2×#21440631C211yesmaximal, characteristic, complemented
3C2×#21440569C211yesmaximal, characteristic, complemented
4#29507726C211yesmaximal, characteristic, complemented
5C2×#21440558C211yesmaximal, characteristic, complemented
6#29507478C211yesmaximal, characteristic, complemented
7C2×#137×C241C211yesmaximal, characteristic, complemented
8C22×D964C2211yescharacteristic
9C2×#16838754C2211yescharacteristic
10#21440638—21no—
11#21440631—22no—
12#21440631—21no—
13#21440631—22no—
14#21440616—22no—
15#21440610—22no—
16#21440601C2211yescharacteristic
17C2×#16838747—22no—
18C2×C4×Dic241C2211yescharacteristic
19#21440569—22no—
20#21440566—22no—
21#21440558—22no—
22C2×#41×C241C2211yescharacteristic
23C22×Q8×C241C2211yescharacteristic
24#137×C241—22no—
25#137×C241—22no—
26C2×Q16×C241—21no—
27C2×SD16×C241—21no—
28C2×M16×C241C2211yescharacteristic
29C23×D241D412yes—
30C2×D964—21no—
31C2×D964—21no—
32C2×D964—21no—
33C2×D964—21no—
34C2×D964—21no—
35C2×D964—21no—
36C2×C22⋊D241—24no—
37C22×Dic241—21no—
38#16838754—21no—
39#16838754—21no—
40C2×#14515349—21no—
41C4⋊Dic241D412yes—
42C22⋊Dic241—21no—
43#16838747—28no—
44#16838747—24no—
45C4×Dic241—21no—
46C4×Dic241—21no—
47C4×Dic241D412yescomplemented
48#41×C241—21no—
49#41×C241—21no—
50#41×C241—21no—
51#41×C241—21no—
52C2×Q8×C241—21no—
53C2×Q8×C241—21no—
54C2×Q8×C241—21no—
55C2×Q8×C241—21no—
56C2×Q8×C241—21no—
57C2×Q8×C241D411yescharacteristic
58C2×Q8×C241D412yescomplemented
59C2×D4×C241—21no—
60C2×D4×C241D411yescharacteristic
61C22×C4×C241—21no—
62C22×C4×C241—21no—
63C22×C4×C241C2311yescharacteristic
64Q16×C241—24no—
65SD16×C241—24no—
66M16×C241—21no—
67M16×C241D412yescomplemented
68C2×C8×C241—21no—
69C22×D241—22no—
70C22×D241—22no—
71C22×D241—22no—
72C22×D241—44no—
73D964—41no—
74D964—41no—
75D964—41no—
76D964—41no—
77C22⋊D241—84no—
78C2×Dic241—22no—
79C2×Dic241—22no—
80C2×Dic241—42no—
81#14515349—41no—
82C23×C241—21no—
83C23×C241C2×D411yescharacteristic
84Q8×C241—22no—
85Q8×C241—21no—
86Q8×C241—22no—
87Q8×C241—21no—
88Q8×C241—41no—
89Q8×C241—41no—
90D4×C241—21no—
91D4×C241—21no—
92D4×C241—41no—
93D4×C241—41no—
94C2×C4×C241—21no—
95C2×C4×C241—22no—
96C2×C4×C241—41no—
97C2×C4×C241—41no—
98C2×C4×C241—41no—
99C2×C4×C241—41no—
100C2×C4×C241C2×D411yescharacteristic
101C2×C4×C241C2×D411yescharacteristic
102C2×C4×C241C2×D411yescharacteristic
103C2×C4×C241C2×D411yescharacteristic
104C2×C4×C241C2×D411yescharacteristic
105C2×C4×C241#4111yescharacteristic
106C8×C241—22no—
107C22×C241—22no—
108C22×C241—41no—
109C22×C241—42no—
110C22×C241#12011yescharacteristic
111C22×C241#12011yescharacteristic
112C22×C241#12111yescharacteristic
113C2×D241—44no—
114C2×D241—42no—
115C2×D241—44no—
116C2×D241—82no—
117C4×C241—41no—
118C4×C241—42no—
119C4×C241#12011yescharacteristic
120C4×C241#12111yescharacteristic
121C4×C241#12111yescharacteristic
122C4×C241#12711yescharacteristic
123Dic241—42no—
124Dic241—42no—
125C2×C241—21no—
126C2×C241—21no—
127C2×C241—81no—
128C2×C241#39211yescharacteristic
129C2×C241#45512yes—
130D241—82no—
131C241#200411yescharacteristic, complemented, Sylow
132#2004—2411nomaximal, Sylow
133C2×#137D24111yescharacteristic, complemented
134#535—2411no—
135C2×#124—2411no—
136C2×#104—2411no—
137#419—2411no—
138C2×#100—2411no—
139#339—2411no—
140C2×#41C2×D24111yescharacteristic
141C22×Q8C2×D24111yescharacteristic
142C22×D4—2411no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—2411no—
148C2×M16C2×D24111yescharacteristic
149#127—4821no—
150#124—4821no—
151#124—4822no—
152#124—4822no—
153#121—4822no—
154#120—4822no—
155#117—2411no—
156C2×C22⋊C4—4822no—
157C2×C42—2411no—
158#104—4822no—
159#103—4822no—
160#100—4822no—
161C24—2412no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D24111yescharacteristic
172C2×Q8C22⋊D24112yescomplemented
173C2×D4—21no—
174C2×D4—4824no—
175C2×D4—4821no—
176C2×D4—4821no—
177C2×D4—4821no—
178C2×D4—4821no—
179C2×D4—4821no—
180C2×D4—4821no—
181C2×D4C22⋊D24111yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—4821no—
185C22×C4C22×D24111yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—4821no—
190M16—4821no—
191M16C22⋊D24112yescomplemented
192C2×C8—21no—
193C2×C8—4821no—
194C4⋊C4—2412no—
195C22⋊C4—4828no—
196C22⋊C4—4824no—
197C22⋊C4—4821no—
198C42—4821no—
199C42—4821no—
200C42—2412no—
201C23—21no—
202C23—9644no—
203C23—4822no—
204C23—4822no—
205C23—4822no—
206C23D4×D24111yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—19284no—
218D4—9641no—
219D4—9641no—
220D4—9641no—
221D4—9641no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—9642no—
229C2×C4—4822no—
230C2×C4—4822no—
231C2×C4D4×D24111yescharacteristic
232C2×C4D4×D24111yescharacteristic
233C2×C4C2×C22⋊D24111yescharacteristic
234C2×C4C2×C22⋊D24111yescharacteristic
235C2×C4C2×C22⋊D24111yescharacteristic
236C2×C4#1683876911yescharacteristic
237C8—22no—
238C8—9641no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—19282no—
243C22—9644no—
244C22—9642no—
245C22—9644no—
246C22#2144061011yescharacteristic
247C22#2144061011yescharacteristic
248C22#2144061311yescharacteristic
249C4—41no—
250C4—42no—
251C4—9642no—
252C4—9642no—
253C4#2144061111yescharacteristic
254C4#2144061411yescharacteristic
255C4#2144061411yescharacteristic
256C4#2144063811yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—19282no—
261C2#2950762511yescharacteristic
262C2#2950789312yes—
subgroupG/H
centerC22#21440610
commutator, cosocleC22×C4×C241C23
FrattiniC22×C4C22×D241
FittingC2×#137×C241C2
socleC22×C241#120
prime radical#29508281C2
hypercenterC2×#137D241
hypocenterC241#2004
LCSG ▷ C22×C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#21440631C2semidirect
3. C2×#21440569C2semidirect
4. #29507726C2semidirect
5. C2×#21440558C2semidirect
6. #29507478C2semidirect
7. C2×#137×C241C2semidirect
47. C4×Dic241D4semidirect
58. C2×Q8×C241D4semidirect
67. M16×C241D4semidirect
131. C241#2004semidirect
133. C2×#137D241semidirect
172. C2×Q8C22⋊D241semidirect
191. M16C22⋊D241semidirect

Maximal subgroups

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

subgroupindex
#295082812
C2×#214406312
C2×#214405692
#295077262
C2×#214405582
#295074782
C2×#137×C2412
#2004241