Finite Group Databasebeta

← #30605339

Subgroups — #30605339

Order 15616.

336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462183C211yesmaximal, characteristic, complemented
2#21461740C211yesmaximal, characteristic, complemented
3#21460808C212yesmaximal, complemented
4#21460805C212yesmaximal
5#21456387C211yesmaximal, characteristic
6#16840708—21no—
7C4⋊#11758187—21no—
8#16840657C2212yes—
9#16840651—21no—
10#16840636—21no—
11#16840636C2212yes—
12#16840294—21no—
13#16840163C411yescharacteristic, complemented
14#16840145C411yescharacteristic, complemented
15#16840115—21no—
16#16840004C2211yescharacteristic
17#16839995C2211yescharacteristic
18#16839990—21no—
19C4×Dic244C411yescharacteristic, complemented
20C4×#11758200—21no—
21C4×C8×D61C2211yescharacteristic
22C4×D8×C61C411yescharacteristic, complemented
23C2×C4⋊#6923—22no—
24C2×C4⋊#6923—41no—
25C2×C4⋊#6923—41no—
26C2×#11758187—22no—
27C2×Dic244C2×C411yescharacteristic
28C2×#11758200—21no—
29C2×C8×D61D411yescharacteristic
30C2×C8×D61C2311yescharacteristic
31#14519770—41no—
32#14519768—21no—
33#14519764—21no—
34#14519759—41no—
35#14519757—21no—
36#14519748—21no—
37C4×Dic122—21no—
38C4×C22⋊D61—41no—
39#14519736—41no—
40C42×D61D411yescharacteristic
41#14519711C2×C411yescharacteristic
42#14519708—21no—
43#14519702—21no—
44#14519700—21no—
45#14519698—21no—
46C4×#92418D411yescharacteristic
47C8×Dic61C2×C411yescharacteristic
48C8×Dic61D411yescharacteristic
49C2×D8×C61C2×C411yescharacteristic
50C4×D4×C61—21no—
51#107×C61C2×C411yescharacteristic
52#102×C61—21no—
53C4×C8×C61C2×C411yescharacteristic
54C22×#6923—42no—
55C2×D4×C61—21no—
56C22×C4×C61—41no—
57C2×C22⋊D61—41no—
58C2×C4×D61D811yescharacteristic
59C2×C4×D61D811yescharacteristic
60C2×C4×D61C2×D411yescharacteristic
61C2×Dic122—21no—
62C4⋊#6923—42no—
63C4⋊#6923—42no—
64C4⋊#6923—82no—
65C4⋊#6923—82no—
66D8×C61—41no—
67C2×C8×C61C22⋊C411yescharacteristic
68C2×C8×C61C22×C411yescharacteristic
69C4⋊C4×C61—21no—
70C22⋊C4×C61—41no—
71C42×C61C22⋊C411yescharacteristic
72C22⋊Dic61—41no—
73C22⋊Dic61—41no—
74C22⋊Dic61—41no—
75#11758200—42no—
76#11758198—41no—
77C4⋊Dic61—21no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80#11758196—41no—
81#11758196—41no—
82C4×Dic61—41no—
83C4×Dic61C22⋊C411yescharacteristic
84C4×Dic61SD1611yescharacteristic
85C4×Dic61SD1611yescharacteristic
86C2×#92418C22⋊C411yescharacteristic
87C2×#92418C2×D411yescharacteristic
88Dic244—41no—
89C8×D61—41no—
90C8×D61—41no—
91#11758187—42no—
92C23×C61—41no—
93C22×D61C2×D811yescharacteristic
94C2×#6923—82no—
95C2×#6923—82no—
96C2×#6923—82no—
97D4×C61—41no—
98D4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—82no—
101C2×C4×C61#10211yescharacteristic
102C2×C4×C61#10211yescharacteristic
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C22⋊D61—81no—
105C22⋊D61—81no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C2×Dic61#10211yescharacteristic
109C2×Dic61#10211yescharacteristic
110C2×Dic61C2×SD1611yescharacteristic
111C4×D61—21no—
112C4×D61—21no—
113C4×D61—22no—
114C4×D61—21no—
115C4×D61—21no—
116Dic122—41no—
117Dic122—41no—
118C8×C61—21no—
119C8×C61—21no—
120#92418—21no—
121#92418—21no—
122#11860—611nomaximal, Sylow
123C22×C61—81no—
124C22×C61—81no—
125C22×C61—81no—
126C22×C61C2×#10211yescharacteristic
127C2×D61—21no—
128C2×D61—21no—
129C2×D61#50911yescharacteristic
130C2×D61#51011yescharacteristic
131#6923—162no—
132C4×C61—21no—
133C4×C61—81no—
134C4×C61#36012yes—
135C4×C61#41011yescharacteristic
136C4×C61#42111yescharacteristic
137Dic61—21no—
138Dic61—21no—
139Dic61—21no—
140Dic61—81no—
141Dic61—81no—
142#3168—611no—
143#2163—611no—
144#1834—611no—
145#1348—612no—
146#1347—612no—
147C2×C61—161no—
148C2×C61#151711yescharacteristic
149C2×C61#213911yescharacteristic
150C2×C61C2×#36011yescharacteristic
151D61—21no—
152D61—21no—
153#529—1221no—
154#499—611no—
155#495—611no—
156#488—1221no—
157#467—611no—
158#466—611no—
159#465—1221no—
160C4×Q16—611no—
161C4×SD16—1221no—
162C4×D8#692311yescharacteristic, complemented
163C2×C4×C8—611no—
164#384—1221no—
165C4⋊C16—1221no—
166C22⋊C16—612no—
167#346—1221no—
168#340—612no—
169#340—1221no—
170C61#1186011yescharacteristic, complemented, Sylow
171C2×Q16—611no—
172C2×SD16—1221no—
173C2×D8C2×#692311yescharacteristic
174C22×C8—611no—
175C22×C8—611no—
176#126—2441no—
177#125—1221no—
178#124—1221no—
179#123—2441no—
180C22⋊Q8—1221no—
181#121—1221no—
182C4×Q8—1221no—
183C4×D4—21no—
184C4×D4—2441no—
185#117—2441no—
186C2×C4⋊C4—1222no—
187C2×C4⋊C4—2441no—
188C2×C4⋊C4—2441no—
189C2×C42—611no—
190C2×C16—1222no—
191#107—611no—
192#107C2×#692311yescharacteristic
193#106—1221no—
194#103—1221no—
195#103—1221no—
196#102—21no—
197#102—1221no—
198C4×C8—611no—
199C4×C8—611no—
200C4×C8—611no—
201C4×C8C2×#692311yescharacteristic
202C2×Q8—1221no—
203C2×D4—21no—
204C2×D4—2441no—
205C22×C4—41no—
206C22×C4—611no—
207C22×C4—611no—
208C22×C4—611no—
209C22×C4—2442no—
210Q16—2441no—
211SD16—2442no—
212D8—41no—
213C2×C8—611no—
214C2×C8—611no—
215C2×C8—2441no—
216C2×C8—2441no—
217C2×C8C22⋊#692311yescharacteristic
218C2×C8C22×#692311yescharacteristic
219C4⋊C4—21no—
220C4⋊C4—1221no—
221C4⋊C4—4882no—
222C4⋊C4—4882no—
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C4⋊C4—2442no—
227C4⋊C4—2442no—
228C4⋊C4—2441no—
229C22⋊C4—41no—
230C22⋊C4—2441no—
231C22⋊C4—2441no—
232C22⋊C4—2441no—
233C22⋊C4—2441no—
234C42—611no—
235C42—611no—
236C42—611no—
237C42—2441no—
238C42C22⋊#692311yescharacteristic
239C16—2442no—
240C23—41no—
241C23—611no—
242Q8—2441no—
243Q8—2441no—
244D4—41no—
245D4—41no—
246D4—4881no—
247D4—4881no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—611no—
251C2×C4—611no—
252C2×C4—611no—
253C2×C4—1221no—
254C2×C4—1221no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1222no—
258C2×C4—4882no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4#1451983911yescharacteristic
264C2×C4#1451983911yescharacteristic
265C2×C4C2×C22⋊#692311yescharacteristic
266C8—21no—
267C8—21no—
268C8—1221no—
269C8—1221no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—611no—
274C22—611no—
275C22—1221no—
276C22—1221no—
277C22C2×#1451983911yescharacteristic
278C4—21no—
279C4—81no—
280C4—1221no—
281C4—1221no—
282C4—1221no—
283C4—9762no—
284C4—4881no—
285C4—4881no—
286C4#1684067812yes—
287C4#1684074111yescharacteristic
288C4#1684075911yescharacteristic
289C2—161no—
290C2—1221no—
291C2—1221no—
292C2#2146124911yescharacteristic
293C2#2146212711yescharacteristic
294C2C2×#1684067811yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC4×D8×C61C4
socleC22×C61C2×#102
prime radical#16839995C22
cosocleC2×C8×D61C23
hypercenterC4×D8#6923
hypocenterC61#11860
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #21462183C2semidirect
2. #21461740C2semidirect
3. #21460808C2semidirect
13. #16840163C4semidirect
14. #16840145C4semidirect
19. C4×Dic244C4semidirect
22. C4×D8×C61C4semidirect
162. C4×D8#6923semidirect
170. C61#11860semidirect

Maximal subgroups

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

subgroupindex
#214621832
#214617402
#214608082
#214608052
#214563872
#1186061