Finite Group Databasebeta

← #29592615

Subgroups — #29592615

Order 15616.

360 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 59 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#21454091C211yesmaximal, characteristic, complemented
2#21447048C211yesmaximal, characteristic
3#21444858C212yesmaximal
4#21443847C212yesmaximal, complemented
5C61×#1324C211yesmaximal, characteristic, complemented
6#16840343C2211yescharacteristic
7#16840081—22no—
8C2×#14519705—21no—
9#16839761—21no—
10#16839704—22no—
11#16839701C2212yes—
12#16839683—22no—
13#16839682—21no—
14#16839628—21no—
15#16839531C2211yescharacteristic
16#16839497—22no—
17C61×#382C2211yescharacteristic
18C61×#324C2212yes—
19C2×#11758222—21no—
20C22×Dic122—21no—
21C2×#11758194—21no—
22#14519757—41no—
23#14519752—41no—
24#14519751—22no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61D412yes—
28C2×#11758196—22no—
29#14519709—42no—
30#14519705—42no—
31#14519700—41no—
32#14519699—42no—
33#14519697—41no—
34#14519684—21no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684D412yes—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C4⋊C4×C61C2311yescharacteristic
41C2×C42×C61D411yescharacteristic
42C22⋊C8×C61—22no—
43#95×C61—22no—
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61C2×D412yes—
48C2×C22⋊D61—41no—
49C22×Dic61—22no—
50C22×Dic61—22no—
51#11758222—41no—
52#11758222—41no—
53#11758222—41no—
54#11758222—41no—
55C2×D244—21no—
56C2×D244—22no—
57C2×C4×D61—41no—
58C2×Dic122—21no—
59C2×Dic122—82no—
60C2×Dic122—81no—
61C2×C8×C61—42no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61—41no—
64C42×C61—21no—
65C42×C61SD1612yes—
66#11758198—42no—
67C4⋊Dic61—22no—
68C4⋊Dic61—42no—
69C4⋊Dic61—81no—
70C4⋊Dic61—81no—
71#11758196—82no—
72#11758194—41no—
73#11758194—41no—
74C2×#92418—41no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—41no—
82C2×C4×C61—41no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—41no—
86C2×C4×C61C2×SD1611yescharacteristic
87C2×C4×C61#13711yescharacteristic
88C22⋊D61—81no—
89C22⋊D61—81no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—41no—
96C2×Dic61—42no—
97D244—41no—
98D244—41no—
99D244—82no—
100C4×D61—81no—
101C4×D61—81no—
102Dic122—41no—
103Dic122—41no—
104Dic122—84no—
105C8×C61—82no—
106#92418—81no—
107#12185—611nomaximal, Sylow
108C22×C61—21no—
109C22×C61—21no—
110C22×C61#44811yescharacteristic
111C22×C61#45411yescharacteristic
112C22×C61B(4,𝔽2)11yescharacteristic
113C2×D61—81no—
114C2×D61—81no—
115C2×D61—81no—
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—42no—
119C4×C61—42no—
120C4×C61—81no—
121Dic61—82no—
122Dic61—81no—
123Dic61—82no—
124#2844—611no—
125#2040—611no—
126#1616—612no—
127#1353—612no—
128#1324D6111yescharacteristic, complemented
129C2×C61—21no—
130C2×C61—21no—
131C2×C61#159911yescharacteristic
132C2×C61#220111yescharacteristic
133C2×C61#220211yescharacteristic
134D61—161no—
135#537—611no—
136#481—1222no—
137C2×#104—1221no—
138#417—1221no—
139#400—612no—
140#400—1222no—
141#395—1222no—
142#395—1221no—
143#382—1221no—
144#382C2×D6111yescharacteristic
145#343—611no—
146#327—1222no—
147#324C2×D6112yes—
148C61#1218511yescharacteristic, complemented, Sylow
149C2×#41—1221no—
150C22×Q8—1221no—
151C2×M16—1221no—
152C22⋊Q8—2441no—
153#121—1222no—
154#121—2441no—
155C2×C4⋊C4—612no—
156C2×C4⋊C4—1221no—
157C2×C4⋊C4—1221no—
158C2×C4⋊C4—1222no—
159C2×C4⋊C4D24411yescharacteristic
160C2×C4⋊C4C22×D6111yescharacteristic
161C2×C42D24411yescharacteristic
162#106—2442no—
163#104—2442no—
164#103—2441no—
165#102—2441no—
166#102—2442no—
167C22⋊C8—22no—
168#95—22no—
169#95—612no—
170#95—1222no—
171#95—1221no—
172#95—1222no—
173#95—1222no—
174#41—2441no—
175#41—2441no—
176#41—2441no—
177#41—2441no—
178C2×Q8—1221no—
179C2×Q8—4882no—
180C2×Q8—4881no—
181C2×D4—1222no—
182C2×D4—1221no—
183C2×D4—2441no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—1222no—
187C22×C4—1222no—
188C22×C4—2441no—
189C22×C4C2×D24411yescharacteristic
190C22×C4D4×D6112yes—
191M16—2441no—
192M16—2441no—
193C2×C8—42no—
194C2×C8—2441no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—1222no—
198C4⋊C4—4882no—
199C4⋊C4—4881no—
200C4⋊C4—4881no—
201C4⋊C4—2442no—
202C22⋊C4—2442no—
203C42—21no—
204C42#1175819012yes—
205C23—2441no—
206C23#1451974711yescharacteristic
207Q8—4884no—
208Q8—2441no—
209Q8—2441no—
210D4—4882no—
211D4—4881no—
212D4—4881no—
213D4—2441no—
214D4—2441no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—4881no—
225C2×C4—4881no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2441no—
232C2×C4—2442no—
233C2×C4C2×#1175819011yescharacteristic
234C2×C4#1451981011yescharacteristic
235C8—82no—
236C8—4881no—
237C22—21no—
238C22—21no—
239C22—4881no—
240C22—4881no—
241C22—4881no—
242C22#1683990311yescharacteristic
243C22#1683993711yescharacteristic
244C22#1683995011yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—4882no—
251C4—4881no—
252C4—4882no—
253C2—21no—
254C2—21no—
255C2—9761no—
256C2#2144476111yescharacteristic
257C2#2144783411yescharacteristic
258C2#2144783711yescharacteristic
subgroupG/H
centerC22#16839950
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1324C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#1324D61
hypocenterC61#12185
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
1. #21454091C2semidirect
4. #21443847C2semidirect
5. C61×#1324C2semidirect
128. #1324D61semidirect
148. C61#12185semidirect

Maximal subgroups

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

subgroupindex
#214540912
#214470482
#214448582
#214438472
C61×#13242
#1218561