Finite Group Databasebeta

← #29592613

Subgroups — #29592613

Order 15616.

360 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447048C211yesmaximal, characteristic
2#21444855C212yesmaximal
3#21443846C212yesmaximal, complemented
4#21443763C211yesmaximal, characteristic, complemented
5C61×#2844C211yesmaximal, characteristic, complemented
6#16840343C2211yescharacteristic
7#16840080—22no—
8C2×#14519705—21no—
9#16839761—21no—
10#16839531C2211yescharacteristic
11#16839495—22no—
12#16839488C2212yes—
13C61×#400—22no—
14C61×#400C2212yes—
15C61×#395—22no—
16C61×#395—21no—
17C61×#382—21no—
18C61×#382C2211yescharacteristic
19C2×#11758222—21no—
20C2×#11758194—21no—
21#14519757—41no—
22#14519752—41no—
23#14519751—22no—
24#14519708—42no—
25#14519705—42no—
26#14519700—41no—
27#14519697—41no—
28#14519697—42no—
29C22⋊#92418—22no—
30C22×Q8×C61—21no—
31C2×C4⋊C4×C61—21no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61D411yescharacteristic
35C2×C4⋊C4×C61D412yes—
36C2×C4⋊C4×C61C2311yescharacteristic
37C2×C42×C61D411yescharacteristic
38#95×C61—22no—
39#95×C61—21no—
40#95×C61—22no—
41#95×C61—22no—
42#95×C61—22no—
43#95×C61D412yes—
44C2×Q8×C61—82no—
45C2×Q8×C61—81no—
46C22×C4×C61—21no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—21no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D412yes—
52C2×C22⋊D61—41no—
53#11758222—41no—
54#11758222—41no—
55#11758222—41no—
56#11758222—41no—
57C2×D244—21no—
58C2×D244—22no—
59C2×C4×D61—41no—
60C2×Dic122—21no—
61C4⋊C4×C61—41no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61—42no—
64C4⋊C4×C61—81no—
65C4⋊C4×C61—81no—
66C4⋊C4×C61—82no—
67C42×C61—21no—
68C42×C61SD1612yes—
69#11758198—42no—
70C4⋊Dic61—22no—
71#11758194—41no—
72#11758194—41no—
73C2×#92418—42no—
74C2×#92418—41no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77Q8×C61—84no—
78C2×C4×C61—21no—
79C2×C4×C61—21no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—41no—
83C2×C4×C61—41no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—41no—
92C2×C4×C61—42no—
93C2×C4×C61C2×SD1611yescharacteristic
94C2×C4×C61#13711yescharacteristic
95C22⋊D61—81no—
96C22⋊D61—81no—
97C2×Dic61—41no—
98D244—41no—
99D244—41no—
100D244—82no—
101C4×D61—81no—
102C4×D61—81no—
103Dic122—41no—
104Dic122—41no—
105#92418—81no—
106#92418—82no—
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—82no—
121C4×C61—81no—
122C4×C61—82no—
123Dic61—81no—
124#2844D6111yescharacteristic, complemented
125#2040—611no—
126#1616—612no—
127#1353—612no—
128#1324—611no—
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—22no—
140#400C2×D6112yes—
141#395—22no—
142#395—21no—
143#382—21no—
144#382C2×D6111yescharacteristic
145#343—611no—
146#327—1222no—
147#324—612no—
148C61#1218511yescharacteristic, complemented, Sylow
149C2×#41—1221no—
150C22×Q8—21no—
151C2×M16—1221no—
152C22⋊Q8—2441no—
153#121—1222no—
154#121—2441no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊D6111yescharacteristic
159C2×C4⋊C4C22⋊D6112yes—
160C2×C4⋊C4C22×D6111yescharacteristic
161C2×C42C22⋊D6111yescharacteristic
162#106—2442no—
163#104—2442no—
164#103—2441no—
165#102—2441no—
166#102—2442no—
167C22⋊C8—1222no—
168#95—22no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C22⋊D6112yes—
174#41—2441no—
175#41—2441no—
176#41—2441no—
177#41—2441no—
178C2×Q8—82no—
179C2×Q8—81no—
180C2×Q8—1221no—
181C2×D4—1222no—
182C2×D4—1221no—
183C2×D4—2441no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—2441no—
189C22×C4C2×C22⋊D6111yescharacteristic
190C22×C4C2×C22⋊D6112yes—
191M16—2441no—
192M16—2441no—
193C2×C8—2442no—
194C2×C8—2441no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—1222no—
202C22⋊C4—2442no—
203C42—21no—
204C42#1175820112yes—
205C23—2441no—
206C23#1451974611yescharacteristic
207Q8—84no—
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—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—4881no—
231C2×C4—4881no—
232C2×C4—2441no—
233C2×C4C2×#1175820111yescharacteristic
234C2×C4#1451980811yescharacteristic
235C8—4881no—
236C8—4882no—
237C22—21no—
238C22—21no—
239C22—4881no—
240C22—4881no—
241C22—4881no—
242C22#1683990111yescharacteristic
243C22#1683993511yescharacteristic
244C22#1683994911yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—4881no—
253C2—21no—
254C2—21no—
255C2—9761no—
256C2#2144475911yescharacteristic
257C2#2144783211yescharacteristic
258C2#2144783611yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2844C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2844D61
hypocenterC61#12185
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
3. #21443846C2semidirect
4. #21443763C2semidirect
5. C61×#2844C2semidirect
124. #2844D61semidirect
148. C61#12185semidirect

Maximal subgroups

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

subgroupindex
#214470482
#214448552
#214438462
#214437632
C61×#28442
#1218561