Finite Group Databasebeta

← #29577557

Subgroups — #29577557

Order 15616.

422 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 89 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#21456366C211yesmaximal, characteristic
2#21448999C211yesmaximal, characteristic
3#21444535C211yesmaximal, characteristic
4#21444370C211yesmaximal, characteristic
5#21443685C211yesmaximal, characteristic
6#21443683C211yesmaximal, characteristic
7C61×#1515C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840172—22no—
10#16840165C412yescomplemented
11#16840158C412yescomplemented
12#16839819C2211yescharacteristic
13#16839805—22no—
14C4×#11758194C2211yescharacteristic
15#16839666—22no—
16#16839647—22no—
17#16839645—22no—
18#16839630C2211yescharacteristic
19#16839507—22no—
20#16839499—22no—
21C61×#533C2211yescharacteristic
22C61×#432C2211yescharacteristic
23C61×#383C2211yescharacteristic
24C61×#328—22no—
25C22×Dic122—21no—
26C2×#11758194—21no—
27#14519777—21no—
28#14519777—21no—
29#14519777C2×C412yes—
30#14519777D412yes—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61D412yes—
33C2×#11758196—22no—
34C2×C4×Dic61—22no—
35#14519710—24no—
36#14519708—24no—
37C4⋊#92418—21no—
38C22⋊#92418—21no—
39#14519688—21no—
40#14519688—21no—
41C4×#92418C2×C412yes—
42#14519684—24no—
43#14519684—21no—
44#14519684—21no—
45#14519684D412yes—
46C4⋊Q8×C61C2×C411yescharacteristic
47C4⋊Q8×C61C2×C411yescharacteristic
48#125×C61—21no—
49C22⋊Q8×C61—21no—
50C22⋊Q8×C61—21no—
51C4×Q8×C61—21no—
52C4×Q8×C61—21no—
53#117×C61—21no—
54C2×C42×C61C2311yescharacteristic
55C4⋊C8×C61—21no—
56C22⋊C8×C61—21no—
57#97×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61D412yes—
61C2×Q8×C61—21no—
62C2×Q8×C61—21no—
63C22×C4×C61—22no—
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C2×Dic122—22no—
70C2×Dic122—22no—
71C2×Dic122—81no—
72C2×Dic122—81no—
73C2×C8×C61—41no—
74C4⋊C4×C61—21no—
75C4⋊C4×C61—21no—
76C4⋊C4×C61—21no—
77C4⋊C4×C61—21no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C42×C61—41no—
82C42×C61C22×C411yescharacteristic
83C42×C61C2×D411yescharacteristic
84C42×C61#4111yescharacteristic
85C42×C61#4111yescharacteristic
86C4⋊Dic61—22no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4⋊Dic61C22⋊C414yes—
90#11758196—82no—
91C4×Dic61—82no—
92#11758194—24no—
93C2×#92418—22no—
94C2×#92418—41no—
95C23×C61#12011yescharacteristic
96Q8×C61—81no—
97Q8×C61—81no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61C4×D411yescharacteristic
108C2×C4×C61C4×D411yescharacteristic
109C2×C4×C61#12111yescharacteristic
110C2×C4×C61#12111yescharacteristic
111C2×C4×C61#12311yescharacteristic
112C2×Dic61—24no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118Dic122—82no—
119Dic122—82no—
120C8×C61—81no—
121#92418—24no—
122#92418—81no—
123#9427—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61#38611yescharacteristic
127C22×C61#45612yes—
128C4×C61—21no—
129C4×C61—21no—
130C4×C61—21no—
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—21no—
134C4×C61—82no—
135C4×C61—81no—
136C4×C61—81no—
137Dic61—44no—
138Dic61—82no—
139#3165—611no—
140#2381—611no—
141#1556—611no—
142#1515—611no—
143#1515D6111yescharacteristic
144#1288—611no—
145#1288—611no—
146C2×C61—22no—
147C2×C61#188611yescharacteristic
148C2×C61#190411yescharacteristic
149C2×C61#190511yescharacteristic
150#533C2×D6111yescharacteristic
151C2×C4⋊Q8—611no—
152#501—1222no—
153#500—612no—
154#498—612no—
155#432—611no—
156#432C2×D6111yescharacteristic
157#428—1222no—
158C4×M16—611no—
159#391—1222no—
160#387—1222no—
161#387—1222no—
162#383—611no—
163#383C2×D6111yescharacteristic
164#332—1222no—
165#328—22no—
166#328—1222no—
167C61#942711yescharacteristic, complemented, Sylow
168C22×Q8—1221no—
169C2×M16—1221no—
170C4⋊Q8—612no—
171C4⋊Q8—612no—
172C4⋊Q8—1221no—
173C4⋊Q8—1221no—
174C4⋊Q8C4×D6111yescharacteristic
175C4⋊Q8C4×D6111yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181#117—21no—
182C2×C4⋊C4—612no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4—1222no—
185C2×C42—1222no—
186C2×C42C22×D6111yescharacteristic
187#107—1224no—
188#106—1224no—
189C4⋊C8—21no—
190C4⋊C8—1221no—
191C22⋊C8—21no—
192C22⋊C8—1221no—
193#97—21no—
194#97—1221no—
195#97—1221no—
196C4×C8—612no—
197#95—21no—
198#95—21no—
199#95—612no—
200#95—1224no—
201#95—1221no—
202#95—1221no—
203#95C22⋊D6112yes—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—1222no—
207C2×Q8—1222no—
208C2×Q8—4881no—
209C2×Q8—4881no—
210C22×C4—22no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4D4×D6111yescharacteristic
214C22×C4D4×D6111yescharacteristic
215C22×C4C2×C22⋊D6111yescharacteristic
216M16—1224no—
217C2×C8—41no—
218C2×C8—1222no—
219C2×C8—2441no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—614no—
227C4⋊C4—1222no—
228C4⋊C4—1222no—
229C4⋊C4—1222no—
230C4⋊C4—4882no—
231C22⋊C4—41no—
232C42—41no—
233C42—4882no—
234C42C2×C4×D6111yescharacteristic
235C42#1175822211yescharacteristic
236C42D4×D6111yescharacteristic
237C42#1175822411yescharacteristic
238C23#1451974511yescharacteristic
239Q8—81no—
240Q8—81no—
241Q8—4882no—
242Q8—4882no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—1224no—
252C2×C4—2442no—
253C2×C4—2442no—
254C2×C4—2442no—
255C2×C4—2442no—
256C2×C4—2442no—
257C2×C4C22⋊C4×D6111yescharacteristic
258C2×C4#1451973711yescharacteristic
259C2×C4C4×C22⋊D6111yescharacteristic
260C2×C4#1451974811yescharacteristic
261C2×C4#1451975011yescharacteristic
262C2×C4#1451975911yescharacteristic
263C8—81no—
264C8—1224no—
265C8—4881no—
266C22—21no—
267C22—21no—
268C22#1683963811yescharacteristic
269C22#1683994312yes—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—82no—
277C4—81no—
278C4—81no—
279C4—4882no—
280C4—2444no—
281C2—22no—
282C2#2144620911yescharacteristic
283C2#2144630311yescharacteristic
284C2#2144631611yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1515C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1515D61
hypocenterC61#9427
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
10. #16840165C4semidirect
11. #16840158C4semidirect
167. C61#9427semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214489992
#214445352
#214443702
#214436852
#214436832
C61×#15152
#942761