Finite Group Databasebeta

← #29577828

Subgroups — #29577828

Order 15616.

610 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 115 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#21448176C211yesmaximal, characteristic
2#21445918C212yesmaximal
3#21444149C211yesmaximal, characteristic
4#21443671C212yesmaximal
5C61×#2484C211yesmaximal, characteristic
6C2×#14519736—21no—
7#16839726C2212yes—
8C4⋊C4×Dic61—24no—
9C4×C22⋊Dic61—21no—
10C4×C22⋊Dic61C2212yes—
11#16839521—24no—
12#16839488—22no—
13C2×#117×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C61×#400C412yescomplemented
16C61×#398C412yescomplemented
17C61×#387—22no—
18C61×#385—21no—
19C61×#381—21no—
20C61×#381C2211yescharacteristic
21C2×#11758194—22no—
22#14519736—28no—
23C2×C4⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C4×Dic61—22no—
26C2×C4×Dic61—22no—
27C2×C4×Dic61D414yes—
28C22⋊#92418—22no—
29C22⋊#92418—22no—
30#14519684—22no—
31#14519684—22no—
32C23×C4×C61D411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C23×C4×C61C2311yescharacteristic
35#117×C61—22no—
36#117×C61—22no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C412yes—
43C2×C4⋊C4×C61D412yes—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C42×C61—21no—
47#95×C61—22no—
48#95×C61—22no—
49#95×C61—22no—
50#95×C61—21no—
51#95×C61C2×C412yes—
52C24×C61C2×D411yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—24no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—22no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D412yes—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61#4112yes—
69C22×Dic61—24no—
70C4⋊C4×C61—24no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61—22no—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—44no—
75C22⋊C4×C61—42no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C42×C61—42no—
79C22⋊Dic61—28no—
80C4⋊Dic61—28no—
81C4×Dic61—28no—
82C4×Dic61—44no—
83C4×Dic61—82no—
84#11758194—44no—
85C2×#92418—42no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—41no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61#12011yescharacteristic
91C23×C61#12311yescharacteristic
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—24no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—24no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—81no—
112C2×C4×C61#10414yes—
113C2×C4×C61C4×D412yes—
114C2×C4×C61#12112yes—
115C2×Dic61—28no—
116C2×Dic61—44no—
117C2×Dic61—44no—
118#92418—82no—
119#9478—611nomaximal, Sylow
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—44no—
128C22×C61#38611yescharacteristic
129C22×C61C2×#10412yes—
130C22×C61#45812yes—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—24no—
134C4×C61—42no—
135C4×C61—82no—
136C4×C61—81no—
137C4×C61—81no—
138Dic61—48no—
139#2484D6111yescharacteristic
140#2274—611no—
141#1826—612no—
142#1463—611no—
143#1284—612no—
144C2×C61—22no—
145C2×C61—22no—
146C2×C61—41no—
147C2×C61#189011yescharacteristic
148C2×C61#189611yescharacteristic
149C2×C61#191211yescharacteristic
150C2×#117—1221no—
151C2×#117C2×D6111yescharacteristic
152C22×C4⋊C4C2×D6111yescharacteristic
153#407—612no—
154#400Dic6112yescomplemented
155#398Dic6112yescomplemented
156#387—22no—
157#385—21no—
158#381—21no—
159#381C2×D6111yescharacteristic
160C4×C4⋊C4—1224no—
161C4×C22⋊C4—612no—
162C4×C22⋊C4—1221no—
163#339—1224no—
164#324—1222no—
165C61#947811yescharacteristic, complemented, Sylow
166C23×C4C22⋊D6111yescharacteristic
167C23×C4C22⋊D6111yescharacteristic
168C23×C4C22×D6111yescharacteristic
169C2×M16—1222no—
170#117—22no—
171#117—22no—
172#117—1228no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—1222no—
177C2×C4⋊C4C2×Dic6111yescharacteristic
178C2×C4⋊C4C2×Dic6111yescharacteristic
179C2×C4⋊C4C2×Dic6112yes—
180C2×C4⋊C4C22⋊D6112yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1222no—
184C2×C42—21no—
185C2×C42—614no—
186C2×C42—1222no—
187C2×C42—1222no—
188C22⋊C8—1222no—
189C22⋊C8—1222no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—1222no—
195#95—1222no—
196#95C2×Dic6112yes—
197C24C2×C22⋊D6111yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—1224no—
209C22×C4C22⋊Dic6112yes—
210C22×C4C22⋊Dic6112yes—
211C22×C4D4×D6112yes—
212C22×C4#1175822412yes—
213C22×C4C22×Dic6111yescharacteristic
214C22×C4C2×C22⋊D6111yescharacteristic
215M16—2444no—
216C2×C8—2442no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—1228no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—1228no—
227C42—42no—
228C42—1228no—
229C42—4882no—
230C42—2444no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×C22⋊Dic6111yescharacteristic
235C23#1451974511yescharacteristic
236C23#1451975811yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—1228no—
258C2×C4—2444no—
259C2×C4—2444no—
260C2×C4#1451970414yes—
261C2×C4D4×Dic6112yes—
262C2×C4#1451974912yes—
263C8—4882no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1683964211yescharacteristic
273C22C2×#1451970412yes—
274C22#1683995112yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4—2448no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2144623311yescharacteristic
287C2#2144626311yescharacteristic
288C2#2144635711yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2484C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#2484D61
hypocenterC61#9478
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
15. C61×#400C4semidirect
16. C61×#398C4semidirect
154. #400Dic61semidirect
155. #398Dic61semidirect
165. C61#9478semidirect

Maximal subgroups

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

subgroupindex
#214481762
#214459182
#214441492
#214436712
C61×#24842
#947861