Finite Group Databasebeta

← #29577840

Subgroups — #29577840

Order 15616.

614 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450093C211yesmaximal, characteristic
2#21448417C211yesmaximal, characteristic
3#21445916C212yesmaximal, complemented
4#21443673C212yesmaximal, complemented
5C61×#1462C211yesmaximal, characteristic
6C2×#14519736C2211yescharacteristic
7#16839698C412yescomplemented
8#16839687C412yescomplemented
9#16839655—24no—
10#16839650—22no—
11#16839635—21no—
12#16839625C2211yescharacteristic
13#16839622—21no—
14#16839618—21no—
15#16839618—21no—
16C4×C22⋊Dic61C2212yes—
17#16839523—24no—
18C22×C22⋊C4×C61C2211yescharacteristic
19C61×#407C2212yes—
20C61×#323—22no—
21#14519736—22no—
22#14519736—22no—
23C2×C4⋊Dic61—22no—
24C2×C4⋊Dic61C2×C411yescharacteristic
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×#11758196—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—21no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—21no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61D414yes—
36#14519684—22no—
37#14519684—22no—
38#14519684—22no—
39#14519684—21no—
40#14519684C2×C412yes—
41C25×C61D411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61C2×C412yes—
48C2×C22⋊C4×C61D412yes—
49C22⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61—21no—
52C24×C61—21no—
53C24×C61—42no—
54C24×C61—42no—
55C24×C61—81no—
56C24×C61C22⋊C412yes—
57C24×C61C2×D411yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—42no—
61C22×C4×C61—81no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D412yes—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61#4112yes—
67C22×Dic61—22no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—24no—
71M16×C61—44no—
72C2×C8×C61—42no—
73C22⋊C4×C61—24no—
74C22⋊C4×C61—24no—
75C22⋊Dic61—44no—
76C22⋊Dic61—42no—
77C22⋊Dic61—44no—
78C22⋊Dic61—41no—
79C22⋊Dic61—41no—
80C22⋊Dic61—82no—
81C22⋊Dic61—82no—
82C22⋊Dic61—82no—
83C4⋊Dic61—22no—
84C4⋊Dic61—22no—
85C4⋊Dic61—82no—
86#11758196—82no—
87C4×Dic61—42no—
88C4×Dic61—44no—
89C23×C61—22no—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61—22no—
93C23×C61—22no—
94C23×C61—41no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—81no—
102C23×C61—81no—
103C23×C61—88no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61#12011yescharacteristic
106C23×C61#12311yescharacteristic
107C2×C4×C61—24no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—81no—
115C2×C4×C61C4×D412yes—
116C2×C4×C61#12112yes—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—44no—
120C2×Dic61—42no—
121C2×Dic61—44no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—44no—
127C8×C61—82no—
128#9481—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—44no—
137C22×C61—44no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—42no—
141C22×C61—82no—
142C22×C61—84no—
143C22×C61—84no—
144C22×C61#38611yescharacteristic
145C22×C61#42112yes—
146C22×C61#45812yes—
147C4×C61—22no—
148C4×C61—22no—
149C4×C61—42no—
150Dic61—84no—
151Dic61—82no—
152Dic61—81no—
153Dic61—81no—
154#2482—611no—
155#2312—611no—
156#1826—612no—
157#1462D6111yescharacteristic
158#1284—612no—
159C2×C61—22no—
160C2×C61—22no—
161C2×C61—41no—
162C2×C61—82no—
163C2×C61#189011yescharacteristic
164C2×C61#189711yescharacteristic
165C2×C61#191111yescharacteristic
166C2×#117—611no—
167C22×C22⋊C4C2×D6111yescharacteristic
168#407C2×D6112yes—
169#399—612no—
170#397—612no—
171#388—1224no—
172#388—1222no—
173#385—1221no—
174#381—611no—
175#380—1221no—
176#379—1221no—
177#379—1221no—
178C4×C22⋊C4—612no—
179#339—1224no—
180#323—22no—
181C61#948111yescharacteristic, complemented, Sylow
182C25D24411yescharacteristic
183C23×C4D24411yescharacteristic
184C23×C4C22×D6111yescharacteristic
185C2×M16—22no—
186#117—1222no—
187#117—1222no—
188C2×C4⋊C4—611no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1222no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1222no—
196C2×C22⋊C4—1222no—
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1221no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4—1221no—
201C2×C22⋊C4C4×D6112yes—
202C2×C22⋊C4C22⋊D6112yes—
203C2×C42—614no—
204C2×C42—1221no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207#95—612no—
208#95—1222no—
209#95—1222no—
210#95—1222no—
211#95—1221no—
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24#1175819812yes—
218C24C2×D24411yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—1222no—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4—1224no—
227C22×C4#1175819812yes—
228C22×C4C2×C4×D6111yescharacteristic
229C22×C4D4×D6112yes—
230C22×C4#1175822412yes—
231C22×C4C2×C22⋊D6111yescharacteristic
232M16—44no—
233C2×C8—42no—
234C4⋊C4—1222no—
235C4⋊C4—1222no—
236C4⋊C4—4882no—
237C4⋊C4—4882no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—4882no—
241C22⋊C4—4882no—
242C22⋊C4—4882no—
243C22⋊C4—2442no—
244C22⋊C4—2444no—
245C22⋊C4—2444no—
246C22⋊C4—2441no—
247C22⋊C4—2441no—
248C42—2444no—
249C42—2442no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×#1175819811yescharacteristic
266C23#1451974711yescharacteristic
267C23#1451976111yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—2442no—
277C2×C4—2444no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2444no—
281C2×C4—2444no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4#1451973712yes—
287C2×C4#1451974912yes—
288C8—82no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1683963911yescharacteristic
305C22#1683977812yes—
306C22#1683995312yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—4884no—
311C4—4882no—
312C4—4881no—
313C4—4881no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#2144623011yescharacteristic
319C2#2144626511yescharacteristic
320C2#2144634911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1462C2
socleC22×C61#386
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenter#1462D61
hypocenterC61#9481
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #21445916C2semidirect
4. #21443673C2semidirect
7. #16839698C4semidirect
8. #16839687C4semidirect
181. C61#9481semidirect

Maximal subgroups

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

subgroupindex
#214500932
#214484172
#214459162
#214436732
C61×#14622
#948161