Finite Group Databasebeta

← #29589371

Subgroups — #29589371

Order 15616.

706 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 81 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#381×D61C211yesmaximal, characteristic
2#21444144C212yesmaximal
3#21443749C211yesmaximal, characteristic, complemented
4#21443748C212yesmaximal, complemented
5C61×#1317C211yesmaximal, characteristic, complemented
6C23×C4×D61C411yescharacteristic
7C2×C22⋊C4×D61C411yescharacteristic
8C2×C22⋊C4×D61C2211yescharacteristic
9#16839724—22no—
10#16839625C411yescharacteristic
11#16839624—22no—
12#16839623C411yescharacteristic
13#16839623C2211yescharacteristic
14#95×D61—22no—
15#16839486C2212yes—
16#16839485—22no—
17C61×#381C2211yescharacteristic
18C61×#323C2212yes—
19C24×D61C2×C411yescharacteristic
20C23×Dic61C2×C411yescharacteristic
21C22×C4×D61—21no—
22C22×C4×D61—21no—
23C22×C4×D61—22no—
24C22×C4×D61—22no—
25C22×C4×D61—21no—
26C22×C4×D61—42no—
27C22×C4×D61—42no—
28C22×C4×D61D412yes—
29C2×#11758189—42no—
30C2×C22⋊Dic61C2×C411yescharacteristic
31C2×C22⋊Dic61C2×C411yescharacteristic
32C2×#11758198—21no—
33C2×#11758198D412yes—
34C22⋊C4×D61—42no—
35C22⋊C4×D61—42no—
36C22⋊#92418—22no—
37#14519689—42no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41C23×C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61C2311yescharacteristic
44C22⋊C8×C61—22no—
45#95×C61—22no—
46C24×C61C22×C411yescharacteristic
47C23×D61—21no—
48C23×D61—21no—
49C23×D61—22no—
50C23×D61—22no—
51C23×D61—44no—
52C23×D61C22⋊C411yescharacteristic
53C23×D61C22⋊C411yescharacteristic
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—41no—
58C22×C4×C61—41no—
59C22×C4×C61M1612yescomplemented
60C22×C4×C61C2×D412yes—
61C22×Dic61—21no—
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61C22⋊C412yes—
67C22×Dic61M1612yescomplemented
68C2×C4×D61—24no—
69C2×C4×D61—24no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—48no—
73C2×C4×D61—42no—
74C2×C4×D61—42no—
75C2×C4×D61—48no—
76C2×C4×D61—42no—
77C2×C4×D61—42no—
78C2×C4×D61—44no—
79C2×C4×D61—44no—
80C2×C4×D61—42no—
81C2×C8×C61—42no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86#11758198—42no—
87#11758198—42no—
88C2×#92418—42no—
89#11758189—84no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—42no—
93C23×C61C2×C22⋊C411yescharacteristic
94C23×C61C2×M1611yescharacteristic
95C23×C61C2×M1611yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—24no—
99C22×D61—24no—
100C22×D61—24no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—48no—
104C22×D61—44no—
105C22×D61—44no—
106C22×D61—44no—
107C22×D61—44no—
108C22×D61—42no—
109C22×D61#9912yes—
110C22×D61#10012yes—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—42no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C4×D61—48no—
130C4×D61—48no—
131C4×D61—44no—
132C4×D61—82no—
133C4×D61—84no—
134C8×C61—82no—
135#92418—82no—
136#11500—611nomaximal, Sylow
137C22×C61—22no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61#40711yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#10011yescharacteristic
147C2×D61—24no—
148C2×D61—24no—
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—48no—
153C2×D61—48no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61—42no—
157C2×D61#35214yes—
158C4×C61—42no—
159C4×C61—42no—
160C4×C61—81no—
161Dic61—42no—
162Dic61—42no—
163Dic61—81no—
164C2×#381—611no—
165#1462—612no—
166#1317—611no—
167#1317—612no—
168#1317D6111yescharacteristic, complemented
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—41no—
172C2×C61—41no—
173C2×C61#146211yescharacteristic
174C2×C61C2×#35212yes—
175D61—24no—
176D61—42no—
177C24×C4—611no—
178C22×C22⋊C4—611no—
179C22×C22⋊C4—611no—
180#407—1222no—
181#381—611no—
182#381—611no—
183#381—611no—
184#381—1222no—
185#381C2×D6111yescharacteristic
186C2×#95—1222no—
187#323—612no—
188#323—1222no—
189#323C2×D6112yes—
190C61#1150011yescharacteristic, complemented, Sylow
191C25—611no—
192C23×C4—611no—
193C23×C4—612no—
194C23×C4—1222no—
195C23×C4—1221no—
196C23×C4—1221no—
197C23×C4—1221no—
198C23×C4—1222no—
199C23×C4—2442no—
200C23×C4—2442no—
201C23×C4C4×D6111yescharacteristic
202C2×M16—2442no—
203C2×C22⋊C4—611no—
204C2×C22⋊C4—612no—
205C2×C22⋊C4—611no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—2442no—
208C2×C22⋊C4—2442no—
209C2×C22⋊C4C4×D6111yescharacteristic
210C2×C22⋊C4C22×D6111yescharacteristic
211C22⋊C8—22no—
212C22⋊C8—1222no—
213C22⋊C8—2442no—
214#95—22no—
215#95—1222no—
216#95—1222no—
217#95—1222no—
218C24—611no—
219C24—611no—
220C24—1222no—
221C24—1222no—
222C24—1221no—
223C24—1221no—
224C24—2444no—
225C24C2×C4×D6111yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—612no—
232C22×C4—612no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—1224no—
236C22×C4—1224no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—2442no—
241C22×C4—2444no—
242C22×C4—2444no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4—2448no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4—2448no—
249C22×C4—2441no—
250C22×C4—2441no—
251C22×C4#1175818912yescomplemented
252C22×C4D4×D6112yes—
253M16—4884no—
254C2×C8—42no—
255C2×C8—2442no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—2442no—
259C22⋊C4—2442no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C23—21no—
263C23—21no—
264C23—42no—
265C23—612no—
266C23—612no—
267C23—1224no—
268C23—1224no—
269C23—1224no—
270C23—1221no—
271C23—1221no—
272C23—1221no—
273C23—1221no—
274C23—2442no—
275C23—2444no—
276C23—2444no—
277C23—2444no—
278C23—2448no—
279C23—2444no—
280C23C22⋊C4×D6111yescharacteristic
281C23M16×D6111yescharacteristic
282C23C2×#1175818911yescharacteristic
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—4882no—
295C2×C4—4884no—
296C2×C4—2444no—
297C2×C4—2448no—
298C2×C4—2448no—
299C2×C4—2442no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2442no—
305C2×C4—2442no—
306C8—82no—
307C8—4882no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—614no—
316C22—1224no—
317C22—1224no—
318C22—1222no—
319C22—1222no—
320C22—1222no—
321C22—2442no—
322C22—2442no—
323C22—2448no—
324C22—2448no—
325C22—2442no—
326C22#1683972411yescharacteristic
327C22#99×D6111yescharacteristic
328C22#100×D6111yescharacteristic
329C4—42no—
330C4—42no—
331C4—81no—
332C4—4881no—
333C4—2442no—
334C4—2442no—
335C2—21no—
336C2—21no—
337C2—41no—
338C2—41no—
339C2—1224no—
340C2—2442no—
341C2#2144414411yescharacteristic
342C2#352×D6112yes—
subgroupG/H
centerC22#99×D61
commutatorC24×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1317C2
socleC22×C61C2×#99
prime radicalC24×D61C2×C4
cosocleC2×C22⋊C4×C61C23
hypercenter#1317D61
hypocenterC61#11500
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1317

Complements

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

NH ≅ G/Ntype
3. #21443749C2semidirect
4. #21443748C2semidirect
5. C61×#1317C2semidirect
59. C22×C4×C61M16semidirect
67. C22×Dic61M16semidirect
168. #1317D61semidirect
190. C61#11500semidirect
251. C22×C4#11758189semidirect

Maximal subgroups

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

subgroupindex
#381×D612
#214441442
#214437492
#214437482
C61×#13172
#1150061