Finite Group Databasebeta

← #29570386

Subgroups — #29570386

Order 15616.

914 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 155 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#407×D61C212yesmaximal
2#21448199C211yesmaximal, characteristic
3#21443728C212yesmaximal, complemented
4#21443727C211yesmaximal, characteristic, complemented
5C61×#1306C211yesmaximal, characteristic, complemented
6C23×C4×D61C2211yescharacteristic
7C2×M16×D61—22no—
8C2×#14519733C412yes—
9#16839726C2212yes—
10#16839725—22no—
11#16839724—22no—
12#16839724C414yescomplemented
13C22⋊C8×D61—22no—
14C22⋊C8×D61C414yescomplemented
15#16839625C2211yescharacteristic
16#16839621—22no—
17#16839617C412yes—
18#16839609—22no—
19C61×#407C2212yes—
20C61×#381C2211yescharacteristic
21C24×D61C2×C411yescharacteristic
22C23×Dic61C2×C411yescharacteristic
23C22×C4×D61—44no—
24C22×C4×D61C2×C412yes—
25C22×C4×D61C2×C412yes—
26C22×C4×D61D412yes—
27C22×C4×D61D412yes—
28C22×C4×D61D412yes—
29C22×C4×D61Q812yes—
30C2×#11758194—22no—
31C2×#11758189—24no—
32M16×D61—48no—
33C2×C8×D61—24no—
34#14519733—44no—
35C2×#11758196—22no—
36C2×C22⋊Dic61C2×C412yes—
37C2×#11758198—24no—
38C2×C4×Dic61—22no—
39C22⋊#92418—22no—
40C22⋊#92418C2×C414yes—
41#14519689—24no—
42#14519689—24no—
43#14519684—22no—
44C23×C4×C61C2311yescharacteristic
45C2×M16×C61—22no—
46C2×C22⋊C4×C61C2×C412yes—
47C22⋊C8×C61—22no—
48C22⋊C8×C61C2×C414yes—
49#95×C61—22no—
50C24×C61C22×C411yescharacteristic
51C23×D61—44no—
52C23×D61—42no—
53C23×D61C4211yescharacteristic
54C23×D61C4211yescharacteristic
55C23×D61C22⋊C412yes—
56C23×D61C4⋊C412yes—
57C22×C4×C61—22no—
58C22×C4×C61—42no—
59C22×C4×C61C22×C412yes—
60C22×C4×C61C2×D412yes—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×Q811yescharacteristic
63C22×Dic61—24no—
64C22×Dic61—22no—
65C22×Dic61C4212yes—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C4⋊C412yes—
69C2×C4×D61—28no—
70C2×C4×D61—44no—
71C2×C4×D61—44no—
72C2×C4×D61—44no—
73C2×C4×D61—44no—
74C2×C4×D61—44no—
75C2×C4×D61C22⋊C412yes—
76C2×C4×D61C22⋊C412yes—
77C2×C4×D61C22⋊C412yes—
78C2×C4×D61C22⋊C412yes—
79C2×C4×D61C4⋊C412yes—
80C2×C4×D61C4⋊C412yes—
81C2×C4×D61C4⋊C412yes—
82C2×C4×D61C4⋊C412yes—
83M16×C61—44no—
84C2×C8×C61—24no—
85C22⋊C4×C61—42no—
86C22⋊Dic61—42no—
87#11758198—48no—
88#11758196—44no—
89C4×Dic61—44no—
90#11758194—44no—
91C2×#92418—24no—
92#11758189—48no—
93C8×D61—48no—
94C23×C61—22no—
95C23×C61—42no—
96C23×C61C2×C4211yescharacteristic
97C23×C61C2×C22⋊C411yescharacteristic
98C23×C61C2×C4⋊C411yescharacteristic
99C22×D61—24no—
100C22×D61—24no—
101C22×D61—24no—
102C22×D61—24no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61—42no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61—42no—
109C22×D61—416no—
110C22×D61#9512yes—
111C22×D61#9512yes—
112C2×C4×C61—24no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—44no—
117C2×C4×C61—42no—
118C2×C4×C61C2×C22⋊C412yes—
119C2×C4×C61C2×C4⋊C412yes—
120C2×Dic61—22no—
121C2×Dic61—24no—
122C2×Dic61—24no—
123C2×Dic61—44no—
124C2×Dic61—42no—
125C2×Dic61#9512yes—
126C2×Dic61#9512yes—
127C2×Dic61#9914yes—
128C4×D61—24no—
129C4×D61—24no—
130C4×D61—24no—
131C4×D61—24no—
132C4×D61—42no—
133C4×D61—42no—
134C4×D61—42no—
135C4×D61—42no—
136C8×C61—44no—
137#92418—44no—
138#8115—611nomaximal, Sylow
139C22×C61—22no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—44no—
145C22×C61C2×#9511yescharacteristic
146C22×C61C2×#9912yes—
147C2×D61—22no—
148C2×D61—22no—
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—44no—
153C2×D61—48no—
154C2×D61—48no—
155C2×D61—44no—
156C2×D61#34414yes—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—82no—
160Dic61—22no—
161Dic61—82no—
162Dic61#34214yes—
163C2×#407—612no—
164#2280—611no—
165#1306—612no—
166#1306—611no—
167#1306D6111yescharacteristic, complemented
168C2×C61—22no—
169C2×C61—41no—
170C2×C61—41no—
171C2×C61C2×#34211yescharacteristic
172C2×C61C2×#34412yes—
173D61—24no—
174D61—42no—
175C24×C4—611no—
176C22×M16—1222no—
177C2×#117—612no—
178#407—612no—
179#407—614no—
180#407—1222no—
181#407—1222no—
182#407C2×D6112yes—
183C2×C22⋊C8—614no—
184C2×C22⋊C8—1222no—
185#381—611no—
186#381C2×D6111yescharacteristic
187#380—1222no—
188#379—612no—
189C4×C22⋊C4—1222no—
190C61#811511yescharacteristic, complemented, Sylow
191C25—611no—
192C23×C4—611no—
193C23×C4—612no—
194C23×C4—612no—
195C23×C4—612no—
196C23×C4—612no—
197C23×C4—612no—
198C23×C4—612no—
199C23×C4—2444no—
200C23×C4C22×D6111yescharacteristic
201C2×M16—22no—
202C2×M16—1224no—
203C2×M16—1222no—
204C2×M16—2448no—
205C22×C8—1224no—
206#117—2444no—
207C2×C4⋊C4—1222no—
208C2×C22⋊C4—612no—
209C2×C22⋊C4—1224no—
210C2×C22⋊C4C4×D6112yes—
211C2×C42—1222no—
212C22⋊C8—22no—
213C22⋊C8—614no—
214C22⋊C8—1224no—
215C22⋊C8—1224no—
216C22⋊C8—1222no—
217C22⋊C8C4×D6114yes—
218#95—22no—
219#95—1222no—
220C24—611no—
221C24—612no—
222C24—612no—
223C24—611no—
224C24—2442no—
225C24—2444no—
226C24C2×C4×D6111yescharacteristic
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—611no—
230C22×C4—612no—
231C22×C4—611no—
232C22×C4—612no—
233C22×C4—612no—
234C22×C4—612no—
235C22×C4—612no—
236C22×C4—612no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—612no—
240C22×C4—612no—
241C22×C4—1228no—
242C22×C4—1224no—
243C22×C4—1222no—
244C22×C4—2444no—
245C22×C4—2444no—
246C22×C4—2444no—
247C22×C4—2444no—
248C22×C4—2444no—
249C22×C4C2×C4×D6112yes—
250C22×C4D4×D6112yes—
251C22×C4D4×D6111yescharacteristic
252C22×C4Q8×D6111yescharacteristic
253M16—44no—
254M16—2448no—
255M16—2444no—
256C2×C8—24no—
257C2×C8—1224no—
258C2×C8—2448no—
259C4⋊C4—2444no—
260C22⋊C4—42no—
261C22⋊C4—2448no—
262C22⋊C4—2442no—
263C42—2444no—
264C23—22no—
265C23—42no—
266C23—612no—
267C23—612no—
268C23—1224no—
269C23—1224no—
270C23—1224no—
271C23—1224no—
272C23—2441no—
273C23—2441no—
274C23—2442no—
275C23—2442no—
276C23—2442no—
277C23—2442no—
278C23—24416no—
279C23C42×D6111yescharacteristic
280C23C22⋊C4×D6111yescharacteristic
281C23C4⋊C4×D6111yescharacteristic
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—614no—
289C2×C4—612no—
290C2×C4—612no—
291C2×C4—1224no—
292C2×C4—1224no—
293C2×C4—1224no—
294C2×C4—1224no—
295C2×C4—1224no—
296C2×C4—1224no—
297C2×C4—1222no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2444no—
303C2×C4—2442no—
304C2×C4C22⋊C4×D6112yes—
305C2×C4C4⋊C4×D6112yes—
306C8—44no—
307C8—2444no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—44no—
314C22—614no—
315C22—1222no—
316C22—1222no—
317C22—1222no—
318C22—1222no—
319C22—1222no—
320C22—2448no—
321C22—2444no—
322C22—2448no—
323C22—2444no—
324C22#95×D6111yescharacteristic
325C22#99×D6112yes—
326C4—22no—
327C4—22no—
328C4—82no—
329C4—614no—
330C4—1222no—
331C4—4882no—
332C2—22no—
333C2—41no—
334C2—41no—
335C2—1224no—
336C2—2442no—
337C2#342×D6111yescharacteristic
338C2#344×D6112yes—
subgroupG/H
centerC22#95×D61
commutatorC23×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1306C2
socleC22×C61C2×#95
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1306D61
hypocenterC61#8115
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. #21443728C2semidirect
4. #21443727C2semidirect
5. C61×#1306C2semidirect
12. #16839724C4semidirect
14. C22⋊C8×D61C4semidirect
167. #1306D61semidirect
190. C61#8115semidirect

Maximal subgroups

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

subgroupindex
#407×D612
#214481992
#214437282
#214437272
C61×#13062
#811561