Finite Group Databasebeta

← #29570338

Subgroups — #29570338

Order 15616.

994 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 159 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#21448197C212yesmaximal
2#21448197C211yesmaximal, characteristic
3#21443723C212yesmaximal
4#21443723C211yesmaximal, characteristic
5C61×#1304C211yesmaximal, characteristic
6C24×Dic61C2211yescharacteristic
7C2×#14519733—22no—
8C2×#14519733C412yescomplemented
9#16839623—22no—
10#16839623C412yescomplemented
11#16839620—22no—
12#16839620—22no—
13#16839620C414yescomplemented
14#16839618C2212yes—
15#16839618C2211yescharacteristic
16C22⋊C4×Dic61—22no—
17C22⋊C4×Dic61—22no—
18C22⋊C4×Dic61C414yescomplemented
19C61×#379C2212yes—
20C61×#379C2211yescharacteristic
21C23×Dic61—44no—
22C23×Dic61C2×C411yescharacteristic
23C23×Dic61C2×C412yes—
24C23×Dic61C2×C412yes—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61D412yes—
27C23×Dic61D412yes—
28C23×Dic61D412yes—
29C23×Dic61Q812yes—
30#14519733—48no—
31#14519733—44no—
32C2×#11758196—24no—
33C2×#11758196—22no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61C2×C412yes—
38C2×C22⋊Dic61C2×C414yes—
39C2×C4×Dic61—24no—
40C2×C4×Dic61—22no—
41#14519684—24no—
42#14519684—24no—
43#14519684—24no—
44C25×C61C2311yescharacteristic
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61C2×C412yes—
49C2×C22⋊C4×C61C2×C414yes—
50C24×C61—42no—
51C24×C61—44no—
52C24×C61C22×C411yescharacteristic
53C24×C61C22×C412yes—
54C24×C61C2×D412yes—
55C24×C61C2×D411yescharacteristic
56C24×C61C2×Q811yescharacteristic
57C22×C4×C61—24no—
58C22×C4×C61—22no—
59C22×Dic61—24no—
60C22×Dic61—28no—
61C22×Dic61—24no—
62C22×Dic61—22no—
63C22×Dic61—44no—
64C22×Dic61—44no—
65C22×Dic61—44no—
66C22×Dic61—44no—
67C22×Dic61—42no—
68C22×Dic61—44no—
69C22×Dic61C4212yes—
70C22×Dic61C4211yescharacteristic
71C22×Dic61C4211yescharacteristic
72C22×Dic61C22⋊C412yes—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C412yes—
75C22×Dic61C22⋊C412yes—
76C22×Dic61C22⋊C412yes—
77C22×Dic61C22⋊C412yes—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C4⋊C412yes—
80C22×Dic61C4⋊C412yes—
81C22×Dic61C4⋊C412yes—
82C22×Dic61C4⋊C412yes—
83C22×Dic61C4⋊C412yes—
84C22×Dic61C4⋊C412yes—
85C22⋊C4×C61—44no—
86C22⋊C4×C61—44no—
87C22⋊C4×C61—44no—
88C22⋊C4×C61—44no—
89C22⋊C4×C61—42no—
90C22⋊Dic61—44no—
91C22⋊Dic61—44no—
92C22⋊Dic61—44no—
93C22⋊Dic61—44no—
94C22⋊Dic61—42no—
95#11758196—48no—
96#11758196—44no—
97C4×Dic61—44no—
98C4×Dic61—48no—
99C23×C61—22no—
100C23×C61—24no—
101C23×C61—24no—
102C23×C61—24no—
103C23×C61—24no—
104C23×C61—42no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—416no—
112C23×C61C2×C4211yescharacteristic
113C23×C61C2×C22⋊C411yescharacteristic
114C23×C61C2×C22⋊C412yes—
115C23×C61C2×C4⋊C411yescharacteristic
116C23×C61C2×C4⋊C412yes—
117C2×C4×C61—44no—
118C2×C4×C61—44no—
119C2×C4×C61—44no—
120C2×C4×C61—42no—
121C2×C4×C61—44no—
122C2×Dic61—22no—
123C2×Dic61—24no—
124C2×Dic61—24no—
125C2×Dic61—24no—
126C2×Dic61—24no—
127C2×Dic61—24no—
128C2×Dic61—24no—
129C2×Dic61—24no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—44no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140C2×Dic61—42no—
141C2×Dic61—44no—
142C2×Dic61#9512yes—
143C2×Dic61#9512yes—
144C2×Dic61#9512yes—
145C2×Dic61#9512yes—
146C2×Dic61#9914yes—
147#8102—611nomaximal, Sylow
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—22no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—22no—
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—48no—
159C22×C61—44no—
160C22×C61—44no—
161C22×C61—48no—
162C22×C61—44no—
163C22×C61C2×#9511yescharacteristic
164C22×C61C2×#9912yes—
165C22×C61#41014yes—
166C4×C61—82no—
167C4×C61—84no—
168Dic61—22no—
169Dic61—22no—
170Dic61—22no—
171Dic61—82no—
172Dic61—84no—
173Dic61#34214yes—
174#2280—611no—
175#2280—612no—
176#1304—612no—
177#1304—611no—
178#1304D6111yescharacteristic
179C2×C61—22no—
180C2×C61—24no—
181C2×C61—41no—
182C2×C61—41no—
183C2×C61—42no—
184C2×C61C2×#34211yescharacteristic
185C2×C61#174012yes—
186C24×C4—611no—
187C2×#117—612no—
188C2×#117—1222no—
189#381—612no—
190#381—1222no—
191#380—614no—
192#380—1222no—
193#380—1222no—
194#379—611no—
195#379—612no—
196#379C2×D6112yes—
197#379C2×D6111yescharacteristic
198C4×C22⋊C4—614no—
199C4×C22⋊C4—1222no—
200C4×C22⋊C4—1222no—
201C61#810211yescharacteristic, complemented, Sylow
202C25C22×D6111yescharacteristic
203C23×C4—611no—
204C23×C4—612no—
205C23×C4—612no—
206C23×C4—612no—
207C23×C4—611no—
208C23×C4—612no—
209C23×C4—612no—
210C23×C4—612no—
211C23×C4—2444no—
212#117—2448no—
213#117—2444no—
214C2×C4⋊C4—1222no—
215C2×C4⋊C4—1224no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—612no—
220C2×C22⋊C4—614no—
221C2×C22⋊C4—1222no—
222C2×C22⋊C4—1222no—
223C2×C22⋊C4—1222no—
224C2×C22⋊C4C4×D6112yes—
225C2×C22⋊C4C4×D6114yes—
226C2×C42—1222no—
227C2×C42—1224no—
228#95—1224no—
229#95—1224no—
230#95—1224no—
231C24—42no—
232C24—44no—
233C24C2×C4×D6111yescharacteristic
234C24C2×C4×D6112yes—
235C24D4×D6112yes—
236C24D4×D6111yescharacteristic
237C24Q8×D6111yescharacteristic
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—612no—
247C22×C4—612no—
248C22×C4—612no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—612no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—1228no—
257C22×C4—1224no—
258C22×C4—1224no—
259C22×C4—1222no—
260C22×C4—2444no—
261C22×C4—2444no—
262C22×C4—2444no—
263C22×C4—2444no—
264C22×C4—2444no—
265C22×C4—2442no—
266C4⋊C4—2444no—
267C4⋊C4—2448no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—44no—
273C22⋊C4—2444no—
274C22⋊C4—2444no—
275C22⋊C4—2444no—
276C22⋊C4—2442no—
277C22⋊C4—2444no—
278C42—2448no—
279C42—2444no—
280C23—22no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—416no—
293C23C42×D6111yescharacteristic
294C23C22⋊C4×D6111yescharacteristic
295C23C22⋊C4×D6112yes—
296C23C4⋊C4×D6111yescharacteristic
297C23C4⋊C4×D6112yes—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—614no—
304C2×C4—612no—
305C2×C4—612no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—1224no—
309C2×C4—1224no—
310C2×C4—1224no—
311C2×C4—1224no—
312C2×C4—1224no—
313C2×C4—1224no—
314C2×C4—1222no—
315C2×C4—1224no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2444no—
324C2×C4—2444no—
325C2×C4—2444no—
326C2×C4—2442no—
327C2×C4—2444no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—48no—
339C22—44no—
340C22—44no—
341C22—48no—
342C22—44no—
343C22#95×D6111yescharacteristic
344C22#99×D6112yes—
345C22#1683973414yes—
346C4—82no—
347C4—84no—
348C4—614no—
349C4—1222no—
350C4—1222no—
351C4—1222no—
352C4—4882no—
353C4—4884no—
354C2—22no—
355C2—24no—
356C2—41no—
357C2—41no—
358C2—42no—
359C2#342×D6111yescharacteristic
360C2#2144555012yes—
subgroupG/H
centerC22#95×D61
commutatorC23×C61C2×C42
FrattiniC25C22×D61
FittingC61×#1304C2
socleC22×C61C2×#95
prime radical, cosocleC25×C61C23
hypercenter#1304D61
hypocenterC61#8102
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1304

Complements

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

NH ≅ G/Ntype
8. C2×#14519733C4semidirect
10. #16839623C4semidirect
13. #16839620C4semidirect
18. C22⋊C4×Dic61C4semidirect
201. C61#8102semidirect

Maximal subgroups

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

subgroupindex
#214481972
#214437232
C61×#13042
#810261