Finite Group Databasebeta

← #29842733

Subgroups — #29842733

Order 15616.

1800 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61⋊#564C211yesmaximal, characteristic
2#21449004C212yesmaximal
3#21449003C214yesmaximal
4#21449002C214yesmaximal
5#21449001C214yesmaximal
6#125×Dic61C211yesmaximal, characteristic
7#21448378C214yesmaximal
8#21448376C214yesmaximal
9#21448376C214yesmaximal
10#21448374C212yesmaximal
11C61×#2382C211yesmaximal, characteristic
12C61⋊#564C4116yescomplemented
13C2×#14519769C2212yes—
14C2×#14519768C2214yes—
15C2×#14519767C2214yes—
16C2×#14519767C2214yes—
17#16839661C2212yes—
18#16839660C2214yes—
19#16839659C2214yes—
20#16839659C2214yes—
21#16839658C2212yes—
22#16839658C2212yes—
23#16839657C2214yes—
24#16839657C2214yes—
25#16839649C2214yes—
26#16839649C2214yes—
27#16839648C2214yes—
28#16839647C2214yes—
29#16839647C2214yes—
30#16839646C2214yes—
31#16839646C2214yes—
32#16839645C2214yes—
33#16839645C2214yes—
34#16839644C2214yes—
35#16839644C2214yes—
36#16839643C2214yes—
37#16839643C2214yes—
38#16839628C2211yescharacteristic
39#16839627C2214yes—
40#16839626C2214yes—
41#16839626C2214yes—
42C4⋊C4×Dic61C2212yes—
43C4⋊C4×Dic61C2212yes—
44C4⋊C4×Dic61C2212yes—
45C4⋊C4×Dic61C2214yes—
46C4⋊C4×Dic61C2214yes—
47#16839613C2214yes—
48#16839613C2214yes—
49#16839613C2214yes—
50#16839613C2214yes—
51C42×Dic61C2211yescharacteristic
52C42×Dic61C2214yes—
53C2×#125×C61C2211yescharacteristic
54C61×#389C2214yes—
55C61×#387C2214yes—
56C61×#387C2214yes—
57C61×#382C2212yes—
58#14519769C2×C4116yes—
59#14519768C2×C4132yes—
60#14519767C2×C4132yes—
61#14519767C2×C4132yes—
62C2×C4⋊Dic61C2314yes—
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2314yes—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196C2314yes—
68C2×#11758196C2314yes—
69C2×#11758196C2314yes—
70C2×#11758196C2314yes—
71C2×#11758196C2314yes—
72C2×#11758196C2314yes—
73C2×#11758196C2314yes—
74C2×#11758196C2314yes—
75C2×C4×Dic61C2314yes—
76C2×C4×Dic61C2311yescharacteristic
77C2×C4×Dic61C2314yes—
78C2×C4×Dic61C2312yes—
79C2×C4×Dic61C2314yes—
80C2×C4×Dic61C2314yes—
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2312yes—
83C2×C4×Dic61C2314yes—
84C2×C4×Dic61C2314yes—
85#14519684C2314yes—
86#14519684C2314yes—
87#14519684C2314yes—
88#14519684C2314yes—
89#14519684C2314yes—
90#14519684C2314yes—
91#14519684C2314yes—
92#14519684C2314yes—
93#14519684C2314yes—
94#14519684C2314yes—
95#14519684C2314yes—
96#125×C61C2×C418yes—
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2314yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2314yes—
102C2×C42×C61C2314yes—
103C2×C42×C61C2311yescharacteristic
104#95×C61C2314yes—
105#95×C61C2314yes—
106#95×C61C2314yes—
107#95×C61C2314yes—
108C22×C4×C61C2414yes—
109C22×C4×C61C2414yes—
110C22×C4×C61C2411yescharacteristic
111C22×C4×C61C2412yes—
112C22×C4×C61C2414yes—
113C22×Dic61C2411yescharacteristic
114C22×Dic61C2412yes—
115C22×Dic61C2414yes—
116C22×Dic61C2414yes—
117C22×Dic61C2414yes—
118C22×Dic61C2411yescharacteristic
119C4⋊C4×C61—42no—
120C4⋊C4×C61—42no—
121C4⋊C4×C61—44no—
122C4⋊C4×C61C22×C418yes—
123C4⋊C4×C61C22×C4116yes—
124C42×C61—44no—
125C42×C61C22×C414yes—
126C4⋊Dic61—42no—
127C4⋊Dic61—44no—
128C4⋊Dic61C22×C418yes—
129C4⋊Dic61C22×C4116yes—
130C4⋊Dic61C22×C418yes—
131#11758196—44no—
132#11758196—44no—
133#11758196—44no—
134#11758196—44no—
135#11758196C22×C4116yes—
136#11758196C22×C4116yes—
137#11758196C22×C4116yes—
138#11758196C22×C4116yes—
139C4×Dic61—28no—
140C4×Dic61—28no—
141C4×Dic61—22no—
142C4×Dic61—24no—
143C4×Dic61—28no—
144C4×Dic61—44no—
145C4×Dic61—44no—
146C4×Dic61—41no—
147C4×Dic61—42no—
148C4×Dic61C22×C4116yes—
149C23×C61C2511yescharacteristic
150C2×C4×C61—21no—
151C2×C4×C61—22no—
152C2×C4×C61—24no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—24no—
156C2×C4×C61—24no—
157C2×C4×C61—24no—
158C2×C4×C61—24no—
159C2×C4×C61—24no—
160C2×C4×C61—24no—
161C2×C4×C61—24no—
162C2×C4×C61C23×C418yes—
163C2×C4×C61C23×C412yes—
164C2×C4×C61C23×C414yes—
165C2×Dic61—24no—
166C2×Dic61—24no—
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—21no—
170C2×Dic61—22no—
171C2×Dic61—24no—
172C2×Dic61—22no—
173C2×Dic61—22no—
174C2×Dic61—24no—
175C2×Dic61—24no—
176C2×Dic61—22no—
177C2×Dic61C23×C418yes—
178C2×Dic61C23×C418yes—
179C2×Dic612−1+412yes—
180C2×Dic612−1+414yes—
181C4×#564—611nomaximal, Sylow
182C22×C61C24×C411yescharacteristic
183C22×C61C2×2−1+411yescharacteristic
184C22×C61C2×2−1+412yes—
185C22×C61#58512yes—
186C22×C61#58511yescharacteristic
187C4×C61—22no—
188C4×C61—24no—
189C4×C61—28no—
190C4×C61—44no—
191C4×C61—44no—
192Dic61—28no—
193Dic61—28no—
194Dic61—41no—
195Dic61—42no—
196Dic61—44no—
197Dic61#56414yescomplemented
198C2×#564—611no—
199#2382—614no—
200#2382—614no—
201#2382—614no—
202#2382—612no—
203#2382D6111yescharacteristic
204C4×#125—614no—
205C4×#125—614no—
206C4×#125—614no—
207C4×#125—612no—
208C4×#125—611no—
209C2×C61C4×2−1+411yescharacteristic
210C2×C61C4×2−1+412yes—
211C2×C61C2×#56411yescharacteristic
212C2×C61#353111yescharacteristic
213C2×C61#353112yes—
214#564—6116no—
215C2×#125—614no—
216C2×#125—614no—
217C2×#125—614no—
218C2×#125—612no—
219C2×#125C2×D6111yescharacteristic
220#389—614no—
221#389—614no—
222#389—612no—
223#389—612no—
224#389—612no—
225#389—614no—
226#389—614no—
227#389—614no—
228#389C2×D6114yes—
229#387—614no—
230#387—614no—
231#387—614no—
232#387—614no—
233#387—614no—
234#387—614no—
235#387—614no—
236#387—614no—
237#387—614no—
238#387—614no—
239#387—614no—
240#387—614no—
241#387—614no—
242#387C2×D6114yes—
243#387C2×D6114yes—
244#382—614no—
245#382—611no—
246#382—614no—
247#382—614no—
248#382C2×D6112yes—
249C4×C4⋊C4—614no—
250C4×C4⋊C4—612no—
251C4×C4⋊C4—612no—
252C4×C4⋊C4—614no—
253C4×C4⋊C4—612no—
254C4×C4⋊C4—614no—
255C4×C4⋊C4—614no—
256C4×C4⋊C4—614no—
257C4×C4⋊C4—614no—
258C43—614no—
259C43—611no—
260C61C4×#56411yescharacteristic, complemented, Sylow
261#125—6132no—
262#125—6132no—
263#125—6132no—
264#125—6116no—
265#125C4×D6118yes—
266C2×C4⋊C4—614no—
267C2×C4⋊C4—614no—
268C2×C4⋊C4—614no—
269C2×C4⋊C4—614no—
270C2×C4⋊C4—612no—
271C2×C4⋊C4—614no—
272C2×C4⋊C4—612no—
273C2×C4⋊C4—614no—
274C2×C4⋊C4—614no—
275C2×C4⋊C4—614no—
276C2×C4⋊C4—614no—
277C2×C4⋊C4—614no—
278C2×C4⋊C4—612no—
279C2×C4⋊C4C22×D6114yes—
280C2×C4⋊C4C22×D6112yes—
281C2×C4⋊C4C22×D6112yes—
282C2×C4⋊C4C22×D6114yes—
283C2×C4⋊C4C22×D6112yes—
284C2×C42—614no—
285C2×C42—614no—
286C2×C42—614no—
287C2×C42—614no—
288C2×C42—612no—
289C2×C42—611no—
290C2×C42—614no—
291C2×C42—614no—
292C2×C42—612no—
293C2×C42—611no—
294C2×C42C22×D6114yes—
295C2×C42C22×D6111yescharacteristic
296#95—614no—
297#95—614no—
298#95—614no—
299#95—614no—
300#95—614no—
301#95—614no—
302#95—614no—
303#95—614no—
304#95—614no—
305#95—614no—
306#95—614no—
307#95C22×D6114yes—
308#95C22×D6114yes—
309#95C22×D6114yes—
310#95C22×D6114yes—
311C22×C4—611no—
312C22×C4—614no—
313C22×C4—614no—
314C22×C4—611no—
315C22×C4—614no—
316C22×C4—612no—
317C22×C4C23×D6114yes—
318C22×C4C23×D6114yes—
319C22×C4C23×D6114yes—
320C22×C4C23×D6112yes—
321C22×C4C23×D6111yescharacteristic
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—6116no—
326C4⋊C4—6116no—
327C4⋊C4—6116no—
328C4⋊C4—6116no—
329C4⋊C4—6116no—
330C4⋊C4—618no—
331C4⋊C4—618no—
332C4⋊C4—2444no—
333C4⋊C4—2444no—
334C4⋊C4—2444no—
335C4⋊C4—2444no—
336C4⋊C4—2444no—
337C4⋊C4—2442no—
338C4⋊C4C2×C4×D61116yes—
339C4⋊C4C2×C4×D6118yes—
340C42—44no—
341C42—6116no—
342C42—1222no—
343C42—1228no—
344C42—1224no—
345C42—1228no—
346C42—1228no—
347C42—2442no—
348C42—2441no—
349C42—2444no—
350C42—2444no—
351C42C2×C4×D6114yes—
352C23C24×D6111yescharacteristic
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—612no—
366C2×C4—614no—
367C2×C4—618no—
368C2×C4—618no—
369C2×C4—1224no—
370C2×C4—1224no—
371C2×C4—1224no—
372C2×C4—1224no—
373C2×C4—1221no—
374C2×C4—1222no—
375C2×C4—1224no—
376C2×C4—1222no—
377C2×C4—1224no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1224no—
381C2×C4C22×C4×D6118yes—
382C2×C4C22×C4×D6112yes—
383C2×C4C22×C4×D6114yes—
384C22C23×C4×D6111yescharacteristic
385C222−1+4×D6111yescharacteristic
386C222−1+4×D6112yes—
387C22#1684059712yes—
388C22#1684059911yescharacteristic
389C4—22no—
390C4—28no—
391C4—24no—
392C4—44no—
393C4—44no—
394C4—614no—
395C4—1228no—
396C4—1228no—
397C4—2441no—
398C4—2442no—
399C4—2444no—
400C2#2145902811yescharacteristic
401C2#2145903012yes—
402C2#564×D6111yescharacteristic
403C2#2145996311yescharacteristic
404C2#2145996512yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2382C2
socle, prime radical, cosocleC23×C61C25
hypercenter#2382D61
hypocenterC61C4×#564
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2382

Complements

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

NH ≅ G/Ntype
12. C61⋊#564C4semidirect
197. Dic61#564semidirect
260. C61C4×#564semidirect

Maximal subgroups

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

subgroupindex
C2×C61⋊#5642
#214490042
#214490032
#214490022
#214490012
#125×Dic612
#214483782
#214483762
#214483742
C61×#23822
C4×#56461