Finite Group Databasebeta

← #29689275

Subgroups — #29689275

Order 15616.

1538 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840324C211yesmaximal, characteristic
2#21448417C211yesmaximal, characteristic, complemented
3#21446655C214yesmaximal, complemented
4#21446654C214yesmaximal, complemented
5#21445733C214yesmaximal, complemented
6C61×#2312C211yesmaximal, characteristic, complemented
7C23×C22⋊D61C2212yes—
8#16840324—24no—
9#16840324—24no—
10C2×#14519761C414yescomplemented
11C2×#14519751C2212yes—
12C2×#14519751C2212yes—
13C2×#14519747C414yescomplemented
14C2×#14519691C22116yescomplemented
15#16839655—24no—
16#16839622C2212yescomplemented
17#16839618C2212yes—
18C4×C22⋊Dic61C2212yescomplemented
19C22×C22⋊C4×C61C2211yescharacteristic
20C61×#388—24no—
21C61×#380C2212yescomplemented
22C61×#379C2212yes—
23C4×C22⋊C4×C61C2212yescomplemented
24C24×D61C2×C412yes—
25C22×D244C2×C412yes—
26C22×D244C2×C412yes—
27C22×C22⋊D61—24no—
28C22×C22⋊D61—24no—
29C22×C22⋊D61—48no—
30C22×C22⋊D61C2×C414yes—
31C22×C22⋊D61C2314yes—
32C23×Dic61C2×C412yes—
33#14519761—216no—
34#14519751—24no—
35#14519751—24no—
36#14519751—24no—
37#14519751—24no—
38#14519747—216no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61C2×C412yes—
41C2×C4⋊Dic61C2×C412yes—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61C2314yes—
45C2×#11758198C2×C414yes—
46C2×#11758198C2×C414yes—
47C2×C4×Dic61—22no—
48#14519691—416no—
49#14519684—22no—
50#14519684—22no—
51C25×C61C2311yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C2×C4⋊C4×C61—22no—
55C2×C22⋊C4×C61—22no—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61C2×C414yes—
58C2×C22⋊C4×C61D414yes—
59C2×C22⋊C4×C61D414yes—
60C2×C22⋊C4×C61C2314yes—
61C2×C42×C61—22no—
62#95×C61—22no—
63#95×C61—22no—
64C24×C61—24no—
65C24×C61—81no—
66C24×C61—81no—
67C24×C61C22×C412yes—
68C24×C61C2×D412yes—
69C24×C61C2×D412yes—
70C24×C61C2411yescharacteristic
71C23×D61—24no—
72C23×D61—44no—
73C23×D61—82no—
74C23×D61—82no—
75C23×D61C22×C414yes—
76C22×C4×C61—24no—
77C22×C4×C61—81no—
78C22×C4×C61—81no—
79C22×C4×C61C22×C412yes—
80C22×C4×C61C22×C412yes—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D412yes—
85C2×C22⋊D61—28no—
86C2×C22⋊D61—24no—
87C2×C22⋊D61—24no—
88C2×C22⋊D61—216no—
89C2×C22⋊D61—48no—
90C2×C22⋊D61—416no—
91C2×C22⋊D61—416no—
92C22×Dic61—24no—
93C22×Dic61—24no—
94C22×Dic61—44no—
95C22×Dic61C22×C414yes—
96C2×D244—24no—
97C2×D244—24no—
98C2×D244—82no—
99C2×D244—82no—
100C4⋊C4×C61—82no—
101C22⋊C4×C61—28no—
102C22⋊C4×C61—82no—
103C22⋊C4×C61—82no—
104C22⋊C4×C61C22⋊C4116yes—
105C42×C61—82no—
106C22⋊Dic61—28no—
107C22⋊Dic61—82no—
108C22⋊Dic61—82no—
109#11758198—28no—
110#11758198—28no—
111C4⋊Dic61—24no—
112C4⋊Dic61—24no—
113C4⋊Dic61—82no—
114C4×Dic61—82no—
115C23×C61—24no—
116C23×C61—24no—
117C23×C61—24no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—42no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—88no—
127C23×C61C2×C22⋊C414yes—
128C23×C61C23×C411yescharacteristic
129C23×C61C22×D411yescharacteristic
130C23×C61C22×D411yescharacteristic
131C23×C61H(5,𝔽2)14yes—
132C22×D61—24no—
133C22×D61—28no—
134C22×D61—48no—
135C22×D61—44no—
136C22×D61—44no—
137C22×D61—84no—
138C22×D61—84no—
139C22×D61—816no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—44no—
147C2×C4×C61—44no—
148C2×C4×C61—44no—
149C2×C4×C61C2×C22⋊C414yes—
150C2×C4×C61C2×C22⋊C414yes—
151C22⋊D61—416no—
152C22⋊D61—48no—
153C22⋊D61—48no—
154C2×Dic61—24no—
155C2×Dic61—28no—
156C2×Dic61—48no—
157C2×Dic61—44no—
158C2×Dic61—44no—
159C2×Dic61—44no—
160D244—84no—
161D244—84no—
162#21795—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—24no—
170C22×C61—24no—
171C22×C61—24no—
172C22×C61—44no—
173C22×C61—48no—
174C22×C61—84no—
175C22×C61—84no—
176C22×C61C22×C22⋊C411yescharacteristic
177C22×C61#51812yes—
178C22×C61#53512yes—
179C22×C61#53512yes—
180C2×D61—44no—
181C2×D61—44no—
182C2×D61—44no—
183C2×D61—816no—
184C2×D61—88no—
185C4×C61—24no—
186C4×C61—24no—
187C4×C61—84no—
188Dic61—44no—
189Dic61—84no—
190C2×#534—611no—
191#2312—611no—
192#2312D6111yescharacteristic, complemented
193#1964—614no—
194#1964—614no—
195#1781—614no—
196C2×C61—22no—
197C2×C61—22no—
198C2×C61—22no—
199C2×C61—44no—
200C2×C61#230911yescharacteristic
201C2×C61#288311yescharacteristic
202C2×C61#288311yescharacteristic
203D61—84no—
204C23×D4—612no—
205#534—1224no—
206#534—1224no—
207C2×#123—614no—
208C2×#121—612no—
209C2×#121—612no—
210C2×#120—614no—
211C22×C22⋊C4C2×D6111yescharacteristic
212C2×#99—6116no—
213#388—24no—
214#388—1224no—
215#380—612no—
216#380C2×D6112yescomplemented
217#379—612no—
218#379C2×D6112yes—
219C4×C22⋊C4—612no—
220C4×C22⋊C4C2×D6112yescomplemented
221C61#2179511yescharacteristic, complemented, Sylow
222C25—612no—
223C25C22×D6111yescharacteristic
224C22×D4—614no—
225C22×D4—614no—
226C22×D4—612no—
227C22×D4—612no—
228C22×D4—1224no—
229C22×D4—1224no—
230C22×D4—2448no—
231C23×C4—612no—
232C23×C4C22×D6111yescharacteristic
233C23×C4C22×D6111yescharacteristic
234#123—12216no—
235#121—1224no—
236#121—1224no—
237#121—1224no—
238#121—1224no—
239#120—12216no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—612no—
242C2×C4⋊C4—612no—
243C2×C4⋊C4—1222no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—614no—
247C2×C22⋊C4—614no—
248C2×C22⋊C4—614no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4C4×D6114yes—
252C2×C22⋊C4D24414yes—
253C2×C22⋊C4C22⋊D6114yes—
254C2×C22⋊C4C22×D6114yes—
255C2×C42—22no—
256C2×C42—1222no—
257#99—24416no—
258#95—22no—
259#95—22no—
260#95—1222no—
261#95—1222no—
262C24—24no—
263C24—81no—
264C24—81no—
265C24—614no—
266C24—1224no—
267C24—4882no—
268C24—4882no—
269C24—2444no—
270C24C2×C4×D6112yes—
271C24C2×D24412yes—
272C24C2×C22⋊D6112yes—
273C24C23×D6111yescharacteristic
274C2×D4—1224no—
275C2×D4—1224no—
276C2×D4—12216no—
277C2×D4—1228no—
278C2×D4—1224no—
279C2×D4—1224no—
280C2×D4—4882no—
281C2×D4—4882no—
282C2×D4—2448no—
283C2×D4—24416no—
284C2×D4—24416no—
285C22×C4—24no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—614no—
289C22×C4—1224no—
290C22×C4—1224no—
291C22×C4—2444no—
292C22×C4C2×C4×D6112yes—
293C22×C4C2×C4×D6112yes—
294C22×C4C2×D24412yes—
295C22×C4C2×D24412yes—
296C22×C4C2×C22⋊D6112yes—
297C22×C4C2×C22⋊D6112yes—
298C4⋊C4—82no—
299C4⋊C4—1224no—
300C4⋊C4—1224no—
301C4⋊C4—4882no—
302C22⋊C4—28no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—1228no—
306C22⋊C4—1228no—
307C22⋊C4—1228no—
308C22⋊C4—4882no—
309C22⋊C4—4882no—
310C22⋊C4#11758198116yes—
311C42—82no—
312C42—4882no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—88no—
325C23—1228no—
326C23—1224no—
327C23—4884no—
328C23—4884no—
329C23—48816no—
330C23—2444no—
331C23—2444no—
332C23—2448no—
333C23C2×#1175819814yes—
334C23C22×C4×D6111yescharacteristic
335C23C22×C22⋊D6111yescharacteristic
336C23C22×D24411yescharacteristic
337C23#1451982314yes—
338D4—4884no—
339D4—4884no—
340D4—24416no—
341D4—2448no—
342D4—2448no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—1224no—
353C2×C4—1228no—
354C2×C4—2444no—
355C2×C4—2444no—
356C2×C4—2444no—
357C2×C4—2448no—
358C2×C4C2×#1175819814yes—
359C2×C4C2×#1175819814yes—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—44no—
370C22—48no—
371C22—84no—
372C22—84no—
373C22—4888no—
374C22—48816no—
375C22—2444no—
376C22—2444no—
377C22—2444no—
378C22C22×#1175819811yescharacteristic
379C22#1684023612yes—
380C22#1684032612yes—
381C22#1684033112yes—
382C4—24no—
383C4—24no—
384C4—84no—
385C4—4884no—
386C4—2444no—
387C2—22no—
388C2—22no—
389C2—22no—
390C2—44no—
391C2—4884no—
392C2#2144839911yescharacteristic
393C2#2145427911yescharacteristic
394C2#2145427911yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#2312C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840324C2
cosocleC24×C61C24
hypercenter#2312D61
hypocenterC61#21795
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21448417C2semidirect
3. #21446655C2semidirect
4. #21446654C2semidirect
5. #21445733C2semidirect
6. C61×#2312C2semidirect
10. C2×#14519761C4semidirect
13. C2×#14519747C4semidirect
14. C2×#14519691C22semidirect
16. #16839622C22semidirect
18. C4×C22⋊Dic61C22semidirect
21. C61×#380C22semidirect
23. C4×C22⋊C4×C61C22semidirect
192. #2312D61semidirect
216. #380C2×D61semidirect
220. C4×C22⋊C4C2×D61semidirect
221. C61#21795semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214484172
#214466552
#214466542
#214457332
C61×#23122
#2179561