Finite Group Databasebeta

← #29652567

Subgroups — #29652567

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455945C212yesmaximal, complemented
2#21451396C211yesmaximal, characteristic
3#21444827C212yesmaximal, complemented
4#21444810C212yesmaximal
5#21444793C212yesmaximal, complemented
6#21444776C212yesmaximal
7#21444370C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic
9C61×#3458C211yesmaximal, characteristic
10C2×C4×Dic122C2211yescharacteristic
11#16840090C2214yes—
12#16840044—22no—
13#16839969—22no—
14#16839917C2214yes—
15#16839867—24no—
16#16839819C2212yes—
17#16839692C2211yescharacteristic
18#16839687C2211yescharacteristic
19#16839682C2211yescharacteristic
20#16839674C2211yescharacteristic
21#16839650—21no—
22#16839646—21no—
23#16839643—21no—
24#16839635—21no—
25#16839630C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#544—22no—
31C61×#544—22no—
32C61×#544—21no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—24no—
37C61×#533—21no—
38C61×#533C2212yes—
39C2×#117×C61—21no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C2×#11758202—24no—
43C4×Dic122—22no—
44C4×Dic122—22no—
45C2×C4⋊Dic61C2311yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519708—24no—
53C4⋊#92418—22no—
54#14519701—24no—
55#14519700—24no—
56C22⋊#92418C2314yes—
57#14519688—22no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C23×C4×C61D411yescharacteristic
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—24no—
71C4⋊Q8×C61D412yes—
72C4⋊Q8×C61D412yes—
73#125×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76#125×C61—22no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61C2314yes—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C42×C61—22no—
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61C2×D414yes—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C2×Dic122—22no—
127C2×Dic122—42no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61C2×D414yes—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C42×C61—24no—
147C42×C61—24no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152C22⋊Dic61—42no—
153C22⋊Dic61—42no—
154#11758202—48no—
155C4⋊Dic61—22no—
156#11758196—42no—
157#11758196—41no—
158#11758196—41no—
159#11758196—42no—
160C4×Dic61—42no—
161C2×#92418—24no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61#13712yes—
167C23×C61C22×D411yescharacteristic
168Q8×C61—44no—
169Q8×C61—82no—
170Q8×C61—82no—
171Q8×C61—82no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—24no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—24no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—82no—
187C2×C4×C61—82no—
188C2×C4×C61#12012yes—
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D412yes—
193C2×Dic61—22no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205Dic122—24no—
206Dic122—42no—
207#92418—44no—
208#18155—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—22no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—81no—
216C22×C61C2×#12011yescharacteristic
217C22×C61C2×#13711yescharacteristic
218C22×C61C2×#13711yescharacteristic
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—41no—
225C4×C61—44no—
226C4×C61—41no—
227C4×C61—44no—
228Dic61—42no—
229Dic61—81no—
230Dic61—81no—
231Dic61—81no—
232#3458D6111yescharacteristic
233#3113—612no—
234#2599—611no—
235#1611—612no—
236#1608—612no—
237#1605—612no—
238#1602—612no—
239#1515—612no—
240#1464—611no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—42no—
244C2×C61#300411yescharacteristic
245C2×C61#302211yescharacteristic
246C2×C61#302511yescharacteristic
247#544—22no—
248#544—22no—
249#544—21no—
250#544C2×D6112yes—
251#543—21no—
252#543C2×D6112yes—
253#533—24no—
254#533—21no—
255#533C2×D6112yes—
256C2×C4×Q8—611no—
257C2×#117—21no—
258C2×#117C2×D6111yescharacteristic
259#483—614no—
260#475—1222no—
261#462—1222no—
262#451—614no—
263#441—1224no—
264#432—612no—
265#398—611no—
266#397—611no—
267#395—611no—
268#393—611no—
269#388—1221no—
270#387—1221no—
271#387—1221no—
272#385—1221no—
273#383—611no—
274#328—614no—
275#328—614no—
276#324—612no—
277#323—612no—
278C61#1815511yescharacteristic, complemented, Sylow
279C22×Q8—611no—
280C23×C4C22⋊D6111yescharacteristic
281C2×Q16—1224no—
282C4⋊Q8—24no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—24no—
286C4⋊Q8C22⋊D6112yes—
287C4⋊Q8C22⋊D6112yes—
288#125—24no—
289#125—22no—
290#125—24no—
291#125—22no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—24no—
299C22⋊Q8C22×D6114yes—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—1222no—
304C4×Q8—1222no—
305C4×Q8C22⋊D6114yes—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4C22⋊D6111yescharacteristic
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4C22⋊D6111yescharacteristic
324C2×C22⋊C4C22×D6111yescharacteristic
325C2×C42—22no—
326C2×C42—1221no—
327C2×C42C22⋊D6111yescharacteristic
328C2×C42C22×D6111yescharacteristic
329#106—1224no—
330C4⋊C8—1222no—
331#103—1224no—
332#103—1224no—
333C22⋊C8—614no—
334#97—1222no—
335#95—611no—
336#95—611no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343C24C2×C22⋊D6111yescharacteristic
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—1222no—
348C2×Q8—2442no—
349C2×Q8C2×C22⋊D6114yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4C2×C22⋊D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C23×D6111yescharacteristic
366Q16—2448no—
367C2×C8—1224no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C4⋊C4—2442no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4C2×C22⋊D6114yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—2442no—
392C22⋊C4—2442no—
393C42—24no—
394C42—24no—
395C42—21no—
396C42—21no—
397C42—2442no—
398C42C2×C22⋊D6112yes—
399C42C2×C22⋊D6112yes—
400C23—21no—
401C23—21no—
402C23—41no—
403C23—41no—
404C23#1451980812yes—
405C23C22×C22⋊D6111yescharacteristic
406Q8—44no—
407Q8—82no—
408Q8—82no—
409Q8—82no—
410Q8—1224no—
411Q8—2442no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—1222no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4#1451974611yescharacteristic
441C2×C4#1451974611yescharacteristic
442C2×C4#1451974612yes—
443C2×C4#1451980812yes—
444C2×C4C22×C22⋊D6112yes—
445C8—2444no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22C2×#1451974611yescharacteristic
454C22C2×#1451980811yescharacteristic
455C22C2×#1451980811yescharacteristic
456C4—22no—
457C4—22no—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—41no—
462C4—44no—
463C4—41no—
464C4—44no—
465C4—4881no—
466C4—4881no—
467C4—4881no—
468C4—2442no—
469C2—21no—
470C2—21no—
471C2—42no—
472C2#2145505311yescharacteristic
473C2#2145519811yescharacteristic
474C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#18155
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21455945C2semidirect
3. #21444827C2semidirect
5. #21444793C2semidirect
7. #21444370C2semidirect
278. C61#18155semidirect

Maximal subgroups

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

subgroupindex
#214559452
#214513962
#214448272
#214448102
#214447932
#214447762
#214443702
#214441532
C61×#34582
#1815561