Finite Group Databasebeta

← #29652300

Subgroups — #29652300

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455823C212yesmaximal, complemented
2#21449055C211yesmaximal, characteristic
3#21444827C212yesmaximal, complemented
4#21444810C212yesmaximal, complemented
5#21444759C212yesmaximal, complemented
6#21444742C212yesmaximal
7#21444318C212yesmaximal, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#3458C211yesmaximal, characteristic, complemented
10C2×C4×D244C2211yescharacteristic
11C2×C4×C22⋊D61—21no—
12#16840090C2214yes—
13#16840049—22no—
14#16839964—22no—
15#16839912C2214yes—
16C4×#11758201—24no—
17#16839815C2212yes—
18#16839692C2211yescharacteristic
19#16839687C2211yescharacteristic
20#16839684—21no—
21#16839679C2211yescharacteristic
22#16839678—21no—
23#16839671C2211yescharacteristic
24C4×C4⋊Dic61C2211yescharacteristic
25C4×#11758198—21no—
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×D244C2311yescharacteristic
42C22×C22⋊D61—21no—
43C22×C4×D61—21no—
44C2×#11758201—24no—
45C4×D244—22no—
46C4×D244—22no—
47C4×C22⋊D61—44no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2311yescharacteristic
50C2×#11758196—21no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×C4×Dic61—21no—
57#14519708—24no—
58C4⋊#92418—22no—
59#14519701—24no—
60#14519697—24no—
61C22⋊#92418C2314yes—
62C4×#92418—22no—
63#14519684—21no—
64#14519684—21no—
65#14519684C2311yescharacteristic
66#14519684C2311yescharacteristic
67C23×C4×C61D411yescharacteristic
68C4⋊Q8×C61—24no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61—24no—
72C4⋊Q8×C61D412yescomplemented
73C4⋊Q8×C61D412yescomplemented
74#125×C61—24no—
75#125×C61—22no—
76#125×C61—24no—
77#125×C61—22no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61C2314yes—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61D414yescomplemented
90#117×C61—22no—
91#117×C61—22no—
92#117×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95#117×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61C2311yescharacteristic
103C2×C42×C61—22no—
104C2×C42×C61D411yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C24×C61C2×D411yescharacteristic
107C23×D61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D414yes—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61SD1614yescomplemented
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C2×D244—22no—
131C2×D244—42no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61C2×D414yes—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C42×C61—24no—
155C42×C61—24no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160C22⋊Dic61—42no—
161#11758201—48no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758198—42no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168#11758196—42no—
169C4×Dic61—42no—
170C2×#92418—24no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61C2×SD1612yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184Q8×C61—44no—
185Q8×C61—82no—
186Q8×C61—82no—
187Q8×C61—82no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—24no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61—82no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61C2×SD1612yes—
206C2×C4×C61C2×SD1612yes—
207C2×C4×C61#13712yes—
208C2×C4×C61C22×D412yes—
209C22⋊D61—48no—
210C2×Dic61—22no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219D244—24no—
220D244—42no—
221C4×D61—82no—
222C4×D61—82no—
223#92418—44no—
224#18124—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—22no—
228C22×C61—42no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—81no—
232C22×C61C2×#12011yescharacteristic
233C22×C61C22×SD1611yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—41no—
246C4×C61—44no—
247C4×C61—41no—
248C4×C61—44no—
249C4×C61#44812yes—
250C4×C61#44812yes—
251Dic61—42no—
252Dic61—81no—
253Dic61—81no—
254#3458D6111yescharacteristic, complemented
255#3101—612no—
256#2392—611no—
257#1611—612no—
258#1608—612no—
259#1599—612no—
260#1596—612no—
261#1503—612no—
262#1458—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—42no—
266C2×C61C2×#44811yescharacteristic
267C2×C61#301811yescharacteristic
268C2×C61#302511yescharacteristic
269D61—82no—
270#544—22no—
271#544—22no—
272#544—21no—
273#544C2×D6112yes—
274#543—21no—
275#543C2×D6112yes—
276#533—24no—
277#533—21no—
278#533C2×D6112yes—
279C2×C4×D4—611no—
280C2×C4×D4—1221no—
281C2×#117—21no—
282C2×#117C2×D6111yescharacteristic
283#483—614no—
284#476—1222no—
285#461—1222no—
286#450—614no—
287C4×SD16—1224no—
288#431—612no—
289#398—611no—
290#397—611no—
291#396—1221no—
292#394—611no—
293#394—1221no—
294#392—611no—
295C4×C4⋊C4—611no—
296C4×C22⋊C4—1221no—
297#328—614no—
298#328—614no—
299#324—612no—
300#323—612no—
301C61#1812411yescharacteristic, complemented, Sylow
302C22×D4—611no—
303C22×D4—1221no—
304C23×C4—1221no—
305C23×C4C22⋊D6111yescharacteristic
306C2×SD16—1224no—
307C4⋊Q8—24no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—24no—
311C4⋊Q8C22⋊D6112yescomplemented
312C4⋊Q8C22⋊D6112yescomplemented
313#125—24no—
314#125—22no—
315#125—24no—
316#125—22no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8C22×D6114yes—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8C22⋊D6114yescomplemented
329C4×D4—1222no—
330C4×D4—1222no—
331C4×D4—2444no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4C22⋊D6111yescharacteristic
352C2×C22⋊C4C22×D6111yescharacteristic
353C2×C42—22no—
354C2×C42—1221no—
355C2×C42C22⋊D6111yescharacteristic
356C2×C42C22×D6111yescharacteristic
357#106—1224no—
358C4⋊C8—1222no—
359#103—1224no—
360#102—1224no—
361C22⋊C8—614no—
362C4×C8—1222no—
363#95—611no—
364#95—611no—
365#95—1221no—
366#95—1221no—
367C24—1221no—
368C24C2×C22⋊D6111yescharacteristic
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8C2×C22⋊D6114yes—
373C2×D4—1222no—
374C2×D4—2442no—
375C2×D4—2442no—
376C2×D4—2442no—
377C2×D4—2442no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—2442no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2442no—
390C22×C4#1175820114yescomplemented
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C23×D6111yescharacteristic
397SD16—2448no—
398C2×C8—1224no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—1222no—
413C4⋊C4—2442no—
414C4⋊C4—2442no—
415C4⋊C4C2×C22⋊D6114yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2442no—
425C42—24no—
426C42—21no—
427C42—24no—
428C42—21no—
429C42—2442no—
430C42C2×C22⋊D6112yes—
431C42C2×C22⋊D6112yes—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—1222no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23—2441no—
442C23—2441no—
443C23C2×#1175820112yes—
444C23C22×C22⋊D6111yescharacteristic
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449D4—1224no—
450D4—2442no—
451D4—2448no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—1222no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4#1451974612yes—
478C2×C4#1451974611yescharacteristic
479C2×C4#1451974611yescharacteristic
480C2×C4C2×#1175820112yes—
481C2×C4C2×#1175820112yes—
482C2×C4#1451980812yes—
483C2×C4C22×C22⋊D6112yes—
484C8—2444no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—81no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451974611yescharacteristic
500C22C22×#1175820111yescharacteristic
501C22C2×#1451980811yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—44no—
507C4—41no—
508C4—44no—
509C4—4881no—
510C4—4881no—
511C4—2442no—
512C4#1683990112yes—
513C4#1683990112yes—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—4882no—
518C2C2×#1683990111yescharacteristic
519C2#2145516811yescharacteristic
520C2#2145521111yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radical#16839671C22
hypercenter#3458D61
hypocenterC61#18124
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21455823C2semidirect
3. #21444827C2semidirect
4. #21444810C2semidirect
5. #21444759C2semidirect
7. #21444318C2semidirect
8. #21444133C2semidirect
9. C61×#3458C2semidirect
72. C4⋊Q8×C61D4semidirect
73. C4⋊Q8×C61D4semidirect
89. C4×Q8×C61D4semidirect
117. C22×C4×C61SD16semidirect
254. #3458D61semidirect
301. C61#18124semidirect
311. C4⋊Q8C22⋊D61semidirect
312. C4⋊Q8C22⋊D61semidirect
328. C4×Q8C22⋊D61semidirect
390. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214490552
#214448272
#214448102
#214447592
#214447422
#214443182
#214441332
C61×#34582
#1812461