Finite Group Databasebeta

← #29721475

Subgroups — #29721475

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455634C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C4×#14519802C211yesmaximal, characteristic
4#21447264C212yesmaximal, complemented
5#21447110C212yesmaximal
6#21446807C211yesmaximal, characteristic, complemented
7#21446599C212yesmaximal, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446481C212yesmaximal, complemented
10#21445808C211yesmaximal, characteristic, complemented
11C61×#2925C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840057—21no—
16#16840053—21no—
17#16840053—21no—
18#16840053—22no—
19#16840052—21no—
20#16840047—21no—
21#16840042—21no—
22#16839872—21no—
23#16839860—21no—
24C4×#11758201—21no—
25C4×#11758199—21no—
26C2×#14519713C2212yes—
27#16839805C2212yes—
28#16839805C2211yescharacteristic
29C2×#14519709C2212yes—
30#16839786C2211yescharacteristic
31#16839766C2211yescharacteristic
32#16839763C2212yes—
33#16839763C2211yescharacteristic
34C2×#14519699C2212yes—
35C2×#14519693C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839530C2212yes—
38#16839530C2212yes—
39#16839530C2212yes—
40#16839527C2212yes—
41C4×#41×C61C2211yescharacteristic
42C61×#433—22no—
43C61×#432—21no—
44C61×#431—21no—
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#408C2212yes—
48C61×#405C2212yes—
49C22×Dic122—21no—
50C22×D244C2311yescharacteristic
51C22×C4×D61—21no—
52#14519802—22no—
53#14519802—22no—
54C2×#11758201—21no—
55C2×#11758199—21no—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758198—21no—
71#14519713—24no—
72#14519710—22no—
73#14519710—21no—
74#14519709—22no—
75#14519709—22no—
76#14519708—22no—
77#14519708—21no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519701—22no—
81#14519701—21no—
82#14519699—22no—
83#14519699—22no—
84#14519698—22no—
85#14519698—21no—
86#14519697—21no—
87#14519697—21no—
88#14519693—24no—
89#14519688—21no—
90C4×#92418—21no—
91C2×#41×C61C2311yescharacteristic
92C2×M16×C61C2312yes—
93C22×C8×C61C2312yes—
94C4×Q8×C61D411yescharacteristic
95C4×Q8×C61D411yescharacteristic
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D411yescharacteristic
99C4×D4×C61D411yescharacteristic
100#117×C61—21no—
101#117×C61C2311yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61—21no—
106C4⋊C8×C61D411yescharacteristic
107C4⋊C8×C61D411yescharacteristic
108C4⋊C8×C61D412yes—
109C22⋊C8×C61—22no—
110C22⋊C8×C61—22no—
111#97×C61—22no—
112C4×C8×C61—22no—
113C23×D61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118C2×Q8×C61C2×D411yescharacteristic
119C2×D4×C61—21no—
120C2×D4×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—22no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244#4112yes—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—82no—
138C2×Dic122—41no—
139C2×Dic122—41no—
140C2×Dic122—82no—
141M16×C61—22no—
142M16×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D412yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D411yescharacteristic
151C4⋊C4×C61C2×D411yescharacteristic
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162#11758201—24no—
163#11758199—24no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—41no—
171C4⋊Dic61—41no—
172#11758194—21no—
173#11758194—22no—
174#11758194—22no—
175#11758194—21no—
176#11758194—21no—
177#11758194#4112yes—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—22no—
181C23×C61—21no—
182C23×C61C2×#4111yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—81no—
187C22×D61—81no—
188Q8×C61—21no—
189Q8×C61—21no—
190D4×C61—21no—
191D4×C61—21no—
192D4×C61—41no—
193D4×C61—41no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—81no—
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—82no—
216C2×Dic61—81no—
217D244—22no—
218D244—22no—
219D244—41no—
220D244—41no—
221C4×D61—82no—
222C4×D61—82no—
223Dic122—82no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—22no—
227C8×C61—42no—
228#92418—22no—
229#92418—42no—
230#92418—41no—
231#24721—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61C2×#12111yescharacteristic
238C22×C61#55211yescharacteristic
239C22×C61#55611yescharacteristic
240C2×D61—42no—
241C2×D61—82no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—82no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61#47112yes—
252C4×C61#52511yescharacteristic
253C4×C61#53611yescharacteristic
254C4×C61#54711yescharacteristic
255C4×C61#54811yescharacteristic
256Dic61—81no—
257Dic61—82no—
258#3079—611no—
259C2×#476—611no—
260C4×AGL(1,ℤ/8)—611no—
261#2925D6111yescharacteristic, complemented
262#2079—612no—
263#2051—612no—
264#1997—611no—
265#1951—612no—
266#1950—611no—
267#1931—612no—
268#1800—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61#241511yescharacteristic
273C2×C61C2×#47111yescharacteristic
274C2×C61#306811yescharacteristic
275D61—82no—
276C2×AGL(1,ℤ/8)—611no—
277C2×C4⋊Q8—611no—
278C4×#41C2×D6111yescharacteristic
279C2×C4×D4—611no—
280#477—1221no—
281#476—1221no—
282#476—1221no—
283#476—1221no—
284#476—1222no—
285#475—1221no—
286#474—1221no—
287#442—1221no—
288#440—1221no—
289C4×SD16—1221no—
290C4×D8—1221no—
291#433—22no—
292#432—21no—
293#431—21no—
294C2×#108—612no—
295#428—612no—
296#428—611no—
297C2×#106—612no—
298#423—611no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8C2×D6111yescharacteristic
301#418—611no—
302#417—612no—
303#417—611no—
304C2×#102—612no—
305C2×#100—612no—
306#408C2×D6112yes—
307#405C2×D6112yes—
308C4×M16—611no—
309#343—612no—
310#343—612no—
311#343—612no—
312#341—612no—
313C61#2472111yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C22×Q8—1221no—
316C22×D4—611no—
317C23×C4—1221no—
318AGL(1,ℤ/8)—1222no—
319AGL(1,ℤ/8)—1222no—
320C2×SD16—1221no—
321C2×D8—1221no—
322C2×M16—611no—
323C2×M16—611no—
324C2×M16—612no—
325C2×M16C22×D6112yes—
326C22×C8C22×D6112yes—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4×Q8D24411yescharacteristic
332C4×Q8D24411yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4—1221no—
339C4×D4D24411yescharacteristic
340C4×D4D24411yescharacteristic
341#117—21no—
342#117C22×D6111yescharacteristic
343C2×C4⋊C4—611no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C42—21no—
348C2×C42C22×D6111yescharacteristic
349#108—1224no—
350#107—1221no—
351#107—1222no—
352#106—1222no—
353#106—1222no—
354#106—1221no—
355#106—1222no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—1221no—
359C4⋊C8—1221no—
360C4⋊C8C22⋊D6111yescharacteristic
361C4⋊C8C22⋊D6111yescharacteristic
362C4⋊C8C22⋊D6112yes—
363#103—1221no—
364#103—1222no—
365#102—1221no—
366#102—1222no—
367#102—1222no—
368#102—1222no—
369#102—1221no—
370#102—1221no—
371#100—1224no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#97—22no—
375#97—1221no—
376C4×C8—22no—
377C4×C8—1221no—
378C24—1221no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—4882no—
384C2×Q8—2441no—
385C2×Q8—2441no—
386C2×Q8C2×D24411yescharacteristic
387C2×D4—21no—
388C2×D4—612no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4C2×D24411yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—4882no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4#1175822211yescharacteristic
404C22×C4#1175822211yescharacteristic
405C22×C4C23×D6111yescharacteristic
406SD16—1224no—
407D8—1224no—
408M16—22no—
409M16—22no—
410M16—612no—
411M16—1221no—
412M16—1222no—
413M16—1222no—
414M16—1221no—
415M16—1221no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—1221no—
420C2×C8—1221no—
421C2×C8—1222no—
422C2×C8C2×C22⋊D6112yes—
423C2×C8C2×C22⋊D6112yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—1222no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1222no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4C2×D24411yescharacteristic
433C4⋊C4C2×D24411yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42C2×D24411yescharacteristic
443C42C2×D24411yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—21no—
447C23—1222no—
448C23—4881no—
449C23—4881no—
450C23—2441no—
451C23—2441no—
452C23C2×#1175822211yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—4882no—
456Q8—4882no—
457D4—21no—
458D4—21no—
459D4—41no—
460D4—41no—
461D4—1222no—
462D4—1222no—
463D4—2441no—
464D4—2441no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—81no—
474C2×C4—4882no—
475C2×C4—4882no—
476C2×C4—4881no—
477C2×C4—4882no—
478C2×C4—2441no—
479C2×C4—2442no—
480C2×C4#1451975111yescharacteristic
481C2×C4#1451975111yescharacteristic
482C2×C4#1451975111yescharacteristic
483C2×C4#1451975111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4C22×D24411yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4#41×D6111yescharacteristic
488C2×C4#1451982411yescharacteristic
489C2×C4#1451982411yescharacteristic
490C8—22no—
491C8—22no—
492C8—42no—
493C8—1222no—
494C8—2442no—
495C8—2441no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—4882no—
502C22—4882no—
503C22—4881no—
504C22—4881no—
505C22—2442no—
506C22C2×#1451975111yescharacteristic
507C22#1684045111yescharacteristic
508C22#1684048011yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—4881no—
516C4—4882no—
517C4#1684002512yes—
518C4#1684027411yescharacteristic
519C4#1684033811yescharacteristic
520C4#1684041111yescharacteristic
521C4#1684041911yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—4882no—
526C2#2144924211yescharacteristic
527C2C2×#1684002511yescharacteristic
528C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2925C2
socleC22×C61C2×#121
prime radicalC2×#14519802C22
hypercenter#2925D61
hypocenterC61#24721
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
1. #21455634C2semidirect
2. C2×#16840053C2semidirect
4. #21447264C2semidirect
6. #21446807C2semidirect
7. #21446599C2semidirect
8. #21446595C2semidirect
9. #21446481C2semidirect
10. #21445808C2semidirect
11. C61×#2925C2semidirect
261. #2925D61semidirect
313. C61#24721semidirect

Maximal subgroups

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

subgroupindex
#214556342
C2×#168400532
C4×#145198022
#214472642
#214471102
#214468072
#214465992
#214465952
#214464812
#214458082
C61×#29252
#2472161