Finite Group Databasebeta

← #29725135

Subgroups — #29725135

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840030C211yesmaximal, characteristic
2#21454669C211yesmaximal, characteristic, complemented
3#21447281C211yesmaximal, characteristic, complemented
4#21447276C211yesmaximal, characteristic, complemented
5#21447093C211yesmaximal, characteristic, complemented
6#21447089C211yesmaximal, characteristic, complemented
7#21446728C211yesmaximal, characteristic, complemented
8#21446608C211yesmaximal, characteristic, complemented
9#21446309C211yesmaximal, characteristic, complemented
10#21445941C211yesmaximal, characteristic, complemented
11#21445930C211yesmaximal, characteristic, complemented
12#21445929C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic, complemented
14#21445621C211yesmaximal, characteristic, complemented
15C61×#1904C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840236C2211yescharacteristic
20#16840030—22no—
21#16840030—22no—
22#16839872—22no—
23C4×#11758199—22no—
24#16839811C2211yescharacteristic
25#16839810C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839798C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839767C2211yescharacteristic
31#16839766C2211yescharacteristic
32#16839766C2211yescharacteristic
33#16839746C2211yescharacteristic
34C2×#14519694C2211yescharacteristic
35C2×#14519694C2211yescharacteristic
36C2×#14519693C2211yescharacteristic
37C2×#14519693C2211yescharacteristic
38#16839718C2211yescharacteristic
39#16839718C2211yescharacteristic
40#16839534C2211yescharacteristic
41#16839533C2211yescharacteristic
42#16839532C2211yescharacteristic
43#16839532C2211yescharacteristic
44#16839532C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839532C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
49C61×#518C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#418C2211yescharacteristic
52C61×#410C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×H(3,ℤ/4)C2211yescharacteristic
55C22×D244C2311yescharacteristic
56#14519802—22no—
57#14519802—41no—
58C2×#11758201—21no—
59C2×#11758199—21no—
60C2×#11758199—21no—
61C2×#11758199—21no—
62C2×#11758199—21no—
63C2×#11758199—21no—
64C2×#11758199D412yes—
65C2×#11758199D412yes—
66#14519784—21no—
67#14519784—21no—
68#14519784—21no—
69#14519784—21no—
70C2×#11758194—21no—
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C22×#92418—21no—
77C22×#92418C2311yescharacteristic
78C4×D244—22no—
79#14519739—41no—
80#14519736C2311yescharacteristic
81C22⋊C4×D61—22no—
82#14519713—41no—
83#14519712—21no—
84#14519712—21no—
85#14519712—21no—
86#14519712—21no—
87#14519712—41no—
88#14519710—21no—
89#14519708—21no—
90#14519704—41no—
91#14519698—22no—
92#14519697—22no—
93#14519697—41no—
94#14519696—21no—
95#14519694—22no—
96#14519694—21no—
97#14519694—41no—
98#14519693—22no—
99#14519693—41no—
100#14519688—21no—
101#14519688—21no—
102C4×#92418—21no—
103C4×#92418—21no—
104C2×#41×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106AGL(1,ℤ/8)×C61—22no—
107AGL(1,ℤ/8)×C61—41no—
108C2×SD16×C61—21no—
109C2×D8×C61—21no—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C4×D4×C61—22no—
113C4×D4×C61—41no—
114#117×C61C2311yescharacteristic
115#117×C61C2311yescharacteristic
116#117×C61C2311yescharacteristic
117C2×C22⋊C4×C61—22no—
118#104×C61—41no—
119#102×C61—41no—
120#99×C61—22no—
121C24×C61—21no—
122C23×D61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126#41×C61—21no—
127C2×Q8×C61#4111yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61#4111yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61C2411yescharacteristic
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C2×D412yes—
145C2×C4×D61—42no—
146SD16×C61—42no—
147D8×C61—42no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—21no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61#4112yes—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C22⋊Dic61—21no—
165#11758201—42no—
166#11758199—22no—
167#11758199—22no—
168#11758199—22no—
169#11758199—22no—
170#11758199—42no—
171#11758198—42no—
172C4⋊Dic61—21no—
173C4×Dic61—21no—
174#11758194—21no—
175#11758194—21no—
176#11758194—21no—
177#11758194—21no—
178#11758194—21no—
179#11758194—21no—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184#11758194—41no—
185#11758194—41no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—41no—
192C2×#92418—41no—
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418#4111yescharacteristic
196C2×#92418#4111yescharacteristic
197C2×#92418#4111yescharacteristic
198C2×#92418#4111yescharacteristic
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—161no—
204C23×C61C2×#4111yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—161no—
209Q8×C61—21no—
210Q8×C61—21no—
211D4×C61—22no—
212D4×C61—21no—
213D4×C61—22no—
214D4×C61—21no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—162no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×Dic61—41no—
233D244—22no—
234D244—22no—
235D244—41no—
236D244—41no—
237C4×D61—162no—
238C8×C61—41no—
239C8×C61—41no—
240#92418—41no—
241#92418—41no—
242#92418—41no—
243#92418—41no—
244#92418—41no—
245#92418#12311yescharacteristic
246#92418#12311yescharacteristic
247#92418#12311yescharacteristic
248#92418#12311yescharacteristic
249#24988—611nomaximal, Sylow
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—42no—
253C22×C61—82no—
254C22×C61—81no—
255C22×C61—161no—
256C22×C61—161no—
257C22×C61#52911yescharacteristic
258C22×C61#54611yescharacteristic
259C22×C61#55511yescharacteristic
260C2×D61—42no—
261C2×D61—82no—
262C2×D61—161no—
263C2×D61—161no—
264C4×C61—42no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61C2×#12311yescharacteristic
269C4×C61#54011yescharacteristic
270C4×C61#54611yescharacteristic
271C4×C61#55311yescharacteristic
272Dic61—81no—
273C2×#472—611no—
274#2948—611no—
275#2081—611no—
276#2081—611no—
277#2048—611no—
278#2048—611no—
279#1978—611no—
280#1953—611no—
281#1904—611no—
282#1904D6111yescharacteristic, complemented
283#1833—611no—
284#1829—611no—
285#1829—611no—
286#1814—611no—
287#1757—611no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—21no—
291C2×C61—81no—
292C2×C61—82no—
293C2×C61#242011yescharacteristic
294D61—82no—
295C2×AGL(1,ℤ/8)—611no—
296C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
297C22×D8—611no—
298C2×#131—611no—
299#518—611no—
300#518C2×D6111yescharacteristic
301#472—1222no—
302#472—1222no—
303#442—1222no—
304C4×D8—1222no—
305#430—611no—
306#430—611no—
307C2×#108—611no—
308C2×#108—611no—
309#427—611no—
310#421—611no—
311#421C2×D6111yescharacteristic
312#418—611no—
313#418—611no—
314#418—611no—
315#418C2×D6111yescharacteristic
316#413—611no—
317C2×#100—611no—
318C2×#100—611no—
319C2×#100—611no—
320C2×#100—611no—
321#410C2×D6111yescharacteristic
322#405—611no—
323#405—611no—
324#344—611no—
325#344—611no—
326#344—611no—
327#344—611no—
328#344—611no—
329#344—611no—
330#344—611no—
331#344C2×D6111yescharacteristic
332H(3,ℤ/4)—611no—
333H(3,ℤ/4)C2×D6111yescharacteristic
334C61#2498811yescharacteristic, complemented, Sylow
335C2×#41C22×D6111yescharacteristic
336C22×D4—611no—
337C22×D4C22×D6111yescharacteristic
338AGL(1,ℤ/8)—22no—
339AGL(1,ℤ/8)—41no—
340AGL(1,ℤ/8)—1222no—
341AGL(1,ℤ/8)—2441no—
342C2×SD16—21no—
343C2×SD16—1221no—
344C2×D8—21no—
345C2×D8—612no—
346C2×D8—612no—
347C2×D8—1221no—
348C2×D8—1221no—
349C2×D8—1221no—
350C2×D8—1221no—
351C2×D8—1221no—
352#131—1221no—
353#131—1221no—
354#131—1221no—
355#131—1221no—
356C2×M16—611no—
357C2×M16—611no—
358C2×M16—611no—
359C2×M16—611no—
360C2×M16—611no—
361C2×M16—1221no—
362C2×M16C22×D6111yescharacteristic
363C2×M16C22×D6111yescharacteristic
364C22×C8—611no—
365C22×C8—1221no—
366C4×D4—22no—
367C4×D4—41no—
368C4×D4—1222no—
369C4×D4—2441no—
370#117—611no—
371#117C22×D6111yescharacteristic
372#117C22×D6111yescharacteristic
373#117C22×D6111yescharacteristic
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1222no—
376#108—1221no—
377#108—1221no—
378#108—1221no—
379#108—1221no—
380#108—2441no—
381#108—2441no—
382#107—1221no—
383#106—1221no—
384#104—41no—
385#104—2441no—
386#102—41no—
387#102—1222no—
388#102—1222no—
389#102—2441no—
390#101—1221no—
391#100—1222no—
392#100—1221no—
393#100—1222no—
394#100—2441no—
395#100—2441no—
396#99—22no—
397#97—1221no—
398#97—1221no—
399C4×C8—1221no—
400C4×C8—1221no—
401C24—21no—
402C24—1221no—
403#41—21no—
404#41—21no—
405#41—21no—
406#41—21no—
407C2×Q8#1175822211yescharacteristic
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—612no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4#1175822211yescharacteristic
421C2×D4D4×D6112yes—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—2442no—
425C22×C4C23×D6111yescharacteristic
426SD16—42no—
427SD16—2442no—
428D8—42no—
429D8—1222no—
430D8—1222no—
431D8—1222no—
432D8—1222no—
433D8—2442no—
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—1221no—
442M16—1221no—
443M16—1221no—
444M16—1221no—
445M16—1221no—
446M16—1221no—
447M16—1221no—
448M16—2441no—
449M16—2441no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—2441no—
464C2×C8—2441no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—1221no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—42no—
472C22⋊C4—1221no—
473C22⋊C4—2442no—
474C22⋊C4#1175822412yes—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—1221no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—161no—
484C23—1222no—
485C23—9761no—
486C23—2441no—
487C23—2441no—
488C23C2×#1175822411yescharacteristic
489Q8—21no—
490Q8—21no—
491D4—21no—
492D4—22no—
493D4—21no—
494D4—22no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—1222no—
500D4—1222no—
501D4—2441no—
502D4—2441no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—162no—
510C2×C4—9762no—
511C2×C4—2441no—
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4#41×D6111yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4C2×#1175822211yescharacteristic
516C2×C4#1451982211yescharacteristic
517C2×C4#1451982211yescharacteristic
518C8—41no—
519C8—41no—
520C8—611no—
521C8—611no—
522C8—611no—
523C8—611no—
524C8—2441no—
525C8—2441no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—82no—
533C22—81no—
534C22—161no—
535C22—161no—
536C22—9761no—
537C22—9761no—
538C22—4882no—
539C22—2442no—
540C22#1684029211yescharacteristic
541C22#1684039911yescharacteristic
542C22#1684047611yescharacteristic
543C4—42no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—4881no—
548C4#123×D6111yescharacteristic
549C4#1684036211yescharacteristic
550C4#1684040211yescharacteristic
551C4#1684046111yescharacteristic
552C2—21no—
553C2—21no—
554C2—21no—
555C2—82no—
556C2—81no—
557C2—4882no—
558C2#2144930811yescharacteristic
subgroupG/H
centerC2#21449308
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1904C2
socleC2×C61#2420
prime radicalC2×#16840030C2
hypercenter#1904D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1904

Complements

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

NH ≅ G/Ntype
2. #21454669C2semidirect
3. #21447281C2semidirect
4. #21447276C2semidirect
5. #21447093C2semidirect
6. #21447089C2semidirect
7. #21446728C2semidirect
8. #21446608C2semidirect
9. #21446309C2semidirect
10. #21445941C2semidirect
11. #21445930C2semidirect
12. #21445929C2semidirect
13. #21445871C2semidirect
14. #21445621C2semidirect
15. C61×#1904C2semidirect
282. #1904D61semidirect
334. C61#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#168400302
#214546692
#214472812
#214472762
#214470932
#214470892
#214467282
#214466082
#214463092
#214459412
#214459302
#214459292
#214458712
#214456212
C61×#19042
#2498861