Finite Group Databasebeta

← #29725120

Subgroups — #29725120

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840023C211yesmaximal, characteristic
2#21454676C211yesmaximal, characteristic, complemented
3#21447288C211yesmaximal, characteristic, complemented
4#21447276C211yesmaximal, characteristic, complemented
5#21447100C211yesmaximal, characteristic, complemented
6#21447089C211yesmaximal, characteristic
7#21446725C211yesmaximal, characteristic, complemented
8#21446608C211yesmaximal, characteristic, complemented
9#21446316C211yesmaximal, characteristic, complemented
10#21445941C211yesmaximal, characteristic, complemented
11#21445932C211yesmaximal, characteristic, complemented
12#21445930C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic, complemented
14#21445621C211yesmaximal, characteristic, complemented
15C61×#1904C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840239C2211yescharacteristic
20#16840023—22no—
21#16840023—22no—
22#16839862—22no—
23C4×#11758200—22no—
24#16839811C2211yescharacteristic
25#16839810C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839798C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839770C2211yescharacteristic
31#16839770C2211yescharacteristic
32#16839767C2211yescharacteristic
33#16839746C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519695C2211yescharacteristic
36C2×#14519694C2211yescharacteristic
37C2×#14519694C2211yescharacteristic
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×Dic122C2311yescharacteristic
56#14519807—22no—
57#14519807—41no—
58C2×#11758202—21no—
59C2×#11758200—21no—
60C2×#11758200—21no—
61C2×#11758200—21no—
62C2×#11758200—21no—
63C2×#11758200—21no—
64C2×#11758200D412yes—
65C2×#11758200D412yes—
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×Dic122—22no—
79#14519742—41no—
80#14519736C2311yescharacteristic
81#14519733—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#14519700—22no—
92#14519700—41no—
93#14519698—22no—
94#14519696—21no—
95#14519695—22no—
96#14519695—41no—
97#14519694—22no—
98#14519694—21no—
99#14519694—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—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126C2×Q8×C61#4111yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C2×D4×C61#4111yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122C2×D412yes—
145SD16×C61—42no—
146D8×C61—42no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61#4112yes—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—21no—
164#11758202—42no—
165#11758200—22no—
166#11758200—22no—
167#11758200—22no—
168#11758200—22no—
169#11758200—42no—
170C4⋊Dic61—21no—
171#11758196—42no—
172C4×Dic61—21no—
173C4×Dic61—42no—
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
205Q8×C61—21no—
206Q8×C61—21no—
207D4×C61—22no—
208D4×C61—21no—
209D4×C61—22no—
210D4×C61—21no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—162no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232Dic122—22no—
233Dic122—22no—
234Dic122—41no—
235Dic122—41no—
236C8×C61—41no—
237C8×C61—41no—
238#92418—41no—
239#92418—41no—
240#92418—41no—
241#92418—41no—
242#92418—41no—
243#92418#12311yescharacteristic
244#92418#12311yescharacteristic
245#92418#12311yescharacteristic
246#92418#12311yescharacteristic
247#24987—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—42no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—161no—
254C22×C61—161no—
255C22×C61#52911yescharacteristic
256C22×C61#54611yescharacteristic
257C22×C61#55511yescharacteristic
258C4×C61—42no—
259C4×C61—41no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C2×#12311yescharacteristic
263C4×C61#54011yescharacteristic
264C4×C61#54611yescharacteristic
265C4×C61#55311yescharacteristic
266Dic61—42no—
267Dic61—81no—
268C2×#471—611no—
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049—611no—
273#2048—611no—
274#1977—611no—
275#1953—611no—
276#1905—611no—
277#1904D6111yescharacteristic
278#1833—611no—
279#1830—611no—
280#1829—611no—
281#1814—611no—
282#1757—611no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—81no—
287C2×C61—82no—
288C2×C61#242011yescharacteristic
289C2×#137—611no—
290C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
291C22×SD16—611no—
292C2×#131—611no—
293#519—611no—
294#518C2×D6111yescharacteristic
295#471—1222no—
296#471—1222no—
297#440—1222no—
298C4×SD16—1222no—
299#430—611no—
300#430—611no—
301C2×#108—611no—
302C2×#108—611no—
303#427—611no—
304#421—611no—
305#421C2×D6111yescharacteristic
306#419—611no—
307#419—611no—
308#418—611no—
309#418C2×D6111yescharacteristic
310#413—611no—
311C2×#101—611no—
312C2×#101—611no—
313C2×#100—611no—
314C2×#100—611no—
315#410C2×D6111yescharacteristic
316#405—611no—
317#405—611no—
318#344—611no—
319#344—611no—
320#344—611no—
321#344—611no—
322#344—611no—
323#344—611no—
324#344—611no—
325#344C2×D6111yescharacteristic
326H(3,ℤ/4)—611no—
327H(3,ℤ/4)C2×D6111yescharacteristic
328C61#2498711yescharacteristic, complemented, Sylow
329C2×#41C22×D6111yescharacteristic
330C22×Q8—611no—
331C22×D4C22×D6111yescharacteristic
332#137—1222no—
333#137—2441no—
334AGL(1,ℤ/8)—22no—
335AGL(1,ℤ/8)—41no—
336C2×Q16—1221no—
337C2×SD16—21no—
338C2×SD16—612no—
339C2×SD16—612no—
340C2×SD16—1221no—
341C2×SD16—1221no—
342C2×SD16—1221no—
343C2×SD16—1221no—
344C2×SD16—1221no—
345C2×D8—21no—
346#131—1221no—
347#131—1221no—
348#131—1221no—
349#131—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—611no—
355C2×M16—1221no—
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6111yescharacteristic
358C22×C8—611no—
359C22×C8—1221no—
360C4×Q8—1222no—
361C4×Q8—2441no—
362C4×D4—22no—
363C4×D4—41no—
364#117—611no—
365#117—1222no—
366#117C22×D6111yescharacteristic
367#117C22×D6111yescharacteristic
368#117C22×D6111yescharacteristic
369C2×C22⋊C4—22no—
370#108—1221no—
371#108—1221no—
372#108—1221no—
373#108—1221no—
374#108—2441no—
375#108—2441no—
376#107—1221no—
377#106—1221no—
378#104—41no—
379#104—2441no—
380#103—1222no—
381#103—2441no—
382#102—41no—
383#102—1222no—
384#101—1222no—
385#101—1221no—
386#101—2441no—
387#100—1221no—
388#100—1222no—
389#100—2441no—
390#99—22no—
391#97—1221no—
392#97—1221no—
393C4×C8—1221no—
394C4×C8—1221no—
395C24—21no—
396#41—21no—
397#41—21no—
398#41—21no—
399#41—21no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8#1175822211yescharacteristic
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4#1175822211yescharacteristic
414C2×D4D4×D6112yes—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—1221no—
418C22×C4C23×D6111yescharacteristic
419Q16—2442no—
420SD16—42no—
421SD16—1222no—
422SD16—1222no—
423SD16—1222no—
424SD16—1222no—
425SD16—2442no—
426D8—42no—
427M16—21no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—1221no—
432M16—1221no—
433M16—1221no—
434M16—1221no—
435M16—1221no—
436M16—1221no—
437M16—1221no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—2441no—
442M16—2441no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—2441no—
457C2×C8—2441no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—1221no—
462C4⋊C4—2442no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—42no—
466C22⋊C4—1221no—
467C22⋊C4#1175822412yes—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1221no—
472C42—2442no—
473C23—21no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—161no—
478C23C2×#1175822411yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—1222no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—21no—
486D4—22no—
487D4—21no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—162no—
500C2×C4—1222no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4#41×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982211yescharacteristic
510C8—41no—
511C8—41no—
512C8—611no—
513C8—611no—
514C8—611no—
515C8—611no—
516C8—2441no—
517C8—2441no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—81no—
526C22—161no—
527C22—161no—
528C22#1684029211yescharacteristic
529C22#1684039911yescharacteristic
530C22#1684047611yescharacteristic
531C4—42no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—4881no—
536C4—2442no—
537C4#123×D6111yescharacteristic
538C4#1684036211yescharacteristic
539C4#1684040211yescharacteristic
540C4#1684046111yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—81no—
546C2#2144930811yescharacteristic
subgroupG/H
centerC2#21449308
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1904C2
socleC2×C61#2420
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenter#1904D61
hypocenterC61#24987
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. #21454676C2semidirect
3. #21447288C2semidirect
4. #21447276C2semidirect
5. #21447100C2semidirect
7. #21446725C2semidirect
8. #21446608C2semidirect
9. #21446316C2semidirect
10. #21445941C2semidirect
11. #21445932C2semidirect
12. #21445930C2semidirect
13. #21445871C2semidirect
14. #21445621C2semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
C2×#168400232
#214546762
#214472882
#214472762
#214471002
#214470892
#214467252
#214466082
#214463162
#214459412
#214459322
#214459302
#214458712
#214456212
C61×#19042
#2498761