Finite Group Databasebeta

← #29722535

Subgroups — #29722535

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455432C212yesmaximal, complemented
2C2×#16839995C211yesmaximal, characteristic, complemented
3#21454511C211yesmaximal, characteristic, complemented
4C2×#108×D61C211yesmaximal, characteristic, complemented
5#21447175C212yesmaximal, complemented
6#21447108C212yesmaximal, complemented
7#21446730C211yesmaximal, characteristic, complemented
8#21446605C211yesmaximal, characteristic, complemented
9#21446279C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C2×C61×#472C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C22×C8×D61C2211yescharacteristic
14#16840269C2211yescharacteristic
15C2×#14519748C2212yes—
16#16840012—22no—
17#16840005—22no—
18#16839995—21no—
19#16839995—21no—
20#16839995—21no—
21#16839995—21no—
22#16839995—22no—
23#16839988—22no—
24#16839831—21no—
25#16839831—21no—
26C8×C22⋊D61—21no—
27C8×C22⋊D61—21no—
28C2×#14519713C2211yescharacteristic
29C2×#14519712C2212yes—
30#108×D61—24no—
31#16839804C2212yes—
32C2×#14519711C2211yescharacteristic
33#16839790C2211yescharacteristic
34#16839767C2212yes—
35#16839763C2212yes—
36C2×#14519698C2212yes—
37C2×#14519694C2212yes—
38#16839729C2211yescharacteristic
39#16839725C2211yescharacteristic
40#16839717C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2212yes—
43C2×AGL(1,ℤ/8)×C61C2212yes—
44C22×D8×C61C2211yescharacteristic
45C2×#131×C61C2211yescharacteristic
46C61×#472—24no—
47C2×#108×C61C2211yescharacteristic
48C2×#100×C61C2212yes—
49C22×C22⋊D61—22no—
50C22×C4×D61C2311yescharacteristic
51C2×#11758194C2311yescharacteristic
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54M16×D61—24no—
55M16×D61—42no—
56C22×#92418C2311yescharacteristic
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59C2×C8×D61—22no—
60C2×C8×D61D411yescharacteristic
61C2×C8×D61D411yescharacteristic
62C2×C8×D61D411yescharacteristic
63C2×C8×D61D411yescharacteristic
64#14519758—21no—
65#14519753D411yescharacteristic
66#14519753D411yescharacteristic
67#14519748—22no—
68#14519748—22no—
69#14519748—22no—
70#14519748—22no—
71#14519748D411yescharacteristic
72#14519748D411yescharacteristic
73#14519745—21no—
74C4×C22⋊D61—21no—
75C4×C22⋊D61—21no—
76#14519736C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×C22⋊Dic61—22no—
79#14519713—22no—
80#14519712—24no—
81#14519711—21no—
82#14519711—21no—
83#14519711—22no—
84#14519709—22no—
85#14519706—21no—
86#14519706—21no—
87#14519701—22no—
88#14519698—22no—
89#14519698—22no—
90#14519698—22no—
91#14519698—22no—
92#14519698—22no—
93#14519694—42no—
94C22⋊#92418—21no—
95C22⋊#92418—21no—
96#14519689—21no—
97#14519689—21no—
98#14519687—21no—
99C8×Dic61—21no—
100C2×#41×C61C2311yescharacteristic
101C22×D4×C61C2312yes—
102AGL(1,ℤ/8)×C61—24no—
103AGL(1,ℤ/8)×C61—42no—
104C2×SD16×C61—22no—
105C2×D8×C61—22no—
106C2×D8×C61—21no—
107C2×D8×C61—21no—
108C2×D8×C61—22no—
109C2×D8×C61—22no—
110#131×C61—22no—
111#131×C61—22no—
112C2×M16×C61—21no—
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C22×C8×C61—21no—
116C22×C8×C61C2311yescharacteristic
117#108×C61D414yescomplemented
118#100×C61—24no—
119C24×C61—22no—
120C23×D61#4111yescharacteristic
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61C2×D411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—21no—
137C2×C22⋊D61—21no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—84no—
141C22×Dic61—22no—
142C22×Dic61#4111yescharacteristic
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151SD16×C61—44no—
152D8×C61—44no—
153D8×C61—42no—
154D8×C61—42no—
155M16×C61—22no—
156M16×C61—21no—
157M16×C61—21no—
158M16×C61—42no—
159M16×C61C2×D414yes—
160C2×C8×C61—22no—
161C2×C8×C61—21no—
162C2×C8×C61—42no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D412yes—
166C2×C8×C61#4112yes—
167C22⋊Dic61—21no—
168C22⋊Dic61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172#11758198—21no—
173#11758198—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178#11758196—21no—
179#11758196—21no—
180C4×Dic61—21no—
181#11758194—22no—
182#11758194—21no—
183#11758194—22no—
184#11758194—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—22no—
189C2×#92418C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191#11758189—44no—
192C8×D61—22no—
193C8×D61—22no—
194C8×D61—22no—
195C8×D61—22no—
196C8×D61—44no—
197C23×C61—21no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—82no—
202C23×C61—82no—
203C23×C61C2×#4111yescharacteristic
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—22no—
207C22×D61—41no—
208C22×D61#12111yescharacteristic
209C22×D61#12111yescharacteristic
210Q8×C61—21no—
211Q8×C61—21no—
212D4×C61—21no—
213D4×C61—21no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—42no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C22⋊D61—42no—
230C22⋊D61—42no—
231C22⋊D61—84no—
232C22⋊D61—84no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—42no—
238C2×Dic61—82no—
239C2×Dic61#12111yescharacteristic
240C2×Dic61#12111yescharacteristic
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C8×C61—24no—
250C8×C61—41no—
251C8×C61#12112yes—
252C8×C61#12112yes—
253#92418—21no—
254#92418—21no—
255#92418—41no—
256#92418—42no—
257#24803—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61—162no—
267C22×C61—162no—
268C22×C61C2×#12111yescharacteristic
269C22×C61#54811yescharacteristic
270C22×C61#55311yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61#47212yescomplemented
279C4×C61—41no—
280C4×C61C2×#12111yescharacteristic
281C4×C61#53711yescharacteristic
282C4×C61D4211yescharacteristic
283C4×C61#54711yescharacteristic
284Dic61—21no—
285Dic61—21no—
286Dic61—41no—
287Dic61—41no—
288Dic61—82no—
289C2×#472D6111yescharacteristic, complemented
290#3055—612no—
291C2×#466—611no—
292#2929—611no—
293C22×#108—611no—
294#2063—612no—
295#2051—612no—
296#1978—611no—
297#1952—611no—
298#1900—612no—
299#1816—611no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—81no—
303C2×C61—162no—
304C2×C61#241611yescharacteristic
305C2×C61C2×#47211yescharacteristic
306C2×C61#306711yescharacteristic
307D61—22no—
308D61—41no—
309C2×AGL(1,ℤ/8)C2×D6112yes—
310C22×D8C2×D6111yescharacteristic
311C2×#131C2×D6111yescharacteristic
312C22×M16—612no—
313C23×C8—611no—
314#525—611no—
315C2×#121—612no—
316#472—24no—
317#469—1222no—
318#468—1222no—
319#466—1222no—
320#466—1221no—
321#466—1221no—
322#466—1221no—
323#466—1221no—
324#465—1222no—
325#435—1221no—
326#435—1221no—
327C8×D4—1221no—
328C8×D4—1221no—
329C2×#108—612no—
330C2×#108—611no—
331C2×#108—1224no—
332C2×#108C2×D6111yescharacteristic
333#428—612no—
334C2×#107—611no—
335#424—611no—
336#418—612no—
337#417—612no—
338C2×#102—612no—
339C2×#100—612no—
340C2×#100C2×D6112yes—
341#408—611no—
342#407—611no—
343#405—611no—
344#343—612no—
345#343—612no—
346C61#2480311yescharacteristic, complemented, Sylow
347C2×#41C22×D6111yescharacteristic
348C22×D4—1222no—
349C22×D4C22×D6112yes—
350C23×C4—611no—
351AGL(1,ℤ/8)—24no—
352AGL(1,ℤ/8)—42no—
353C2×SD16—22no—
354C2×D8—22no—
355C2×D8—21no—
356C2×D8—21no—
357C2×D8—22no—
358C2×D8—22no—
359#131—22no—
360#131—22no—
361C2×M16—21no—
362C2×M16—612no—
363C2×M16—611no—
364C2×M16—1222no—
365C2×M16—1224no—
366C2×M16—2442no—
367C2×M16C22×D6111yescharacteristic
368C2×M16C22×D6112yes—
369C22×C8—21no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—1222no—
376C22×C8—1222no—
377C22×C8—1222no—
378C22×C8C22×D6111yescharacteristic
379#123—1221no—
380C22⋊Q8—611no—
381C22⋊Q8—611no—
382#121—611no—
383#121—611no—
384#121—1222no—
385#121—1222no—
386#121—1222no—
387#121—1222no—
388#120—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391#117—611no—
392C2×C4⋊C4—612no—
393C2×C22⋊C4—1222no—
394#108—1224no—
395#108—1222no—
396#108C22⋊D6114yescomplemented
397#107—1222no—
398#107—1221no—
399#107—1221no—
400#106—1222no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403#103—1222no—
404#102—1222no—
405#102—1222no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409#100—24no—
410#100—2442no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415#97—1221no—
416C4×C8—1221no—
417C24—22no—
418C24—611no—
419#41—21no—
420#41—21no—
421#41—21no—
422#41—21no—
423C2×Q8D4×D6111yescharacteristic
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—4884no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4D4×D6111yescharacteristic
437C22×C4—21no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1222no—
448C22×C4C23×D6111yescharacteristic
449SD16—44no—
450D8—44no—
451D8—42no—
452D8—42no—
453M16—22no—
454M16—21no—
455M16—21no—
456M16—42no—
457M16—1222no—
458M16—1221no—
459M16—1222no—
460M16—1221no—
461M16—2444no—
462M16C2×C22⋊D6114yes—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—42no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1221no—
475C2×C8—1222no—
476C2×C8—2444no—
477C2×C8D4×D6111yescharacteristic
478C2×C8D4×D6111yescharacteristic
479C2×C8#1175822412yes—
480C2×C8C2×C22⋊D6112yes—
481C4⋊C4—611no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—1221no—
495C23—21no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—82no—
500C23—82no—
501C23—611no—
502C23—611no—
503C23—1222no—
504C23—1221no—
505C23—1221no—
506C23—2441no—
507C23#41×D6111yescharacteristic
508Q8—21no—
509Q8—21no—
510D4—21no—
511D4—21no—
512D4—41no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—4884no—
519D4—4884no—
520D4—2442no—
521D4—2442no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—2442no—
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4#1451982211yescharacteristic
546C2×C4#1451982411yescharacteristic
547C8—24no—
548C8—41no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2442no—
553C8#1451974812yes—
554C8#1451974812yes—
555C22—21no—
556C22—21no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—162no—
564C22—162no—
565C22—612no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—2442no—
572C22—2442no—
573C22#121×D6111yescharacteristic
574C22#1684041911yescharacteristic
575C22#1684045911yescharacteristic
576C4—41no—
577C4—1221no—
578C4—1221no—
579C4—4882no—
580C4—2441no—
581C4—2441no—
582C4C2×#1451974811yescharacteristic
583C4#1684034111yescharacteristic
584C4D4×C22⋊D6111yescharacteristic
585C4#1684041211yescharacteristic
586C2—21no—
587C2—21no—
588C2—81no—
589C2—162no—
590C2—1222no—
591C2—2441no—
592C2#2144925511yescharacteristic
593C2#472×D6111yescharacteristic
594C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#472C2
socleC22×C61C2×#121
hypercenterC2×#472D61
hypocenterC61#24803
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #21455432C2semidirect
2. C2×#16839995C2semidirect
3. #21454511C2semidirect
4. C2×#108×D61C2semidirect
5. #21447175C2semidirect
6. #21447108C2semidirect
7. #21446730C2semidirect
8. #21446605C2semidirect
9. #21446279C2semidirect
10. #21445883C2semidirect
11. C2×C61×#472C2semidirect
117. #108×C61D4semidirect
278. C2×D61#472semidirect
289. C2×#472D61semidirect
346. C61#24803semidirect
396. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214554322
C2×#168399952
#214545112
C2×#108×D612
#214471752
#214471082
#214467302
#214466052
#214462792
#214458832
C2×C61×#4722
#2480361