Finite Group Databasebeta

← #29730394

Subgroups — #29730394

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21447262C212yesmaximal, complemented
5#21447108C212yesmaximal, complemented
6#21446807C211yesmaximal, characteristic, complemented
7#21446592C211yesmaximal, characteristic, complemented
8#21446329C212yesmaximal, complemented
9#21445881C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C61×#3036C211yesmaximal, characteristic
12C2×#14519784C2212yes—
13C2×#14519777C2211yescharacteristic
14#16840121—22no—
15#16840112—21no—
16#16840108—21no—
17#16840042—21no—
18#16840042—22no—
19#16840042—21no—
20#16839840—21no—
21#16839834—22no—
22D4×#92418—21no—
23C2×#14519712C2212yes—
24C2×#14519710C2212yes—
25C2×C4⋊#92418C2211yescharacteristic
26#16839763C2212yes—
27#16839763C2212yes—
28C2×#14519698C2212yes—
29C2×#14519694C2211yescharacteristic
30C2×#14519694C2211yescharacteristic
31#16839729C2212yes—
32C4×#11758194C2211yescharacteristic
33C4×#11758194C2211yescharacteristic
34#16839531C2211yescharacteristic
35#16839531C2211yescharacteristic
36#16839530C2212yes—
37#16839530C2212yes—
38#16839526C2212yes—
39C2×AGL(1,ℤ/8)×C61C2212yes—
40C61×#532C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#463—22no—
43C61×#461—21no—
44C61×#459—21no—
45C61×#423C2211yescharacteristic
46C61×#417C2212yes—
47C22×Dic122—21no—
48#14519784—24no—
49#14519784—42no—
50C2×#11758194—22no—
51C2×#11758194C2311yescharacteristic
52C2×#11758194C2311yescharacteristic
53C2×#11758194C2311yescharacteristic
54C2×#11758194C2311yescharacteristic
55C22×#92418—22no—
56C22×#92418C2312yes—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63#14519712—42no—
64#14519710—22no—
65#14519710—22no—
66C4⋊#92418D411yescharacteristic
67C4⋊#92418D412yes—
68C4⋊#92418D411yescharacteristic
69#14519701—22no—
70#14519701—22no—
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74#14519698—22no—
75#14519694—22no—
76#14519694—22no—
77C22⋊#92418—22no—
78C22⋊#92418—22no—
79#14519688—21no—
80#14519688—21no—
81C4×#92418—21no—
82C4×#92418—21no—
83C2×#41×C61C2312yes—
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86AGL(1,ℤ/8)×C61—24no—
87AGL(1,ℤ/8)×C61—42no—
88C2×SD16×C61—22no—
89C2×D8×C61—22no—
90C2×M16×C61C2312yes—
91C4⋊Q8×C61D411yescharacteristic
92#127×C61D411yescharacteristic
93#124×C61D412yes—
94#121×C61—21no—
95#121×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107#103×C61—22no—
108#102×C61—22no—
109C24×C61—21no—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—22no—
113#41×C61—22no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61C2×D412yes—
124C2×D4×C61#4112yes—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61—82no—
130C22×C4×C61#4111yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—22no—
134C2×Dic122—41no—
135C2×Dic122—41no—
136C2×Dic122—82no—
137SD16×C61—44no—
138D8×C61—44no—
139M16×C61—22no—
140M16×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158#11758194—21no—
159#11758194—21no—
160#11758194—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—21no—
165#11758194—44no—
166#11758194#4112yes—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—44no—
174C2×#92418C2×D412yes—
175C2×#92418C2×D412yes—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—81no—
181C23×C61—81no—
182C23×C61C2×#4111yescharacteristic
183Q8×C61—22no—
184Q8×C61—22no—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—42no—
194D4×C61—81no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—82no—
202C2×C4×C61—82no—
203C2×C4×C61—81no—
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C2×Dic61—42no—
215C2×Dic61—82no—
216Dic122—82no—
217Dic122—82no—
218C8×C61—42no—
219#92418—22no—
220#92418—22no—
221#92418—22no—
222#92418—41no—
223#92418—41no—
224#92418—41no—
225#25358—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61C2×#12111yescharacteristic
237C22×C61#52911yescharacteristic
238C22×C61#54211yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—42no—
243C4×C61—81no—
244C4×C61#47212yes—
245C4×C61#54611yescharacteristic
246C4×C61#54611yescharacteristic
247C4×C61#54711yescharacteristic
248C4×C61#54811yescharacteristic
249Dic61—82no—
250#3141—611no—
251C2×#474—611no—
252#3036D6111yescharacteristic
253#2935—611no—
254#2078—612no—
255#2051—612no—
256#1997—611no—
257#1950—611no—
258#1907—612no—
259#1816—612no—
260C4×#100—611no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—82no—
264C2×C61—82no—
265C2×C61#242711yescharacteristic
266C2×C61C2×#47211yescharacteristic
267C2×C61#306911yescharacteristic
268C2×AGL(1,ℤ/8)C2×D6112yes—
269C2×#131—612no—
270#532C2×D6111yescharacteristic
271C2×C4⋊Q8—611no—
272C2×C4×D4C2×D6111yescharacteristic
273#489—1222no—
274#487—1221no—
275#486—1221no—
276#474—1221no—
277#474—1222no—
278#474—1221no—
279#463—22no—
280#461—21no—
281#459—21no—
282#436—1221no—
283#435—1222no—
284C8×D4—1221no—
285C2×#108—612no—
286C2×#107—612no—
287#423C2×D6111yescharacteristic
288C2×C4⋊C8—611no—
289#417—612no—
290#417—612no—
291#417C2×D6112yes—
292C2×#102—612no—
293C2×#100—611no—
294C2×#100—611no—
295#408—612no—
296C4×M16—611no—
297C4×M16—611no—
298#343—612no—
299#343—612no—
300#343—611no—
301#343—611no—
302#341—612no—
303C61#2535811yescharacteristic, complemented, Sylow
304C2×#41C22×D6112yes—
305C22×Q8—1221no—
306C22×D4C22×D6111yescharacteristic
307C23×C4—21no—
308AGL(1,ℤ/8)—24no—
309AGL(1,ℤ/8)—42no—
310C2×SD16—22no—
311C2×D8—22no—
312#131—1224no—
313#131—2442no—
314C2×M16—611no—
315C2×M16—611no—
316C2×M16—611no—
317C2×M16—611no—
318C2×M16—1222no—
319C2×M16C22×D6112yes—
320C22×C8—612no—
321C22×C8—1222no—
322C4⋊Q8—1221no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8C22⋊D6111yescharacteristic
327#127C22⋊D6111yescharacteristic
328#124C22⋊D6112yes—
329#121—21no—
330#121—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D6111yescharacteristic
342#108—2442no—
343#107—1222no—
344#107—1222no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—611no—
348C4⋊C8—612no—
349C4⋊C8—611no—
350#103—22no—
351#103—1222no—
352#103—1222no—
353#102—22no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1222no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362#97—1221no—
363#97—1221no—
364C4×C8—1221no—
365C4×C8—1221no—
366C24—21no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371C2×Q8—4882no—
372C2×Q8—2441no—
373C2×Q8—2441no—
374C2×Q8C2×C22⋊D6112yes—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4#1175822412yes—
384C2×D4C2×C22⋊D6112yes—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—82no—
390C22×C4—1222no—
391C22×C4#1175822411yescharacteristic
392C22×C4#1175822411yescharacteristic
393C22×C4C23×D6111yescharacteristic
394SD16—44no—
395D8—44no—
396M16—22no—
397M16—22no—
398M16—612no—
399M16—1221no—
400M16—1221no—
401M16—1221no—
402M16—1221no—
403M16—1221no—
404M16—1221no—
405M16—1221no—
406M16—2444no—
407C2×C8—22no—
408C2×C8—612no—
409C2×C8—612no—
410C2×C8—1221no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—2444no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—81no—
438C23—81no—
439C23C2×#1175822411yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—4882no—
443Q8—4882no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—41no—
452D4—41no—
453D4—81no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—4882no—
464C2×C4—2442no—
465C2×C4#1451974911yescharacteristic
466C2×C4#1451974911yescharacteristic
467C2×C4#1451974911yescharacteristic
468C2×C4#1451974911yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4C2×#1175822411yescharacteristic
473C2×C4#1451982211yescharacteristic
474C2×C4#1451982411yescharacteristic
475C8—42no—
476C8—1222no—
477C8—1222no—
478C8—1222no—
479C8—2441no—
480C8—2441no—
481C8—2441no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—81no—
491C22—81no—
492C22C2×#1451974911yescharacteristic
493C22#1684029311yescharacteristic
494C22#1684037911yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—81no—
500C4—4882no—
501C4#1684003012yes—
502C4#1684040611yescharacteristic
503C4#1684040911yescharacteristic
504C4#1684041111yescharacteristic
505C4#1684041811yescharacteristic
506C2—21no—
507C2—21no—
508C2—82no—
509C2—82no—
510C2#2144939811yescharacteristic
511C2C2×#1684003011yescharacteristic
512C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#121
hypercenter#3036D61
hypocenterC61#25358
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
2. C2×#16840042C2semidirect
3. #21454563C2semidirect
4. #21447262C2semidirect
5. #21447108C2semidirect
6. #21446807C2semidirect
7. #21446592C2semidirect
8. #21446329C2semidirect
9. #21445881C2semidirect
10. C4×#14519694C2semidirect
303. C61#25358semidirect

Maximal subgroups

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

subgroupindex
#214562122
C2×#168400422
#214545632
#214472622
#214471082
#214468072
#214465922
#214463292
#214458812
C4×#145196942
C61×#30362
#2535861