Finite Group Databasebeta

← #29837063

Subgroups — #29837063

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458246C211yesmaximal, characteristic, complemented
2#21457498C211yesmaximal, characteristic
3#21448057C211yesmaximal, characteristic, complemented
4#21448026C211yesmaximal, characteristic, complemented
5#21448011C211yesmaximal, characteristic, complemented
6#21447964C211yesmaximal, characteristic, complemented
7#21447924C211yesmaximal, characteristic, complemented
8#21447917C211yesmaximal, characteristic, complemented
9#21447883C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12#21447641C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic, complemented
14#21447526C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840483—21no—
18#16840400—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519716C2211yescharacteristic
21C22×#11758185C2211yescharacteristic, complemented
22#16840166C2211yescharacteristic, complemented
23#16840147C2211yescharacteristic
24#16840101—21no—
25#16840100—21no—
26#16840045—21no—
27#16840000C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16839998C2211yescharacteristic
30#16839979—21no—
31#16839911—21no—
32#16839907—21no—
33Q16×Dic61C2211yescharacteristic, complemented
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839800C2211yescharacteristic, complemented
37#16839799C2211yescharacteristic
38#16839590C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839566C2211yescharacteristic, complemented
44#16839566C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839541C2211yescharacteristic
51#16839537C2211yescharacteristic, complemented
52C2×#135×C61C2211yescharacteristic
53C2×Q32×C61C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C61×#467C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D61—21no—
60C2×#11758201—21no—
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63D8×D61—22no—
64D8×D61—41no—
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519772—41no—
68#14519770—41no—
69#14519766—21no—
70#14519763—41no—
71#14519760—41no—
72#14519753—21no—
73#14519753—21no—
74#14519751—21no—
75#14519749—41no—
76#14519745—41no—
77Q8×Dic61—21no—
78#14519742—41no—
79D4×Dic61—21no—
80C4×C22⋊D61D411yescharacteristic
81#14519736—21no—
82#14519735—21no—
83C2×#11758198—41no—
84#14519720—41no—
85#14519716—41no—
86C2×#11758185—41no—
87C2×#11758185C2311yescharacteristic
88C2×#11758185C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519710—21no—
93#14519709—21no—
94#14519708—21no—
95#14519706D411yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519701—21no—
99#14519699—21no—
100#14519698—21no—
101C22⋊#92418D411yescharacteristic
102#14519689D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C2×#41×C61—21no—
105#135×C61—22no—
106#135×C61—41no—
107C2×Q16×C61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C2×SD16×C61—21no—
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22⋊Q8×C61—21no—
113Q32×C61—41no—
114SD32×C61—22no—
115C2×C16×C61C2311yescharacteristic
116C2×C16×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119C23×D61—41no—
120#41×C61—42no—
121#41×C61—81no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—41no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61D811yescharacteristic
131C22×Dic61—41no—
132D4×D61—42no—
133D4×D61—81no—
134C2×D244—21no—
135C2×C4×D61C2×D411yescharacteristic
136C2×Dic122—41no—
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61—21no—
140Q16×C61—21no—
141SD16×C61—42no—
142D8×C61—21no—
143D8×C61#4112yes—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2411yescharacteristic
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153C22⋊Dic61D811yescharacteristic
154#11758201—81no—
155#11758199—42no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198D811yescharacteristic
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196—41no—
170#11758196D811yescharacteristic
171C4×Dic61—41no—
172C4×Dic61—41no—
173C4×Dic61—41no—
174C4×Dic61C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177D488—21no—
178D488—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61—82no—
189C22×D61—82no—
190C22×D61—161no—
191C22×D61C2×D811yescharacteristic
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—41no—
195Q8×C61—41no—
196D4×C61—22no—
197D4×C61—41no—
198D4×C61—82no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—82no—
205C2×C4×C61C2×D811yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C22⋊D61—41no—
208C22⋊D61—82no—
209C22⋊D61—161no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—161no—
214C2×Dic61C2×D811yescharacteristic
215C2×Dic61C2×D811yescharacteristic
216C2×Dic61C2×D811yescharacteristic
217D244—41no—
218D244—41no—
219C4×D61—21no—
220C4×D61—21no—
221Dic122—82no—
222C8×C61—21no—
223C8×C61C2×#4111yescharacteristic
224C8×C61H(5,𝔽2)11yescharacteristic
225#92418—21no—
226#92418—21no—
227#33878—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—42no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61C22×D811yescharacteristic
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C4×C61—21no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61#54611yescharacteristic
247C4×C61#57711yescharacteristic
248Dic61—21no—
249Dic61—21no—
250Dic61—21no—
251Dic61—81no—
252Dic61—81no—
253Dic61—81no—
254#3342—611no—
255#3280—611no—
256#2243—611no—
257#2237—611no—
258#2234—611no—
259#2226—611no—
260#2219—611no—
261#2217—611no—
262#2211—611no—
263#2189D6111yescharacteristic, complemented
264#2187—611no—
265#2175—611no—
266#2168—611no—
267#2140—611no—
268#2138—611no—
269C2×C61—41no—
270C2×C61—82no—
271C2×C61#329411yescharacteristic
272C2×C61#341411yescharacteristic
273C2×C61#341611yescharacteristic
274D61—41no—
275D61—161no—
276C2×#135C2×D6111yescharacteristic
277C22×D8—611no—
278#556—1221no—
279#546—1221no—
280C2×Q32C2×D6111yescharacteristic, complemented
281C2×SD32—611no—
282C2×SD32C2×D6111yescharacteristic
283C2×M32—611no—
284C22×C16—611no—
285#500—611no—
286#495—611no—
287#485—1221no—
288#485—1221no—
289#475—1221no—
290#467—611no—
291#467—611no—
292#467C2×D6111yescharacteristic, complemented
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16—611no—
298C4×D8—611no—
299C8×D4—611no—
300#427—611no—
301#427—611no—
302#367—611no—
303#367—611no—
304#366—611no—
305C4⋊C16—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#358C2×D6111yescharacteristic, complemented
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357C2×D6111yescharacteristic
316C22⋊C16—611no—
317C22⋊C16C2×D6111yescharacteristic, complemented
318#346—611no—
319C61#3387811yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—1221no—
322#135—22no—
323#135—41no—
324C2×Q16C22×D6111yescharacteristic
325C2×Q16C22×D6111yescharacteristic
326C2×SD16—21no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—1221no—
330C2×D8—1222no—
331C2×D8—2441no—
332C2×D8C22×D6111yescharacteristic
333C22×C8—611no—
334C22×C8C22×D6111yescharacteristic
335C4⋊Q8—1221no—
336#126—2441no—
337#126—2441no—
338#124—1221no—
339#124—2441no—
340#123—2441no—
341C22⋊Q8—21no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—1221no—
348C4×Q8—2441no—
349C4×D4—611no—
350C4×D4—1221no—
351#117—1221no—
352#117—1221no—
353C2×C22⋊C4—2441no—
354Q32—41no—
355SD32—22no—
356SD32—2441no—
357M32—2441no—
358C2×C16—611no—
359C2×C16—611no—
360C2×C16—2441no—
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—611no—
364#107—611no—
365#107—611no—
366#107—1221no—
367#107C22×D6111yescharacteristic
368#106—1221no—
369#106—1221no—
370C4⋊C8—611no—
371#103—21no—
372#103—1221no—
373#103—1221no—
374#103—1221no—
375#102—1221no—
376#102—1221no—
377C22⋊C8—611no—
378C22⋊C8—611no—
379C4×C8—611no—
380C24—2441no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—2441no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—4881no—
391C2×D4—2441no—
392C2×D4—2442no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—611no—
396C22×C4—2441no—
397C22×C4D4×D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—21no—
401Q16—21no—
402SD16—42no—
403SD16—4881no—
404D8—21no—
405D8—1221no—
406D8—1221no—
407D8—2442no—
408D8#1175822412yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—611no—
412C2×C8—611no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8D4×D6111yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—611no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C22⋊C4—41no—
428C22⋊C4—611no—
429C22⋊C4—611no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C42—611no—
439C42—2441no—
440C42—2441no—
441C42—2441no—
442C16—21no—
443C16—21no—
444C16—1221no—
445C16—1221no—
446C23—41no—
447C23—611no—
448C23—9761no—
449C23—4882no—
450C23—4882no—
451C23—2441no—
452C23D8×D6111yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—4882no—
458D4—22no—
459D4—41no—
460D4—82no—
461D4—9761no—
462D4—4882no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—9761no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4D8×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—1221no—
485C8—1221no—
486C8C2×#1175822411yescharacteristic
487C8#1451982211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—1221no—
495C22—1221no—
496C22—1221no—
497C22—9764no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22C2×D8×D6111yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—1221no—
507C4—1221no—
508C4—1221no—
509C4—4881no—
510C4—4881no—
511C4—4881no—
512C4#1684039911yescharacteristic
513C4#1684057211yescharacteristic
514C2—41no—
515C2—82no—
516C2—9761no—
517C2—2441no—
518C2#2145762511yescharacteristic
519C2#2145895211yescharacteristic
520C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radical#21457498C2
hypercenter#2189D61
hypocenterC61#33878
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #21458246C2semidirect
3. #21448057C2semidirect
4. #21448026C2semidirect
5. #21448011C2semidirect
6. #21447964C2semidirect
7. #21447924C2semidirect
8. #21447917C2semidirect
9. #21447883C2semidirect
10. #21447746C2semidirect
11. #21447684C2semidirect
12. #21447641C2semidirect
13. #21447533C2semidirect
14. #21447526C2semidirect
15. C61×#2189C2semidirect
21. C22×#11758185C22semidirect
22. #16840166C22semidirect
33. Q16×Dic61C22semidirect
36. #16839800C22semidirect
38. #16839590C22semidirect
39. #16839589C22semidirect
40. #16839588C22semidirect
41. #16839580C22semidirect
42. #16839566C22semidirect
43. #16839566C22semidirect
44. #16839566C22semidirect
51. #16839537C22semidirect
53. C2×Q32×C61C22semidirect
55. C61×#467C22semidirect
56. C61×#358C22semidirect
58. C61×C22⋊C16C22semidirect
263. #2189D61semidirect
280. C2×Q32C2×D61semidirect
292. #467C2×D61semidirect
309. #358C2×D61semidirect
317. C22⋊C16C2×D61semidirect
319. C61#33878semidirect

Maximal subgroups

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

subgroupindex
#214582462
#214574982
#214480572
#214480262
#214480112
#214479642
#214479242
#214479172
#214478832
#214477462
#214476842
#214476412
#214475332
#214475262
C61×#21892
#3387861