Finite Group Databasebeta

← #29837097

Subgroups — #29837097

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458208C211yesmaximal, characteristic, complemented
2#21457498C211yesmaximal, characteristic
3#21448057C211yesmaximal, characteristic, complemented
4#21448020C211yesmaximal, characteristic, complemented
5#21448000C211yesmaximal, characteristic, complemented
6#21447954C211yesmaximal, characteristic, complemented
7#21447934C211yesmaximal, characteristic, complemented
8#21447907C211yesmaximal, characteristic, complemented
9#21447883C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12#21447647C211yesmaximal, characteristic, complemented
13#21447531C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840470—21no—
18#16840400—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519717C2211yescharacteristic, complemented
21C2×C16×D61C2211yescharacteristic
22#16840166C2211yescharacteristic, complemented
23#16840147C2211yescharacteristic
24#16840096—21no—
25#16840095—21no—
26#16840055—21no—
27#16840000C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16839998C2211yescharacteristic
30#16839979—21no—
31#16839911—21no—
32#16839907—21no—
33Q16×Dic61C2211yescharacteristic, complemented
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36C2×#14519711C2211yescharacteristic, complemented
37#107×D61C2211yescharacteristic
38#16839588C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, 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#14519767—41no—
68#14519766—21no—
69#14519760—41no—
70#14519757—41no—
71#14519754—41no—
72#14519753—21no—
73#14519753—21no—
74#14519751—21no—
75#14519749—41no—
76#14519745—41no—
77Q8×Dic61—21no—
78D4×Dic61—21no—
79C4×C22⋊D61D411yescharacteristic
80#14519737—41no—
81C2×C4⋊Dic61—21no—
82C4⋊C4×D61—21no—
83C2×#11758198—41no—
84#14519720—41no—
85#14519717—41no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88C16×D61—41no—
89#14519711—21no—
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—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—
132C22×Dic61—41no—
133D4×D61—42no—
134D4×D61—81no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×C4×D61C2×D411yescharacteristic
138C2×Dic122—41no—
139Q16×C61—21no—
140Q16×C61—21no—
141Q16×C61—21no—
142Q16×C61—21no—
143SD16×C61—42no—
144D8×C61—21no—
145D8×C61#4112yes—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2411yescharacteristic
150C4⋊C4×C61—21no—
151C4⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊Dic61—41no—
154C22⋊Dic61D811yescharacteristic
155#11758201—81no—
156#11758199—42no—
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—
167C4⋊Dic61—41no—
168#11758196—41no—
169#11758196—41no—
170#11758196—41no—
171#11758196—41no—
172#11758196D811yescharacteristic
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—22no—
208C22⋊D61—82no—
209C22⋊D61—161no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—161no—
214C2×Dic61—161no—
215C2×Dic61C2×D811yescharacteristic
216C2×Dic61C2×D811yescharacteristic
217C2×Dic61C2×D811yescharacteristic
218D244—41no—
219D244—41no—
220C4×D61—21no—
221C4×D61—21no—
222C4×D61—161no—
223Dic122—161no—
224C8×C61—21no—
225C8×C61C2×#4111yescharacteristic
226C8×C61H(5,𝔽2)11yescharacteristic
227#92418—21no—
228#92418—21no—
229#33881—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61C22×D811yescharacteristic
237C2×D61—21no—
238C2×D61—21no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—164no—
243C2×D61#51012yes—
244C4×C61—21no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61#54611yescharacteristic
249C4×C61#57711yescharacteristic
250Dic61—21no—
251Dic61—21no—
252Dic61—21no—
253Dic61—81no—
254Dic61—81no—
255Dic61—81no—
256#3338—611no—
257#3280—611no—
258#2243—611no—
259#2236—611no—
260#2232—611no—
261#2224—611no—
262#2220—611no—
263#2216—611no—
264#2211—611no—
265#2189D6111yescharacteristic, complemented
266#2187—611no—
267#2175—611no—
268#2168—611no—
269#2139—611no—
270C2×#357—611no—
271C2×C61—41no—
272C2×C61—82no—
273C2×C61#329411yescharacteristic
274C2×C61C2×#51011yescharacteristic
275C2×C61#341511yescharacteristic
276D61—22no—
277D61—161no—
278C2×#135C2×D6111yescharacteristic
279C22×D8—611no—
280#554—1221no—
281#546—1221no—
282C2×Q32C2×D6111yescharacteristic, complemented
283C2×SD32—611no—
284C2×SD32C2×D6111yescharacteristic
285C2×M32—611no—
286C22×C16—611no—
287#500—611no—
288#495—611no—
289#484—1221no—
290#484—1221no—
291#477—1221no—
292#467—611no—
293#467—611no—
294#467C2×D6111yescharacteristic, complemented
295#466—611no—
296#463—1221no—
297#450—1221no—
298#449—1221no—
299C4×Q16—611no—
300C4×D8—611no—
301C8×D4—611no—
302C2×#107—611no—
303C2×#107—611no—
304#366—611no—
305#366—611no—
306#366—611no—
307C4⋊C16—611no—
308#358—611no—
309#358—611no—
310#358—611no—
311#358C2×D6111yescharacteristic, complemented
312#357—611no—
313#357—611no—
314#357—611no—
315#357—611no—
316#357—611no—
317#357C2×D6111yescharacteristic
318C22⋊C16—611no—
319C22⋊C16C2×D6111yescharacteristic, complemented
320#346—611no—
321C61#3388111yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C22×D4—1221no—
324#135—22no—
325#135—41no—
326C2×Q16C22×D6111yescharacteristic
327C2×Q16C22×D6111yescharacteristic
328C2×SD16—21no—
329C2×SD16—1221no—
330C2×D8—611no—
331C2×D8—1221no—
332C2×D8—1222no—
333C2×D8—2441no—
334C2×D8C22×D6111yescharacteristic
335C22×C8—611no—
336C22×C8C22×D6111yescharacteristic
337C4⋊Q8—1221no—
338#125—2441no—
339#124—1221no—
340#123—2441no—
341C22⋊Q8—21no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344C22⋊Q8—2441no—
345C22⋊Q8—2441no—
346#121—1221no—
347#121—2441no—
348#120—2441no—
349C4×Q8—1221no—
350C4×D4—611no—
351C4×D4—1221no—
352C4×D4—2441no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C22⋊C4—2441no—
356Q32—41no—
357SD32—22no—
358SD32—2441no—
359M32—2441no—
360C2×C16—611no—
361C2×C16—611no—
362C2×C16—2441no—
363C2×C16C22×D6111yescharacteristic
364C2×C16C22×D6111yescharacteristic
365#107—611no—
366#107—611no—
367#107—611no—
368#107—1221no—
369#107—1221no—
370#107—1221no—
371#107C22×D6111yescharacteristic
372C4⋊C8—611no—
373#103—21no—
374#103—1221no—
375#103—1221no—
376#103—1221no—
377#102—1221no—
378#102—1221no—
379C22⋊C8—611no—
380C22⋊C8—611no—
381C4×C8—611no—
382C24—2441no—
383#41—42no—
384#41—81no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—2441no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—611no—
391C2×D4—1221no—
392C2×D4—4881no—
393C2×D4—2441no—
394C2×D4—2442no—
395C2×D4—2441no—
396C22×C4—41no—
397C22×C4—611no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4—2441no—
401C22×C4D4×D6111yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—21no—
405Q16—21no—
406SD16—42no—
407SD16—4881no—
408D8—21no—
409D8—1221no—
410D8—1221no—
411D8—2442no—
412D8#1175822412yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—611no—
416C2×C8—611no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8D4×D6111yescharacteristic
420C2×C8C23×D6111yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—611no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C22⋊C4—41no—
434C22⋊C4—611no—
435C22⋊C4—611no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C22⋊C4—2441no—
442C42—611no—
443C42—2441no—
444C16—21no—
445C16—21no—
446C16—1221no—
447C16—1221no—
448C23—41no—
449C23—611no—
450C23—9761no—
451C23—4882no—
452C23—4882no—
453C23—2441no—
454C23D8×D6111yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—9761no—
460D4—22no—
461D4—41no—
462D4—82no—
463D4—1222no—
464D4—9761no—
465D4—4882no—
466D4—2441no—
467D4—2441no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—9761no—
480C2×C4—9761no—
481C2×C4—9761no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4D8×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C8—21no—
488C8—1221no—
489C8—1221no—
490C8C2×#1175822411yescharacteristic
491C8#1451982211yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—612no—
499C22—1221no—
500C22—1221no—
501C22—9764no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22C2×D8×D6111yescharacteristic
506C4—21no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—1221no—
511C4—1221no—
512C4—1221no—
513C4—4881no—
514C4—4881no—
515C4—4881no—
516C4#1684039911yescharacteristic
517C4#1684057211yescharacteristic
518C2—41no—
519C2—82no—
520C2—1222no—
521C2—9761no—
522C2#2145762511yescharacteristic
523C2#510×D6111yescharacteristic
524C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radical#21457498C2
hypercenter#2189D61
hypocenterC61#33881
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. #21458208C2semidirect
3. #21448057C2semidirect
4. #21448020C2semidirect
5. #21448000C2semidirect
6. #21447954C2semidirect
7. #21447934C2semidirect
8. #21447907C2semidirect
9. #21447883C2semidirect
10. #21447746C2semidirect
11. #21447684C2semidirect
12. #21447647C2semidirect
13. #21447531C2semidirect
14. #357×D61C2semidirect
15. C61×#2189C2semidirect
20. C2×#14519717C22semidirect
22. #16840166C22semidirect
33. Q16×Dic61C22semidirect
36. C2×#14519711C22semidirect
38. #16839588C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. #16839580C22semidirect
42. #16839566C22semidirect
43. #16839566C22semidirect
44. #16839566C22semidirect
51. #16839537C22semidirect
53. C2×Q32×C61C22semidirect
55. C61×#467C22semidirect
56. C61×#358C22semidirect
58. C61×C22⋊C16C22semidirect
265. #2189D61semidirect
282. C2×Q32C2×D61semidirect
294. #467C2×D61semidirect
311. #358C2×D61semidirect
319. C22⋊C16C2×D61semidirect
321. C61#33881semidirect

Maximal subgroups

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

subgroupindex
#214582082
#214574982
#214480572
#214480202
#214480002
#214479542
#214479342
#214479072
#214478832
#214477462
#214476842
#214476472
#214475312
#357×D612
C61×#21892
#3388161