Finite Group Databasebeta

← #29836980

Subgroups — #29836980

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458164C211yesmaximal, characteristic, complemented
2#21457504C211yesmaximal, characteristic, complemented
3#21448054C211yesmaximal, characteristic, complemented
4#21448016C211yesmaximal, characteristic, complemented
5#21448003C211yesmaximal, characteristic, complemented
6#21447964C211yesmaximal, characteristic, complemented
7#21447921C211yesmaximal, characteristic, complemented
8#21447915C211yesmaximal, characteristic, complemented
9#21447873C211yesmaximal, characteristic, complemented
10#21447743C211yesmaximal, characteristic, complemented
11#21447731C211yesmaximal, characteristic, complemented
12#21447692C211yesmaximal, characteristic, complemented
13#21447644C211yesmaximal, characteristic, complemented
14#21447528C211yesmaximal, characteristic, complemented
15C2×C61×#357C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17#16840413—21no—
18#16840406—21no—
19C2×#14519722C2211yescharacteristic, complemented
20C2×#14519718C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519716C2211yescharacteristic, complemented
23#16840166C2211yescharacteristic, complemented
24#16840145C2211yescharacteristic
25#16840077—21no—
26#16840075—21no—
27#16840041—21no—
28#16840003C2211yescharacteristic
29#16839999C2211yescharacteristic, complemented
30#16839998C2211yescharacteristic
31#16839995C2211yescharacteristic
32#16839975—21no—
33#16839913—21no—
34#16839904—21no—
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic, complemented
37C8×C22⋊D61C2211yescharacteristic
38#107×D61C2211yescharacteristic, complemented
39#16839590C2211yescharacteristic, complemented
40#16839589C2211yescharacteristic, complemented
41#16839589C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839565C2211yescharacteristic, complemented
45#16839565C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic, complemented
51#16839537C2211yescharacteristic, complemented
52C22×D8×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C61×#357C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758222—41no—
60C2×#11758224—21no—
61#14519796—22no—
62#14519796—41no—
63C2×Dic244C2311yescharacteristic
64C2×#11758200—21no—
65C2×#11758200—21no—
66C2×D488C2311yescharacteristic
67C2×C8×D61C2311yescharacteristic
68#14519776—21no—
69#14519764—21no—
70#14519762—41no—
71#14519760—41no—
72#14519758—41no—
73#14519756—21no—
74#14519755—41no—
75#14519752—21no—
76#14519751—21no—
77#14519749—41no—
78#14519748—21no—
79#14519746—41no—
80D4×Dic61—21no—
81#14519739—21no—
82C4×C22⋊D61D411yescharacteristic
83C4⋊C4×D61—21no—
84C2×C22⋊Dic61—41no—
85#14519722—41no—
86#14519718—22no—
87#14519718—41no—
88#14519716—41no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519706D411yescharacteristic
96#14519702—21no—
97#14519699—21no—
98#14519698—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102C22⋊#92418D411yescharacteristic
103#14519689D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C22×D4×C61—21no—
106C2×D8×C61—22no—
107C2×D8×C61—21no—
108C2×D8×C61—41no—
109C2×D8×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C16×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61—21no—
117#107×C61C2311yescharacteristic
118#107×C61C2311yescharacteristic
119C24×C61—41no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—42no—
123C2×D4×C61—41no—
124C2×D4×C61—81no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61D811yescharacteristic
131C22×Dic61—41no—
132C22×Dic61—41no—
133#11758224—42no—
134#11758224—81no—
135#11758222—84no—
136C2×D244—21no—
137C2×D244—41no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C2×D411yescharacteristic
141C2×Dic122—21no—
142C2×Dic122—41no—
143D8×C61—21no—
144D8×C61—21no—
145D8×C61—21no—
146D8×C61—42no—
147D8×C61#4112yescomplemented
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2411yescharacteristic
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—41no—
155C22⋊Dic61—41no—
156C22⋊Dic61—41no—
157C22⋊Dic61—41no—
158C22⋊Dic61—41no—
159C22⋊Dic61—41no—
160C22⋊Dic61D811yescharacteristic
161#11758200—42no—
162#11758200—81no—
163#11758198—41no—
164#11758198—41no—
165#11758198—41no—
166#11758198D811yescharacteristic
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169#11758196—41no—
170#11758196—41no—
171#11758196—41no—
172#11758196D811yescharacteristic
173C4×Dic61C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176Dic244—21no—
177Dic244—21no—
178D488—21no—
179D488—21no—
180C8×D61—21no—
181C8×D61—21no—
182C16×C61—21no—
183C16×C61—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61—161no—
191C23×C61C2×D811yescharacteristic
192C22×D61—41no—
193C22×D61C2×D811yescharacteristic
194D4×C61—22no—
195D4×C61—41no—
196D4×C61—41no—
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—82no—
204C2×C4×C61C2×D811yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C22⋊D61—22no—
207C22⋊D61—82no—
208C22⋊D61—82no—
209C22⋊D61—161no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—82no—
214C2×Dic61—161no—
215C2×Dic61C2×D811yescharacteristic
216C2×Dic61C2×D811yescharacteristic
217C2×Dic61C2×D811yescharacteristic
218D244—41no—
219D244—41no—
220D244—82no—
221C4×D61—21no—
222C4×D61—21no—
223C4×D61—82no—
224C4×D61—82no—
225Dic122—41no—
226Dic122—41no—
227Dic122—82no—
228C8×C61—21no—
229C8×C61C2×#4111yescharacteristic
230C8×C61H(5,𝔽2)11yescharacteristic
231#92418—21no—
232#92418—21no—
233#33873—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—42no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—164no—
243C22×C61#50912yes—
244C22×C61C22×D811yescharacteristic
245C2×D61—21no—
246C2×D61—21no—
247C2×D61—21no—
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—81no—
251C4×C61—21no—
252C4×C61—81no—
253C4×C61—81no—
254C4×C61#54611yescharacteristic
255C4×C61#57711yescharacteristic
256Dic61—21no—
257Dic61—21no—
258Dic61—21no—
259Dic61—81no—
260Dic61—81no—
261Dic61—81no—
262#3334—611no—
263#3280—611no—
264#2243—611no—
265#2235—611no—
266#2233—611no—
267#2226—611no—
268#2218—611no—
269#2217—611no—
270#2210—611no—
271#2186—611no—
272#2184—611no—
273#2177—611no—
274#2168—611no—
275#2139—611no—
276C2×#357D6111yescharacteristic, complemented
277C2×C61—22no—
278C2×C61—82no—
279C2×C61—161no—
280C2×C61#329411yescharacteristic
281C2×C61C2×#50911yescharacteristic
282C2×C61#341611yescharacteristic
283D61—41no—
284D61—161no—
285C2×#135—611no—
286C22×D8C2×D6111yescharacteristic
287#547—1221no—
288#546—1221no—
289C2×SD32—611no—
290C2×D16—611no—
291C2×D16—611no—
292C2×M32—611no—
293C22×C16C2×D6111yescharacteristic
294#500—611no—
295#495—611no—
296#480—1221no—
297#480—1221no—
298#474—1221no—
299#467—611no—
300#466—611no—
301#466—611no—
302#466—611no—
303#463—1221no—
304#450—1221no—
305#449—1221no—
306C4×D8—611no—
307C4×D8—611no—
308C8×D4—611no—
309C2×#107—611no—
310C2×#107C2×D6111yescharacteristic
311#367—611no—
312#367—611no—
313#367—611no—
314C4⋊C16—611no—
315#358—611no—
316#358—611no—
317#358—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357C2×D6111yescharacteristic
322#357C2×D6111yescharacteristic
323#357C2×D6111yescharacteristic
324#357C2×D6111yescharacteristic
325C22⋊C16—611no—
326C22⋊C16—611no—
327#346—611no—
328C61#3387311yescharacteristic, complemented, Sylow
329C2×#41—1221no—
330C2×#41—2441no—
331C22×D4—21no—
332#135—1222no—
333#135—2441no—
334C2×Q16—611no—
335C2×SD16—1221no—
336C2×SD16—1221no—
337C2×D8—22no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—611no—
341C2×D8C22×D6111yescharacteristic
342C2×D8C22×D6111yescharacteristic
343C22×C8—611no—
344C22×C8C22×D6111yescharacteristic
345C4⋊Q8—1221no—
346#124—1221no—
347#123—2441no—
348#123—2441no—
349#123—2441no—
350C22⋊Q8—1221no—
351C22⋊Q8—2441no—
352#121—1221no—
353#121—1221no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357C4×D4—611no—
358C4×D4—1221no—
359C4×D4—1221no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—1221no—
362C2×C22⋊C4—2441no—
363SD32—2441no—
364D16—1222no—
365D16—2441no—
366M32—2441no—
367C2×C16—41no—
368C2×C16—611no—
369C2×C16—611no—
370C2×C16C22×D6111yescharacteristic
371C2×C16C22×D6111yescharacteristic
372#107—21no—
373#107—611no—
374#107—611no—
375#107—1221no—
376#107—1221no—
377#107C22×D6111yescharacteristic
378#107C22×D6111yescharacteristic
379C4⋊C8—611no—
380#103—1221no—
381#102—1221no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386C22⋊C8—611no—
387C22⋊C8—611no—
388C4×C8—611no—
389C24—41no—
390#41—4884no—
391#41—4881no—
392#41—2442no—
393C2×Q8—1221no—
394C2×Q8—2441no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—81no—
400C2×D4—611no—
401C2×D4—1221no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2441no—
405C2×D4—2441no—
406C22×C4—41no—
407C22×C4—611no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4D4×D6111yescharacteristic
413Q16—1221no—
414Q16—1221no—
415SD16—4881no—
416SD16—2442no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—42no—
421D8—1221no—
422D8—1221no—
423D8#1175822212yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—611no—
427C2×C8—611no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8D4×D6111yescharacteristic
431C2×C8C23×D6111yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—611no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C22⋊C4—611no—
442C22⋊C4—611no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C42—611no—
452C16—21no—
453C16—21no—
454C16—1221no—
455C16—1221no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23—82no—
460C23—161no—
461C23—611no—
462C23—2441no—
463C23D8×D6111yescharacteristic
464Q8—4882no—
465Q8—2441no—
466Q8—2441no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—82no—
472D4—1222no—
473D4—9761no—
474D4—4882no—
475D4—4882no—
476D4—4882no—
477D4—2441no—
478D4—2441no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—9761no—
490C2×C4—4882no—
491C2×C4—4882no—
492C2×C4—4882no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4D8×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C8—21no—
499C8—1221no—
500C8—1221no—
501C8C2×#1175822211yescharacteristic
502C8#1451982211yescharacteristic
503C22—21no—
504C22—21no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—164no—
512C22—1221no—
513C22—1221no—
514C22—1221no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22#1684019612yes—
519C22C2×D8×D6111yescharacteristic
520C4—21no—
521C4—81no—
522C4—81no—
523C4—1221no—
524C4—1221no—
525C4—1221no—
526C4—4881no—
527C4—4881no—
528C4—4881no—
529C4#1684040211yescharacteristic
530C4#1684056811yescharacteristic
531C2—22no—
532C2—82no—
533C2—161no—
534C2—9761no—
535C2—2441no—
536C2#2145762811yescharacteristic
537C2C2×#1684019611yescharacteristic
538C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#357C2
socleC22×C61C22×D8
hypercenterC2×#357D61
hypocenterC61#33873
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #21458164C2semidirect
2. #21457504C2semidirect
3. #21448054C2semidirect
4. #21448016C2semidirect
5. #21448003C2semidirect
6. #21447964C2semidirect
7. #21447921C2semidirect
8. #21447915C2semidirect
9. #21447873C2semidirect
10. #21447743C2semidirect
11. #21447731C2semidirect
12. #21447692C2semidirect
13. #21447644C2semidirect
14. #21447528C2semidirect
15. C2×C61×#357C2semidirect
19. C2×#14519722C22semidirect
22. C2×#14519716C22semidirect
23. #16840166C22semidirect
29. #16839999C22semidirect
36. #16839844C22semidirect
38. #107×D61C22semidirect
39. #16839590C22semidirect
40. #16839589C22semidirect
41. #16839589C22semidirect
42. #16839580C22semidirect
43. #16839567C22semidirect
44. #16839565C22semidirect
45. #16839565C22semidirect
46. #16839564C22semidirect
50. #16839541C22semidirect
51. #16839537C22semidirect
147. D8×C61#41semidirect
276. C2×#357D61semidirect
328. C61#33873semidirect
423. D8#11758222semidirect

Maximal subgroups

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

subgroupindex
#214581642
#214575042
#214480542
#214480162
#214480032
#214479642
#214479212
#214479152
#214478732
#214477432
#214477312
#214476922
#214476442
#214475282
C2×C61×#3572
#3387361