Finite Group Databasebeta

← #29834990

Subgroups — #29834990

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456393C211yesmaximal, characteristic, complemented
2#135×Dic61C211yesmaximal, characteristic, complemented
3#21448011C211yesmaximal, characteristic, complemented
4#21448007C211yesmaximal, characteristic, complemented
5#21447969C211yesmaximal, characteristic, complemented
6#21447964C211yesmaximal, characteristic, complemented
7#21447857C211yesmaximal, characteristic, complemented
8#21447852C211yesmaximal, characteristic, complemented
9#21447742C211yesmaximal, characteristic, complemented
10SD32×Dic61C211yesmaximal, characteristic, complemented
11#21447668C211yesmaximal, characteristic, complemented
12D16×Dic61C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447615C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic, complemented
16#16840307—21no—
17#41×Dic61—21no—
18C2×#14519720C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20#16840166C2211yescharacteristic
21#16840137C2211yescharacteristic
22#16840107—21no—
23#16839998C2211yescharacteristic
24#16839996—21no—
25Q16×Dic61C2211yescharacteristic
26Q16×Dic61C2211yescharacteristic
27SD16×Dic61—21no—
28SD16×Dic61—21no—
29D8×Dic61C2211yescharacteristic
30D8×Dic61C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32#16839844C2211yescharacteristic
33#16839842—21no—
34#16839800C2211yescharacteristic
35#16839758—21no—
36C2×C8×Dic61C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50C16×Dic61C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×D16×C61C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758224—41no—
60C2×D488C2311yescharacteristic
61C2×#11758199—21no—
62C22×#92418D411yescharacteristic
63#14519776—21no—
64#14519773—21no—
65#14519763—41no—
66#14519751—21no—
67#14519749—41no—
68Q8×Dic61—21no—
69Q8×Dic61—21no—
70D4×Dic61—21no—
71D4×Dic61—21no—
72D4×Dic61—21no—
73D4×Dic61—41no—
74D4×Dic61—41no—
75#14519739—21no—
76#14519737—41no—
77#14519736—21no—
78#14519736—41no—
79C2×C4×Dic61—41no—
80C2×C4×Dic61D411yescharacteristic
81#14519720—22no—
82#14519718—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519710—21no—
89#14519709—21no—
90#14519701—21no—
91#14519701—21no—
92#14519699—21no—
93#14519698—21no—
94#14519698—21no—
95#14519698—21no—
96#14519697—21no—
97C8×Dic61D411yescharacteristic
98C8×Dic61D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C2×#41×C61—21no—
102#135×C61—22no—
103#135×C61—22no—
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×D8×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—21no—
110SD32×C61—22no—
111D16×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#102×C61—21no—
116#41×C61—24no—
117#41×C61—42no—
118C2×Q8×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61D811yescharacteristic
132C22×Dic61D811yescharacteristic
133#11758224—84no—
134C2×D244—21no—
135C2×C4×D61—41no—
136C2×Dic122—41no—
137Q16×C61—21no—
138Q16×C61#4112yescomplemented
139SD16×C61—24no—
140D8×C61—21no—
141D8×C61—21no—
142D8×C61—21no—
143D8×C61#4112yescomplemented
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2411yescharacteristic
148C4⋊C4×C61—21no—
149C22⋊C4×C61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153C22⋊Dic61—41no—
154#11758199—42no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—41no—
161C4⋊Dic61—41no—
162#11758196—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61D811yescharacteristic
167C4×Dic61D811yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C2×#92418—41no—
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173D488—21no—
174D488—21no—
175C16×C61—21no—
176C16×C61—21no—
177#11758185—21no—
178#11758185—21no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C2×D811yescharacteristic
182C22×D61—41no—
183Q8×C61—22no—
184Q8×C61—41no—
185D4×C61—22no—
186D4×C61—44no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—82no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—82no—
199C22⋊D61—82no—
200C22⋊D61—82no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—82no—
207C2×Dic61—82no—
208C2×Dic61—82no—
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211C2×Dic61C2×D811yescharacteristic
212C2×Dic61C2×D811yescharacteristic
213D244—41no—
214D244—41no—
215C4×D61—82no—
216Dic122—82no—
217C8×C61—21no—
218C8×C61C2×#4111yescharacteristic
219C8×C61C2×#4111yescharacteristic
220#92418—21no—
221#92418—21no—
222#33666—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—42no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61C22×D811yescharacteristic
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—81no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—81no—
239C4×C61#52511yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—21no—
242Dic61—21no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246Dic61#50812yes—
247Dic61#50912yes—
248#3169—611no—
249C4×#135—611no—
250#2234—611no—
251#2234—611no—
252#2227—611no—
253#2226—611no—
254#2207—611no—
255#2206—611no—
256#2186—611no—
257#2186D6111yescharacteristic, complemented
258C4×SD32—611no—
259C4×SD32—611no—
260C4×D16—611no—
261C4×D16—611no—
262#2162—611no—
263C2×C61—41no—
264C2×C61—82no—
265C2×C61—161no—
266C2×C61#310011yescharacteristic
267C2×C61C2×#50811yescharacteristic
268C2×C61C2×#50911yescharacteristic
269D61—161no—
270C2×#135C2×D6111yescharacteristic
271#532—1221no—
272C4×#41—1221no—
273C2×SD32—611no—
274C2×SD32C2×D6111yescharacteristic
275C2×D16—611no—
276C2×D16C2×D6111yescharacteristic
277#500—611no—
278#493—611no—
279#486—1221no—
280#466—611no—
281#466—1221no—
282#466C2×D6111yescharacteristic
283C4×Q16—611no—
284C4×Q16—611no—
285C4×SD16—1221no—
286C4×SD16—1221no—
287C4×D8—611no—
288C4×D8—611no—
289C4×D8—611no—
290C4×D8—611no—
291C4×D8—1221no—
292#427—611no—
293#416—1221no—
294C2×C4×C8—611no—
295#367—611no—
296#367—611no—
297#366—611no—
298#366—611no—
299C4⋊C16—611no—
300C4⋊C16—611no—
301#358—611no—
302#358C2×D6111yescharacteristic
303#357—611no—
304#357—611no—
305#357—611no—
306#357—611no—
307#357—611no—
308#357C2×D6111yescharacteristic
309C22⋊C16—611no—
310C22⋊C16C2×D6111yescharacteristic
311C4×C16—611no—
312C4×C16—611no—
313C61#3366611yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C2×#41—2441no—
316#135—22no—
317#135—22no—
318C2×Q16C22×D6111yescharacteristic
319C2×SD16—21no—
320C2×D8—611no—
321C2×D8—1221no—
322C2×D8C22×D6111yescharacteristic
323C2×D8C22×D6111yescharacteristic
324C22×C8—611no—
325C22×C8C22×D6111yescharacteristic
326C4⋊Q8—1221no—
327#127—1221no—
328#124—2441no—
329#121—21no—
330#121—1221no—
331#121—2441no—
332C4×Q8—1221no—
333C4×Q8—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4—2441no—
339C4×D4—2441no—
340C4×D4—2441no—
341#117—1221no—
342#117—2441no—
343C2×C42—611no—
344C2×C42—2441no—
345SD32—22no—
346SD32—1222no—
347D16—22no—
348D16—1222no—
349C2×C16—611no—
350C2×C16—611no—
351C2×C16C22×D6111yescharacteristic
352C2×C16C22×D6111yescharacteristic
353#107—611no—
354#107—611no—
355#107—611no—
356#107—1221no—
357#107C22×D6111yescharacteristic
358#106—1221no—
359#103—1221no—
360#103—1221no—
361#102—21no—
362#102—1221no—
363#102—1221no—
364#102—1221no—
365#102—1221no—
366#102—1221no—
367C4×C8—611no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8—611no—
371#41—24no—
372#41—42no—
373#41—4884no—
374C2×Q8—21no—
375C2×Q8—2441no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—1221no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2441no—
384C22×C4—41no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4D4×D6111yescharacteristic
392Q16—21no—
393Q16#1175822412yescomplemented
394SD16—24no—
395D8—21no—
396D8—21no—
397D8—21no—
398D8—1221no—
399D8—1221no—
400D8—2442no—
401D8#1175822412yescomplemented
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—611no—
405C2×C8—611no—
406C2×C8—2441no—
407C2×C8D4×D6111yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C22⋊C4—41no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C42—611no—
424C42—611no—
425C42—611no—
426C42—611no—
427C42—2441no—
428C42—2441no—
429C42—2441no—
430C16—21no—
431C16—21no—
432C16—1221no—
433C16—1221no—
434C23—41no—
435C23—41no—
436C23—2441no—
437C23D8×D6111yescharacteristic
438Q8—22no—
439Q8—41no—
440Q8—4882no—
441D4—22no—
442D4—44no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—82no—
447D4—4882no—
448D4—4882no—
449D4—4882no—
450D4—2441no—
451D4—2441no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4D8×D6111yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C8—21no—
473C8—1221no—
474C8—1221no—
475C8C2×#1175822411yescharacteristic
476C8C2×#1175822411yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22C2×D8×D6111yescharacteristic
490C4—21no—
491C4—42no—
492C4—81no—
493C4—612no—
494C4—612no—
495C4—1221no—
496C4—1221no—
497C4—4881no—
498C4—4881no—
499C4—4881no—
500C4#1684026811yescharacteristic
501C4#1684055811yescharacteristic
502C2—41no—
503C2—82no—
504C2—161no—
505C2—9761no—
506C2#2145581411yescharacteristic
507C2#508×D6111yescharacteristic
508C2#509×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33666
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #21456393C2semidirect
2. #135×Dic61C2semidirect
3. #21448011C2semidirect
4. #21448007C2semidirect
5. #21447969C2semidirect
6. #21447964C2semidirect
7. #21447857C2semidirect
8. #21447852C2semidirect
9. #21447742C2semidirect
10. SD32×Dic61C2semidirect
11. #21447668C2semidirect
12. D16×Dic61C2semidirect
13. #21447663C2semidirect
14. #21447615C2semidirect
15. C61×#2186C2semidirect
138. Q16×C61#41semidirect
143. D8×C61#41semidirect
257. #2186D61semidirect
313. C61#33666semidirect
393. Q16#11758224semidirect
401. D8#11758224semidirect

Maximal subgroups

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

subgroupindex
#214563932
#135×Dic612
#214480112
#214480072
#214479692
#214479642
#214478572
#214478522
#214477422
SD32×Dic612
#214476682
D16×Dic612
#214476632
#214476152
C61×#21862
#3366661