Finite Group Databasebeta

← #29835129

Subgroups — #29835129

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456462C211yesmaximal, characteristic, complemented
2#21454695C211yesmaximal, characteristic, complemented
3#21448002C211yesmaximal, characteristic, complemented
4#21448000C211yesmaximal, characteristic, complemented
5#21447985C211yesmaximal, characteristic, complemented
6#21447974C211yesmaximal, characteristic, complemented
7#21447857C211yesmaximal, characteristic, complemented
8#21447852C211yesmaximal, characteristic, complemented
9#21447742C211yesmaximal, characteristic, complemented
10#21447692C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12#21447668C211yesmaximal, characteristic, complemented
13D16×Dic61C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic, complemented
16#16840365—21no—
17#16840245—21no—
18C2×#14519720C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20#16840163C2211yescharacteristic
21#16840137C2211yescharacteristic
22#16840125—21no—
23#16840013—21no—
24#16839998C2211yescharacteristic
25#16839871—21no—
26#16839863—21no—
27#16839863—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30D8×Dic61C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33#16839844C2211yescharacteristic
34C2×#14519711C2211yescharacteristic
35#16839763—21no—
36#16839717C2211yescharacteristic
37#16839589C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839588C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×D16×C61C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×D488C2311yescharacteristic
60C2×#11758199—21no—
61C2×#11758194D411yescharacteristic
62#14519773—21no—
63#14519768—21no—
64#14519761—41no—
65#14519759—41no—
66#14519751—21no—
67#14519749—41no—
68#14519748—41no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71D4×Dic61—21no—
72D4×Dic61—21no—
73D4×Dic61—21no—
74D4×Dic61—41no—
75D4×Dic61—41no—
76#14519739—21no—
77#14519736—41no—
78#14519736D411yescharacteristic
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—41no—
81#14519720—22no—
82#14519718—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711—21no—
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519708—21no—
90#14519701—21no—
91#14519701—21no—
92#14519699—21no—
93#14519698—21no—
94#14519698—21no—
95#14519698—21no—
96#14519697—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
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×Dic61—41no—
132C2×D244—21no—
133C2×C4×D61—41no—
134Q16×C61—21no—
135Q16×C61#4112yes—
136SD16×C61—24no—
137D8×C61—21no—
138D8×C61—21no—
139D8×C61—21no—
140D8×C61#4112yes—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—21no—
146C22⋊C4×C61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61D811yescharacteristic
151C22⋊Dic61D811yescharacteristic
152#11758199—42no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—41no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C4⋊Dic61—41no—
162C4⋊Dic61D811yescharacteristic
163C4⋊Dic61D811yescharacteristic
164#11758196—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—41no—
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172D488—21no—
173D488—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C2×D811yescharacteristic
181C22×D61—41no—
182Q8×C61—22no—
183Q8×C61—41no—
184D4×C61—22no—
185D4×C61—44no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—161no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—44no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C22⋊D61—161no—
198C22⋊D61—161no—
199C22⋊D61—161no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—161no—
204C2×Dic61—161no—
205C2×Dic61—161no—
206C2×Dic61—161no—
207C2×Dic61C2×D811yescharacteristic
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—41no—
212D244—41no—
213C4×D61—161no—
214C8×C61—21no—
215C8×C61C2×#4111yescharacteristic
216C8×C61C2×#4111yescharacteristic
217#92418—21no—
218#92418—21no—
219#33679—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—42no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61C22×D811yescharacteristic
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—81no—
236C4×C61#52511yescharacteristic
237C4×C61#57511yescharacteristic
238Dic61—21no—
239Dic61—21no—
240Dic61—21no—
241Dic61—21no—
242Dic61—81no—
243Dic61—81no—
244Dic61—81no—
245#3176—611no—
246#2951—611no—
247#2233—611no—
248#2232—611no—
249#2229—611no—
250#2228—611no—
251#2207—611no—
252#2206—611no—
253#2186—611no—
254#2186D6111yescharacteristic, complemented
255#2177—611no—
256#2175—611no—
257C4×SD32—611no—
258C4×D16—611no—
259#2164—611no—
260C2×C61—41no—
261C2×C61—82no—
262C2×C61—161no—
263C2×C61#310011yescharacteristic
264C2×C61#341511yescharacteristic
265C2×C61#341611yescharacteristic
266D61—161no—
267C2×#135C2×D6111yescharacteristic
268#540—1221no—
269#520—1221no—
270C2×SD32—611no—
271C2×SD32C2×D6111yescharacteristic
272C2×D16—611no—
273C2×D16C2×D6111yescharacteristic
274#499—611no—
275#493—611no—
276#490—1221no—
277#469—1221no—
278#466—611no—
279#466C2×D6111yescharacteristic
280#442—1221no—
281#440—1221no—
282#440—1221no—
283C4×Q16—611no—
284C4×Q16—611no—
285C4×D8—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289C2×#107—611no—
290#417—1221no—
291#405—611no—
292#367—611no—
293#367—611no—
294#366—611no—
295#366—611no—
296C4⋊C16—611no—
297C4⋊C16—611no—
298#358—611no—
299#358C2×D6111yescharacteristic
300#357—611no—
301#357—611no—
302#357—611no—
303#357—611no—
304#357—611no—
305#357C2×D6111yescharacteristic
306C22⋊C16—611no—
307C22⋊C16C2×D6111yescharacteristic
308#346—611no—
309C4×C16—611no—
310C61#3367911yescharacteristic, complemented, Sylow
311C2×#41—21no—
312#135—22no—
313#135—22no—
314C2×Q16C22×D6111yescharacteristic
315C2×SD16—21no—
316C2×D8—611no—
317C2×D8—1221no—
318C2×D8C22×D6111yescharacteristic
319C2×D8C22×D6111yescharacteristic
320C2×M16—611no—
321C22×C8C22×D6111yescharacteristic
322#127—1221no—
323#125—1221no—
324#123—2441no—
325#123—2441no—
326#121—21no—
327#121—1221no—
328#121—2441no—
329#121—2441no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—2441no—
337C4×D4—2441no—
338#117—611no—
339#117—2441no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—2441no—
342SD32—22no—
343SD32—1222no—
344D16—22no—
345D16—1222no—
346C2×C16—611no—
347C2×C16—611no—
348C2×C16C22×D6111yescharacteristic
349C2×C16C22×D6111yescharacteristic
350#107—611no—
351#107—611no—
352#107—611no—
353#107—1221no—
354#107C22×D6111yescharacteristic
355#106—1221no—
356#103—1221no—
357#103—1221no—
358#102—21no—
359#102—1221no—
360#102—1221no—
361#102—1221no—
362#102—1221no—
363#102—1221no—
364#97—611no—
365#97—611no—
366C4×C8—611no—
367C4×C8—611no—
368#41—24no—
369#41—42no—
370C2×Q8—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—1221no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C22×C4—41no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4D4×D6111yescharacteristic
386Q16—21no—
387Q16#1175822412yes—
388SD16—24no—
389D8—21no—
390D8—21no—
391D8—21no—
392D8—1221no—
393D8—1221no—
394D8—2442no—
395D8#1175822412yes—
396M16—2441no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—611no—
400C2×C8—611no—
401C2×C8D4×D6111yescharacteristic
402C2×C8C23×D6111yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—611no—
405C4⋊C4—611no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—41no—
415C22⋊C4—611no—
416C22⋊C4—611no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C42—611no—
423C42—611no—
424C42—2441no—
425C42—2441no—
426C16—21no—
427C16—21no—
428C16—1221no—
429C16—1221no—
430C23—41no—
431C23—41no—
432C23—2441no—
433C23D8×D6111yescharacteristic
434Q8—22no—
435Q8—41no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—161no—
442D4—9761no—
443D4—9761no—
444D4—9761no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—611no—
456C2×C4—9761no—
457C2×C4—9761no—
458C2×C4—9761no—
459C2×C4—9761no—
460C2×C4—9761no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D8×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C8—21no—
467C8—1221no—
468C8—1221no—
469C8C2×#1175822411yescharacteristic
470C8C2×#1175822411yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22C2×D8×D6111yescharacteristic
484C4—21no—
485C4—42no—
486C4—81no—
487C4—1221no—
488C4—1221no—
489C4—1221no—
490C4—1221no—
491C4—4881no—
492C4—4881no—
493C4—4881no—
494C4#1684026811yescharacteristic
495C4#1684055811yescharacteristic
496C2—41no—
497C2—82no—
498C2—161no—
499C2—9761no—
500C2#2145581411yescharacteristic
501C2#2145895711yescharacteristic
502C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33679
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. #21456462C2semidirect
2. #21454695C2semidirect
3. #21448002C2semidirect
4. #21448000C2semidirect
5. #21447985C2semidirect
6. #21447974C2semidirect
7. #21447857C2semidirect
8. #21447852C2semidirect
9. #21447742C2semidirect
10. #21447692C2semidirect
11. #21447684C2semidirect
12. #21447668C2semidirect
13. D16×Dic61C2semidirect
14. #21447623C2semidirect
15. C61×#2186C2semidirect
254. #2186D61semidirect
310. C61#33679semidirect

Maximal subgroups

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

subgroupindex
#214564622
#214546952
#214480022
#214480002
#214479852
#214479742
#214478572
#214478522
#214477422
#214476922
#214476842
#214476682
D16×Dic612
#214476232
C61×#21862
#3367961