Finite Group Databasebeta

← #29836405

Subgroups — #29836405

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456462C211yesmaximal, characteristic, complemented
2#21456459C211yesmaximal, characteristic, complemented
3#21447985C211yesmaximal, characteristic, complemented
4#21447975C211yesmaximal, characteristic, complemented
5#21447965C211yesmaximal, characteristic, complemented
6#21447954C211yesmaximal, characteristic, complemented
7#21447934C211yesmaximal, characteristic, complemented
8#21447920C211yesmaximal, characteristic, complemented
9#21447905C211yesmaximal, characteristic, complemented
10#21447877C211yesmaximal, characteristic, complemented
11#21447876C211yesmaximal, characteristic, complemented
12#21447857C211yesmaximal, characteristic, complemented
13#21447853C211yesmaximal, characteristic, complemented
14#21447577C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16#16840385—21no—
17#16840365—21no—
18C2×#14519720C2211yescharacteristic
19C2×D976C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21C2×#14519717C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840163C2211yescharacteristic
24#16840147C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840125—21no—
27#16840123—21no—
28#16840013—21no—
29#16840010—21no—
30#16839998C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839871—21no—
33#16839865—21no—
34Q16×Dic61C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839844C2211yescharacteristic
38#16839717C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839588C2211yescharacteristic
41#16839587C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839562C2211yescharacteristic
51#16839562C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#367C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758201—21no—
60C2×D488C2311yescharacteristic
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63C2×#11758194D411yescharacteristic
64#14519776—21no—
65#14519773—21no—
66#14519772—41no—
67#14519768—21no—
68#14519766—21no—
69#14519763—41no—
70#14519761—41no—
71#14519759—41no—
72#14519759—41no—
73#14519757—41no—
74#14519751—21no—
75#14519751—21no—
76#14519749—41no—
77#14519748—41no—
78Q8×Dic61—21no—
79D4×Dic61—21no—
80#14519739—21no—
81#14519739—21no—
82#14519736D411yescharacteristic
83#14519720—41no—
84D976—41no—
85#14519718—41no—
86#14519717—41no—
87C2×#11758185C2311yescharacteristic
88C2×#11758185C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519708—21no—
93#14519701—21no—
94#14519699—21no—
95#14519699—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519687D411yescharacteristic
100#14519687D411yescharacteristic
101C8×Dic61C2311yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×D8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22⋊Q8×C61—21no—
107#121×C61—21no—
108SD32×C61—41no—
109C2×C16×C61—41no—
110C2×C16×C61C2311yescharacteristic
111C2×C16×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#103×C61—21no—
115#102×C61—21no—
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C22×Dic61—41no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×Dic122—41no—
130Q16×C61—21no—
131Q16×C61—21no—
132D8×C61—21no—
133D8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2411yescharacteristic
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61D811yescharacteristic
145C22⋊Dic61D811yescharacteristic
146#11758201—42no—
147#11758199—42no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158C4⋊Dic61D811yescharacteristic
159C4⋊Dic61D811yescharacteristic
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163C4×Dic61C2×D411yescharacteristic
164C4×Dic61C2×D411yescharacteristic
165#11758194—41no—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168D488—21no—
169D488—21no—
170D488—21no—
171D488—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C23×C61—41no—
177C23×C61C2×D811yescharacteristic
178C22×D61—41no—
179C22×D61—41no—
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—161no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C22⋊D61—161no—
193C22⋊D61—161no—
194C22⋊D61—161no—
195C22⋊D61—161no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—161no—
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203D244—41no—
204D244—41no—
205D244—41no—
206D244—41no—
207C4×D61—161no—
208C4×D61—161no—
209Dic122—161no—
210C8×C61—21no—
211C8×C61H(5,𝔽2)11yescharacteristic
212C8×C612−1+411yescharacteristic
213#92418—21no—
214#92418—21no—
215#33823—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61C22×D811yescharacteristic
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C4×C61—21no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61#53711yescharacteristic
234C4×C61#57511yescharacteristic
235Dic61—21no—
236Dic61—21no—
237Dic61—21no—
238Dic61—21no—
239Dic61—81no—
240Dic61—81no—
241#3176—611no—
242#3175—611no—
243#2229—611no—
244#2228—611no—
245#2226—611no—
246#2224—611no—
247#2220—611no—
248#2218—611no—
249#2217D6111yescharacteristic, complemented
250#2215—611no—
251#2210—611no—
252#2210—611no—
253#2207—611no—
254#2206—611no—
255#2151—611no—
256C2×C61—41no—
257C2×C61—161no—
258C2×C61#324511yescharacteristic
259C2×C61#341511yescharacteristic
260C2×C61#341611yescharacteristic
261D61—161no—
262D61—161no—
263#542—1221no—
264#540—1221no—
265C2×SD32—611no—
266C2×SD32C2×D6111yescharacteristic
267C2×D16—611no—
268C2×D16—611no—
269C2×M32—611no—
270C22×C16C2×D6111yescharacteristic
271#500—611no—
272#499—611no—
273#495—611no—
274#493—611no—
275#490—1221no—
276#489—1221no—
277#469—1221no—
278#468—1221no—
279#467C2×D6111yescharacteristic
280#466—611no—
281#466—611no—
282#466C2×D6111yescharacteristic
283#442—1221no—
284#440—1221no—
285C4×Q16—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289#405—611no—
290#367—611no—
291#367C2×D6111yescharacteristic
292#366—611no—
293#366—611no—
294C4⋊C16—611no—
295C4⋊C16—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
306C61#3382311yescharacteristic, complemented, Sylow
307C2×Q16C22×D6111yescharacteristic
308C2×SD16—1221no—
309C2×D8—611no—
310C2×D8—611no—
311C2×D8—1221no—
312C2×D8C22×D6111yescharacteristic
313C2×M16—611no—
314C22×C8C22×D6111yescharacteristic
315C4⋊Q8—1221no—
316#127—1221no—
317#126—2441no—
318#125—1221no—
319#124—1221no—
320#124—2441no—
321#123—2441no—
322#123—2441no—
323#123—2441no—
324C22⋊Q8—21no—
325C22⋊Q8—2441no—
326#121—21no—
327#121—1221no—
328#121—1221no—
329#121—2441no—
330#121—2441no—
331C4×Q8—1221no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335#117—611no—
336SD32—41no—
337SD32—2441no—
338D16—2441no—
339D16—2441no—
340M32—2441no—
341C2×C16—41no—
342C2×C16—611no—
343C2×C16—611no—
344C2×C16C22×D6111yescharacteristic
345C2×C16C22×D6111yescharacteristic
346#107—611no—
347#107—611no—
348#107—1221no—
349#107C22×D6111yescharacteristic
350#107C22×D6111yescharacteristic
351#106—1221no—
352#103—21no—
353#103—1221no—
354#102—21no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#102—1221no—
359#102—1221no—
360#97—611no—
361#97—611no—
362C4×C8—611no—
363C4×C8—611no—
364C2×Q8—21no—
365C2×Q8—2441no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4D4×D6111yescharacteristic
378Q16—21no—
379Q16—21no—
380SD16—2442no—
381D8—21no—
382D8—21no—
383D8—1221no—
384D8—1221no—
385D8—1221no—
386D8—1221no—
387D8—2442no—
388M16—2441no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8D4×D6111yescharacteristic
394C2×C8C23×D6111yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—611no—
399C4⋊C4—611no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—611no—
409C22⋊C4—611no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C42—611no—
418C42—611no—
419C42—2441no—
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—2441no—
426C23—2441no—
427C23D8×D6111yescharacteristic
428Q8—41no—
429Q8—41no—
430Q8—9761no—
431D4—41no—
432D4—41no—
433D4—161no—
434D4—9761no—
435D4—9761no—
436D4—9761no—
437D4—9761no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441D4—2441no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—9761no—
452C2×C4—9761no—
453C2×C4—9761no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4D8×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C8—21no—
459C8—1221no—
460C8—1221no—
461C8#1451982211yescharacteristic
462C8#1451982411yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22C2×D8×D6111yescharacteristic
476C4—21no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—1221no—
481C4—1221no—
482C4—1221no—
483C4—1221no—
484C4—4881no—
485C4—4881no—
486C4#1684033911yescharacteristic
487C4#1684055911yescharacteristic
488C2—41no—
489C2—161no—
490C2—9761no—
491C2—9761no—
492C2#2145712511yescharacteristic
493C2#2145895711yescharacteristic
494C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
hypercenter#2217D61
hypocenterC61#33823
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #21456462C2semidirect
2. #21456459C2semidirect
3. #21447985C2semidirect
4. #21447975C2semidirect
5. #21447965C2semidirect
6. #21447954C2semidirect
7. #21447934C2semidirect
8. #21447920C2semidirect
9. #21447905C2semidirect
10. #21447877C2semidirect
11. #21447876C2semidirect
12. #21447857C2semidirect
13. #21447853C2semidirect
14. #21447577C2semidirect
15. C61×#2217C2semidirect
249. #2217D61semidirect
306. C61#33823semidirect

Maximal subgroups

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

subgroupindex
#214564622
#214564592
#214479852
#214479752
#214479652
#214479542
#214479342
#214479202
#214479052
#214478772
#214478762
#214478572
#214478532
#214475772
C61×#22172
#3382361