Finite Group Databasebeta

← #29836406

Subgroups — #29836406

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#21456470C211yesmaximal, characteristic, complemented
2#21456454C211yesmaximal, characteristic, complemented
3#21447980C211yesmaximal, characteristic, complemented
4#21447965C211yesmaximal, characteristic, complemented
5#21447955C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447921C211yesmaximal, characteristic, complemented
8#21447916C211yesmaximal, characteristic, complemented
9#21447904C211yesmaximal, characteristic, complemented
10#21447873C211yesmaximal, characteristic, complemented
11#21447873C211yesmaximal, characteristic, complemented
12#21447860C211yesmaximal, characteristic, complemented
13#21447854C211yesmaximal, characteristic, complemented
14#21447576C211yesmaximal, characteristic, complemented
15C61×#2228C211yesmaximal, characteristic, complemented
16#16840377—21no—
17#16840368—21no—
18C2×#14519722C2211yescharacteristic
19C2×#14519722C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519716C2211yescharacteristic
23C2×C16×D61C2211yescharacteristic
24#16840166C2211yescharacteristic
25#16840145C2211yescharacteristic
26#16840139C2211yescharacteristic
27#16840128—21no—
28#16840120—21no—
29#16840016—21no—
30#16840007—21no—
31#16840004C2211yescharacteristic
32#16839999C2211yescharacteristic
33#16839999C2211yescharacteristic
34#16839995C2211yescharacteristic
35#16839873—21no—
36#16839862—21no—
37C4×Dic244C2211yescharacteristic
38#16839844C2211yescharacteristic
39#16839844C2211yescharacteristic
40#16839716C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839587C2211yescharacteristic
43C4⋊#11758185C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839564C2211yescharacteristic
50#16839562C2211yescharacteristic
51#16839562C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758200—21no—
61C2×D488—21no—
62C2×D488C2311yescharacteristic
63C2×D488C2311yescharacteristic
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519774—21no—
68#14519770—41no—
69#14519764—21no—
70#14519763—41no—
71#14519762—41no—
72#14519762—41no—
73#14519760—41no—
74#14519757—21no—
75#14519756—41no—
76#14519752—21no—
77#14519752—21no—
78#14519752—41no—
79#14519752—41no—
80#14519748—21no—
81C4×Dic122—21no—
82#14519739—21no—
83#14519739—21no—
84#14519734D411yescharacteristic
85#14519722—41no—
86#14519722—41no—
87#14519718—41no—
88#14519718—41no—
89#14519716—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519710—21no—
95#14519702—21no—
96#14519700—21no—
97#14519698—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519688D411yescharacteristic
103#14519687D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106#125×C61—21no—
107C4×D4×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#106×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×D4×C61—21no—
117C22×C4×C61—41no—
118C2×C22⋊D61—41no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—41no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C2×C4×D61—41no—
126C2×C4×D61—41no—
127C2×C4×D61C2×D411yescharacteristic
128C2×Dic122—21no—
129C2×Dic122—41no—
130D8×C61—21no—
131D8×C61—21no—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2411yescharacteristic
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C42×C61C2×D411yescharacteristic
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144#11758200—42no—
145#11758198—41no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149#11758198—41no—
150#11758198D811yescharacteristic
151#11758198D811yescharacteristic
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154#11758196—41no—
155#11758196—41no—
156#11758196D811yescharacteristic
157#11758196D811yescharacteristic
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244—21no—
164D488—21no—
165D488—21no—
166D488—21no—
167D488—21no—
168D488—42no—
169#11758189—41no—
170C8×D61—21no—
171C8×D61—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C23×C61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61C2×D811yescharacteristic
180D4×C61—41no—
181D4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—161no—
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61C2×D811yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C22⋊D61—161no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61C2×D811yescharacteristic
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195D244—41no—
196D244—41no—
197D244—41no—
198D244—41no—
199D244—161no—
200D244—161no—
201D244—161no—
202D244—161no—
203C4×D61—21no—
204C4×D61—21no—
205C4×D61—161no—
206C4×D61—161no—
207Dic122—41no—
208Dic122—41no—
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—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61C22×D811yescharacteristic
220C2×D61—21no—
221C2×D61—21no—
222C2×D61—21no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61#53711yescharacteristic
235C4×C61#57511yescharacteristic
236Dic61—21no—
237Dic61—21no—
238Dic61—21no—
239Dic61—81no—
240Dic61—81no—
241#3176—611no—
242#3175—611no—
243#2229—611no—
244#2228D6111yescharacteristic, complemented
245#2226—611no—
246#2224—611no—
247#2220—611no—
248#2218—611no—
249#2217—611no—
250#2215—611no—
251#2210—611no—
252#2210—611no—
253#2207—611no—
254#2206—611no—
255#2151—611no—
256C2×C61—161no—
257C2×C61#324511yescharacteristic
258C2×C61#341511yescharacteristic
259C2×C61#341611yescharacteristic
260D61—41no—
261D61—161no—
262D61—161no—
263#542—1221no—
264#540—1221no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×D16—611no—
268C2×D16—611no—
269C2×M32—611no—
270C22×C16—611no—
271#500—611no—
272#499C2×D6111yescharacteristic
273#495—611no—
274#493—611no—
275#490—1221no—
276#489—1221no—
277#469—1221no—
278#468—1221no—
279#467—611no—
280#466—611no—
281#466—611no—
282#466—611no—
283#442—1221no—
284#440—1221no—
285C4×Q16—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8C2×D6111yescharacteristic
289#405—611no—
290#367—611no—
291#367C2×D6111yescharacteristic
292#366—611no—
293#366C2×D6111yescharacteristic
294C4⋊C16—611no—
295C4⋊C16—611no—
296C4⋊C16—611no—
297C4⋊C16C2×D6111yescharacteristic
298#358—611no—
299#358—611no—
300#357—611no—
301#357—611no—
302#357—611no—
303#357—611no—
304#357C2×D6111yescharacteristic
305#357C2×D6111yescharacteristic
306C61#3382311yescharacteristic, complemented, Sylow
307C2×Q16—611no—
308C2×SD16—1221no—
309C2×D8—611no—
310C2×D8—611no—
311C2×D8—1221no—
312C2×D8C22×D6111yescharacteristic
313C2×M16—611no—
314C22×C8—611no—
315C4⋊Q8—1221no—
316#127—1221no—
317#126—2441no—
318#125—21no—
319#124—1221no—
320#124—2441no—
321#123—2441no—
322#123—2441no—
323#123—2441no—
324C22⋊Q8—1221no—
325C22⋊Q8—2441no—
326#121—1221no—
327#121—1221no—
328#121—1221no—
329#121—2441no—
330#121—2441no—
331C4×Q8—1221no—
332C4×D4—21no—
333C4×D4—1221no—
334C4×D4—1221no—
335#117—611no—
336SD32—2441no—
337SD32—2441no—
338D16—2441no—
339D16—2441no—
340M32—2441no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16—2441no—
344C2×C16C22×D6111yescharacteristic
345C2×C16C22×D6111yescharacteristic
346#107—611no—
347#107—1221no—
348#107C22×D6111yescharacteristic
349#107C22×D6111yescharacteristic
350#107C22×D6111yescharacteristic
351#106—21no—
352#103—1221no—
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×C8C22×D6111yescharacteristic
364C2×Q8—1221no—
365C2×Q8—2441no—
366C2×D4—21no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C22×C4—41no—
375C22×C4—611no—
376C22×C4—2441no—
377C22×C4—2441no—
378Q16—1221no—
379Q16—1221no—
380SD16—2442no—
381D8—21no—
382D8—21no—
383D8—1221no—
384D8—1221no—
385D8—1221no—
386D8—1221no—
387D8—2442no—
388M16—2441no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8—1221no—
392C2×C8—1221no—
393C2×C8D4×D6111yescharacteristic
394C2×C8C23×D6111yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—611no—
401C4⋊C4—611no—
402C4⋊C4—1221no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—611no—
408C22⋊C4—611no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C42—611no—
418C42—2441no—
419C42D4×D6111yescharacteristic
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—611no—
426C23—2441no—
427C23—2441no—
428Q8—9761no—
429Q8—2441no—
430Q8—2441no—
431D4—41no—
432D4—41no—
433D4—9761no—
434D4—9761no—
435D4—9761no—
436D4—9761no—
437D4—9761no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441D4—2441no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—161no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—9761no—
452C2×C4—9761no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4D8×D6111yescharacteristic
456C2×C4D8×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C8—21no—
459C8—1221no—
460C8—1221no—
461C8#1451982311yescharacteristic
462C8#1451982511yescharacteristic
463C22—81no—
464C22—81no—
465C22—81no—
466C22—1221no—
467C22—1221no—
468C22—1221no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22C2×D8×D6111yescharacteristic
476C4—21no—
477C4—21no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—1221no—
482C4—1221no—
483C4—1221no—
484C4—4881no—
485C4—4881no—
486C4#1684034211yescharacteristic
487C4#1684056111yescharacteristic
488C2—161no—
489C2—9761no—
490C2—9761no—
491C2—2441no—
492C2#2145712811yescharacteristic
493C2#2145895911yescharacteristic
494C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2228C2
socleC22×C61C22×D8
hypercenter#2228D61
hypocenterC61#33823
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #21456470C2semidirect
2. #21456454C2semidirect
3. #21447980C2semidirect
4. #21447965C2semidirect
5. #21447955C2semidirect
6. #21447933C2semidirect
7. #21447921C2semidirect
8. #21447916C2semidirect
9. #21447904C2semidirect
10. #21447873C2semidirect
11. #21447873C2semidirect
12. #21447860C2semidirect
13. #21447854C2semidirect
14. #21447576C2semidirect
15. C61×#2228C2semidirect
244. #2228D61semidirect
306. C61#33823semidirect

Maximal subgroups

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

subgroupindex
#214564702
#214564542
#214479802
#214479652
#214479552
#214479332
#214479212
#214479162
#214479042
#214478732
#214478602
#214478542
#214475762
C61×#22282
#3382361