Finite Group Databasebeta

← #29835196

Subgroups — #29835196

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456455C211yesmaximal, characteristic, complemented
2#21454689C211yesmaximal, characteristic, complemented
3#21448020C211yesmaximal, characteristic, complemented
4#21447998C211yesmaximal, characteristic, complemented
5#21447972C211yesmaximal, characteristic, complemented
6#21447956C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447756C211yesmaximal, characteristic, complemented
9#21447739C211yesmaximal, characteristic
10#21447684C211yesmaximal, characteristic, complemented
11#21447680C211yesmaximal, characteristic, complemented
12#21447675C211yesmaximal, characteristic, complemented
13D16×Dic61C211yesmaximal, characteristic, complemented
14#21447625C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17#16840381—21no—
18#16840241—21no—
19C2×#14519723C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21#16840167C2211yescharacteristic
22#16840121—21no—
23#16840005—21no—
24#16840000C2211yescharacteristic
25#16839995C2211yescharacteristic
26#16839863—21no—
27#16839862—21no—
28#16839859—21no—
29C4×Dic244C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31#16839856C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34#107×D61C2211yescharacteristic
35#16839760—21no—
36#16839716C2211yescharacteristic
37#16839588C2211yescharacteristic
38#16839588C2211yescharacteristic
39#16839587C2211yescharacteristic
40#16839587C2211yescharacteristic
41C4⋊#11758185C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C2×D16×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758224—21no—
60#14519796—22no—
61#14519796—22no—
62C2×Dic244C2311yescharacteristic
63C2×#11758200—21no—
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519777—21no—
67#14519770—41no—
68#14519763—41no—
69#14519761—41no—
70#14519754—41no—
71#14519753—21no—
72#14519748—21no—
73Q8×Dic61—21no—
74C4×Dic122—21no—
75#14519742—21no—
76D4×Dic61—21no—
77D4×Dic61—21no—
78C4×C22⋊D61—41no—
79#14519737—41no—
80#14519734D411yescharacteristic
81#14519733—41no—
82C2×#11758196—41no—
83C4⋊C4×D61—21no—
84#14519723—22no—
85#14519721—22no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711—21no—
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519710—21no—
93#14519702—21no—
94#14519701—21no—
95#14519701—21no—
96#14519700—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519688D411yescharacteristic
101#14519687D411yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×SD16×C61—21no—
105C2×D8×C61C2311yescharacteristic
106C2×D8×C61C2311yescharacteristic
107#124×C61—21no—
108C4×D4×C61—21no—
109SD32×C61—22no—
110D16×C61—22no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C22×C4×C61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125#11758224—24no—
126#11758224—42no—
127C2×C4×D61—41no—
128C2×C4×D61C2×D411yescharacteristic
129C2×Dic122—21no—
130C2×Dic122—41no—
131Q16×C61—21no—
132Q16×C61—21no—
133SD16×C61—42no—
134D8×C61—21no—
135D8×C61—21no—
136D8×C61—21no—
137D8×C61#4112yes—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2411yescharacteristic
140C4⋊C4×C61—21no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61C2×D411yescharacteristic
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148#11758200—24no—
149#11758198—41no—
150#11758198D811yescharacteristic
151#11758198D811yescharacteristic
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160#11758196—41no—
161#11758196D811yescharacteristic
162#11758196D811yescharacteristic
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169Dic244—21no—
170Dic244#4112yes—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61—41no—
180C22×D61C2×D811yescharacteristic
181Q8×C61—41no—
182Q8×C61—41no—
183D4×C61—22no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—161no—
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61C2×D811yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C22⋊D61—44no—
194C22⋊D61—161no—
195C2×Dic61—44no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—161no—
201C2×Dic61—161no—
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204C2×Dic61C2×D811yescharacteristic
205C4×D61—21no—
206C4×D61—21no—
207C4×D61—161no—
208Dic122—22no—
209Dic122—41no—
210Dic122—161no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33684—611nomaximal, Sylow
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#52511yescharacteristic
232C4×C61#57511yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—42no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240#3175—611no—
241#2951—611no—
242#2236—611no—
243#2232—611no—
244#2227—611no—
245#2224—611no—
246#2211—611no—
247#2210D6111yescharacteristic, complemented
248#2189—611no—
249#2186—611no—
250#2175—611no—
251#2175—611no—
252C4×Q32—611no—
253C4×D16—611no—
254#2164—611no—
255C2×C61—82no—
256C2×C61—161no—
257C2×C61#310011yescharacteristic
258C2×C61#341511yescharacteristic
259C2×C61#341711yescharacteristic
260D61—41no—
261C2×#135—611no—
262#542—1221no—
263#520—1221no—
264C2×Q32—611no—
265C2×SD32—611no—
266C2×SD32C2×D6111yescharacteristic
267C2×D16C2×D6111yescharacteristic
268#500—611no—
269#495C2×D6111yescharacteristic
270#489—1221no—
271#468—1221no—
272#467—611no—
273#466—611no—
274#440—1221no—
275#440—1221no—
276#440—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×D8—611no—
281C4×D8—611no—
282C4×D8C2×D6111yescharacteristic
283C2×#107—611no—
284#417—1221no—
285#405—611no—
286#366—611no—
287#366—611no—
288#366—611no—
289#366—611no—
290C4⋊C16—611no—
291C4⋊C16C2×D6111yescharacteristic
292#358—611no—
293#358—611no—
294#358—611no—
295#358C2×D6111yescharacteristic
296#357—611no—
297#357—611no—
298#357—611no—
299#357C2×D6111yescharacteristic
300C22⋊C16—611no—
301C22⋊C16—611no—
302#346—611no—
303C4×C16—611no—
304C61#3368411yescharacteristic, complemented, Sylow
305C2×#41—1221no—
306#135—1222no—
307#135—1222no—
308C2×Q16—611no—
309C2×Q16C22×D6111yescharacteristic
310C2×SD16—21no—
311C2×SD16—1221no—
312C2×D8C22×D6111yescharacteristic
313C2×D8C22×D6111yescharacteristic
314C2×M16—611no—
315C22×C8—611no—
316C4⋊Q8—1221no—
317#126—2441no—
318#124—21no—
319#124—2441no—
320#123—2441no—
321C22⋊Q8—1221no—
322C22⋊Q8—2441no—
323#121—1221no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—2441no—
331C4×D4—2441no—
332#117—611no—
333#117—2441no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—2441no—
336Q32—1222no—
337SD32—22no—
338SD32—1222no—
339D16—22no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16C22×D6111yescharacteristic
343C2×C16C22×D6111yescharacteristic
344#107—611no—
345#107—611no—
346#107—611no—
347#107—1221no—
348#107—1221no—
349#107C22×D6111yescharacteristic
350#103—1221no—
351#103—1221no—
352#103—1221no—
353#103—1221no—
354#102—21no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#97—611no—
359#97—611no—
360C4×C8—611no—
361C4×C8C22×D6111yescharacteristic
362#41—1224no—
363#41—2442no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×Q8—2441no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—2441no—
370C2×D4—2441no—
371C22×C4—41no—
372C22×C4—611no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2441no—
377Q16—21no—
378Q16—21no—
379Q16—612no—
380Q16—1221no—
381SD16—42no—
382SD16—1224no—
383D8—21no—
384D8—21no—
385D8—21no—
386D8#1175822412yes—
387M16—2441no—
388C2×C8—611no—
389C2×C8—611no—
390C2×C8—1221no—
391C2×C8—1221no—
392C2×C8D4×D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—611no—
396C4⋊C4—611no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—611no—
410C22⋊C4—611no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C42—611no—
416C42—2441no—
417C42—2441no—
418C42—2441no—
419C42D4×D6111yescharacteristic
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—41no—
426C23—611no—
427Q8—41no—
428Q8—41no—
429Q8—1222no—
430Q8—9761no—
431Q8—2441no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—9761no—
437D4—2444no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—161no—
441C2×C4—611no—
442C2×C4—611no—
443C2×C4—611no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—9761no—
447C2×C4—9761no—
448C2×C4—9761no—
449C2×C4—2444no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4D8×D6111yescharacteristic
455C2×C4D8×D6111yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C8—21no—
458C8—1221no—
459C8—1221no—
460C8#41×D6111yescharacteristic
461C8C2×#1175822411yescharacteristic
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—1221no—
469C22—1221no—
470C22—1221no—
471C22C2×D8×D6111yescharacteristic
472C4—21no—
473C4—21no—
474C4—81no—
475C4—81no—
476C4—1221no—
477C4—1221no—
478C4—1221no—
479C4—4881no—
480C4—4881no—
481C4—4881no—
482C4—2442no—
483C4#1684027211yescharacteristic
484C4#1684055811yescharacteristic
485C2—82no—
486C2—161no—
487C2—2441no—
488C2#2145581811yescharacteristic
489C2#2145895911yescharacteristic
490C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radical#21447739C2
hypercenter#2210D61
hypocenterC61#33684
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. #21456455C2semidirect
2. #21454689C2semidirect
3. #21448020C2semidirect
4. #21447998C2semidirect
5. #21447972C2semidirect
6. #21447956C2semidirect
7. #21447878C2semidirect
8. #21447756C2semidirect
10. #21447684C2semidirect
11. #21447680C2semidirect
12. #21447675C2semidirect
13. D16×Dic61C2semidirect
14. #21447625C2semidirect
15. C61×#2210C2semidirect
247. #2210D61semidirect
304. C61#33684semidirect

Maximal subgroups

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

subgroupindex
#214564552
#214546892
#214480202
#214479982
#214479722
#214479562
#214478782
#214477562
#214477392
#214476842
#214476802
#214476752
D16×Dic612
#214476252
C61×#22102
#3368461