Finite Group Databasebeta

← #29835570

Subgroups — #29835570

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456400C211yesmaximal, characteristic, complemented
2#21456393C211yesmaximal, characteristic
3#21448100C211yesmaximal, characteristic, complemented
4#21448073C211yesmaximal, characteristic, complemented
5#21448045C211yesmaximal, characteristic, complemented
6#21448041C211yesmaximal, characteristic, complemented
7#21447917C211yesmaximal, characteristic, complemented
8#21447915C211yesmaximal, characteristic, complemented
9#21447913C211yesmaximal, characteristic, complemented
10#21447674C211yesmaximal, characteristic, complemented
11SD32×Dic61C211yesmaximal, characteristic, complemented
12#21447670C211yesmaximal, characteristic, complemented
13#21447668C211yesmaximal, characteristic, complemented
14C2×C16×Dic61C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16#16840314—21no—
17#16840307—21no—
18C2×#14519722C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C22×#11758185C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840166C2211yescharacteristic
24#16840140C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840111—21no—
27#16840107—21no—
28#16840003C2211yescharacteristic
29#16840001—21no—
30#16839998C2211yescharacteristic
31#16839996—21no—
32Q16×Dic61C2211yescharacteristic
33#16839856C2211yescharacteristic
34#16839854—21no—
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839842—21no—
38C2×C8×Dic61C2211yescharacteristic
39#16839590C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839562C2211yescharacteristic
48C16×Dic61C2211yescharacteristic
49C16×Dic61C2211yescharacteristic
50C16×Dic61C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#367C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758224—41no—
60C2×Dic244C2311yescharacteristic
61C2×#11758202—21no—
62C2×D488C2311yescharacteristic
63C2×#11758199—21no—
64C22×#92418D411yescharacteristic
65#14519776—21no—
66#14519776—21no—
67#14519775—21no—
68#14519773—21no—
69#14519767—41no—
70#14519763—41no—
71#14519756—21no—
72#14519754—41no—
73#14519751—21no—
74#14519749—41no—
75Q8×Dic61—21no—
76#14519742—21no—
77#14519742—41no—
78D4×Dic61—21no—
79#14519739—21no—
80#14519737—41no—
81#14519733—41no—
82C2×C4×Dic61D411yescharacteristic
83#14519722—22no—
84#14519721—22no—
85#14519720—22no—
86C2×#11758185—22no—
87C2×#11758185C2311yescharacteristic
88C2×#11758185C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519710—21no—
93#14519702—21no—
94#14519701—21no—
95#14519700—21no—
96#14519699—21no—
97#14519698—21no—
98#14519697—21no—
99C8×Dic61D411yescharacteristic
100C8×Dic61D411yescharacteristic
101C8×Dic61C2311yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×D8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22⋊Q8×C61—21no—
107#121×C61—21no—
108SD32×C61—22no—
109C2×C16×C61—22no—
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—
123C22×Dic61—41no—
124C22×Dic61D811yescharacteristic
125C22×Dic61D811yescharacteristic
126#11758224—84no—
127C2×D244—21no—
128C2×C4×D61—41no—
129C2×Dic122—21no—
130C2×Dic122—41no—
131C2×Dic122—41no—
132Q16×C61—21no—
133Q16×C61—21no—
134D8×C61—21no—
135D8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2411yescharacteristic
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147#11758202—42no—
148#11758199—42no—
149#11758198—41no—
150#11758198—41no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61D811yescharacteristic
163C4×Dic61D811yescharacteristic
164C4×Dic61C2×D411yescharacteristic
165C4×Dic61C2×D411yescharacteristic
166C2×#92418—22no—
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169Dic244—21no—
170Dic244—21no—
171D488—21no—
172D488—21no—
173C16×C61—21no—
174C16×C61#4112yescomplemented
175#11758185—21no—
176#11758185#4112yescomplemented
177C23×C61—41no—
178C23×C61C2×D811yescharacteristic
179C22×D61—41no—
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—82no—
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—82no—
193C22⋊D61—82no—
194C22⋊D61—82no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—82no—
200C2×Dic61SD3214yescomplemented
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204C2×Dic61C2×D811yescharacteristic
205D244—41no—
206D244—41no—
207C4×D61—82no—
208Dic122—41no—
209Dic122—41no—
210Dic122—82no—
211Dic122—82no—
212C8×C61—21no—
213C8×C61C2×#4111yescharacteristic
214C8×C61C2×#4111yescharacteristic
215#92418—21no—
216#92418#13512yes—
217#33736—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C2×SD3212yes—
224C22×C61C22×D811yescharacteristic
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C4×C61—21no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#53211yescharacteristic
233C4×C61C2×#13511yescharacteristic
234Dic61—21no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238Dic61C2×SD3212yes—
239Dic61C2×SD3212yes—
240Dic61#50812yes—
241#3170—611no—
242#3169—611no—
243#2254—611no—
244#2248—611no—
245#2241—611no—
246#2240—611no—
247#2217—611no—
248#2217—611no—
249#2217—611no—
250#2217D6111yescharacteristic, complemented
251C4×SD32—611no—
252C4×SD32—611no—
253C4×SD32—611no—
254C4×SD32—611no—
255C2×C4×C16—611no—
256C2×C61—22no—
257C2×C61—161no—
258C2×C61#316911yescharacteristic
259C2×C61C22×SD3211yescharacteristic
260C2×C61C2×#50811yescharacteristic
261D61—161no—
262#533—1221no—
263#532—1221no—
264C2×SD32—611no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×SD32C2×D6111yescharacteristic
268C22×C16—611no—
269C22×C16C2×D6111yescharacteristic
270#500—611no—
271#500—611no—
272#494—611no—
273#493—611no—
274#487—1221no—
275#486—1221no—
276#467—611no—
277#467—1221no—
278#467C2×D6111yescharacteristic
279#466—611no—
280#466—1221no—
281#466C2×D6111yescharacteristic
282C4×Q16—611no—
283C4×Q16—611no—
284C4×Q16—1221no—
285C4×D8—611no—
286C4×D8—611no—
287C4×D8—1221no—
288C2×C4×C8—611no—
289#367—611no—
290#367—611no—
291#367—611no—
292#367C2×D6111yescharacteristic
293#358—611no—
294#358—611no—
295#358—611no—
296#358C2×D6111yescharacteristic
297#357—611no—
298#357—611no—
299#357—611no—
300#357C2×D6111yescharacteristic
301C4×C16—611no—
302C4×C16—611no—
303C4×C16—611no—
304C4×C16—611no—
305C61#3373611yescharacteristic, complemented, Sylow
306C2×#41—2441no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16C22×D6111yescharacteristic
310C2×D8—611no—
311C2×D8—1221no—
312C2×D8C22×D6111yescharacteristic
313C22×C8—611no—
314C22×C8C22×D6111yescharacteristic
315C4⋊Q8—1221no—
316C4⋊Q8—1221no—
317C4⋊Q8—1221no—
318#127—1221no—
319#125—2441no—
320#124—2441no—
321C22⋊Q8—21no—
322C22⋊Q8—1221no—
323C22⋊Q8—2441no—
324#121—21no—
325#121—1221no—
326#121—2441no—
327C4×Q8—1221no—
328C4×Q8—1221no—
329C4×Q8—2441no—
330C4×D4—1221no—
331C4×D4—1221no—
332C4×D4—2441no—
333#117—2441no—
334C2×C42—611no—
335SD32—22no—
336SD32—1222no—
337SD32—1222no—
338SD32—1222no—
339C2×C16—22no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16—1222no—
343C2×C16C22×D6111yescharacteristic
344C2×C16C22×D6111yescharacteristic
345#107—611no—
346#107—611no—
347#107—1221no—
348#107—1221no—
349#107C22×D6111yescharacteristic
350#107C22×D6111yescharacteristic
351#103—21no—
352#103—1221no—
353#103—1221no—
354#103—1221no—
355#102—21no—
356#102—1221no—
357#102—1221no—
358#102—1221no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—611no—
362C4×C8—611no—
363#41—4884no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×Q8—2441no—
367C2×Q8—2441no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—1221no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4D4×D6111yescharacteristic
379Q16—21no—
380Q16—21no—
381Q16—1221no—
382Q16—1221no—
383Q16—2442no—
384D8—21no—
385D8—21no—
386D8—1221no—
387D8—1221no—
388D8—2442no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—1222no—
394C2×C8D4×D6111yescharacteristic
395C2×C8C23×D6111yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—611no—
415C42—611no—
416C42—611no—
417C42—611no—
418C42—2441no—
419C42—2441no—
420C16—21no—
421C16—612no—
422C16—1221no—
423C16#1175822412yescomplemented
424C23—41no—
425C23—2441no—
426C23D8×D6111yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—4882no—
430Q8—4882no—
431Q8—2441no—
432Q8—2441no—
433D4—41no—
434D4—41no—
435D4—82no—
436D4—4882no—
437D4—4882no—
438D4—4882no—
439D4—2441no—
440D4—2441no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—614no—
450C2×C4—611no—
451C2×C4—1222no—
452C2×C4—4882no—
453C2×C4—4882no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4D8×D6111yescharacteristic
458C2×C4C2×D4×D6111yescharacteristic
459C8—21no—
460C8—612no—
461C8—1221no—
462C8#41×D6111yescharacteristic
463C8C2×#1175822411yescharacteristic
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22SD32×D6112yes—
473C22C2×D8×D6111yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—612no—
479C4—612no—
480C4—612no—
481C4—1221no—
482C4—4881no—
483C4—4881no—
484C4—4881no—
485C4#1684030711yescharacteristic
486C4#135×D6111yescharacteristic
487C2—22no—
488C2—161no—
489C2—9761no—
490C2#2145639311yescharacteristic
491C2C2×SD32×D6111yescharacteristic
492C2#508×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
prime radical#21456393C2
hypercenter#2217D61
hypocenterC61#33736
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. #21456400C2semidirect
3. #21448100C2semidirect
4. #21448073C2semidirect
5. #21448045C2semidirect
6. #21448041C2semidirect
7. #21447917C2semidirect
8. #21447915C2semidirect
9. #21447913C2semidirect
10. #21447674C2semidirect
11. SD32×Dic61C2semidirect
12. #21447670C2semidirect
13. #21447668C2semidirect
14. C2×C16×Dic61C2semidirect
15. C61×#2217C2semidirect
174. C16×C61#41semidirect
176. #11758185#41semidirect
200. C2×Dic61SD32semidirect
250. #2217D61semidirect
305. C61#33736semidirect
423. C16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214564002
#214563932
#214481002
#214480732
#214480452
#214480412
#214479172
#214479152
#214479132
#214476742
SD32×Dic612
#214476702
#214476682
C2×C16×Dic612
C61×#22172
#3373661