Finite Group Databasebeta

← #29834561

Subgroups — #29834561

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2C2×#16840178C211yesmaximal, characteristic, complemented
3#21455484C211yesmaximal, characteristic
4#21455469C211yesmaximal, characteristic, complemented
5#21447930C211yesmaximal, characteristic, complemented
6#21447929C211yesmaximal, characteristic, complemented
7#21447928C211yesmaximal, characteristic, complemented
8#21447925C211yesmaximal, characteristic, complemented
9#21447914C211yesmaximal, characteristic, complemented
10#21447910C211yesmaximal, characteristic, complemented
11#21447909C211yesmaximal, characteristic, complemented
12#21447606C211yesmaximal, characteristic, complemented
13#21447523C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840341—21no—
20#16840335—21no—
21#16840201—24no—
22C2×Dic488C2211yescharacteristic
23C2×#14519723C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25#16840178—24no—
26C2×#14519717C2211yescharacteristic
27C2×#14519716C2211yescharacteristic
28C2×#14519716C2211yescharacteristic
29C2×C16×D61C2211yescharacteristic
30C2×C16×D61C2211yescharacteristic
31#16840019—21no—
32#16840018—21no—
33#16840007—21no—
34#16840006—21no—
35#16840004C2211yescharacteristic
36#16840004C2211yescharacteristic
37#16840003C2211yescharacteristic
38#16840003C2211yescharacteristic
39#16840000C2211yescharacteristic
40#16839995C2211yescharacteristic
41#16839799C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839769—21no—
44#16839760—21no—
45#16839589C2211yescharacteristic
46#16839588C2211yescharacteristic
47#16839587C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839567C2211yescharacteristic
50#16839566C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839565C2211yescharacteristic
53#16839563C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C22×C16×C61C2211yescharacteristic
56C61×#467C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#367C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×#357C2211yescharacteristic
61C2×#11758222D411yescharacteristic
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64#14519796—22no—
65#14519796—22no—
66C2×Dic244C2311yescharacteristic
67C2×Dic244C2311yescharacteristic
68C2×#11758202—21no—
69Q16×D61—22no—
70Q16×D61—22no—
71C2×#11758200—21no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194D411yescharacteristic
75C2×#11758189D411yescharacteristic
76C2×#11758189D411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519776—41no—
80#14519763—41no—
81#14519762—41no—
82#14519759—41no—
83#14519757—21no—
84#14519757—21no—
85#14519756—21no—
86#14519756—21no—
87#14519754—41no—
88#14519753—21no—
89#14519750—41no—
90#14519749—41no—
91#14519748—21no—
92#14519748—41no—
93#14519735—21no—
94C4⋊C4×D61—21no—
95Dic488—22no—
96#14519723—22no—
97#14519721—22no—
98#14519717—22no—
99#14519716—22no—
100#14519716D414yescomplemented
101C2×#11758185C2311yescharacteristic
102C2×#11758185C2311yescharacteristic
103C16×D61—22no—
104C16×D61—22no—
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519710—21no—
108#14519708—21no—
109#14519702—21no—
110#14519702—21no—
111#14519701—21no—
112#14519700—21no—
113#14519700—21no—
114#14519698—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121C2×C16×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C2×Q8×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—41no—
131C22×C4×C61C2×D411yescharacteristic
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61D811yescharacteristic
135C2×C22⋊D61D811yescharacteristic
136C22×Dic61—41no—
137Q8×D61—42no—
138Q8×D61—42no—
139#11758224—42no—
140#11758224—42no—
141#11758222—22no—
142#11758222—22no—
143C2×D244D811yescharacteristic
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—41no—
151C2×Dic122D811yescharacteristic
152Q16×C61—21no—
153Q16×C61—21no—
154D8×C61—21no—
155D8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2411yescharacteristic
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊Dic61—41no—
166C22⋊Dic61—41no—
167C22⋊Dic61—41no—
168#11758202—42no—
169#11758200—42no—
170#11758198—41no—
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—41no—
177C4⋊Dic61—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183#11758194—22no—
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186Dic244—21no—
187Dic244—21no—
188Dic244—21no—
189Dic244—21no—
190#11758189—22no—
191#11758189—22no—
192C8×D61—21no—
193C8×D61—21no—
194C8×D61C2×D412yes—
195C8×D61#4112yes—
196C16×C61—21no—
197C16×C61C2×D412yes—
198#11758185—21no—
199#11758185C2×D412yes—
200C23×C61—41no—
201C23×C61C2×D811yescharacteristic
202C22×D61C2×D811yescharacteristic
203C22×D61C2×D811yescharacteristic
204Q8×C61—41no—
205Q8×C61—41no—
206D4×C61—41no—
207D4×C61—41no—
208D4×C61—161no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61C2×D811yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C22⋊D61—22no—
217C22⋊D61—22no—
218C22⋊D61—82no—
219C22⋊D61—161no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—82no—
225C2×Dic61C2×D811yescharacteristic
226C2×Dic61C2×D811yescharacteristic
227D244—22no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—82no—
231C4×D61—161no—
232C4×D61#12112yes—
233C4×D61#13512yes—
234Dic122—22no—
235Dic122—41no—
236Dic122—41no—
237Dic122—41no—
238Dic122—41no—
239Dic122—82no—
240C8×C61—21no—
241C8×C61C22×D411yescharacteristic
242C8×C61C2×#4111yescharacteristic
243#92418—21no—
244#92418#12112yes—
245#33613—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61C22×D811yescharacteristic
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61#46612yes—
259C4×C61—21no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C2×#12111yescharacteristic
264C4×C61C2×#13511yescharacteristic
265Dic61—21no—
266Dic61—81no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270Dic61#46612yes—
271C2×#510—611no—
272C2×#504—611no—
273#3061—611no—
274#3059—611no—
275#2220—611no—
276#2220—611no—
277#2219—611no—
278#2219—611no—
279#2217—611no—
280#2217D6111yescharacteristic, complemented
281#2216—611no—
282#2216—611no—
283#2159—611no—
284#2138—611no—
285C2×#358—611no—
286C2×C61—41no—
287C2×C61—161no—
288C2×C61C2×#46611yescharacteristic
289C2×C61#341611yescharacteristic
290C2×C61#341711yescharacteristic
291D61—22no—
292D61—41no—
293C2×#135—611no—
294C22×Q16—611no—
295C2×#131—611no—
296#537—1221no—
297#536—1221no—
298#510—1224no—
299C2×Q32—611no—
300C2×Q32—611no—
301C2×SD32—611no—
302C2×SD32C2×D6111yescharacteristic
303#504—1224no—
304C2×M32—611no—
305C2×M32—611no—
306C2×M32—611no—
307C22×C16—611no—
308C22×C16—611no—
309C22×C16C2×D6111yescharacteristic
310#470—1221no—
311#470—1221no—
312#468—1221no—
313#468—1221no—
314#467—611no—
315#467—611no—
316#467—611no—
317#467—611no—
318#467—611no—
319#467C2×D6111yescharacteristic
320#466—611no—
321#466C2×D6111yescharacteristic
322#427—611no—
323C2×#107—611no—
324#419—1221no—
325#417—1221no—
326#367—611no—
327#367C2×D6111yescharacteristic
328#366—611no—
329#366—611no—
330#358—611no—
331#358—611no—
332#358—611no—
333#358—611no—
334#358—611no—
335#358C2×D6111yescharacteristic
336#357—611no—
337#357C2×D6111yescharacteristic
338C61#3361311yescharacteristic, complemented, Sylow
339C2×#41—611no—
340C2×#41—1221no—
341C22×Q8—1221no—
342#135—1222no—
343#135—1222no—
344C2×Q16—611no—
345C2×Q16—611no—
346C2×Q16—1222no—
347C2×Q16—1222no—
348C2×Q16—1221no—
349C2×Q16C22×D6111yescharacteristic
350C2×SD16—1221no—
351C2×D8C22×D6111yescharacteristic
352#131—1222no—
353#131—1222no—
354C2×M16—611no—
355C2×M16—611no—
356C2×M16—611no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8C22×D6111yescharacteristic
360C4⋊Q8—2441no—
361#124—2441no—
362#123—2441no—
363#123—2441no—
364C22⋊Q8—21no—
365C22⋊Q8—1221no—
366C22⋊Q8—1221no—
367C22⋊Q8—1221no—
368C22⋊Q8—1221no—
369C22⋊Q8—1221no—
370C22⋊Q8—2441no—
371#121—21no—
372#121—1221no—
373#121—2441no—
374#121—2441no—
375#121—2441no—
376#117—1221no—
377C2×C4⋊C4—1221no—
378Q32—1222no—
379Q32—1222no—
380SD32—22no—
381SD32—1222no—
382M32—614no—
383M32—1222no—
384M32—1222no—
385C2×C16—22no—
386C2×C16—611no—
387C2×C16—611no—
388C2×C16—1222no—
389C2×C16—1222no—
390C2×C16C22×D6111yescharacteristic
391C2×C16C22×D6111yescharacteristic
392#107—611no—
393#107—611no—
394#107—1221no—
395#107C22×D6111yescharacteristic
396#107C22×D6111yescharacteristic
397#106—1221no—
398#103—21no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#102—21no—
405#102—1221no—
406#41—1222no—
407#41—1222no—
408#41—2442no—
409#41—2442no—
410C2×Q8—21no—
411C2×Q8—611no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—2441no—
415C2×Q8—2442no—
416C2×Q8—2442no—
417C2×D4—21no—
418C2×D4—41no—
419C2×D4—611no—
420C2×D4—611no—
421C2×D4—611no—
422C2×D4—2441no—
423C2×D4—2441no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—2441no—
427C22×C4—2441no—
428C22×C4—2441no—
429C22×C4D4×D6111yescharacteristic
430Q16—21no—
431Q16—21no—
432Q16—1221no—
433Q16—1221no—
434Q16—1221no—
435Q16—1221no—
436Q16—2442no—
437SD16—2442no—
438D8—21no—
439D8—21no—
440M16—1222no—
441M16—1222no—
442M16—1222no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—611no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8D4×D6111yescharacteristic
452C2×C8C23×D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—2441no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—2441no—
474C16—21no—
475C16—612no—
476C16—1221no—
477C16D4×D6112yes—
478C23—41no—
479C23—611no—
480C23—611no—
481C23D8×D6111yescharacteristic
482Q8—41no—
483Q8—41no—
484Q8—1222no—
485Q8—4882no—
486Q8—2441no—
487Q8—2441no—
488Q8—2441no—
489Q8—2441no—
490D4—41no—
491D4—41no—
492D4—161no—
493D4—1222no—
494D4—1222no—
495D4—1222no—
496D4—9761no—
497D4—4882no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—611no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—9761no—
510C2×C4—4882no—
511C2×C4—4882no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4D8×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C8—21no—
519C8—612no—
520C8—1221no—
521C8C2×D4×D6111yescharacteristic
522C8#41×D6111yescharacteristic
523C22—21no—
524C22—21no—
525C22—21no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—612no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1221no—
534C22—1221no—
535C22C2×D8×D6111yescharacteristic
536C4—21no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—612no—
541C4—1221no—
542C4—4881no—
543C4—4881no—
544C4—4881no—
545C4—4881no—
546C4#121×D6111yescharacteristic
547C4#135×D6111yescharacteristic
548C2—41no—
549C2—161no—
550C2—1222no—
551C2—2441no—
552C2#466×D6111yescharacteristic
553C2#2145896111yescharacteristic
554C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
prime radical#21455484C2
hypercenter#2217D61
hypocenterC61#33613
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. C2×#16840201C2semidirect
2. C2×#16840178C2semidirect
4. #21455469C2semidirect
5. #21447930C2semidirect
6. #21447929C2semidirect
7. #21447928C2semidirect
8. #21447925C2semidirect
9. #21447914C2semidirect
10. #21447910C2semidirect
11. #21447909C2semidirect
12. #21447606C2semidirect
13. #21447523C2semidirect
14. #358×D61C2semidirect
15. C61×#2217C2semidirect
100. #14519716D4semidirect
280. #2217D61semidirect
338. C61#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C2×#168401782
#214554842
#214554692
#214479302
#214479292
#214479282
#214479252
#214479142
#214479102
#214479092
#214476062
#214475232
#358×D612
C61×#22172
#3361361