Finite Group Databasebeta

← #29834278

Subgroups — #29834278

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840194C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic, complemented
3#21456243C211yesmaximal, characteristic, complemented
4#21454590C211yesmaximal, characteristic, complemented
5#21447884C211yesmaximal, characteristic, complemented
6#21447880C211yesmaximal, characteristic, complemented
7#21447877C211yesmaximal, characteristic, complemented
8#21447873C211yesmaximal, characteristic, complemented
9#21447868C211yesmaximal, characteristic, complemented
10#21447863C211yesmaximal, characteristic, complemented
11#21447858C211yesmaximal, characteristic, complemented
12#21447853C211yesmaximal, characteristic, complemented
13C2×#16839580C211yesmaximal, characteristic, complemented
14#21447524C211yesmaximal, characteristic, complemented
15C61×#2138C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519799C2211yescharacteristic
18C2×#14519797—21no—
19#16840308—21no—
20#16840230—21no—
21#16840194—24no—
22#16840192—24no—
23C2×Dic488C2211yescharacteristic
24C2×#14519723C2211yescharacteristic
25C2×#14519722C2211yescharacteristic
26C2×#14519722C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×D976C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C22×#11758185C2211yescharacteristic
32#16840147C2211yescharacteristic
33#16840145C2211yescharacteristic
34#16840143—21no—
35#16840140C2211yescharacteristic
36#16840137C2211yescharacteristic
37#16840133—21no—
38#16839856C2211yescharacteristic
39#16839856C2211yescharacteristic
40#16839849—21no—
41#16839849—21no—
42#16839844C2211yescharacteristic
43#16839844C2211yescharacteristic
44#16839756—21no—
45C2×C8×Dic61C2211yescharacteristic
46#16839580C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839580C2211yescharacteristic
49#16839580C2211yescharacteristic
50#16839565C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839562C2211yescharacteristic
53#16839562C2211yescharacteristic
54C2×#135×C61C2211yescharacteristic
55C22×C16×C61C2211yescharacteristic
56C61×#427C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C61×#357C2211yescharacteristic
60C61×#357C2211yescharacteristic
61C2×#11758222—21no—
62C2×#11758222—21no—
63#14519799—22no—
64#14519799—22no—
65#14519799—22no—
66#14519799D414yescomplemented
67#14519797—44no—
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C22×#92418D411yescharacteristic
79#14519775—21no—
80#14519773—21no—
81#14519766—21no—
82#14519764—21no—
83#14519749—41no—
84#14519749—41no—
85#14519742—21no—
86#14519742—21no—
87#14519739—21no—
88#14519739—21no—
89C4×C22⋊D61—41no—
90C4×C22⋊D61—41no—
91#14519737—41no—
92#14519737—41no—
93#14519734—41no—
94C2×C4×Dic61D411yescharacteristic
95C42×D61—41no—
96Dic488—22no—
97#14519723—22no—
98#14519722—22no—
99#14519722—22no—
100#14519721—22no—
101#14519720—22no—
102D976—22no—
103#14519718—22no—
104C2×#11758185—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107#14519700—21no—
108#14519700—21no—
109#14519697—21no—
110#14519697—21no—
111C8×Dic61D411yescharacteristic
112C8×Dic61D411yescharacteristic
113C8×Dic61C2311yescharacteristic
114C8×Dic61C2311yescharacteristic
115C2×#41×C61—21no—
116#135×C61—22no—
117#135×C61—22no—
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122#117×C61—21no—
123C2×C16×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#106×C61—21no—
129#41×C61—42no—
130#41×C61—42no—
131C2×Q8×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—41no—
134C22×C4×C61—41no—
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61D811yescharacteristic
141C22×Dic61D811yescharacteristic
142#11758222—24no—
143#11758222—42no—
144#11758222—42no—
145#11758222—42no—
146C2×D244—21no—
147C2×D244—21no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154Q16×C61—21no—
155Q16×C61—21no—
156SD16×C61—42no—
157D8×C61—21no—
158D8×C61—21no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D412yes—
161C2×C8×C61#4112yes—
162C2×C8×C61C2411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C22⋊C4×C61—41no—
166C42×C61—41no—
167C22⋊Dic61—41no—
168C22⋊Dic61—41no—
169#11758202—42no—
170#11758201—42no—
171#11758200—42no—
172#11758199—42no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758196—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183C4×Dic61D811yescharacteristic
184C4×Dic61D811yescharacteristic
185C4×Dic61C2×D411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C2×#92418—22no—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244C2×D412yes—
194D488—21no—
195D488—21no—
196D488—21no—
197D488C2×D412yes—
198#11758190—24no—
199#11758190—42no—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C23×C61C2×D811yescharacteristic
206C22×D61—41no—
207C22×D61—41no—
208Q8×C61—41no—
209Q8×C61—41no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—82no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—82no—
217C2×C4×C61#12112yes—
218C2×C4×C61C2×D811yescharacteristic
219C2×C4×C61AGL(1,ℤ/8)12yes—
220C2×C4×C61C22×D411yescharacteristic
221C22⋊D61—44no—
222C22⋊D61—82no—
223C22⋊D61—82no—
224C22⋊D61—82no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C2×D811yescharacteristic
230C2×Dic61C2×D811yescharacteristic
231C2×Dic61C2×D811yescharacteristic
232C2×Dic61C2×D811yescharacteristic
233D244—22no—
234D244—41no—
235D244—41no—
236D244—41no—
237C4×D61—44no—
238C4×D61—82no—
239C4×D61—82no—
240C4×D61—82no—
241Dic122—22no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245C8×C61#12112yes—
246C8×C61C22×D411yescharacteristic
247C8×C61C2×#4111yescharacteristic
248#92418—21no—
249#92418—21no—
250#33579—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61#45912yes—
257C22×C61C22×D811yescharacteristic
258C2×D61—42no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61#45912yes—
268C4×C61C2×#12111yescharacteristic
269C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
270Dic61—21no—
271Dic61—21no—
272Dic61—21no—
273Dic61—42no—
274Dic61—81no—
275Dic61#50812yescomplemented
276C2×#508—611no—
277C2×#508—611no—
278#3147—611no—
279C4×#135—611no—
280#2211—611no—
281#2211—611no—
282#2210—611no—
283#2210—611no—
284#2209—611no—
285#2208—611no—
286#2207—611no—
287#2206—611no—
288C2×C4⋊C16—611no—
289#2138—611no—
290#2138D6111yescharacteristic, complemented
291C2×C61—22no—
292C2×C61—161no—
293C2×C61C2×#45911yescharacteristic
294C2×C61C2×#50811yescharacteristic
295C2×C61#341411yescharacteristic
296D61—82no—
297D61—161no—
298C2×#135—611no—
299C2×#135—611no—
300C2×#135—1221no—
301C2×#135C2×D6111yescharacteristic
302#532—1221no—
303C4×#41—1221no—
304#508—1224no—
305#508—1224no—
306C2×Q32—611no—
307C2×Q32—611no—
308C2×SD32—611no—
309C2×SD32—611no—
310C2×SD32—611no—
311C2×SD32—611no—
312C2×D16—611no—
313C2×D16—611no—
314C22×C16—611no—
315C22×C16C2×D6111yescharacteristic
316#495—611no—
317#495—611no—
318#495—1221no—
319#494—611no—
320#493—611no—
321#492—1221no—
322C4×Q16—611no—
323C4×Q16—611no—
324C4×SD16—1221no—
325C4×SD16—1221no—
326C4×D8—611no—
327C4×D8—611no—
328#427C2×D6111yescharacteristic
329#416—1221no—
330C2×C4×C8—611no—
331C4⋊C16—611no—
332C4⋊C16—611no—
333C4⋊C16—611no—
334C4⋊C16—611no—
335#358—611no—
336#358—611no—
337#358C2×D6111yescharacteristic
338#358C2×D6111yescharacteristic
339#357—611no—
340#357—611no—
341#357C2×D6111yescharacteristic
342#357C2×D6111yescharacteristic
343C61#3357911yescharacteristic, complemented, Sylow
344C2×#41—21no—
345C2×#41—1221no—
346C2×#41—1221no—
347#135—22no—
348#135—22no—
349#135—614no—
350#135—1222no—
351#135—1222no—
352#135—1222no—
353#135—2444no—
354C2×Q16—611no—
355C2×Q16—611no—
356C2×Q16—1221no—
357C2×Q16C22×D6111yescharacteristic
358C2×SD16—21no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×SD16—1221no—
363C2×D8—611no—
364C2×D8—611no—
365C2×D8—1221no—
366C2×D8C22×D6111yescharacteristic
367C22×C8—611no—
368C22×C8C22×D6111yescharacteristic
369C4⋊Q8—1221no—
370#127—1221no—
371#124—1221no—
372#124—1221no—
373#121—2441no—
374#121—2441no—
375C4×Q8—1221no—
376C4×Q8—1221no—
377C4×D4—1221no—
378C4×D4—1221no—
379C4×D4—2441no—
380C4×D4—2441no—
381C4×D4—2441no—
382C4×D4—2441no—
383#117—21no—
384#117—2441no—
385C2×C42—611no—
386C2×C42—2441no—
387Q32—1222no—
388Q32—1222no—
389SD32—1222no—
390SD32—1222no—
391SD32—1222no—
392SD32—1222no—
393D16—1222no—
394D16—1222no—
395C2×C16—22no—
396C2×C16—611no—
397C2×C16—611no—
398C2×C16—1222no—
399C2×C16C22×D6111yescharacteristic
400C2×C16C22×D6111yescharacteristic
401#107C22×D6111yescharacteristic
402#107C22×D6111yescharacteristic
403#106—21no—
404#103—1221no—
405#103—1221no—
406#102—1221no—
407#102—1221no—
408C4×C8—611no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8—611no—
412#41—42no—
413#41—42no—
414#41—1224no—
415#41—2442no—
416#41—2442no—
417#41—2442no—
418C2×Q8—21no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×D4—21no—
422C2×D4—41no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—2441no—
426C2×D4—2441no—
427C2×D4—2441no—
428C2×D4—2441no—
429C22×C4—41no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4D4×D6111yescharacteristic
437Q16—21no—
438Q16—21no—
439Q16—612no—
440Q16—1221no—
441Q16—1221no—
442Q16—1221no—
443Q16—2442no—
444SD16—42no—
445SD16—1224no—
446SD16—2442no—
447SD16—2442no—
448SD16—2442no—
449D8—21no—
450D8—21no—
451D8—612no—
452D8—1221no—
453D8—1221no—
454D8—1221no—
455D8—2442no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1222no—
459C2×C8#1175822212yes—
460C2×C8D4×D6112yes—
461C2×C8D4×D6111yescharacteristic
462C2×C8C23×D6111yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C22⋊C4—41no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C42—41no—
477C42—611no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—2441no—
482C42—2441no—
483C16—21no—
484C16—21no—
485C16—1221no—
486C16—1221no—
487C23—41no—
488C23—2441no—
489C23—2441no—
490C23D8×D6111yescharacteristic
491Q8—41no—
492Q8—41no—
493Q8—1222no—
494Q8—2441no—
495Q8—2441no—
496Q8—2441no—
497D4—41no—
498D4—41no—
499D4—82no—
500D4—1222no—
501D4—4882no—
502D4—4882no—
503D4—4882no—
504D4—2444no—
505D4—2441no—
506D4—2441no—
507D4—2441no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—1222no—
517C2×C4—4882no—
518C2×C4—4882no—
519C2×C4—4882no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2444no—
524C2×C4#1451975012yes—
525C2×C4D8×D6111yescharacteristic
526C2×C4#1451980412yes—
527C2×C4C2×D4×D6111yescharacteristic
528C8—1221no—
529C8—1221no—
530C8#1451975012yes—
531C8C2×D4×D6111yescharacteristic
532C8C2×#1175822211yescharacteristic
533C22—21no—
534C22—21no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—2442no—
544C22#1683995612yes—
545C22C2×D8×D6111yescharacteristic
546C4—81no—
547C4—81no—
548C4—81no—
549C4—612no—
550C4—1221no—
551C4—1221no—
552C4—1221no—
553C4—4881no—
554C4—2442no—
555C4#1683995612yes—
556C4C2×#1451975011yescharacteristic
557C4C2×#1451980411yescharacteristic
558C2—22no—
559C2—161no—
560C2—9761no—
561C2—4882no—
562C2C2×#1683995611yescharacteristic
563C2#508×D6111yescharacteristic
564C2#2145895211yescharacteristic
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
hypercenter#2138D61
hypocenterC61#33579
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#16840194C2semidirect
2. C2×#16840192C2semidirect
3. #21456243C2semidirect
4. #21454590C2semidirect
5. #21447884C2semidirect
6. #21447880C2semidirect
7. #21447877C2semidirect
8. #21447873C2semidirect
9. #21447868C2semidirect
10. #21447863C2semidirect
11. #21447858C2semidirect
12. #21447853C2semidirect
13. C2×#16839580C2semidirect
14. #21447524C2semidirect
15. C61×#2138C2semidirect
66. #14519799D4semidirect
275. Dic61#508semidirect
290. #2138D61semidirect
343. C61#33579semidirect

Maximal subgroups

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

subgroupindex
C2×#168401942
C2×#168401922
#214562432
#214545902
#214478842
#214478802
#214478772
#214478732
#214478682
#214478632
#214478582
#214478532
C2×#168395802
#214475242
C61×#21382
#3357961