Finite Group Databasebeta

← #29834295

Subgroups — #29834295

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C2×#16840201C211yesmaximal, characteristic, complemented
3#21456244C211yesmaximal, characteristic
4C4×Q16×D61C211yesmaximal, characteristic, complemented
5#21447882C211yesmaximal, characteristic, complemented
6#21447881C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447868C211yesmaximal, characteristic, complemented
9#21447867C211yesmaximal, characteristic, complemented
10#21447858C211yesmaximal, characteristic, complemented
11#21447857C211yesmaximal, characteristic, complemented
12#21447573C211yesmaximal, characteristic, complemented
13#21447530C211yesmaximal, characteristic, complemented
14#21447528C211yesmaximal, characteristic, complemented
15C61×#2211C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519798—21no—
18C2×#14519796C2211yescharacteristic
19C2×Q16×D61C2211yescharacteristic
20#16840311—21no—
21C4×Q8×D61—21no—
22#16840204—24no—
23#16840201—24no—
24C2×Dic488C2211yescharacteristic
25C2×#14519723C2211yescharacteristic
26C2×#14519723C2211yescharacteristic
27C2×#14519722C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32#16840146C2211yescharacteristic
33#16840144—21no—
34#16840140C2211yescharacteristic
35#16840137C2211yescharacteristic
36#16840133—21no—
37C4×Dic244C2211yescharacteristic
38Q16×Dic61C2211yescharacteristic
39#16839856C2211yescharacteristic
40C4×#11758202—21no—
41#16839854—21no—
42#107×D61C2211yescharacteristic
43#103×D61—21no—
44C4×C8×D61C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic
46#16839580C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839566C2211yescharacteristic
50#16839565C2211yescharacteristic
51#16839564C2211yescharacteristic
52#16839563C2211yescharacteristic
53#16839562C2211yescharacteristic
54C2×Q32×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×#357C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64#14519800—22no—
65#14519800—22no—
66#14519798—44no—
67#14519796—22no—
68#14519796—22no—
69C2×Dic244C2311yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—22no—
74Q16×D61D414yescomplemented
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×#11758201—21no—
78C2×D488C2311yescharacteristic
79C2×#11758199—21no—
80C2×C8×D61D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519775—21no—
83#14519773—21no—
84#14519765—21no—
85#14519750—41no—
86#14519750—41no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89C4×Dic122—41no—
90#14519742—21no—
91#14519742—41no—
92#14519737—41no—
93#14519737—41no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—41no—
96C42×D61—41no—
97C42×D61D411yescharacteristic
98Dic488—22no—
99#14519723—22no—
100#14519723—22no—
101#14519722—22no—
102#14519721—22no—
103#14519720—22no—
104#14519716—22no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108#14519711C2311yescharacteristic
109#14519710—21no—
110#14519702—21no—
111#14519701—21no—
112#14519700—21no—
113C4×#92418D411yescharacteristic
114C8×Dic61D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×Q16×C61C2311yescharacteristic
118C2×SD16×C61—21no—
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×Q8×C61—21no—
122Q32×C61—22no—
123SD32×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#103×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×D4×C61—21no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C22×Dic61—41no—
136#11758226—42no—
137#11758226—42no—
138Q8×D61—24no—
139Q8×D61—42no—
140#11758224—42no—
141#11758224—42no—
142C2×D244—21no—
143C2×D244—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61D811yescharacteristic
148C2×C4×D61D811yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—41no—
153Q16×C61—21no—
154Q16×C61—21no—
155Q16×C61—21no—
156Q16×C61C2×D412yes—
157SD16×C61—42no—
158D8×C61—21no—
159D8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2411yescharacteristic
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C42×C61—41no—
167C42×C61C2×D411yescharacteristic
168#11758202—24no—
169#11758202—42no—
170#11758201—42no—
171#11758200—42no—
172#11758199—42no—
173#11758198—41no—
174#11758198—41no—
175C4⋊Dic61—21no—
176C4⋊Dic61—41no—
177#11758196—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61D811yescharacteristic
186C4×Dic61D811yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244C2×D412yes—
194D488—21no—
195D488—21no—
196#11758190—42no—
197C8×D61—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C22×D61—41no—
206C22×D61C2×D811yescharacteristic
207Q8×C61—22no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61C2×D811yescharacteristic
217C2×C4×C61C2×D811yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C22⋊D61—82no—
220C22⋊D61—82no—
221C22⋊D61—82no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—82no—
226C2×Dic61C2×D811yescharacteristic
227C2×Dic61C2×D811yescharacteristic
228C2×Dic61C2×D811yescharacteristic
229D244—41no—
230D244—41no—
231D244—82no—
232C4×D61—22no—
233C4×D61—41no—
234C4×D61—44no—
235C4×D61—82no—
236C4×D61—82no—
237C4×D61#12112yes—
238C4×D61AGL(1,ℤ/8)12yes—
239Dic122—22no—
240Dic122—41no—
241Dic122—44no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—21no—
245C8×C61C22×D411yescharacteristic
246C8×C61C2×#4111yescharacteristic
247#92418—21no—
248#92418#12112yes—
249#33581—611nomaximal, Sylow
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61C22×D811yescharacteristic
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61#45912yes—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—42no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C2×#12111yescharacteristic
266C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
267Dic61—21no—
268Dic61—42no—
269Dic61—81no—
270Dic61—81no—
271Dic61#45912yes—
272Dic61#51012yescomplemented
273C2×#510—611no—
274C2×#510—611no—
275#3147—611no—
276C2×C4×Q16—611no—
277#2211—611no—
278#2211—611no—
279#2211—611no—
280#2211D6111yescharacteristic, complemented
281#2209—611no—
282#2209—611no—
283#2207—611no—
284#2207—611no—
285#2150—611no—
286#2139—611no—
287#2139—611no—
288C2×C61—161no—
289C2×C61C2×#45911yescharacteristic
290C2×C61C2×#51011yescharacteristic
291C2×C61#341411yescharacteristic
292D61—22no—
293D61—161no—
294C2×#135—611no—
295C2×#135—611no—
296C2×#135—1221no—
297C22×Q16—611no—
298#532—1221no—
299C2×C4×Q8—1221no—
300#510—1224no—
301#510—1224no—
302C2×Q32—611no—
303C2×Q32—611no—
304C2×Q32—611no—
305C2×Q32C2×D6111yescharacteristic
306C2×SD32—611no—
307C2×SD32—611no—
308C2×SD32—611no—
309C2×SD32C2×D6111yescharacteristic
310C2×M32—611no—
311C2×M32—611no—
312#495—611no—
313#495—1221no—
314#495C2×D6111yescharacteristic
315#494—611no—
316#493—611no—
317#492—1221no—
318C4×Q16—611no—
319C4×Q16—611no—
320C4×Q16—611no—
321C4×Q16—1221no—
322C4×Q16—1221no—
323C4×Q16C2×D6111yescharacteristic
324C2×#107—611no—
325C2×#103—1221no—
326C2×C4×C8—611no—
327C4⋊C16—611no—
328C4⋊C16—611no—
329C4⋊C16—611no—
330C4⋊C16C2×D6111yescharacteristic
331#358—611no—
332#358—611no—
333#358—611no—
334#358C2×D6111yescharacteristic
335#357—611no—
336#357—611no—
337#357—611no—
338#357C2×D6111yescharacteristic
339C61#3358111yescharacteristic, complemented, Sylow
340C2×#41—1221no—
341C2×#41—1221no—
342C22×Q8—1221no—
343#135—1222no—
344#135—1222no—
345#135—1222no—
346#135—1222no—
347#135—2444no—
348C2×Q16—614no—
349C2×Q16—611no—
350C2×Q16—611no—
351C2×Q16—1222no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16C22×D6111yescharacteristic
355C2×Q16C22×D6111yescharacteristic
356C2×SD16—21no—
357C2×SD16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×D8—611no—
361C2×D8—1221no—
362C2×D8C22×D6111yescharacteristic
363C22×C8—611no—
364C22×C8—611no—
365C4⋊Q8—1221no—
366#127—1221no—
367#124—21no—
368#124—1221no—
369#121—2441no—
370#121—2441no—
371C4×Q8—21no—
372C4×Q8—1221no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8—2441no—
376C4×Q8—2441no—
377C4×D4—2441no—
378C4×D4—2441no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—2441no—
381C2×C42—611no—
382C2×C42—2441no—
383Q32—22no—
384Q32—1222no—
385Q32—1222no—
386Q32—1222no—
387SD32—22no—
388SD32—1222no—
389SD32—1222no—
390SD32—1222no—
391M32—1222no—
392M32—1222no—
393C2×C16—611no—
394C2×C16—611no—
395C2×C16C22×D6111yescharacteristic
396C2×C16C22×D6111yescharacteristic
397#107—611no—
398#107—1221no—
399#107C22×D6111yescharacteristic
400#103—21no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8—611no—
407C4×C8C22×D6111yescharacteristic
408#41—2442no—
409#41—2442no—
410#41—2442no—
411#41—2442no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—1224no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—2441no—
418C2×Q8—2442no—
419C2×D4—21no—
420C2×D4—1221no—
421C2×D4—2441no—
422C2×D4—2441no—
423C2×D4—2441no—
424C2×D4—2441no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—2441no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2441no—
432Q16—21no—
433Q16—21no—
434Q16—21no—
435Q16—612no—
436Q16—1224no—
437Q16—1221no—
438Q16—1221no—
439Q16—1221no—
440Q16—2442no—
441Q16D4×D6112yes—
442SD16—42no—
443SD16—2442no—
444SD16—2442no—
445SD16—2442no—
446D8—21no—
447D8—21no—
448D8—1221no—
449D8—1221no—
450D8—2442no—
451C2×C8—611no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—611no—
455C2×C8—1222no—
456C2×C8D4×D6111yescharacteristic
457C2×C8C23×D6111yescharacteristic
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—41no—
472C42—611no—
473C42—611no—
474C42—611no—
475C42—2441no—
476C42—2441no—
477C42—2441no—
478C42D4×D6111yescharacteristic
479C16—21no—
480C16—21no—
481C16—1221no—
482C16—1221no—
483C23—41no—
484C23—611no—
485C23—2441no—
486Q8—22no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—1222no—
491Q8—2444no—
492Q8—2441no—
493Q8—2441no—
494Q8—2441no—
495D4—41no—
496D4—41no—
497D4—4882no—
498D4—4882no—
499D4—4882no—
500D4—4882no—
501D4—2441no—
502D4—2441no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—611no—
511C2×C4—1222no—
512C2×C4—4882no—
513C2×C4—4882no—
514C2×C4—4882no—
515C2×C4—2441no—
516C2×C4—2444no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4D8×D6111yescharacteristic
521C2×C4D8×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C8—21no—
524C8—612no—
525C8—1221no—
526C8C2×D4×D6111yescharacteristic
527C8#41×D6111yescharacteristic
528C22—81no—
529C22—81no—
530C22—81no—
531C22—612no—
532C22—1221no—
533C22—1221no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22C2×D8×D6111yescharacteristic
538C4—21no—
539C4—21no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4—612no—
544C4—612no—
545C4—1221no—
546C4—4881no—
547C4—4881no—
548C4—2442no—
549C4#121×D6111yescharacteristic
550C4AGL(1,ℤ/8)×D6111yescharacteristic
551C2—161no—
552C2—1222no—
553C2—9761no—
554C2#459×D6111yescharacteristic
555C2#510×D6111yescharacteristic
556C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
prime radical#21456244C2
hypercenter#2211D61
hypocenterC61#33581
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2211

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
2. C2×#16840201C2semidirect
4. C4×Q16×D61C2semidirect
5. #21447882C2semidirect
6. #21447881C2semidirect
7. #21447878C2semidirect
8. #21447868C2semidirect
9. #21447867C2semidirect
10. #21447858C2semidirect
11. #21447857C2semidirect
12. #21447573C2semidirect
13. #21447530C2semidirect
14. #21447528C2semidirect
15. C61×#2211C2semidirect
74. Q16×D61D4semidirect
272. Dic61#510semidirect
280. #2211D61semidirect
339. C61#33581semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
C2×#168402012
#214562442
C4×Q16×D612
#214478822
#214478812
#214478782
#214478682
#214478672
#214478582
#214478572
#214475732
#214475302
#214475282
C61×#22112
#3358161