Finite Group Databasebeta

← #29709318

Subgroups — #29709318

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2#21448535C211yesmaximal, characteristic, complemented
3#21448462C211yesmaximal, characteristic
4#21447304C211yesmaximal, characteristic, complemented
5#21447235C211yesmaximal, characteristic
6#21447210C211yesmaximal, characteristic, complemented
7#21447195C211yesmaximal, characteristic, complemented
8#21447135C211yesmaximal, characteristic
9#21447132C211yesmaximal, characteristic
10#21447064C211yesmaximal, characteristic, complemented
11#21446900C211yesmaximal, characteristic
12#21445677C211yesmaximal, characteristic, complemented
13#21445649C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C61C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19#16839862—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839655—21no—
37#16839655—21no—
38#16839646—21no—
39#16839646—21no—
40#16839643—21no—
41#16839643—21no—
42#16839627C2211yescharacteristic
43#16839625—21no—
44#16839622—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839608C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×SD16×C61—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519775—22no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519710—22no—
88#14519706—22no—
89#14519706—22no—
90#14519702—22no—
91#14519702—22no—
92#14519701—22no—
93#14519701—22no—
94#14519700—22no—
95#14519698—22no—
96#14519688—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C2×SD16×C61—22no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61D414yes—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#106×C61—22no—
129#103×C61—22no—
130#102×C61—22no—
131C4×C8×C61—22no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61#4112yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—21no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122#4112yes—
168SD16×C61—28no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61#4112yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C42×C61—42no—
182C42×C61C2×D412yes—
183C42×C61#4112yes—
184C22⋊Dic61—81no—
185C22⋊Dic61—81no—
186C22⋊Dic61—81no—
187C22⋊Dic61—81no—
188#11758200—28no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—81no—
196C4⋊Dic61—81no—
197#11758196—24no—
198#11758196—24no—
199#11758196—42no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418#4112yes—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—41no—
227D4×C61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—84no—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#23689—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C2×#12111yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61#53211yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—81no—
302C4×C61#52511yescharacteristic
303C4×C61#52511yescharacteristic
304C4×C61#52911yescharacteristic
305C4×C61#52911yescharacteristic
306Dic61—41no—
307Dic61—41no—
308Dic61—42no—
309Dic61—41no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314C2×#440—611no—
315C2×C4×SD16D6111yescharacteristic
316#2327—611no—
317#2319—611no—
318#2085—611no—
319#2073—611no—
320#2068—611no—
321#2066—611no—
322#2055—611no—
323#2055—611no—
324#2043—611no—
325#2013—611no—
326#1768—611no—
327#1763—611no—
328C4×#103—611no—
329C2×C61—82no—
330C2×C61#239211yescharacteristic
331C2×C61#304211yescharacteristic
332C2×C61#304611yescharacteristic
333C2×C61#312111yescharacteristic
334C2×C61#312611yescharacteristic
335C2×C61#317411yescharacteristic
336C2×C61#317811yescharacteristic
337C22×SD16—611no—
338C22×SD16C2×D6111yescharacteristic
339C2×C4⋊Q8—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8C2×D6111yescharacteristic
342C2×C4×D4C2×D6111yescharacteristic
343#440—1224no—
344C4×SD16—24no—
345C2×#107—611no—
346C2×#106C2×D6111yescharacteristic
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102C2×D6111yescharacteristic
357C2×#97—611no—
358C2×C4×C8C2×D6111yescharacteristic
359#391—611no—
360#391—611no—
361#390—611no—
362#390—611no—
363#389—611no—
364#389—611no—
365#388—1221no—
366#388—1221no—
367#387—1221no—
368#387—1221no—
369#387—1221no—
370#387—1221no—
371#382—611no—
372#381—1221no—
373#380—1221no—
374C4×C4⋊C4—611no—
375#376—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336—611no—
381#336—611no—
382C61#2368911yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8C22×D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386C23×C4—21no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16—22no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4⋊Q8—1222no—
398C4⋊Q8—1222no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×D4—22no—
404C4×D4C22⋊D6114yes—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—21no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427#107—1222no—
428#106—22no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#103—22no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#103—1222no—
437#102—22no—
438#102—1222no—
439#97—1222no—
440C4×C8—22no—
441#95—611no—
442#95—611no—
443#95—611no—
444#95—611no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454C24—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—612no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8#1175822212yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C22⋊D6112yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490SD16—28no—
491SD16—1228no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—612no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8#1175822212yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—2442no—
519C4⋊C4C2×C22⋊D6112yes—
520C22⋊C4—24no—
521C22⋊C4—4881no—
522C22⋊C4—4881no—
523C22⋊C4—4881no—
524C22⋊C4—4881no—
525C42—42no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—4881no—
531C42—2442no—
532C42#1175822412yes—
533C42C2×C22⋊D6112yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—1222no—
547Q8—1222no—
548Q8—2441no—
549Q8—2441no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4#1451974912yes—
593C2×C4#1451974912yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4C2×#1175822211yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1451974911yescharacteristic
612C22#1684027111yescharacteristic
613C22#1684030911yescharacteristic
614C22#1684056811yescharacteristic
615C22#1684056811yescharacteristic
616C22#1684056911yescharacteristic
617C22#1684057611yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—81no—
623C4—4881no—
624C4—4881no—
625C4—4881no—
626C4—2441no—
627C4—2441no—
628C4—2442no—
629C4—2441no—
630C4—2441no—
631C4#1684026911yescharacteristic
632C4#1684027311yescharacteristic
633C4#1684029211yescharacteristic
634C4#1684029411yescharacteristic
635C2—82no—
636C2#2144906111yescharacteristic
637C2#2145532811yescharacteristic
638C2#2145536711yescharacteristic
639C2#2145604411yescharacteristic
640C2#2145610711yescharacteristic
641C2#2145644311yescharacteristic
642C2#2145648311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×C4×SD16D61
hypocenterC61#23689
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. #21448535C2semidirect
4. #21447304C2semidirect
6. #21447210C2semidirect
7. #21447195C2semidirect
10. #21447064C2semidirect
12. #21445677C2semidirect
13. #21445649C2semidirect
14. C4×#14519702C2semidirect
382. C61#23689semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214485352
#214484622
#214473042
#214472352
#214472102
#214471952
#214471352
#214471322
#214470642
#214469002
#214456772
#214456492
C4×#145197022
C2×C4×SD16×C612
#2368961