Finite Group Databasebeta

← #29709414

Subgroups — #29709414

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
2C2×C4×Dic244C211yesmaximal, characteristic, complemented
3#21448531C211yesmaximal, characteristic
4#21448465C211yesmaximal, characteristic, complemented
5#21447294C211yesmaximal, characteristic, complemented
6#21447231C211yesmaximal, characteristic, complemented
7#21447215C211yesmaximal, characteristic, complemented
8#21447205C211yesmaximal, characteristic
9#21447143C211yesmaximal, characteristic, complemented
10#21447135C211yesmaximal, characteristic
11#21447058C211yesmaximal, characteristic
12#21446915C211yesmaximal, characteristic
13#21445677C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16839862—24no—
22C4×Dic244—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839667C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839650—21no—
41#16839650—21no—
42#16839646—21no—
43#16839646—21no—
44#16839635—21no—
45#16839635—21no—
46#16839631C2211yescharacteristic
47#16839629—21no—
48#16839626—21no—
49#16839608C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×Dic244—22no—
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519764—22no—
72#14519764—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122D414yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95#14519706—22no—
96#14519706—22no—
97#14519702—22no—
98#14519702—22no—
99#14519700—22no—
100#14519700—22no—
101#14519700—22no—
102#14519698—22no—
103#14519688—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—81no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×D412yes—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61#4112yes—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—42no—
181C42×C61—81no—
182C42×C61C2×D412yes—
183C42×C61#4112yes—
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C22⋊Dic61—81no—
187C22⋊Dic61—81no—
188#11758200—28no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C2×D412yes—
193#11758196—24no—
194#11758196—42no—
195#11758196—81no—
196#11758196—81no—
197#11758196—81no—
198#11758196—81no—
199C4×Dic61—24no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418#4112yes—
211Dic244—28no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—84no—
238C2×C4×C61#12112yes—
239C2×C4×C61#12112yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
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—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270Dic122—22no—
271Dic122—22no—
272Dic122—22no—
273Dic122—22no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#23695—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#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#52511yescharacteristic
303C4×C61#52511yescharacteristic
304C4×C61#52911yescharacteristic
305C4×C61#52911yescharacteristic
306Dic61—41no—
307Dic61—41no—
308Dic61—42no—
309Dic61—42no—
310Dic61—41no—
311Dic61—41no—
312Dic61—42no—
313Dic61—81no—
314C2×#440—611no—
315C2×C4×Q16—611no—
316#2326—611no—
317#2320—611no—
318#2083—611no—
319#2072—611no—
320#2068—611no—
321#2067—611no—
322#2056—611no—
323#2055—611no—
324#2042—611no—
325#2015—611no—
326#1768—611no—
327#1763—611no—
328C4×#102D6111yescharacteristic
329C2×C61—82no—
330C2×C61#239211yescharacteristic
331C2×C61#304411yescharacteristic
332C2×C61#304511yescharacteristic
333C2×C61#312011yescharacteristic
334C2×C61#312411yescharacteristic
335C2×C61#317511yescharacteristic
336C2×C61#317811yescharacteristic
337C22×Q16—611no—
338C22×SD16—611no—
339C2×#124—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×D4C2×D6111yescharacteristic
343#440—1224no—
344C4×Q16—1224no—
345C2×#107—611no—
346C2×#107—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×#102C2×D6111yescharacteristic
357C2×#97—611no—
358C2×C4×C8C2×D6111yescharacteristic
359#391—611no—
360#391—611no—
361#391—611no—
362#390—611no—
363#389—611no—
364#389—611no—
365#388—1221no—
366#388—1221no—
367#387—1221no—
368#387—1221no—
369#385—1221no—
370#385—1221no—
371#384—611no—
372#383—1221no—
373#382—1221no—
374C4×C4⋊C4C2×D6111yescharacteristic
375#376—611no—
376#340—611no—
377#340—611no—
378#340C2×D6111yescharacteristic
379#340C2×D6111yescharacteristic
380#336—611no—
381#336—611no—
382C61#2369511yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×D4C22×D6111yescharacteristic
386C23×C4—21no—
387C2×Q16—1222no—
388C2×Q16—1222no—
389C2×Q16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—1222no—
398#124—1222no—
399C4×Q8—614no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×D4—22no—
404C4×D4—22no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22×D6111yescharacteristic
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#107—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#102—22no—
437#102—22no—
438#102—1222no—
439#97—1222no—
440C4×C8—22no—
441#95—21no—
442#95—611no—
443#95—611no—
444#95—611no—
445#95—611no—
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—612no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1175822212yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
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
490Q16—1228no—
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—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—612no—
511C4⋊C4—1221no—
512C4⋊C4—1224no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—2442no—
520C22⋊C4—42no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C22⋊C4—4881no—
524C22⋊C4—4881no—
525C42—81no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1224no—
531C42—2442no—
532C42#1175822212yes—
533C42D4×D6112yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—1222no—
545Q8—1222no—
546Q8—2441no—
547Q8—2441no—
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—41no—
559C2×C4—41no—
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—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1222no—
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—1221no—
582C2×C4—1221no—
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#1451975012yes—
593C2×C4#1451975012yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
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×#1451975011yescharacteristic
612C22#1684027111yescharacteristic
613C22#1684030911yescharacteristic
614C22#1684056311yescharacteristic
615C22#1684056811yescharacteristic
616C22#1684057611yescharacteristic
617C22#1684057611yescharacteristic
618C4—22no—
619C4—22no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—4881no—
624C4—2442no—
625C4—2441no—
626C4—2441no—
627C4—2442no—
628C4—2442no—
629C4—2441no—
630C4—2441no—
631C4#1684027111yescharacteristic
632C4#1684027111yescharacteristic
633C4#1684029011yescharacteristic
634C4#1684029611yescharacteristic
635C2—82no—
636C2#2144906311yescharacteristic
637C2#2145535211yescharacteristic
638C2#2145536311yescharacteristic
639C2#2145602911yescharacteristic
640C2#2145607711yescharacteristic
641C2#2145645411yescharacteristic
642C2#2145648311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC4×#102D61
hypocenterC61#23695
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic244C2semidirect
4. #21448465C2semidirect
5. #21447294C2semidirect
6. #21447231C2semidirect
7. #21447215C2semidirect
9. #21447143C2semidirect
13. #21445677C2semidirect
14. #21445652C2semidirect
382. C61#23695semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
C2×C4×Dic2442
#214485312
#214484652
#214472942
#214472312
#214472152
#214472052
#214471432
#214471352
#214470582
#214469152
#214456772
#214456522
C4×#102×C612
#2369561