Finite Group Databasebeta

← #29709401

Subgroups — #29709401

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×#16839863C211yesmaximal, characteristic
2C2×Q16×Dic61C211yesmaximal, characteristic, complemented
3#21448532C211yesmaximal, characteristic
4#21448464C211yesmaximal, characteristic, complemented
5#21447292C211yesmaximal, characteristic, complemented
6#21447227C211yesmaximal, characteristic, complemented
7#21447212C211yesmaximal, characteristic, complemented
8#21447205C211yesmaximal, characteristic
9#21447139C211yesmaximal, characteristic, complemented
10#21447132C211yesmaximal, characteristic
11#21447055C211yesmaximal, characteristic
12#21445675C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2015C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16839863—24no—
20Q16×Dic61—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839655—21no—
39#16839655—21no—
40#16839647—21no—
41#16839645—21no—
42#16839637—21no—
43#16839637—21no—
44#16839632C2211yescharacteristic
45#16839629—21no—
46#16839626—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48#16839608C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×#124×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61D414yes—
68D4×Dic61—22no—
69D4×Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519711—22no—
89#14519711—22no—
90#14519706—22no—
91#14519706—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701—22no—
96#14519698—22no—
97#14519698—22no—
98#14519687—22no—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C2×Q16×C61—22no—
115C2×Q16×C61—22no—
116C2×Q16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122#124×C61—22no—
123#124×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#102×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
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—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61#4112yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168Q16×C61—28no—
169SD16×C61—28no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61#4112yes—
175C2×C8×C61#4112yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C42×C61—21no—
181C42×C61—21no—
182C22⋊Dic61—42no—
183C22⋊Dic61—81no—
184C22⋊Dic61—81no—
185C4⋊Dic61—21no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—81no—
195C4⋊Dic61—81no—
196C4⋊Dic61C2×D412yes—
197#11758196—81no—
198#11758196—81no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—24no—
202C4×Dic61—42no—
203C4×Dic61—81no—
204C4×Dic61C2×D412yes—
205C4×Dic61#4112yes—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
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—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
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—
274C2×Dic61—41no—
275C2×Dic61—84no—
276C2×Dic61#12112yes—
277C2×Dic61#12112yes—
278C8×C61—22no—
279C8×C61—22no—
280#92418—41no—
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—41no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—81no—
302C4×C61#52511yescharacteristic
303C4×C61#52511yescharacteristic
304C4×C61#52911yescharacteristic
305C4×C61#52911yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—41no—
309Dic61—41no—
310Dic61—42no—
311Dic61—81no—
312Dic61—81no—
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#2015D6111yescharacteristic
326#1768—611no—
327#1763—611no—
328C4×#102—611no—
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×Q16C2×D6111yescharacteristic
338C22×SD16C2×D6111yescharacteristic
339C2×#124C2×D6111yescharacteristic
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×D4—611no—
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×#103C2×D6111yescharacteristic
354C2×#102—611no—
355C2×#102—611no—
356C2×#102C2×D6111yescharacteristic
357C2×#97—611no—
358C2×C4×C8—611no—
359#391—611no—
360#391—611no—
361#391C2×D6111yescharacteristic
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⋊C4—611no—
375#376—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336—611no—
381#336C2×D6111yescharacteristic
382C61#2369511yescharacteristic, complemented, Sylow
383C22×Q8C22×D6111yescharacteristic
384C22×Q8C22×D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386C23×C4—1221no—
387C2×Q16—22no—
388C2×Q16—22no—
389C2×Q16—22no—
390C2×SD16—22no—
391C2×SD16—22no—
392C2×SD16—22no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—22no—
398#124—22no—
399C4×Q8—614no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C2×C4⋊C4—611no—
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⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427#107—1222no—
428#107—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#103—22no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#102—22no—
437#102—1222no—
438#102—1222no—
439#97—1222no—
440C4×C8—1222no—
441#95—21no—
442#95—21no—
443#95—611no—
444#95—611no—
445#95—611no—
446#95—611no—
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—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8D4×D6112yes—
466C2×Q8#1175822412yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1175822412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—4881no—
485C22×C4—4881no—
486C22×C4—2442no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—28no—
491SD16—28no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8#1175822412yes—
503C2×C8#1175822412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—612no—
507C4⋊C4—1224no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—2442no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—4881no—
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C42—21no—
526C42—21no—
527C42—612no—
528C42—612no—
529C42—1224no—
530C42—1221no—
531C42—1221no—
532C42—4881no—
533C42—2442no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—4884no—
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—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4C2×#1175822411yescharacteristic
599C2×C4C2×#1175822411yescharacteristic
600C8—22no—
601C8—22no—
602C8—2441no—
603C8—2441no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#121×D6111yescharacteristic
612C22#1684026811yescharacteristic
613C22#1684031011yescharacteristic
614C22#1684056411yescharacteristic
615C22#1684057111yescharacteristic
616C22#1684057611yescharacteristic
617C22#1684057811yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—1222no—
624C4—1222no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—2442no—
629C4—2441no—
630C4—2441no—
631C4#1684026811yescharacteristic
632C4#1684026811yescharacteristic
633C4#1684029311yescharacteristic
634C4#1684029311yescharacteristic
635C2—82no—
636C2#2144905411yescharacteristic
637C2#2145534711yescharacteristic
638C2#2145536111yescharacteristic
639C2#2145603411yescharacteristic
640C2#2145608211yescharacteristic
641C2#2145645511yescharacteristic
642C2#2145648411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2015C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2015D61
hypocenterC61#23695
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2015

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
4. #21448464C2semidirect
5. #21447292C2semidirect
6. #21447227C2semidirect
7. #21447212C2semidirect
9. #21447139C2semidirect
12. #21445675C2semidirect
13. #21445651C2semidirect
382. C61#23695semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
C2×Q16×Dic612
#214485322
#214484642
#214472922
#214472272
#214472122
#214472052
#214471392
#214471322
#214470552
#214456752
#214456512
#102×Dic612
C61×#20152
#2369561