Finite Group Databasebeta

← #29730103

Subgroups — #29730103

Order 15616.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic
2C2×#16839981C211yesmaximal, characteristic
3#21451034C211yesmaximal, characteristic, complemented
4#21447237C211yesmaximal, characteristic, complemented
5#21447232C211yesmaximal, characteristic
6#21447141C211yesmaximal, characteristic, complemented
7#21447136C211yesmaximal, characteristic
8#21446767C211yesmaximal, characteristic, complemented
9#21446543C211yesmaximal, characteristic, complemented
10#21446160C211yesmaximal, characteristic, complemented
11#21446153C211yesmaximal, characteristic
12#21445834C211yesmaximal, characteristic, complemented
13#21445829C211yesmaximal, characteristic
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2367C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840121—24no—
21#16839981—24no—
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839702—21no—
35#16839702—21no—
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839667—21no—
39#16839667—21no—
40#16839628C2211yescharacteristic
41#16839628C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#124×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#391C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#388—21no—
56C61×#388—21no—
57C61×#385—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C22⋊C4×C61—21no—
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—41no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519777—22no—
74#14519777—22no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×C4×Dic61—21no—
83C4⋊#92418—22no—
84C4⋊#92418—41no—
85C4⋊#92418—41no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519700—22no—
90#14519700—22no—
91#14519698—22no—
92#14519698—22no—
93#14519698—22no—
94#14519688—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106#124×C61—22no—
107#124×C61D414yes—
108C4×Q8×C61—22no—
109C4×Q8×C61—41no—
110C4×D4×C61—22no—
111C4×D4×C61—41no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
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×C61C2×D412yes—
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—21no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122#4112yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61#4112yes—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—81no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188C42×C61—81no—
189C42×C61C2×D412yes—
190C42×C61#4112yes—
191#11758202—44no—
192#11758200—44no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200#11758196—42no—
201C4×Dic61—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—84no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—22no—
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—41no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280Dic122—84no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#25337—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57811yescharacteristic
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61#52911yescharacteristic
310C4×C61#54211yescharacteristic
311C4×C61#54611yescharacteristic
312C4×C61#54711yescharacteristic
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#489—611no—
318C2×#464—611no—
319#2568—611no—
320#2367D6111yescharacteristic
321#2074—611no—
322#2073—611no—
323#2056—611no—
324#2055—611no—
325#1987—611no—
326#1940—611no—
327#1878—611no—
328#1877—611no—
329#1805—611no—
330#1804—611no—
331#1763—611no—
332C2×C61—82no—
333C2×C61#242711yescharacteristic
334C2×C61#304011yescharacteristic
335C2×C61#304611yescharacteristic
336C2×C61#331111yescharacteristic
337C2×C61#331311yescharacteristic
338C2×C61#337111yescharacteristic
339C2×C61#337711yescharacteristic
340C22×Q16—611no—
341C22×SD16—611no—
342C2×C4⋊Q8—611no—
343C2×#124C2×D6111yescharacteristic
344C2×C4×Q8—611no—
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×D4C2×D6111yescharacteristic
347#489—1224no—
348#464—1224no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#97—611no—
361#400—1221no—
362#400—1221no—
363#391—611no—
364#391—611no—
365#391—1221no—
366#391—1221no—
367#391C2×D6111yescharacteristic
368#390C2×D6111yescharacteristic
369#388—21no—
370#388—21no—
371#385—21no—
372#382—611no—
373#382—611no—
374C4×C4⋊C4C2×D6111yescharacteristic
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—21no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2533711yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8—1221no—
388C22×Q8C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C2×Q16—1222no—
392C2×Q16—1222no—
393C2×Q16—2441no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C4⋊Q8—1222no—
402C4⋊Q8—1222no—
403#124—22no—
404#124D24414yes—
405C4×Q8—22no—
406C4×Q8—41no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×D4—22no—
410C4×D4—41no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22⋊D6111yescharacteristic
418C2×C4⋊C4C22⋊D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1221no—
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431C2×C42C22×D6111yescharacteristic
432C4⋊C8—1222no—
433C4⋊C8—2441no—
434C4⋊C8—2441no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443#97—1222no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95C22⋊D6111yescharacteristic
456#95C22⋊D6111yescharacteristic
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—2442no—
470C2×Q8—2442no—
471C2×Q8—2442no—
472C2×Q8C2×D24412yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D24412yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496Q16—2444no—
497SD16—2444no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4#1175822412yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C42—2442no—
537C42C2×D24412yes—
538C42#1175822212yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—1222no—
552Q8—1222no—
553Q8—4884no—
554Q8—2441no—
555Q8—2441no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—1222no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451975112yes—
599C2×C4#1451975112yes—
600C2×C4C22×D24411yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1451975111yescharacteristic
618C22#1684040611yescharacteristic
619C22#1684041911yescharacteristic
620C22#1684056011yescharacteristic
621C22#1684056011yescharacteristic
622C22#1684057311yescharacteristic
623C22#1684057711yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4#1684029411yescharacteristic
638C4#1684037711yescharacteristic
639C4#1684040711yescharacteristic
640C4#1684041611yescharacteristic
641C2—82no—
642C2#2144939011yescharacteristic
643C2#2145531311yescharacteristic
644C2#2145536911yescharacteristic
645C2#2145785511yescharacteristic
646C2#2145788511yescharacteristic
647C2#2145858111yescharacteristic
648C2#2145865511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2367C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2367D61
hypocenterC61#25337
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #21451034C2semidirect
4. #21447237C2semidirect
6. #21447141C2semidirect
8. #21446767C2semidirect
9. #21446543C2semidirect
10. #21446160C2semidirect
12. #21445834C2semidirect
14. #21445652C2semidirect
385. C61#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168399812
#214510342
#214472372
#214472322
#214471412
#214471362
#214467672
#214465432
#214461602
#214461532
#214458342
#214458292
#214456522
C61×#23672
#2533761