Finite Group Databasebeta

← #29721349

Subgroups — #29721349

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840041C211yesmaximal, characteristic, complemented
2C2×#16839821C211yesmaximal, characteristic, complemented
3#21451646C211yesmaximal, characteristic, complemented
4#21448510C211yesmaximal, characteristic
5#21447251C211yesmaximal, characteristic, complemented
6#21447243C211yesmaximal, characteristic, complemented
7#21447011C211yesmaximal, characteristic
8#21447003C211yesmaximal, characteristic
9#21446765C211yesmaximal, characteristic, complemented
10#21446584C211yesmaximal, characteristic, complemented
11#21446559C211yesmaximal, characteristic, complemented
12#21446515C211yesmaximal, characteristic, complemented
13#21446502C211yesmaximal, characteristic, complemented
14#21445828C211yesmaximal, characteristic, complemented
15C61×#1942C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519752C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519734C2211yescharacteristic
20#16840041—24no—
21#16839821—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839703—21no—
34#16839697—21no—
35#16839682—21no—
36#16839675—21no—
37#16839664C2211yescharacteristic
38#16839663—21no—
39#16839660C2211yescharacteristic
40#16839651—21no—
41#16839640—21no—
42#16839633—21no—
43#16839627C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—22no—
70#14519752—22no—
71#14519752—22no—
72#14519739—22no—
73#14519739—22no—
74#14519734—22no—
75#14519734—22no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198D411yescharacteristic
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519710—22no—
95#14519708—22no—
96#14519706—22no—
97#14519706D414yescomplemented
98#14519699—22no—
99#14519699—22no—
100#14519697—22no—
101#14519697—22no—
102#14519689—22no—
103#14519689—22no—
104#14519688—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#107×C61—22no—
123C4×C8×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C23×D61C2×D411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—24no—
147C2×D244—81no—
148C2×D244—81no—
149C2×D244#4112yes—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×Dic122—42no—
160C2×Dic122—81no—
161C2×Dic122—81no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—81no—
176C4⋊C4×C61—81no—
177C4⋊C4×C61#4112yes—
178C42×C61—21no—
179C42×C61—21no—
180#11758198—24no—
181#11758198—22no—
182#11758198—22no—
183#11758198—42no—
184#11758198—81no—
185#11758198—81no—
186#11758198—81no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—22no—
196#11758196—22no—
197#11758196—42no—
198#11758196—81no—
199C4×Dic61—81no—
200C4×Dic61C2×D412yes—
201C4×Dic61#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D412yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
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—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61#12112yes—
264C2×Dic61#12112yes—
265D244—22no—
266D244—22no—
267D244—41no—
268D244—41no—
269D244—84no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—84no—
275Dic122—84no—
276C8×C61—22no—
277C8×C61—41no—
278#92418—22no—
279#92418—41no—
280#24712—611nomaximal, Sylow
281C22×C61C2×#12111yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61#53611yescharacteristic
284C22×C61#57611yescharacteristic
285C22×C61#57611yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57811yescharacteristic
288C2×D61—41no—
289C2×D61—42no—
290C2×D61—41no—
291C2×D61—42no—
292C2×D61—41no—
293C2×D61—42no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61#54711yescharacteristic
309C4×C61#54811yescharacteristic
310C4×C61#55211yescharacteristic
311C4×C61#55611yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318C2×#474—611no—
319C2×#433—611no—
320#2628—611no—
321#2324—611no—
322#2076—611no—
323#2075—611no—
324#2034—611no—
325#2033—611no—
326#1987—611no—
327#1949—611no—
328#1944—611no—
329#1942D6111yescharacteristic, complemented
330#1936—611no—
331#1934—611no—
332#1804—611no—
333C2×C61#241511yescharacteristic
334C2×C61#305811yescharacteristic
335C2×C61#306311yescharacteristic
336C2×C61#311811yescharacteristic
337C2×C61#312411yescharacteristic
338C2×C61#324111yescharacteristic
339C2×C61#324311yescharacteristic
340D61—81no—
341D61—82no—
342C2×C4⋊Q8—611no—
343C2×#121—611no—
344C2×C4×D4—611no—
345C2×#117—611no—
346#474—1224no—
347#433—1224no—
348C2×#107—611no—
349C2×#107C2×D6111yescharacteristic
350C2×#107C2×D6111yescharacteristic
351C2×#106—611no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×C22⋊C8—611no—
359C2×C22⋊C8—611no—
360C2×#97—611no—
361C2×C4×C8C2×D6111yescharacteristic
362#400—1221no—
363#399—1221no—
364#395—1221no—
365#393—1221no—
366#390—611no—
367#390—1221no—
368#389—611no—
369#389C2×D6111yescharacteristic
370#389C2×D6111yescharacteristic
371#388—1221no—
372#386—1221no—
373#385—1221no—
374#382—611no—
375#380—611no—
376#380—611no—
377#380—611no—
378C4×C4⋊C4—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340C2×D6111yescharacteristic
386#340C2×D6111yescharacteristic
387C61#2471211yescharacteristic, complemented, Sylow
388C22×Q8—1221no—
389C22×D4—611no—
390C22×D4—1221no—
391C23×C4—611no—
392C23×C4—1221no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4⋊Q8—1222no—
398C4⋊Q8—1222no—
399#121—1222no—
400#121—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—611no—
428C2×C42—1221no—
429C2×C42C22×D6111yescharacteristic
430#107—22no—
431#107—22no—
432#107—1222no—
433#106—1222no—
434C4⋊C8—614no—
435C4⋊C8—1222no—
436#102—1222no—
437#102—1222no—
438#102—1222no—
439#102—1222no—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442#97—1222no—
443C4×C8—22no—
444#95—21no—
445#95—21no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—611no—
454C24—1221no—
455C2×Q8—4881no—
456C2×Q8—4881no—
457C2×Q8—2442no—
458C2×D4—612no—
459C2×D4—1224no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—4881no—
466C2×D4—4881no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—4881no—
484C22×C4—4881no—
485C22×C4—2442no—
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—612no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8D4×D6112yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—4881no—
519C4⋊C4—2442no—
520C4⋊C4#1175822212yes—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1224no—
524C22⋊C4—4881no—
525C22⋊C4—4881no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—612no—
533C42—612no—
534C42—4881no—
535C23—1221no—
536C23—1222no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23C2×D4×D6111yescharacteristic
550Q8—4884no—
551D4—1222no—
552D4—1222no—
553D4—4884no—
554D4—2441no—
555D4—2441no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—4884no—
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×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C2×C4#1451982511yescharacteristic
602C2×C4#1451982511yescharacteristic
603C8—22no—
604C8—41no—
605C8—1222no—
606C8—2441no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—2441no—
612C22—2442no—
613C22—2441no—
614C22—2441no—
615C22—2442no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2442no—
621C22#121×D6111yescharacteristic
622C22#1684027111yescharacteristic
623C22#1684033711yescharacteristic
624C22#1684056311yescharacteristic
625C22#1684056411yescharacteristic
626C22#1684056911yescharacteristic
627C22#1684057511yescharacteristic
628C4—41no—
629C4—42no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—1222no—
635C4—1222no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—2441no—
640C4#1684041311yescharacteristic
641C4#1684042211yescharacteristic
642C4#1684044411yescharacteristic
643C4#1684048411yescharacteristic
644C2—4882no—
645C2—4881no—
646C2#2144924711yescharacteristic
647C2#2145546211yescharacteristic
648C2#2145550911yescharacteristic
649C2#2145599911yescharacteristic
650C2#2145608711yescharacteristic
651C2#2145707511yescharacteristic
652C2#2145710511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1942C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1942D61
hypocenterC61#24712
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
1. C2×#16840041C2semidirect
2. C2×#16839821C2semidirect
3. #21451646C2semidirect
5. #21447251C2semidirect
6. #21447243C2semidirect
9. #21446765C2semidirect
10. #21446584C2semidirect
11. #21446559C2semidirect
12. #21446515C2semidirect
13. #21446502C2semidirect
14. #21445828C2semidirect
15. C61×#1942C2semidirect
97. #14519706D4semidirect
329. #1942D61semidirect
387. C61#24712semidirect

Maximal subgroups

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

subgroupindex
C2×#168400412
C2×#168398212
#214516462
#214485102
#214472512
#214472432
#214470112
#214470032
#214467652
#214465842
#214465592
#214465152
#214465022
#214458282
C61×#19422
#2471261