Finite Group Databasebeta

← #29729423

Subgroups — #29729423

Order 15616.

822 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×#16840120C211yesmaximal, characteristic
2C2×#16840056C211yesmaximal, characteristic, complemented
3#21450801C211yesmaximal, characteristic
4#21448878C211yesmaximal, characteristic, complemented
5#21447209C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic
7#21447066C211yesmaximal, characteristic, complemented
8#21447059C211yesmaximal, characteristic
9#21446750C211yesmaximal, characteristic
10#21446750C211yesmaximal, characteristic
11#21446742C211yesmaximal, characteristic, complemented
12#21446541C211yesmaximal, characteristic, complemented
13#21446458C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1925C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×#14519742C2211yescharacteristic
22#16840120—24no—
23#16840056—24no—
24C2×#14519710C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519698C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519687C2211yescharacteristic
39#16839685—21no—
40#16839685—21no—
41#16839667C2211yescharacteristic
42#16839666—21no—
43#16839661C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839655—21no—
46#16839655—21no—
47#16839649—21no—
48#16839646—21no—
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51C4×#11758196C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519777—22no—
67#14519777—41no—
68#14519776—22no—
69#14519776D414yes—
70#14519764—22no—
71#14519764—22no—
72#14519764—22no—
73#14519764—41no—
74C4×Dic122—44no—
75#14519742—22no—
76#14519742—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196D411yescharacteristic
86C2×#11758196D411yescharacteristic
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97C4⋊#92418—41no—
98#14519706—22no—
99#14519706—41no—
100#14519702—22no—
101#14519702—22no—
102#14519702—22no—
103#14519702—41no—
104#14519700—22no—
105#14519700—22no—
106#14519700—22no—
107#14519700—22no—
108#14519698—22no—
109#14519698—41no—
110#14519687—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×D4×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#107×C61—22no—
130#102×C61—22no—
131#102×C61—22no—
132#95×C61—21no—
133C24×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—24no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122#4112yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61#4112yes—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61C2×D412yes—
197#11758196—22no—
198#11758196—24no—
199#11758196—41no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C2×D412yes—
208C4×Dic61#4112yes—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
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×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
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—
270C2×Dic61#12112yes—
271C2×Dic61#12112yes—
272Dic122—22no—
273Dic122—22no—
274Dic122—48no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281C8×C61—41no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#25286—611nomaximal, Sylow
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57811yescharacteristic
299C22×C61#57811yescharacteristic
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61#52911yescharacteristic
306C4×C61#54211yescharacteristic
307C4×C61#54811yescharacteristic
308C4×C61#55011yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—42no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317C2×#489—611no—
318C2×#477—611no—
319#2545—611no—
320#2365—611no—
321#2068—611no—
322#2067—611no—
323#2043—611no—
324#2042—611no—
325#1983—611no—
326#1983—611no—
327#1981—611no—
328#1940—611no—
329#1928—611no—
330#1925D6111yescharacteristic
331#1763—611no—
332C2×C61—161no—
333C2×C61#242611yescharacteristic
334C2×C61#303611yescharacteristic
335C2×C61#304411yescharacteristic
336C2×C61#330811yescharacteristic
337C2×C61#331311yescharacteristic
338C2×C61#336711yescharacteristic
339C2×C61#337711yescharacteristic
340C2×C4⋊Q8—611no—
341C2×C4⋊Q8—611no—
342C2×#124—611no—
343C2×#124—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346#489—1224no—
347#477—1224no—
348C2×#107—611no—
349C2×#107C2×D6111yescharacteristic
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102C2×D6111yescharacteristic
364C2×#102C2×D6111yescharacteristic
365C2×#97—611no—
366#396—1221no—
367#396—1221no—
368#391—611no—
369#391—1221no—
370#390C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#388—1221no—
374#388—1221no—
375#387—1221no—
376#387—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340C2×D6111yescharacteristic
383#336—611no—
384#336C2×D6111yescharacteristic
385C61#2528611yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8—611no—
388C22×Q8—1221no—
389C22×D4C22×D6111yescharacteristic
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—614no—
395C4⋊Q8—1222no—
396C4⋊Q8—1222no—
397C4⋊Q8—2441no—
398#124—1222no—
399#124—1222no—
400#124—1222no—
401#124—2441no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—2444no—
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×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—21no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431#107—22no—
432#107—1222no—
433C4⋊C8—1222no—
434C4⋊C8—2441no—
435C4⋊C8—2441no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—22no—
445#102—22no—
446#102—1222no—
447#102—2441no—
448#97—1222no—
449#95—21no—
450#95—611no—
451#95—611no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C2×Q8—612no—
460C2×Q8—1224no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—2442no—
473C2×Q8—2442no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—612no—
511C4⋊C4—1224no—
512C4⋊C4—1221no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2441no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4#1175822212yes—
523C4⋊C4D4×D6112yes—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—612no—
534C42—612no—
535C42—1221no—
536C42—1221no—
537C42—2442no—
538C42—2442no—
539C42—2442no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D6111yescharacteristic
548Q8—1222no—
549Q8—1222no—
550Q8—2448no—
551Q8—2441no—
552Q8—2441no—
553Q8—2441no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
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—2441no—
599C2×C4#1451980112yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—41no—
607C8—41no—
608C8—2441no—
609C8—2441no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#121×D6111yescharacteristic
618C22#1684040411yescharacteristic
619C22#1684042111yescharacteristic
620C22C2×#1451980111yescharacteristic
621C22#1684055811yescharacteristic
622C22#1684057611yescharacteristic
623C22#1684057811yescharacteristic
624C4—22no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—1222no—
630C4—1222no—
631C4—4881no—
632C4—4881no—
633C4—2441no—
634C4—2442no—
635C4—2442no—
636C4—2441no—
637C4#1684029011yescharacteristic
638C4#1684037811yescharacteristic
639C4#1684042011yescharacteristic
640C4#1684043311yescharacteristic
641C2—161no—
642C2#2144938411yescharacteristic
643C2#2145527811yescharacteristic
644C2#2145534711yescharacteristic
645C2#2145780811yescharacteristic
646C2#2145788811yescharacteristic
647C2#2145853511yescharacteristic
648C2#2145866211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1925D61
hypocenterC61#25286
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. C2×#16840056C2semidirect
4. #21448878C2semidirect
5. #21447209C2semidirect
7. #21447066C2semidirect
11. #21446742C2semidirect
12. #21446541C2semidirect
13. #21446458C2semidirect
14. #21445648C2semidirect
385. C61#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
C2×#168400562
#214508012
#214488782
#214472092
#214472072
#214470662
#214470592
#214467502
#214467422
#214465412
#214464582
#214456482
C61×#19252
#2528661