Finite Group Databasebeta

← #30619828

Subgroups — #30619828

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461940C211yesmaximal, characteristic
2#21461918C211yesmaximal, characteristic, complemented
3#21461892C211yesmaximal, characteristic, complemented
4#21461892C211yesmaximal, characteristic, complemented
5#21461889C211yesmaximal, characteristic, complemented
6#21461889C211yesmaximal, characteristic, complemented
7#21461873C211yesmaximal, characteristic, complemented
8#21461872C211yesmaximal, characteristic, complemented
9#21461796C211yesmaximal, characteristic
10#21461766C211yesmaximal, characteristic, complemented
11#21461677C211yesmaximal, characteristic, complemented
12#21461619C211yesmaximal, characteristic, complemented
13#21461561C211yesmaximal, characteristic, complemented
14#21461549C211yesmaximal, characteristic, complemented
15#21449075C211yesmaximal, characteristic
16C22×#11758217C2211yescharacteristic
17C2×#14519844C2211yescharacteristic
18C2×#14519844C2211yescharacteristic
19C2×#14519844C2211yescharacteristic
20C2×C4⋊#92420C2211yescharacteristic
21C2×C4⋊#92420C2211yescharacteristic
22C2×C22⋊#92420C2211yescharacteristic
23C2×C22⋊#92420C2211yescharacteristic
24C2×C22⋊#92420C2211yescharacteristic
25C2×C22⋊#92420C2211yescharacteristic
26C2×C22⋊#92420C2211yescharacteristic
27C2×C22⋊#92420C2211yescharacteristic
28C2×C22⋊#92420C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×C4×#92420C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34#16840628C2211yescharacteristic
35#16840628C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44C2×D4×Dic61C2211yescharacteristic
45C2×#14519733C2211yescharacteristic
46#16839704C411yescharacteristic
47#16839698C411yescharacteristic
48#16839695C411yescharacteristic
49#16839690C411yescharacteristic
50#16839690C411yescharacteristic
51#16839687C411yescharacteristic
52#16839676C411yescharacteristic
53C4⋊C4×Dic61C2211yescharacteristic
54#16839613C2211yescharacteristic
55C22⋊C4×Dic61C2211yescharacteristic
56C22⋊C4×Dic61C2211yescharacteristic
57C22⋊C4×Dic61C2211yescharacteristic
58C61×#394C411yescharacteristic
59C2×#11758217—22no—
60C2×#11758217—41no—
61C2×#11758217—41no—
62C22×#92420C2311yescharacteristic
63C22×#92420C2311yescharacteristic
64C22×#92420C2311yescharacteristic
65C22×#92420C2311yescharacteristic
66C22×#92420C2311yescharacteristic
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70#14519844—22no—
71#14519844—41no—
72#14519844—41no—
73C4⋊#92420—41no—
74C4⋊#92420—41no—
75C22⋊#92420—22no—
76C22⋊#92420—22no—
77C22⋊#92420—22no—
78C22⋊#92420—41no—
79C22⋊#92420—41no—
80C22⋊#92420—41no—
81C22⋊#92420—41no—
82#14519832—22no—
83#14519832—41no—
84C4×#92420—41no—
85C4×#92420—41no—
86C4×#92420—41no—
87C23×Dic61C2311yescharacteristic
88C23×Dic61C2311yescharacteristic
89D4×Dic61—41no—
90D4×Dic61—41no—
91#14519733—41no—
92#14519733—41no—
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×C22⋊Dic61C2×C411yescharacteristic
100C2×C22⋊Dic61C2×C411yescharacteristic
101C2×C22⋊Dic61C2×C411yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×C22⋊Dic61C2×C411yescharacteristic
104C2×C22⋊Dic61C2×C411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110#14519684C2×C411yescharacteristic
111#14519684C2×C411yescharacteristic
112#14519684C2×C411yescharacteristic
113#14519684C2×C411yescharacteristic
114#14519684C2×C411yescharacteristic
115#14519684C2×C411yescharacteristic
116#14519684C2×C411yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119C22×D4×C61C2×C411yescharacteristic
120C2×C4⋊C4×C61C2×C411yescharacteristic
121C2×C22⋊C4×C61C2×C411yescharacteristic
122C2×C22⋊C4×C61C2×C411yescharacteristic
123C2×C22⋊C4×C61C2×C411yescharacteristic
124C2×C22⋊C4×C61C2×C411yescharacteristic
125#95×C61C2×C411yescharacteristic
126C24×C61C22×C411yescharacteristic
127C24×C61C22×C411yescharacteristic
128C2×D4×C61—22no—
129C2×D4×C61—41no—
130C2×D4×C61—41no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C2411yescharacteristic
156#11758217—44no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420#4112yes—
180C2×#92420#4112yes—
181C4⋊C4×C61—22no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61—41no—
194#11758196—22no—
195#11758196—22no—
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61C23×C411yescharacteristic
223D4×C61—44no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61#13112yes—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
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—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61#13112yes—
268C2×Dic61C2×#4111yescharacteristic
269C2×Dic61C2×#4111yescharacteristic
270C2×Dic61H(5,𝔽2)11yescharacteristic
271C2×Dic61H(5,𝔽2)11yescharacteristic
272C2×Dic61H(5,𝔽2)11yescharacteristic
273C2×Dic612−1+411yescharacteristic
274#92420—22no—
275#92420—22no—
276#92420—41no—
277#92420—41no—
278#92420—41no—
279#92420—41no—
280#92420—41no—
281#92420—41no—
282#22360—611nomaximal, Sylow
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61C4×#4111yescharacteristic
298C22×C61#51811yescharacteristic
299C22×C61#52011yescharacteristic
300C22×C61C2×#13111yescharacteristic
301C22×C61C2×#13111yescharacteristic
302C22×C61#56811yescharacteristic
303C22×C61#56811yescharacteristic
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309Dic61—22no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—41no—
315Dic61—41no—
316Dic61#53811yescharacteristic
317Dic61#53911yescharacteristic
318Dic61#54011yescharacteristic
319Dic61#54211yescharacteristic
320#2395—611no—
321#1954—611no—
322#1939—611no—
323#1924—611no—
324#1924—611no—
325#1923—611no—
326#1923—611no—
327#1916—611no—
328#1916—611no—
329#1870—611no—
330#1851—611no—
331#1798—611no—
332#1771—611no—
333#1753—611no—
334#1750—611no—
335C2×C61—81no—
336C2×C61—81no—
337C2×C61#233511yescharacteristic
338C2×C61#296911yescharacteristic
339C2×C61#296911yescharacteristic
340C2×C61#297611yescharacteristic
341C2×C61#298111yescharacteristic
342C2×C61#299411yescharacteristic
343C2×C61#299511yescharacteristic
344C22×M16—611no—
345C2×C4×D4—611no—
346C2×#117—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—611no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×#97—611no—
360C2×#97—611no—
361C2×C4×C8—611no—
362C2×C4×C8—611no—
363C2×C4×C8—611no—
364#400—611no—
365#399—611no—
366#398—611no—
367#397—611no—
368#397—611no—
369#397—611no—
370#394—611no—
371#394#692311yescharacteristic
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C22⋊C4—611no—
375C4×C22⋊C4—611no—
376C4×C22⋊C4—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387C61#2236011yescharacteristic, complemented, Sylow
388C22×D4C2×#692311yescharacteristic
389C23×C4—611no—
390C23×C4—611no—
391C2×M16—1222no—
392C2×M16—2441no—
393C2×M16—2441no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—611no—
402C4×D4—2441no—
403C4×D4—2441no—
404#117—2441no—
405#117—2441no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4C2×#692311yescharacteristic
413C2×C22⋊C4—611no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—611no—
419C2×C22⋊C4C2×#692311yescharacteristic
420C2×C22⋊C4C2×#692311yescharacteristic
421C2×C22⋊C4C2×#692311yescharacteristic
422C2×C22⋊C4C2×#692311yescharacteristic
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C4⋊C8—1222no—
429C4⋊C8—2441no—
430C4⋊C8—2441no—
431C4⋊C8—2441no—
432C4⋊C8—2441no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—2441no—
437C22⋊C8—2441no—
438C22⋊C8—2441no—
439C22⋊C8—2441no—
440#97—1222no—
441#97—2441no—
442C4×C8—2441no—
443C4×C8—2441no—
444C4×C8—2441no—
445#95—611no—
446#95—611no—
447#95—611no—
448#95—611no—
449#95—611no—
450#95—611no—
451#95—611no—
452#95—611no—
453#95—611no—
454#95C2×#692311yescharacteristic
455C24C22×#692311yescharacteristic
456C24C22×#692311yescharacteristic
457C2×D4—22no—
458C2×D4—41no—
459C2×D4—41no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4C22×#692311yescharacteristic
481C22×C4C22×#692311yescharacteristic
482C22×C4C22×#692311yescharacteristic
483C22×C4C22×#692311yescharacteristic
484C22×C4C22×#692311yescharacteristic
485M16—2444no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
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—22no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C23×#692311yescharacteristic
552D4—44no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—612no—
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—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451986412yes—
603C8—1222no—
604C8—1222no—
605C8—2441no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8—2441no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22#41×#692311yescharacteristic
626C22#1684082911yescharacteristic
627C22#1684083111yescharacteristic
628C22C2×#1451986411yescharacteristic
629C22D61⋊#13111yescharacteristic
630C22#1684084011yescharacteristic
631C22#1684084011yescharacteristic
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—611no—
638C4—611no—
639C4—611no—
640C4—611no—
641C4—1222no—
642C4—2441no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4—2441no—
647C4—2441no—
648C2—81no—
649C2—81no—
650C2#2146231811yescharacteristic
651C2#2146265011yescharacteristic
652C2#2146265311yescharacteristic
653C2#2146268511yescharacteristic
654C2#2146270711yescharacteristic
655C2#2146276911yescharacteristic
656C2#2146277211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#394C4
prime radicalC22×D4×C61C2×C4
cosocleC22×Dic61C24
hypercenter#394#6923
hypocenterC61#22360
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. #21461918C2semidirect
3. #21461892C2semidirect
4. #21461892C2semidirect
5. #21461889C2semidirect
6. #21461889C2semidirect
7. #21461873C2semidirect
8. #21461872C2semidirect
10. #21461766C2semidirect
11. #21461677C2semidirect
12. #21461619C2semidirect
13. #21461561C2semidirect
14. #21461549C2semidirect
387. C61#22360semidirect

Maximal subgroups

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

subgroupindex
#214619402
#214619182
#214618922
#214618892
#214618732
#214618722
#214617962
#214617662
#214616772
#214616192
#214615612
#214615492
#214490752
#2236061