Finite Group Databasebeta

← #29729469

Subgroups — #29729469

Order 15616.

792 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×#16840131C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic
3#21450811C211yesmaximal, characteristic
4#21448915C211yesmaximal, characteristic, complemented
5#21447215C211yesmaximal, characteristic, complemented
6#21447201C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic
8#21447058C211yesmaximal, characteristic
9#21446750C211yesmaximal, characteristic
10#21446750C211yesmaximal, characteristic
11#21446742C211yesmaximal, characteristic, complemented
12#21446542C211yesmaximal, characteristic, complemented
13#21446458C211yesmaximal, characteristic, complemented
14#21446451C211yesmaximal, characteristic, complemented
15C61×#1762C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20#16840131—24no—
21#16840062—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519698C2211yescharacteristic
37#16839687—21no—
38#16839687—21no—
39#16839667C2211yescharacteristic
40#16839667C2211yescharacteristic
41#16839659—21no—
42#16839658C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839649—21no—
45#16839646—21no—
46#16839622—21no—
47#16839622—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49C4⋊C4×Dic61C2211yescharacteristic
50C4×#11758196—21no—
51C4×C4⋊Dic61C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777—22no—
67#14519769—22no—
68#14519769D414yes—
69#14519764—22no—
70#14519764—22no—
71#14519764—41no—
72#14519764—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—22no—
92#14519708—22no—
93C4⋊#92418—22no—
94#14519706—41no—
95#14519706—41no—
96#14519702—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702—22no—
100#14519700—22no—
101#14519700—22no—
102#14519700—41no—
103#14519700—41no—
104#14519698—22no—
105#14519698—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#102×C61—22no—
129#102×C61—22no—
130#97×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—42no—
181C42×C61—81no—
182C42×C61C2×D412yes—
183C42×C61#4112yes—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61—41no—
192C4⋊Dic61—42no—
193C4⋊Dic61—42no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61C2×D412yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—42no—
199#11758196—81no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—81no—
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
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
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—84no—
238C2×C4×C61#12112yes—
239C2×C4×C61#12112yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
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—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—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—41no—
281#92418—41no—
282#25289—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#54611yescharacteristic
292C22×C61#54811yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57711yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—42no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#52911yescharacteristic
303C4×C61#54211yescharacteristic
304C4×C61#54811yescharacteristic
305C4×C61#55011yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—81no—
313Dic61—81no—
314C2×#491—611no—
315C2×#478—611no—
316#2547—611no—
317#2370—611no—
318#2068—611no—
319#2066—611no—
320#2042—611no—
321#2042—611no—
322#1983—611no—
323#1983—611no—
324#1981—611no—
325#1940—611no—
326#1928—611no—
327#1927—611no—
328#1762D6111yescharacteristic
329C2×C61—82no—
330C2×C61#242611yescharacteristic
331C2×C61#304011yescharacteristic
332C2×C61#304611yescharacteristic
333C2×C61#330711yescharacteristic
334C2×C61#331511yescharacteristic
335C2×C61#337611yescharacteristic
336C2×C61#337711yescharacteristic
337C2×C4⋊Q8—611no—
338C2×#125—611no—
339C2×#124—611no—
340C2×#124—611no—
341C2×C4×D4C2×D6111yescharacteristic
342#491—1224no—
343#478—1224no—
344C2×#107—611no—
345C2×#106—611no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×#102C2×D6111yescharacteristic
361C2×#97C2×D6111yescharacteristic
362#397—1221no—
363#397—1221no—
364#391—611no—
365#391—611no—
366#389—611no—
367#389—611no—
368#389—1221no—
369#387—1221no—
370#387—1221no—
371#380—1221no—
372#380—1221no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378#340C2×D6111yescharacteristic
379#340C2×D6111yescharacteristic
380#336—611no—
381#336—611no—
382C61#2528911yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×D4C22×D6111yescharacteristic
386C23×C4—21no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391C4⋊Q8—1222no—
392C4⋊Q8—1222no—
393#125—614no—
394#125—1222no—
395#124—1222no—
396#124—1222no—
397#124—2441no—
398#124—2441no—
399C4×D4—22no—
400C4×D4—22no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C42—21no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42C22×D6111yescharacteristic
425#107—1222no—
426#106—1222no—
427C4⋊C8—1222no—
428C4⋊C8—2441no—
429C4⋊C8—2441no—
430#103—1222no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#103—2441no—
437#103—2441no—
438#102—22no—
439#102—22no—
440#102—1222no—
441#102—1222no—
442#97—22no—
443#95—21no—
444#95—611no—
445#95—611no—
446#95—1221no—
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—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1175822212yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490C2×C8—21no—
491C2×C8—21no—
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—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—612no—
508C4⋊C4—612no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1224no—
512C4⋊C4—1224no—
513C4⋊C4—4881no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4#1175822212yes—
520C22⋊C4—42no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C42—81no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—4881no—
531C42—2442no—
532C42#1175822212yes—
533C42D4×D6112yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—2441no—
543Q8—2441no—
544Q8—2441no—
545Q8—2441no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
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—2441no—
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#1451975012yes—
593C2×C4#1451975012yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×#1175822211yescharacteristic
596C2×C4C2×#1175822211yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4#1451982311yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—41no—
601C8—41no—
602C8—2441no—
603C8—2441no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1451975011yescharacteristic
612C22#1684040811yescharacteristic
613C22#1684042111yescharacteristic
614C22#1684055811yescharacteristic
615C22#1684055811yescharacteristic
616C22#1684057211yescharacteristic
617C22#1684057811yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—4881no—
624C4—4881no—
625C4—2441no—
626C4—2441no—
627C4—2442no—
628C4—2441no—
629C4—2441no—
630C4—2442no—
631C4#1684029611yescharacteristic
632C4#1684038311yescharacteristic
633C4#1684042111yescharacteristic
634C4#1684043311yescharacteristic
635C2—82no—
636C2#2144938511yescharacteristic
637C2#2145531211yescharacteristic
638C2#2145536811yescharacteristic
639C2#2145780011yescharacteristic
640C2#2145792011yescharacteristic
641C2#2145864711yescharacteristic
642C2#2145866211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1762D61
hypocenterC61#25289
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. C2×#16840131C2semidirect
4. #21448915C2semidirect
5. #21447215C2semidirect
6. #21447201C2semidirect
11. #21446742C2semidirect
12. #21446542C2semidirect
13. #21446458C2semidirect
14. #21446451C2semidirect
382. C61#25289semidirect

Maximal subgroups

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

subgroupindex
C2×#168401312
C2×#168400622
#214508112
#214489152
#214472152
#214472012
#214470582
#214467502
#214467422
#214465422
#214464582
#214464512
C61×#17622
#2528961