Finite Group Databasebeta

← #29748501

Subgroups — #29748501

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450196C211yesmaximal, characteristic
2#21448848C211yesmaximal, characteristic
3#21447142C211yesmaximal, characteristic, complemented
4#21447018C211yesmaximal, characteristic
5#21446957C211yesmaximal, characteristic, complemented
6#21446830C211yesmaximal, characteristic
7#21446510C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446431C211yesmaximal, characteristic
10#21446407C211yesmaximal, characteristic, complemented
11#21446161C211yesmaximal, characteristic, complemented
12#21446130C211yesmaximal, characteristic
13#21445913C211yesmaximal, characteristic
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1987C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C2×#14519757C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839701—21no—
32#16839673—21no—
33#16839669—21no—
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839654—21no—
39#16839653—21no—
40#16839652—21no—
41#16839647—21no—
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×C4⋊Dic61—21no—
45C4×#11758198C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×Dic244—22no—
63C2×Dic244—41no—
64C2×Dic244—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519757—22no—
68#14519757—22no—
69#14519757—22no—
70#14519757—41no—
71#14519742—22no—
72#14519742—41no—
73#14519735—22no—
74#14519735—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198D411yescharacteristic
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91#14519708—22no—
92#14519702—22no—
93#14519702—41no—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—41no—
98#14519689—41no—
99#14519689—41no—
100#14519687—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4⋊Q8×C61—22no—
116C4⋊Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—41no—
125#95×C61—21no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C2×Q8×C61—81no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122#4112yes—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61#4112yes—
175C4⋊C4×C61#4112yes—
176C4⋊C4×C61#4112yes—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—42no—
180#11758198—24no—
181#11758198—22no—
182#11758198—41no—
183#11758198—42no—
184#11758198—42no—
185#11758198—81no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—42no—
192C4⋊Dic61—42no—
193#11758196—42no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—81no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205Dic244—44no—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214Q8×C61—84no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#13712yes—
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—22no—
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—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264Dic122—22no—
265Dic122—22no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272C8×C61—41no—
273C8×C61—41no—
274#92418—41no—
275#92418—41no—
276#26750—611nomaximal, Sylow
277C22×C61#52511yescharacteristic
278C22×C61#54611yescharacteristic
279C22×C61#54611yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C22×C61#57511yescharacteristic
282C22×C61#57711yescharacteristic
283C22×C61#57811yescharacteristic
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—41no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—81no—
297C4×C61#52911yescharacteristic
298C4×C61#53811yescharacteristic
299C4×C61#55511yescharacteristic
300C4×C61#55911yescharacteristic
301Dic61—41no—
302Dic61—42no—
303Dic61—41no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308#2492—611no—
309#2362—611no—
310#2056—611no—
311#2035—611no—
312#2024—611no—
313#2001—611no—
314#1987D6111yescharacteristic, complemented
315#1935—611no—
316#1927—611no—
317#1924—611no—
318#1920—611no—
319#1878—611no—
320#1873—611no—
321#1825—611no—
322#1763—611no—
323C2×C61#245911yescharacteristic
324C2×C61#311611yescharacteristic
325C2×C61#312111yescharacteristic
326C2×C61#331111yescharacteristic
327C2×C61#331211yescharacteristic
328C2×C61#331311yescharacteristic
329C2×C61#333111yescharacteristic
330D61—81no—
331C22×Q16—611no—
332C2×C4⋊Q8C2×D6111yescharacteristic
333C2×C22⋊Q8—611no—
334C2×C22⋊Q8—611no—
335C2×C4×Q8—611no—
336C2×#117—611no—
337C2×#106—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×C22⋊C8—611no—
346C2×C22⋊C8—611no—
347C2×#97—611no—
348#400—1221no—
349#393—1221no—
350#391—611no—
351#391—611no—
352#391—611no—
353#391—1221no—
354#389—611no—
355#388—1221no—
356#388—1221no—
357#388—1221no—
358#387—1221no—
359#380—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C4⋊C4C2×D6111yescharacteristic
363C4×C22⋊C4—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340C2×D6111yescharacteristic
369#340C2×D6111yescharacteristic
370#340C2×D6111yescharacteristic
371#340C2×D6111yescharacteristic
372#336—611no—
373#336—611no—
374C61#2675011yescharacteristic, complemented, Sylow
375C22×Q8—21no—
376C22×Q8—611no—
377C22×Q8—611no—
378C23×C4—611no—
379C2×Q16—1222no—
380C2×Q16—2441no—
381C2×Q16—2441no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386C4⋊Q8—22no—
387C4⋊Q8—41no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—2441no—
392C4×Q8—1222no—
393C4×Q8—2441no—
394#117—1222no—
395#117—2441no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C22⋊C4—611no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C42—21no—
414C2×C42—611no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42C22×D6111yescharacteristic
419#106—1222no—
420C4⋊C8—41no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#103—1222no—
425#103—2441no—
426#103—2441no—
427C22⋊C8—2441no—
428C22⋊C8—2441no—
429#97—2441no—
430#95—21no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C24—611no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8—612no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478Q16—2444no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4#1175822212yes—
506C4⋊C4#1175822612yes—
507C4⋊C4#1175822612yes—
508C22⋊C4—1224no—
509C22⋊C4—1222no—
510C22⋊C4—4881no—
511C22⋊C4—2441no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—1221no—
518C42—1221no—
519C42—4881no—
520C42—2442no—
521C42—2442no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23C2×D4×D6111yescharacteristic
530Q8—84no—
531Q8—1222no—
532Q8—1222no—
533Q8—2441no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
538Q8—2441no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4#1451980612yes—
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4C2×#1175822611yescharacteristic
585C2×C4C2×#1175822611yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982311yescharacteristic
588C8—41no—
589C8—41no—
590C8—2441no—
591C8—2441no—
592C22—2441no—
593C22—2441no—
594C22—2441no—
595C22—2441no—
596C22—2441no—
597C22—2441no—
598C22—2441no—
599C22#1684027511yescharacteristic
600C22#1684040211yescharacteristic
601C22#1684040311yescharacteristic
602C22C2×#1451980611yescharacteristic
603C22#1684055811yescharacteristic
604C22#1684057211yescharacteristic
605C22#1684057511yescharacteristic
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—4881no—
613C4—4881no—
614C4—4881no—
615C4—2441no—
616C4—2442no—
617C4—2441no—
618C4—2441no—
619C4#1684029911yescharacteristic
620C4#1684034611yescharacteristic
621C4#1684047811yescharacteristic
622C4#1684049711yescharacteristic
623C2—4881no—
624C2#2144976311yescharacteristic
625C2#2145598311yescharacteristic
626C2#2145605311yescharacteristic
627C2#2145785611yescharacteristic
628C2#2145786611yescharacteristic
629C2#2145788211yescharacteristic
630C2#2145813211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1987D61
hypocenterC61#26750
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #21447142C2semidirect
5. #21446957C2semidirect
7. #21446510C2semidirect
8. #21446453C2semidirect
10. #21446407C2semidirect
11. #21446161C2semidirect
14. #21445648C2semidirect
15. C61×#1987C2semidirect
314. #1987D61semidirect
374. C61#26750semidirect

Maximal subgroups

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

subgroupindex
#214501962
#214488482
#214471422
#214470182
#214469572
#214468302
#214465102
#214464532
#214464312
#214464072
#214461612
#214461302
#214459132
#214456482
C61×#19872
#2675061