Finite Group Databasebeta

← #30613169

Subgroups — #30613169

Order 15616.

740 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
1#21462561C211yesmaximal, characteristic, complemented
2D4×#11758217C211yesmaximal, characteristic, complemented
3#21461469C211yesmaximal, characteristic, complemented
4#21461467C211yesmaximal, characteristic, complemented
5#21461441C211yesmaximal, characteristic, complemented
6#21461440C211yesmaximal, characteristic, complemented
7#21461440C211yesmaximal, characteristic, complemented
8#21461438C211yesmaximal, characteristic, complemented
9#21461416C211yesmaximal, characteristic, complemented
10#21461415C211yesmaximal, characteristic, complemented
11#21461386C211yesmaximal, characteristic, complemented
12#21461164C211yesmaximal, characteristic, complemented
13C2×D4⋊#92420C211yesmaximal, characteristic, complemented
14#21461030C211yesmaximal, characteristic, complemented
15#21455814C211yesmaximal, characteristic
16C2×#14519864—21no—
17C22×#11758217—21no—
18#16840797C2211yescharacteristic
19#16840797C2211yescharacteristic
20#16840797C2211yescharacteristic, complemented
21#16840796C2211yescharacteristic, complemented
22#16840794—21no—
23#16840793—21no—
24#16840793—21no—
25#16840791—21no—
26#16840790C2211yescharacteristic, complemented
27D4×#92420C2211yescharacteristic
28D4×#92420C2211yescharacteristic
29D4×#92420C2211yescharacteristic, complemented
30#16840765C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32#16840735—21no—
33#16840732—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840624C2211yescharacteristic, complemented
37#16840624C2211yescharacteristic, complemented
38#16840623C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840610C2211yescharacteristic, complemented
41#16840610C2211yescharacteristic, complemented
42D4⋊#92420C2211yescharacteristic
43D4⋊#92420C2211yescharacteristic
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic, complemented
47D4⋊#92420C2211yescharacteristic, complemented
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51#16840603C2211yescharacteristic, complemented
52#16840307C2211yescharacteristic
53C2×D4×Dic61C2211yescharacteristic
54#16840076C411yescharacteristic
55#16840073C411yescharacteristic
56#16840040C411yescharacteristic
57#16840040C411yescharacteristic
58#16839955C411yescharacteristic
59#16839955C411yescharacteristic
60#16839896C411yescharacteristic
61D8×Dic61C2211yescharacteristic
62D8×Dic61C2211yescharacteristic
63#16839842C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#447C411yescharacteristic
67#14519864—44no—
68C2×#11758213—21no—
69C2×#11758217—21no—
70C2×#11758217—21no—
71C2×#11758217—21no—
72C2×#11758217—21no—
73C2×#11758217—44no—
74C2×#11758217D411yescharacteristic
75C2×#11758217D411yescharacteristic
76C2×D61⋊C8—21no—
77C22×#92420—21no—
78C22×#92420—21no—
79C22×#92420—21no—
80C22×#92420—21no—
81C22×#92420—21no—
82#14519844—21no—
83C4⋊#92420—21no—
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86C4⋊#92420C2311yescharacteristic
87C4⋊#92420C2311yescharacteristic
88#14519834—21no—
89#14519834—21no—
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420—21no—
93C22⋊#92420—21no—
94C22⋊#92420—21no—
95C22⋊#92420—21no—
96#14519832D411yescharacteristic
97#14519832D411yescharacteristic
98C4×#92420—21no—
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C4×#92420C2311yescharacteristic
102C4×#92420C2311yescharacteristic
103C2×#11758224—21no—
104C23×Dic61—21no—
105C2×#11758199C2×C411yescharacteristic
106C2×#11758199C2×C411yescharacteristic
107#14519776C2×C411yescharacteristic
108#14519773C2×C411yescharacteristic
109#14519763—21no—
110#14519751C2×C411yescharacteristic
111#14519749—21no—
112D4×Dic61—21no—
113D4×Dic61—21no—
114D4×Dic61C2×C411yescharacteristic
115D4×Dic61C2×C411yescharacteristic
116D4×Dic61C2311yescharacteristic
117D4×Dic61C2311yescharacteristic
118D4×Dic61C2311yescharacteristic
119#14519739C2311yescharacteristic
120#14519737—21no—
121C2×C4⋊Dic61C2×C411yescharacteristic
122C2×C22⋊Dic61—21no—
123C2×C4×Dic61C2311yescharacteristic
124#14519711C2×C411yescharacteristic
125#14519711C2×C411yescharacteristic
126#14519710C2×C411yescharacteristic
127#14519710C2×C411yescharacteristic
128#14519706C2×C411yescharacteristic
129#14519706C2×C411yescharacteristic
130#14519699C2×C411yescharacteristic
131#14519699C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134#14519698C2×C411yescharacteristic
135#14519698C2×C411yescharacteristic
136C22⋊#92418C2×C411yescharacteristic
137C8×Dic61C2311yescharacteristic
138C8×Dic61C2311yescharacteristic
139C22×D4×C61C2×C411yescharacteristic
140C2×D8×C61C2×C411yescharacteristic
141C2×D8×C61C2×C411yescharacteristic
142#121×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144#102×C61C2×C411yescharacteristic
145C22⋊C8×C61C2×C411yescharacteristic
146C24×C61—21no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61C22×C411yescharacteristic
152C2×D4×C61C22×C411yescharacteristic
153C2×D4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C2×C22⋊D61—21no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61#4111yescharacteristic
168#11758224—42no—
169#11758224—42no—
170C2×D244C22×C411yescharacteristic
171C2×C4×D61—21no—
172C2×Dic122—21no—
173#11758217—22no—
174#11758217—22no—
175#11758217—42no—
176#11758217—42no—
177#11758217—42no—
178#11758217—42no—
179C2×#92420—21no—
180C2×#92420—21no—
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—42no—
186C2×#92420—42no—
187C2×#92420—42no—
188C2×#92420—42no—
189C2×#92420—42no—
190C2×#92420C2×D411yescharacteristic
191C2×#92420C2×D411yescharacteristic
192C2×#92420C2×D411yescharacteristic
193C2×#92420C2×D411yescharacteristic
194#11758213—42no—
195D61⋊C8—42no—
196D8×C61—22no—
197D8×C61—41no—
198C2×C8×C61C22×C411yescharacteristic
199C2×C8×C61C22×C411yescharacteristic
200C4⋊C4×C61C22×C411yescharacteristic
201C22⋊C4×C61—21no—
202C22⋊Dic61—21no—
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C22⋊Dic61—21no—
206C22⋊Dic61—21no—
207#11758199—22no—
208#11758199—41no—
209#11758198—21no—
210#11758198—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C22×C411yescharacteristic
213C4⋊Dic61C22×C411yescharacteristic
214C4⋊Dic61C22×C411yescharacteristic
215#11758196—21no—
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61#4111yescharacteristic
219C4×Dic61C2411yescharacteristic
220C2×#92418C22×C411yescharacteristic
221C2×#92418C22×C411yescharacteristic
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61—81no—
228C23×C61C4×D411yescharacteristic
229C22×D61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61#13112yescomplemented
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61C4×D411yescharacteristic
242C2×C4×C61C23×C411yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—81no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61C4×D411yescharacteristic
260C2×Dic61C4×D411yescharacteristic
261C2×Dic61AGL(1,ℤ/8)12yes—
262C2×Dic61AGL(1,ℤ/8)12yes—
263C2×Dic61C22×D411yescharacteristic
264C2×Dic61C2×#4111yescharacteristic
265D244—21no—
266D244—21no—
267C4×D61—41no—
268C4×D61—41no—
269Dic122—42no—
270#92420—22no—
271#92420—21no—
272#92420—21no—
273#92420—21no—
274#92420—21no—
275#92420—41no—
276#92420—41no—
277#92420—41no—
278#92420—41no—
279#92420—41no—
280C8×C61—21no—
281C8×C61—21no—
282#92418—21no—
283#92418—21no—
284#18001—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—22no—
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—84no—
297C22×C61#44212yes—
298C22×C61C2×C4×D411yescharacteristic
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C4×C61—21no—
303C4×C61—41no—
304C4×C61C2×#13111yescharacteristic
305C4×C61#56811yescharacteristic
306Dic61—21no—
307Dic61—21no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61#44212yes—
312Dic61C2×AGL(1,ℤ/8)11yescharacteristic
313Dic61C2×AGL(1,ℤ/8)11yescharacteristic
314#3100—611no—
315#2935—611no—
316D4×M16—611no—
317#1594—611no—
318#1594—611no—
319#1586—611no—
320#1586—611no—
321#1586—611no—
322#1586—611no—
323#1580—611no—
324#1580—611no—
325#1570—611no—
326#1492—611no—
327C2×D4⋊C8—611no—
328#1451—611no—
329C2×C61—22no—
330C2×C61—42no—
331C2×C61—81no—
332C2×C61—81no—
333C2×C61#293311yescharacteristic
334C2×C61C2×#44211yescharacteristic
335C2×C61#295611yescharacteristic
336D61—81no—
337C2×#131—1221no—
338C22×M16—1221no—
339#532—611no—
340C2×C4×D4—611no—
341#480—611no—
342#480—611no—
343#474—611no—
344#474—611no—
345#459—611no—
346#459—611no—
347#447—611no—
348#447#692311yescharacteristic
349C4×D8—611no—
350C4×D8—611no—
351C4×D8—611no—
352C4×D8—611no—
353#436—611no—
354#436—611no—
355#436—611no—
356#436—611no—
357#435—1221no—
358#435—1221no—
359#435—1221no—
360#435—1221no—
361C8×D4—611no—
362C8×D4—611no—
363C8×D4—611no—
364C8×D4—611no—
365#431—611no—
366#423—611no—
367C2×C4⋊C8—611no—
368#408—1221no—
369#407—1221no—
370C4×M16—611no—
371C2×C4×C8—611no—
372#334—611no—
373#334—611no—
374#334—611no—
375#334—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384#322—611no—
385#322—611no—
386#322—611no—
387#322—611no—
388C61#1800111yescharacteristic, complemented, Sylow
389C2×#41—1221no—
390C22×D4C2×#692311yescharacteristic
391C23×C4—1221no—
392C2×D8—611no—
393C2×D8—611no—
394C2×D8C2×#692311yescharacteristic
395C2×D8C2×#692311yescharacteristic
396#131—2444no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16—1221no—
400C2×M16—1221no—
401C2×M16—1221no—
402C2×M16—1221no—
403C2×M16—1221no—
404C2×M16—2444no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—1221no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—611no—
412#127—611no—
413#124—1221no—
414#121—611no—
415#121—1221no—
416#121C2×#692311yescharacteristic
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—611no—
420C4×D4—611no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4—1221no—
426C2×C4⋊C4—611no—
427C2×C22⋊C4—1221no—
428C2×C42—611no—
429#107—611no—
430#107—611no—
431#107—611no—
432#107—611no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—1221no—
440C4⋊C8—1221no—
441#102—611no—
442#102—611no—
443#102—611no—
444#102—611no—
445#102—611no—
446#102—611no—
447#102C2×#692311yescharacteristic
448#102C2×#692311yescharacteristic
449C22⋊C8—611no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8C2×#692311yescharacteristic
459#97—611no—
460#97—611no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8—611no—
467C4×C8—1221no—
468C24—21no—
469#41—2442no—
470#41—2442no—
471C2×Q8—1221no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—41no—
476C2×D4—611no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4C22×#692311yescharacteristic
481C2×D4C22×#692311yescharacteristic
482C2×D4C22×#692311yescharacteristic
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C22×#692311yescharacteristic
495D8—22no—
496D8—41no—
497D8—1222no—
498D8—2441no—
499M16—1222no—
500M16—1222no—
501M16—2442no—
502M16—2442no—
503M16—2442no—
504M16—2442no—
505M16—2442no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—2442no—
519C2×C8—2442no—
520C2×C8—2442no—
521C2×C8—2442no—
522C2×C8—2442no—
523C2×C8—2442no—
524C2×C8C22×#692311yescharacteristic
525C2×C8C22×#692311yescharacteristic
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4C22×#692311yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—42no—
548C23—42no—
549C23—81no—
550C23—1221no—
551C23D4×#692311yescharacteristic
552Q8—2442no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—21no—
558D4—42no—
559D4—42no—
560D4—1221no—
561D4—1221no—
562D4—4881no—
563D4—2442no—
564D4—2442no—
565D4#1451986412yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2442no—
590C2×C4D4×#692311yescharacteristic
591C2×C4C23×#692311yescharacteristic
592C8—21no—
593C8—21no—
594C8—1221no—
595C8—1221no—
596C8—1221no—
597C8—1221no—
598C8—1222no—
599C8—1221no—
600C8—1221no—
601C8—2441no—
602C8—2441no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—84no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22#1684080312yes—
622C22C2×D4×#692311yescharacteristic
623C4—21no—
624C4—41no—
625C4—611no—
626C4—611no—
627C4—612no—
628C4—1221no—
629C4—1221no—
630C4—2441no—
631C4—2441no—
632C4—2441no—
633C4C2×#1451986411yescharacteristic
634C4#1684084011yescharacteristic
635C2—22no—
636C2—42no—
637C2—81no—
638C2—81no—
639C2—4881no—
640C2#2146255111yescharacteristic
641C2C2×#1684080311yescharacteristic
642C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#447C4
socleC22×C61C2×C4×D4
prime radical#21455814C2
cosocleC4×Dic61C24
hypercenter#447#6923
hypocenterC61#18001
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #21462561C2semidirect
2. D4×#11758217C2semidirect
3. #21461469C2semidirect
4. #21461467C2semidirect
5. #21461441C2semidirect
6. #21461440C2semidirect
7. #21461440C2semidirect
8. #21461438C2semidirect
9. #21461416C2semidirect
10. #21461415C2semidirect
11. #21461386C2semidirect
12. #21461164C2semidirect
13. C2×D4⋊#92420C2semidirect
14. #21461030C2semidirect
20. #16840797C22semidirect
21. #16840796C22semidirect
26. #16840790C22semidirect
29. D4×#92420C22semidirect
36. #16840624C22semidirect
37. #16840624C22semidirect
38. #16840623C22semidirect
39. #16840623C22semidirect
40. #16840610C22semidirect
41. #16840610C22semidirect
46. D4⋊#92420C22semidirect
47. D4⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. #16840603C22semidirect
237. D4×C61#131semidirect
388. C61#18001semidirect
565. D4#14519864semidirect

Maximal subgroups

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

subgroupindex
#214625612
D4×#117582172
#214614692
#214614672
#214614412
#214614402
#214614382
#214614162
#214614152
#214613862
#214611642
C2×D4⋊#924202
#214610302
#214558142
#1800161