Finite Group Databasebeta

← #29743105

Subgroups — #29743105

Order 15616.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840055C211yesmaximal, characteristic, complemented
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21449672C211yesmaximal, characteristic
4#21448740C211yesmaximal, characteristic, complemented
5#21447340C211yesmaximal, characteristic
6#21447302C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic
8#21447151C211yesmaximal, characteristic, complemented
9#21446551C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic
12#21446085C211yesmaximal, characteristic, complemented
13#21446073C211yesmaximal, characteristic, complemented
14#21446068C211yesmaximal, characteristic
15C61×#1880C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519754—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16840055—24no—
22SD16×Dic61—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839685—21no—
35#16839674—21no—
36#16839665C2211yescharacteristic
37#16839661—21no—
38#16839661—21no—
39#16839661C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839634—21no—
44#16839628—21no—
45#16839628C2211yescharacteristic
46C4×C4⋊Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519776—22no—
66#14519776—41no—
67#14519754—44no—
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61—41no—
71Q8×Dic61—41no—
72D4×Dic61—22no—
73D4×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519711—22no—
95#14519710—22no—
96#14519710—22no—
97#14519709—22no—
98#14519708—41no—
99#14519706—22no—
100#14519701—22no—
101#14519701D414yes—
102#14519698—22no—
103#14519698—41no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61Q814yescomplemented
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×Q8×C61—22no—
121C4×Q8×C61—22no—
122C2×C4⋊C4×C61—21no—
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#103×C61—22no—
130#102×C61—22no—
131C24×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61C2×Q812yes—
137C2×Q8×C61#4112yes—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×Q812yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—81no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×Dic122—81no—
164SD16×C61—28no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×Q812yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61#4112yes—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—81no—
178C42×C61—24no—
179C42×C61—21no—
180C42×C61—21no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61C2×D412yes—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D412yes—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C22×D411yescharacteristic
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61C22⋊Q814yescomplemented
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
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—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61#13512yes—
274C2×Dic61AGL(1,ℤ/8)12yes—
275Dic122—84no—
276C8×C61—22no—
277C8×C61—41no—
278#92418—22no—
279#92418—41no—
280#26377—611nomaximal, Sylow
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61#52511yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61D4×Q811yescharacteristic
291C22×C61C2×#13511yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57811yescharacteristic
295C4×C61—24no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61C2×C22⋊Q811yescharacteristic
301C4×C61#53911yescharacteristic
302C4×C61#54711yescharacteristic
303C4×C61#55411yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—41no—
307Dic61—42no—
308Dic61—41no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312C2×#477—611no—
313C2×C4×SD16—611no—
314#2452—611no—
315#2351—611no—
316#2092—611no—
317#2085—611no—
318#2067—611no—
319#2058—611no—
320#1942—611no—
321#1916—611no—
322#1880—611no—
323#1880D6111yescharacteristic
324#1865—611no—
325#1863—611no—
326#1862—611no—
327C2×C61—82no—
328C2×C61#244911yescharacteristic
329C2×C61#310611yescharacteristic
330C2×C61#312111yescharacteristic
331C2×C61#326511yescharacteristic
332C2×C61#328511yescharacteristic
333C2×C61#338111yescharacteristic
334C2×C61#338611yescharacteristic
335C22×SD16C2×D6111yescharacteristic
336C2×C4⋊Q8—611no—
337C2×C22⋊Q8—1221no—
338C2×C4×Q8—611no—
339C2×C4×Q8—611no—
340C2×C4×Q8C2×D6111yescharacteristic
341C2×C4×D4—611no—
342#477—1224no—
343C4×SD16—1224no—
344C2×#107—611no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103C2×D6111yescharacteristic
353C2×#102—611no—
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×C4×C8—611no—
357#396—1221no—
358#393—1221no—
359#390—611no—
360#390C2×D6111yescharacteristic
361#389—611no—
362#389—611no—
363#389—611no—
364#389—611no—
365#389—1221no—
366#389—1221no—
367#385—1221no—
368#382—611no—
369#382—1221no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—1221no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340C2×D6111yescharacteristic
376#336—611no—
377#336—611no—
378#336—611no—
379#336C2×D6111yescharacteristic
380C61#2637711yescharacteristic, complemented, Sylow
381C22×Q8—1221no—
382C22×Q8C22×D6111yescharacteristic
383C22×D4C22×D6111yescharacteristic
384C23×C4—1221no—
385C2×SD16—22no—
386C2×SD16—22no—
387C2×SD16Dic12214yescomplemented
388C22×C8—611no—
389C22×C8—611no—
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C4⋊Q8—1222no—
393C4⋊Q8—2441no—
394C22⋊Q8—2444no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—2441no—
400C4×Q8—2441no—
401C4×D4—1222no—
402C4×D4—2441no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
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×C42C22×D6111yescharacteristic
430#107—1222no—
431#107—1222no—
432#107—1222no—
433#106—1222no—
434#106—2441no—
435C4⋊C8—1222no—
436#103—22no—
437#103—614no—
438#103—1222no—
439#102—22no—
440#102—1222no—
441#102—2441no—
442C4×C8—1222no—
443#95—611no—
444#95—611no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451C24—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—4881no—
456C2×Q8—2442no—
457C2×Q8—2442no—
458C2×Q8C2×Dic12212yes—
459C2×Q8D4×D6112yes—
460C2×Q8#1175822412yes—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×Dic12212yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—28no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—612no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8C2×Dic12212yes—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—612no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2442no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4#1175822412yes—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—2442no—
532C42—2442no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×D4×D6111yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—4884no—
545Q8#1451975414yescomplemented
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—4884no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
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×C4C2×D4×D6111yescharacteristic
591C2×C4C22×Dic12211yescharacteristic
592C2×C4C2×#1175822411yescharacteristic
593C2×C4C2×#1175822411yescharacteristic
594C2×C4#1451982311yescharacteristic
595C2×C4#1451982411yescharacteristic
596C8—22no—
597C8—41no—
598C8—1222no—
599C8—2441no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1684026811yescharacteristic
608C22#1684039611yescharacteristic
609C22D4×Dic12211yescharacteristic
610C22#135×D6111yescharacteristic
611C22AGL(1,ℤ/8)×D6111yescharacteristic
612C22#1684056811yescharacteristic
613C22#1684057811yescharacteristic
614C4—24no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—1222no—
620C4—1222no—
621C4—4881no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2441no—
626C4—2442no—
627C4C2×#1451975411yescharacteristic
628C4#1684035611yescharacteristic
629C4#1684041211yescharacteristic
630C4#1684046911yescharacteristic
631C2—82no—
632C2#2144963411yescharacteristic
633C2#2145587211yescharacteristic
634C2#2145604111yescharacteristic
635C2#2145730911yescharacteristic
636C2#2145755511yescharacteristic
637C2#2145869511yescharacteristic
638C2#2145874111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#26377
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#16840055C2semidirect
4. #21448740C2semidirect
6. #21447302C2semidirect
8. #21447151C2semidirect
9. #21446551C2semidirect
10. #21446375C2semidirect
12. #21446085C2semidirect
13. #21446073C2semidirect
117. C2×SD16×C61Q8semidirect
224. Q8×C61C22⋊Q8semidirect
380. C61#26377semidirect
387. C2×SD16Dic122semidirect
545. Q8#14519754semidirect

Maximal subgroups

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

subgroupindex
C2×#168400552
C2×SD16×Dic612
#214496722
#214487402
#214473402
#214473022
#214472032
#214471512
#214465512
#214463752
#214461762
#214460852
#214460732
#214460682
C61×#18802
#2637761