Finite Group Databasebeta

← #29743063

Subgroups — #29743063

Order 15616.

820 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
1C2×#16840067C211yesmaximal, characteristic, complemented
2#21449685C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21447342C211yesmaximal, characteristic
5#21447304C211yesmaximal, characteristic, complemented
6#21447191C211yesmaximal, characteristic
7#21447159C211yesmaximal, characteristic, complemented
8#21446547C211yesmaximal, characteristic, complemented
9#21446376C211yesmaximal, characteristic, complemented
10#21446176C211yesmaximal, characteristic
11#21446176C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic, complemented
13#21446073C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic
15C2×C4×SD16×C61C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19#16840067—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29#16839698—21no—
30#16839685—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660—21no—
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839658C2211yescharacteristic
39#16839635—21no—
40#16839627—21no—
41#16839627C2211yescharacteristic
42#16839622—21no—
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×SD16×C61—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519768—22no—
64#14519768—41no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
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#14519710—41no—
93#14519709—22no—
94#14519709—22no—
95#14519706—22no—
96#14519701—22no—
97#14519701—22no—
98#14519698—22no—
99#14519698—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C2×SD16×C61—22no—
112C2×SD16×C61—22no—
113C2×SD16×C61Q814yes—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C4×Q8×C61—22no—
117C4×Q8×C61—41no—
118C4×D4×C61—22no—
119C4×D4×C61—41no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—22no—
127#103×C61D414yes—
128#102×C61—22no—
129C4×C8×C61—22no—
130C24×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×Q812yes—
136C2×Q8×C61#4112yes—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160SD16×C61—28no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D412yes—
166C2×C8×C61C2×Q812yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—42no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—81no—
178C22⋊Dic61—81no—
179C22⋊Dic61—81no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61—42no—
190C4⋊Dic61—81no—
191C4⋊Dic61#4112yes—
192#11758196—22no—
193#11758196—41no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197#11758196—81no—
198C4×Dic61—24no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C22×D411yescharacteristic
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61C22⋊Q814yes—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—84no—
241C2×C4×C61#13512yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
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—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—
272C8×C61—22no—
273C8×C61—22no—
274#92418—41no—
275#92418—41no—
276#26375—611nomaximal, Sylow
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61#52511yescharacteristic
285C22×C61D4211yescharacteristic
286C22×C61D4×Q811yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57711yescharacteristic
291C4×C61—24no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—81no—
296C4×C61C2×C22⋊Q811yescharacteristic
297C4×C61#53911yescharacteristic
298C4×C61#54711yescharacteristic
299C4×C61#55411yescharacteristic
300Dic61—41no—
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308C2×#479—611no—
309C2×C4×SD16D6111yescharacteristic
310#2454—611no—
311#2351—611no—
312#2092—611no—
313#2085—611no—
314#2065—611no—
315#2059—611no—
316#1941—611no—
317#1916—611no—
318#1880—611no—
319#1880—611no—
320#1865—611no—
321#1863—611no—
322#1862—611no—
323C2×C61—82no—
324C2×C61#244911yescharacteristic
325C2×C61#310211yescharacteristic
326C2×C61#312611yescharacteristic
327C2×C61#327111yescharacteristic
328C2×C61#328211yescharacteristic
329C2×C61#338111yescharacteristic
330C2×C61#338811yescharacteristic
331C22×SD16C2×D6111yescharacteristic
332C2×#125—611no—
333C2×C4×Q8—611no—
334C2×C4×Q8—611no—
335C2×C4×Q8C2×D6111yescharacteristic
336C2×C4×D4C2×D6111yescharacteristic
337#479—1224no—
338C4×SD16—24no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×#106C2×D6111yescharacteristic
344C2×C4⋊C8—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103C2×D6111yescharacteristic
348C2×#102—611no—
349C2×#102—611no—
350C2×#102C2×D6111yescharacteristic
351C2×C4×C8C2×D6111yescharacteristic
352#399—1221no—
353#396—1221no—
354#390—611no—
355#390—611no—
356#389—611no—
357#389—611no—
358#389—611no—
359#389—611no—
360#389—1221no—
361#389—1221no—
362#385—1221no—
363#382—611no—
364#382—1221no—
365#380—1221no—
366C4×C4⋊C4—611no—
367C4×C4⋊C4—1221no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376C61#2637511yescharacteristic, complemented, Sylow
377C22×Q8C22×D6111yescharacteristic
378C22×D4C22×D6111yescharacteristic
379C23×C4—21no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16Dic12214yes—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387#125—1222no—
388#125—2441no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—2441no—
395C4×D4—22no—
396C4×D4—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1221no—
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⋊C4—1221no—
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#107—2441no—
427#106—22no—
428#106—1222no—
429#106—1222no—
430C4⋊C8—1222no—
431#103—1222no—
432#103—1222no—
433#103C22⋊D6114yes—
434#102—22no—
435#102—1222no—
436#102—2441no—
437C4×C8—22no—
438#95—611no—
439#95—611no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446C24—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8C2×Dic12212yes—
451C2×Q8#1175822212yes—
452C2×Q8C2×C22⋊D6112yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4C2×Dic12212yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476SD16—28no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8C2×Dic12212yes—
488C2×C8C2×C22⋊D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—612no—
494C4⋊C4—1224no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—4881no—
505C4⋊C4—2441no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4C2×C22⋊D6112yes—
512C22⋊C4—42no—
513C22⋊C4—4881no—
514C22⋊C4—4881no—
515C22⋊C4—4881no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—1224no—
521C42—1221no—
522C42—1221no—
523C42—1221no—
524C42—1221no—
525C42—2442no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×D4×D6111yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1451975614yes—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
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#1451979812yes—
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×Dic12211yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982411yescharacteristic
588C8—22no—
589C8—22no—
590C8—2441no—
591C8—2441no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1684027111yescharacteristic
600C22D4×C22⋊D6111yescharacteristic
601C22D4×Dic12211yescharacteristic
602C22C2×#1451979811yescharacteristic
603C22#1684055911yescharacteristic
604C22#1684056811yescharacteristic
605C22#1684056911yescharacteristic
606C4—22no—
607C4—24no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—4881no—
612C4—4881no—
613C4—4881no—
614C4—2441no—
615C4—2441no—
616C4—2442no—
617C4—2441no—
618C4—2441no—
619C4C2×#1451975611yescharacteristic
620C4#1684035411yescharacteristic
621C4#1684041111yescharacteristic
622C4#1684047111yescharacteristic
623C2—82no—
624C2#2144962211yescharacteristic
625C2#2145583511yescharacteristic
626C2#2145610711yescharacteristic
627C2#2145740411yescharacteristic
628C2#2145752611yescharacteristic
629C2#2145869211yescharacteristic
630C2#2145876011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×C4×SD16D61
hypocenterC61#26375
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16840067C2semidirect
3. #21448737C2semidirect
5. #21447304C2semidirect
7. #21447159C2semidirect
8. #21446547C2semidirect
9. #21446376C2semidirect
12. #21446089C2semidirect
13. #21446073C2semidirect
376. C61#26375semidirect

Maximal subgroups

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

subgroupindex
C2×#168400672
#214496852
#214487372
#214473422
#214473042
#214471912
#214471592
#214465472
#214463762
#214461762
#214460892
#214460732
#214460662
C2×C4×SD16×C612
#2637561