Finite Group Databasebeta

← #29743117

Subgroups — #29743117

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840061C211yesmaximal, characteristic
2C2×#16839847C211yesmaximal, characteristic
3#21449683C211yesmaximal, characteristic, complemented
4#21448727C211yesmaximal, characteristic
5#21447301C211yesmaximal, characteristic, complemented
6#21447198C211yesmaximal, characteristic, complemented
7#21447155C211yesmaximal, characteristic, complemented
8#21446545C211yesmaximal, characteristic, complemented
9#21446377C211yesmaximal, characteristic, complemented
10#21446156C211yesmaximal, characteristic
11#21446150C211yesmaximal, characteristic
12#21446083C211yesmaximal, characteristic
13#21446071C211yesmaximal, characteristic, complemented
14#21446063C211yesmaximal, characteristic
15C61×#2097C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22#16840061—24no—
23#16839847—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839704—21no—
34#16839694—21no—
35#16839669C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839659—21no—
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839649—21no—
42#16839636—21no—
43#16839636—21no—
44#16839621—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519768—22no—
71#14519768—41no—
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74C4×D244—22no—
75C4×D244—22no—
76#14519739—22no—
77#14519739—22no—
78#14519739—41no—
79#14519739—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519710—22no—
99#14519709—22no—
100#14519708—22no—
101#14519706—22no—
102#14519701—22no—
103#14519699—22no—
104#14519699D414yes—
105#14519697—22no—
106C8×Dic61—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#107×C61—41no—
123#106×C61—22no—
124#103×C61—22no—
125#103×C61—41no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130C23×D61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×Q812yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C2×D412yes—
155C2×D244C2×Q812yes—
156C2×D244#4112yes—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180#11758201—28no—
181#11758198—24no—
182#11758198—42no—
183#11758198—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—22no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—42no—
191C4⋊Dic61—42no—
192C4⋊Dic61C2×D412yes—
193C4⋊Dic61#4112yes—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197#11758196—81no—
198#11758196—81no—
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×#92418C2×Q812yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—41no—
221Q8×C61—41no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
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—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61#13512yes—
267D244—22no—
268D244—22no—
269D244—22no—
270D244C22⋊Q814yes—
271C4×D61—84no—
272C4×D61—84no—
273C4×D61—84no—
274C8×C61—22no—
275C8×C61—41no—
276#92418—22no—
277#92418—41no—
278#26378—611nomaximal, Sylow
279C22×C61#52511yescharacteristic
280C22×C61D4211yescharacteristic
281C22×C61D4×Q811yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57711yescharacteristic
285C22×C61#57711yescharacteristic
286C2×D61—24no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61C2×C22⋊Q811yescharacteristic
300C4×C61#53911yescharacteristic
301C4×C61#54711yescharacteristic
302C4×C61#55411yescharacteristic
303Dic61—22no—
304Dic61—42no—
305Dic61—41no—
306Dic61—41no—
307Dic61—42no—
308Dic61—41no—
309Dic61—81no—
310C2×#478—611no—
311C2×C4×SD16—611no—
312#2453—611no—
313#2349—611no—
314#2097D6111yescharacteristic, complemented
315#2085—611no—
316#2066—611no—
317#2059—611no—
318#1941—611no—
319#1916—611no—
320#1877—611no—
321#1877—611no—
322#1864—611no—
323#1863—611no—
324#1862—611no—
325C2×C61#244911yescharacteristic
326C2×C61#310211yescharacteristic
327C2×C61#312311yescharacteristic
328C2×C61#326711yescharacteristic
329C2×C61#328211yescharacteristic
330C2×C61#338111yescharacteristic
331C2×C61#338811yescharacteristic
332D61—44no—
333C22×SD16—611no—
334C2×#125—611no—
335C2×C4×Q8—611no—
336C2×C4×D4—611no—
337C2×C4×D4—611no—
338C2×C4×D4—611no—
339#478—1224no—
340C4×SD16—1224no—
341C2×#107—611no—
342C2×#107C2×D6111yescharacteristic
343C2×#106—611no—
344C2×#106—611no—
345C2×#106C2×D6111yescharacteristic
346C2×C4⋊C8—611no—
347C2×#103—611no—
348C2×#103C2×D6111yescharacteristic
349C2×#103C2×D6111yescharacteristic
350C2×#102—611no—
351C2×#102—611no—
352C2×#102—611no—
353C2×C4×C8—611no—
354#400—1221no—
355#398—1221no—
356#391—611no—
357#391C2×D6111yescharacteristic
358#389—611no—
359#389—611no—
360#389—611no—
361#389—611no—
362#389—1221no—
363#387—1221no—
364#385—1221no—
365#385—1221no—
366#382C2×D6111yescharacteristic
367#380—1221no—
368C4×C4⋊C4—611no—
369C2×#95—1221no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336C2×D6111yescharacteristic
378C61#2637811yescharacteristic, complemented, Sylow
379C22×Q8C22×D6111yescharacteristic
380C22×D4—611no—
381C23×C4—1221no—
382C23×C4—1221no—
383C23×C4—1221no—
384C2×SD16—614no—
385C2×SD16—1222no—
386C2×SD16—1222no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391#125—1222no—
392#125—2441no—
393C4×Q8—1222no—
394C4×Q8—2441no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—2441no—
400C4×D4—2441no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
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⋊C4—1221no—
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#107—41no—
424#107—1222no—
425#106—22no—
426#106—1222no—
427#106—1222no—
428C4⋊C8—1222no—
429#103—22no—
430#103—41no—
431#103—1222no—
432#102—614no—
433#102—1222no—
434#102—1222no—
435C4×C8—1222no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—611no—
441#95—611no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—1221no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8Q8×D6112yes—
455C2×D4—612no—
456C2×D4—612no—
457C2×D4—612no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—4881no—
474C22×C4—4881no—
475C22×C4—4881no—
476C22×C4—4881no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—1228no—
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×C8D4×D6112yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—612no—
504C4⋊C4—612no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—4881no—
511C4⋊C4—4881no—
512C4⋊C4—2441no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C22⋊C4—1224no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C42—21no—
522C42—21no—
523C42—1221no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—2442no—
528C23—1222no—
529C23—1222no—
530C23—1222no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23C2×D4×D6111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—614no—
541D4—1222no—
542D4—1222no—
543D4—1222no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—612no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1221no—
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—1222no—
576C2×C4—4884no—
577C2×C4—4884no—
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×C4C2×D4×D6111yescharacteristic
587C2×C4C2×Q8×D6111yescharacteristic
588C2×C4#41×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—22no—
593C8—41no—
594C8—1222no—
595C8—2441no—
596C22—1224no—
597C22—2442no—
598C22—2442no—
599C22—2442no—
600C22—2442no—
601C22—2442no—
602C22—2442no—
603C22#1684027011yescharacteristic
604C22#1684039711yescharacteristic
605C22#1684042511yescharacteristic
606C22#135×D6111yescharacteristic
607C22#1684056111yescharacteristic
608C22#1684056911yescharacteristic
609C22#1684057111yescharacteristic
610C4—42no—
611C4—41no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—1222no—
617C4—4881no—
618C4—2441no—
619C4—2441no—
620C4—2441no—
621C4—2442no—
622C4—2442no—
623C4C22⋊Q8×D6111yescharacteristic
624C4#1684035511yescharacteristic
625C4#1684041011yescharacteristic
626C4#1684046611yescharacteristic
627C2—2444no—
628C2#2144963111yescharacteristic
629C2#2145584011yescharacteristic
630C2#2145606711yescharacteristic
631C2#2145734311yescharacteristic
632C2#2145752511yescharacteristic
633C2#2145869711yescharacteristic
634C2#2145876711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2097C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2097D61
hypocenterC61#26378
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
3. #21449683C2semidirect
5. #21447301C2semidirect
6. #21447198C2semidirect
7. #21447155C2semidirect
8. #21446545C2semidirect
9. #21446377C2semidirect
13. #21446071C2semidirect
15. C61×#2097C2semidirect
314. #2097D61semidirect
378. C61#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#168400612
C2×#168398472
#214496832
#214487272
#214473012
#214471982
#214471552
#214465452
#214463772
#214461562
#214461502
#214460832
#214460712
#214460632
C61×#20972
#2637861