Finite Group Databasebeta

← #29753208

Subgroups — #29753208

Order 15616.

762 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
1#21450405C211yesmaximal, characteristic
2#21448875C211yesmaximal, characteristic, complemented
3#21447361C211yesmaximal, characteristic, complemented
4#21447137C211yesmaximal, characteristic
5#21447066C211yesmaximal, characteristic, complemented
6#21446904C211yesmaximal, characteristic
7#21446673C211yesmaximal, characteristic
8#21446454C211yesmaximal, characteristic, complemented
9#21446447C211yesmaximal, characteristic
10#21446398C211yesmaximal, characteristic, complemented
11#21446173C211yesmaximal, characteristic
12#21446173C211yesmaximal, characteristic
13#21445825C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1916C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×#14519772—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839689—21no—
33#16839684—21no—
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839658—21no—
41#16839654—21no—
42#16839645—21no—
43#16839645—21no—
44#16839632C2211yescharacteristic
45C4⋊C4×Dic61—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C4×C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#336C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C2×#11758190—22no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190—41no—
66C2×#11758190—41no—
67C2×#11758190—41no—
68#14519777—22no—
69#14519777—22no—
70#14519772—44no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74C4×Dic122—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97#14519709—22no—
98#14519709—41no—
99#14519702—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702—41no—
103#14519702—41no—
104#14519702—41no—
105#14519699—22no—
106#14519699—41no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C4⋊C4×C61D411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—41no—
124C4×C8×C61—41no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C23×D61—21no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×Dic122—21no—
149C2×Dic122—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#4112yes—
159C2×Dic122#4112yes—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61#4112yes—
172C4⋊C4×C61—41no—
173C4⋊C4×C61SD1614yescomplemented
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—81no—
185#11758198—81no—
186C4⋊Dic61—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—81no—
199C4⋊Dic61#4112yes—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207#11758190—44no—
208#11758190—44no—
209C23×C61C22×D411yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61C2×SD1612yes—
233C2×C4×C61C2×SD1612yes—
234C2×C4×C61C2×SD1612yes—
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
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—
260C2×Dic61—41no—
261C2×Dic61—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266Dic122—22no—
267Dic122—22no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—22no—
275C8×C61—41no—
276C8×C61—41no—
277C8×C61—41no—
278#27066—611nomaximal, Sylow
279C22×C61#52511yescharacteristic
280C22×C61#54611yescharacteristic
281C22×C61#55011yescharacteristic
282C22×C61C22×SD1611yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57711yescharacteristic
285C22×C61#57811yescharacteristic
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61#52911yescharacteristic
300C4×C61#53811yescharacteristic
301C4×C61#55111yescharacteristic
302C4×C61#55311yescharacteristic
303Dic61—42no—
304Dic61—42no—
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308Dic61—81no—
309Dic61—81no—
310#2512—611no—
311#2365—611no—
312#2096—611no—
313#2055—611no—
314#2043—611no—
315#2013—611no—
316#1968—611no—
317#1927—611no—
318#1926—611no—
319#1919—611no—
320#1916D6111yescharacteristic, complemented
321#1880—611no—
322#1880—611no—
323#1803—611no—
324C4×#103—611no—
325C2×C61#246711yescharacteristic
326C2×C61#310311yescharacteristic
327C2×C61#312111yescharacteristic
328C2×C61#329811yescharacteristic
329C2×C61#330711yescharacteristic
330C2×C61#339211yescharacteristic
331C2×C61#340211yescharacteristic
332D61—161no—
333C22×SD16—611no—
334C22×SD16—611no—
335C2×C4⋊Q8—611no—
336C2×#126—1221no—
337C2×C4×Q8—611no—
338C2×C4×Q8—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×C4⋊C8C2×D6111yescharacteristic
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×C4×C8C2×D6111yescharacteristic
351#397—1221no—
352#396—1221no—
353#391—611no—
354#391—611no—
355#390—611no—
356#390—611no—
357#390—611no—
358#389—611no—
359#389—1221no—
360#388—1221no—
361#387—1221no—
362#387—1221no—
363#384—611no—
364C4×C4⋊C4—611no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—1221no—
367C4×C4⋊C4C2×D6111yescharacteristic
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336C2×D6111yescharacteristic
373#336C2×D6111yescharacteristic
374#336C2×D6111yescharacteristic
375#336C2×D6111yescharacteristic
376C61#2706611yescharacteristic, complemented, Sylow
377C22×Q8—611no—
378C22×Q8—611no—
379C22×D4—611no—
380C2×SD16—1222no—
381C2×SD16—1222no—
382C2×SD16—1222no—
383C2×SD16—2441no—
384C2×SD16—2441no—
385C2×SD16—2441no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C4⋊Q8—1222no—
391C4⋊Q8—1222no—
392#126—2444no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—2441no—
396C4×Q8—2441no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4D24411yescharacteristic
410C2×C4⋊C4D24411yescharacteristic
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—1221no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#106—1222no—
425#106—2441no—
426C4⋊C8—41no—
427#103—1222no—
428#103—1222no—
429#103—1222no—
430#103—2441no—
431#103—2441no—
432#103—2441no—
433#102—1222no—
434#102—2441no—
435C4×C8—41no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95D24411yescharacteristic
444#95D24411yescharacteristic
445C24—1221no—
446C2×Q8—612no—
447C2×Q8—612no—
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—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×D24411yescharacteristic
472C22×C4C2×D24411yescharacteristic
473C22×C4C2×D24411yescharacteristic
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×D24411yescharacteristic
476C22×C4C2×D24411yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—2444no—
479SD16—2444no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8#1175822612yes—
492C4⋊C4—41no—
493C4⋊C4—612no—
494C4⋊C4—1224no—
495C4⋊C4—1224no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4881no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4#1175819014yescomplemented
511C22⋊C4—4881no—
512C22⋊C4—4881no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—2442no—
524C42—2442no—
525C42—2442no—
526C42—2442no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23C22×D24411yescharacteristic
535Q8—1222no—
536Q8—1222no—
537Q8—1222no—
538Q8—1222no—
539Q8—2441no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—2441no—
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—2441no—
582C2×C4—2441no—
583C2×C4C2×#1175819012yes—
584C2×C4C2×#1175819012yes—
585C2×C4C2×#1175819012yes—
586C2×C4#41×D6111yescharacteristic
587C2×C4#41×D6111yescharacteristic
588C2×C4#41×D6111yescharacteristic
589C2×C4C2×#1175822611yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982311yescharacteristic
592C8—22no—
593C8—41no—
594C8—41no—
595C8—41no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22#1684027411yescharacteristic
604C22#1684040711yescharacteristic
605C22#1684043411yescharacteristic
606C22C22×#1175819011yescharacteristic
607C22#1684056211yescharacteristic
608C22#1684057411yescharacteristic
609C22#1684057911yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—4881no—
617C4—4881no—
618C4—4881no—
619C4—4881no—
620C4—2442no—
621C4—2442no—
622C4—2442no—
623C4#1684029711yescharacteristic
624C4#1684034811yescharacteristic
625C4#1684043911yescharacteristic
626C4#1684046311yescharacteristic
627C2—9761no—
628C2#2144986811yescharacteristic
629C2#2145585111yescharacteristic
630C2#2145605011yescharacteristic
631C2#2145768511yescharacteristic
632C2#2145779711yescharacteristic
633C2#2145880111yescharacteristic
634C2#2145890011yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1916C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1916D61
hypocenterC61#27066
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
2. #21448875C2semidirect
3. #21447361C2semidirect
5. #21447066C2semidirect
8. #21446454C2semidirect
10. #21446398C2semidirect
13. #21445825C2semidirect
14. C4×#14519702C2semidirect
15. C61×#1916C2semidirect
173. C4⋊C4×C61SD16semidirect
320. #1916D61semidirect
376. C61#27066semidirect
510. C4⋊C4#11758190semidirect

Maximal subgroups

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

subgroupindex
#214504052
#214488752
#214473612
#214471372
#214470662
#214469042
#214466732
#214464542
#214464472
#214463982
#214461732
#214458252
C4×#145197022
C61×#19162
#2706661