Finite Group Databasebeta

← #29729567

Subgroups — #29729567

Order 15616.

800 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
1C2×#16839985C211yesmaximal, characteristic
2#21451307C211yesmaximal, characteristic
3#21451185C211yesmaximal, characteristic, complemented
4#21447206C211yesmaximal, characteristic
5#21447199C211yesmaximal, characteristic, complemented
6#21447142C211yesmaximal, characteristic, complemented
7#21447137C211yesmaximal, characteristic
8#21446680C211yesmaximal, characteristic, complemented
9#21446673C211yesmaximal, characteristic
10#21446630C211yesmaximal, characteristic
11#21446589C211yesmaximal, characteristic, complemented
12#21446502C211yesmaximal, characteristic
13#21446495C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839985—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31#16839700—21no—
32#16839688—21no—
33#16839681—21no—
34#16839673—21no—
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839652—21no—
44#16839652—21no—
45#16839643—21no—
46#16839643—21no—
47#16839628C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C8×Q8×C61—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×Dic244—41no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190—41no—
70#14519765—22no—
71#14519765—22no—
72C4×Dic122—22no—
73C4×Dic122—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90#14519711—41no—
91#14519709—41no—
92#14519702—22no—
93#14519702—22no—
94#14519702—22no—
95#14519702—22no—
96#14519702—41no—
97#14519699—22no—
98#14519699—22no—
99#14519699—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×Q8×C61D411yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127C4⋊C8×C61—22no—
128C4⋊C8×C61D414yes—
129C4×C8×C61—22no—
130C4×C8×C61—22no—
131C4×C8×C61—22no—
132C23×D61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C2×C8×C61C2×D412yes—
179C2×C8×C61#4112yes—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—81no—
194#11758198—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—42no—
209C4×Dic61—81no—
210Dic244—44no—
211#11758190—44no—
212C23×C61C22×D411yescharacteristic
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220Q8×C61—28no—
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×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#13512yes—
238C2×C4×C61#13512yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—22no—
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—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266D244—41no—
267D244—41no—
268D244—41no—
269D244—41no—
270Dic122—22no—
271Dic122—22no—
272Dic122—41no—
273Dic122—41no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—22no—
280C8×C61—22no—
281C8×C61—41no—
282#25297—611nomaximal, Sylow
283C22×C61C2×#12111yescharacteristic
284C22×C61#54811yescharacteristic
285C22×C61#55011yescharacteristic
286C22×C61C2×#13511yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61#57711yescharacteristic
289C22×C61#57811yescharacteristic
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61#52911yescharacteristic
304C4×C61#54211yescharacteristic
305C4×C61#54611yescharacteristic
306C4×C61#54811yescharacteristic
307Dic61—42no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314C2×#464—611no—
315C2×C8×Q8D6111yescharacteristic, complemented
316#2592—611no—
317#2581—611no—
318#2067—611no—
319#2066—611no—
320#2056—611no—
321#2055—611no—
322#1969—611no—
323#1968—611no—
324#1958—611no—
325#1949—611no—
326#1934—611no—
327#1933—611no—
328C4×#103—611no—
329C2×C61#242611yescharacteristic
330C2×C61#305011yescharacteristic
331C2×C61#306311yescharacteristic
332C2×C61#336411yescharacteristic
333C2×C61#336511yescharacteristic
334C2×C61#339311yescharacteristic
335C2×C61#339411yescharacteristic
336D61—161no—
337C22×Q16—611no—
338C22×SD16—611no—
339C2×#124—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8C2×D6111yescharacteristic
342#464—1224no—
343C8×Q8—24no—
344C2×#107—611no—
345C2×#106—611no—
346C2×C4⋊C8C2×D6111yescharacteristic
347C2×C4⋊C8C2×D6111yescharacteristic
348C2×C4⋊C8C2×D6111yescharacteristic
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×C4×C8C2×D6111yescharacteristic
358C2×C4×C8C2×D6111yescharacteristic
359C2×C4×C8C2×D6111yescharacteristic
360#400—1221no—
361#397—1221no—
362#395—1221no—
363#393—1221no—
364#391—611no—
365#391—611no—
366#391—611no—
367#391—611no—
368#390—611no—
369#390—611no—
370#389—611no—
371#389—611no—
372#388—1221no—
373#388—1221no—
374#387—1221no—
375#387—1221no—
376#382—611no—
377C4×C4⋊C4—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382C61#2529711yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×Q8D24411yescharacteristic
386C22×D4—611no—
387C2×Q16—1222no—
388C2×Q16—1222no—
389C2×Q16—2441no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—2441no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—1222no—
398#124—1222no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4D24411yescharacteristic
412C2×C4⋊C4D24411yescharacteristic
413C2×C4⋊C4D24411yescharacteristic
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425#107—2441no—
426#106—2441no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8C22⋊D6114yes—
430#103—1222no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—2441no—
435#102—1222no—
436#102—1222no—
437#102—2441no—
438C4×C8—22no—
439C4×C8—22no—
440C4×C8—22no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454C24—1221no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—612no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—2444no—
491SD16—2444no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8#1175822612yes—
502C2×C8C2×C22⋊D6112yes—
503C2×C8C2×C22⋊D6112yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—4881no—
521C22⋊C4—4881no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—4881no—
531C42—2442no—
532C42#1175822212yes—
533C42C2×C22⋊D6112yes—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23C22×D24411yescharacteristic
542Q8—28no—
543Q8—1222no—
544Q8—1222no—
545Q8—2441no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550Q8—2441no—
551D4—2441no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—1222no—
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×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4#1451975112yes—
591C2×C4#1451975112yes—
592C2×C4#1451979912yes—
593C2×C4#1451979912yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822211yescharacteristic
597C2×C4C2×#1175822611yescharacteristic
598C2×C4#1451982311yescharacteristic
599C2×C4#1451982511yescharacteristic
600C8—22no—
601C8—22no—
602C8—22no—
603C8—41no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22C2×#1451975111yescharacteristic
612C22#1684042411yescharacteristic
613C22#1684043411yescharacteristic
614C22C2×#1451979911yescharacteristic
615C22C2×#1451979911yescharacteristic
616C22#1684057411yescharacteristic
617C22#1684057911yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—4881no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—4881no—
629C4—4881no—
630C4—2442no—
631C4#1684029911yescharacteristic
632C4#1684038411yescharacteristic
633C4#1684040011yescharacteristic
634C4#1684042311yescharacteristic
635C2—9761no—
636C2#2144937911yescharacteristic
637C2#2145540011yescharacteristic
638C2#2145551011yescharacteristic
639C2#2145850611yescharacteristic
640C2#2145851711yescharacteristic
641C2#2145880811yescharacteristic
642C2#2145881711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×C8×Q8D61
hypocenterC61#25297
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
3. #21451185C2semidirect
5. #21447199C2semidirect
6. #21447142C2semidirect
8. #21446680C2semidirect
11. #21446589C2semidirect
13. #21446495C2semidirect
14. C4×#14519702C2semidirect
15. C2×C8×Q8×C61C2semidirect
315. C2×C8×Q8D61semidirect
382. C61#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#168399852
#214513072
#214511852
#214472062
#214471992
#214471422
#214471372
#214466802
#214466732
#214466302
#214465892
#214465022
#214464952
C4×#145197022
C2×C8×Q8×C612
#2529761