Finite Group Databasebeta

← #29646492

Subgroups — #29646492

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519823C211yesmaximal, characteristic
2#21454936C211yesmaximal, characteristic, complemented
3#21444263C212yesmaximal, complemented
4#21444259C212yesmaximal, complemented
5#21444254C212yesmaximal, complemented
6#21444245C212yesmaximal, complemented
7#21444211C211yesmaximal, characteristic, complemented
8#21444207C211yesmaximal, characteristic, complemented
9#21444202C211yesmaximal, characteristic, complemented
10C2×#16839491C211yesmaximal, characteristic, complemented
11C61×#2965C211yesmaximal, characteristic, complemented
12C2×#14519823C411yescharacteristic
13#16840245C411yescharacteristic
14#16840244—21no—
15#16840242C412yes—
16#16840242C411yescharacteristic
17#16840236C412yes—
18#16840236C411yescharacteristic
19#16840234—22no—
20#16840234—21no—
21C4×#11758226—21no—
22C4×#11758222C2211yescharacteristic
23C4×#11758222C2212yes—
24C2×C4×D244C2211yescharacteristic
25C2×C4×D244C2212yes—
26C4×D4×D61—22no—
27C4×D4×D61—21no—
28#16839893—21no—
29#16839840—21no—
30#16839834—22no—
31#16839819C2212yescomplemented
32#16839815C2211yescharacteristic, complemented
33#16839786C2212yescomplemented
34C2×C4⋊#92418C2211yescharacteristic, complemented
35#16839494C2211yescharacteristic, complemented
36#16839494C2211yescharacteristic, complemented
37#16839494C2212yescomplemented
38#16839491C2212yescomplemented
39#16839491C2212yescomplemented
40#16839491C2211yescharacteristic, complemented
41#16839491C2211yescharacteristic, complemented
42#16839491C2212yescomplemented
43#16839491C2212yescomplemented
44#16839491C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#436—21no—
48C61×#435—22no—
49C61×#431C2211yescharacteristic, complemented
50C61×#431C2212yescomplemented
51C4×M16×C61C2212yescomplemented
52C2×C4×C8×C61C2211yescharacteristic, complemented
53#14519823—24no—
54#14519823—24no—
55C2×#11758226—21no—
56C2×#11758222C2×C412yes—
57C2×#11758222C2×C411yescharacteristic
58C22×D244C2×C412yes—
59C22×D244C2×C411yescharacteristic
60C2×D4×D61—22no—
61C2×D4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—22no—
64C2×#11758194—22no—
65C22×#92418—21no—
66Q8×Dic61—21no—
67C4×Dic122C2×C411yescharacteristic
68C4×Dic122C2×C412yes—
69C4×Dic122C2311yescharacteristic
70C4×D244—21no—
71C4×D244—22no—
72C4×D244—21no—
73C4×D244—22no—
74C4×D244C2×C412yes—
75C4×D244C2×C412yes—
76C4×D244C2×C412yes—
77C4×D244C2×C411yescharacteristic
78C4×D244C2×C412yes—
79C4×D244C2312yes—
80C4×D244C2311yescharacteristic
81D4×Dic61—22no—
82D4×Dic61—21no—
83#14519739—22no—
84#14519739—21no—
85#14519739—22no—
86#14519739—21no—
87C4×C22⋊D61—21no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—21no—
90C4×C22⋊D61—22no—
91#14519737—22no—
92#14519737—22no—
93#14519737—22no—
94#14519736C2×C412yes—
95#14519736C2×C411yescharacteristic
96#14519735—22no—
97#14519735—21no—
98#14519734—21no—
99#14519734—22no—
100C2×C4⋊Dic61C2×C412yes—
101C2×C4⋊Dic61C2×C411yescharacteristic
102C4⋊C4×D61—22no—
103C4⋊C4×D61—21no—
104C2×#11758198—21no—
105C2×#11758198—22no—
106C22⋊C4×D61—22no—
107C22⋊C4×D61—22no—
108C22⋊C4×D61—22no—
109C42×D61—21no—
110C42×D61—22no—
111C4⋊#92418—21no—
112C4⋊#92418—22no—
113C4⋊#92418C2312yes—
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116C22⋊#92418—21no—
117C22⋊#92418—22no—
118#14519688—22no—
119C4×#92418—21no—
120C2×#41×C61C2×C411yescharacteristic
121C2×M16×C61—22no—
122C22×C8×C61—21no—
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61D411yescharacteristic
125C4×D4×C61D412yes—
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128C4×D4×C61D411yescharacteristic
129#117×C61C2×C412yes—
130#117×C61C2×C411yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2312yes—
133C4⋊C8×C61—22no—
134C4⋊C8×C61—21no—
135C22⋊C8×C61—21no—
136C22⋊C8×C61—22no—
137#97×C61—22no—
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C4×C8×C61C2312yes—
142C23×D61—22no—
143C23×D61—21no—
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61C22⋊C411yescharacteristic
147C2×D4×C61C22⋊C412yes—
148C2×D4×C61C22⋊C411yescharacteristic
149C22×C4×C61C22×C412yes—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—21no—
157C22×Dic61—22no—
158C22×Dic61—21no—
159#11758226—28no—
160D4×D61—216no—
161D4×D61—28no—
162#11758222—22no—
163#11758222—24no—
164#11758222—24no—
165#11758222M1614yescomplemented
166C2×D244—22no—
167C2×D244—21no—
168C2×D244—22no—
169C2×D244—24no—
170C2×D244—24no—
171C2×D244M1614yescomplemented
172C2×D244C22×C411yescharacteristic
173C2×D244C22×C412yes—
174C2×C4×D61—22no—
175C2×C4×D61—21no—
176C2×C4×D61—22no—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—21no—
182C2×C4×D61—22no—
183C2×C4×D61—81no—
184C2×C4×D61—82no—
185C2×C4×D61—81no—
186C2×C4×D61—82no—
187C2×Dic122C22×C411yescharacteristic
188M16×C61—44no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—22no—
192C2×C8×C61—42no—
193C4⋊C4×C61C22⋊C412yes—
194C4⋊C4×C61C22⋊C411yescharacteristic
195C4⋊C4×C61C22⋊C412yes—
196C4⋊C4×C61C22⋊C411yescharacteristic
197C22⋊C4×C61C22⋊C412yes—
198C22⋊C4×C61C22⋊C412yes—
199C22⋊C4×C61C22⋊C412yes—
200C42×C61C22×C411yescharacteristic
201C42×C61C22×C412yes—
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D412yes—
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D412yes—
206C42×C61C2411yescharacteristic
207C22⋊Dic61—22no—
208C22⋊Dic61—21no—
209#11758198—21no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—21no—
216C4⋊Dic61—22no—
217C4⋊Dic61—21no—
218C4⋊Dic61C22×C411yescharacteristic
219C4⋊Dic61C22×C412yes—
220C4⋊Dic61C22×C411yescharacteristic
221#11758196—21no—
222#11758196—22no—
223#11758196—21no—
224C4×Dic61—22no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C4×Dic61—22no—
228C4×Dic61—21no—
229#11758194—82no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—22no—
233C2×#92418—81no—
234C23×C61C2×C22⋊C412yes—
235C23×C61C2×C22⋊C411yescharacteristic
236C22×D61—22no—
237C22×D61—21no—
238C22×D61—28no—
239C22×D61—24no—
240C22×D61—81no—
241C22×D61—82no—
242C22×D61—81no—
243C22×D61—82no—
244Q8×C61—22no—
245D4×C61—24no—
246D4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×M1612yes—
262C2×C4×C61C2×M1612yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C22⋊D61—28no—
267C22⋊D61—24no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—81no—
273C2×Dic61—82no—
274D244—28no—
275D244—21no—
276D244—22no—
277D244—24no—
278D244C2×M1612yes—
279D244C2×M1614yes—
280C4×D61—28no—
281C4×D61—24no—
282C4×D61—42no—
283C4×D61—41no—
284C4×D61—42no—
285C4×D61—41no—
286C4×D61—81no—
287C4×D61—82no—
288C4×D61—82no—
289C4×D61—82no—
290C4×D61—82no—
291Dic122—21no—
292Dic122C2×M1612yes—
293C8×C61—22no—
294C8×C61—41no—
295C8×C61—42no—
296#92418—42no—
297#92418—41no—
298#92418—41no—
299#17453—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61#40712yes—
306C22×C61C22×C22⋊C411yescharacteristic
307C2×D61—24no—
308C2×D61—22no—
309C2×D61—42no—
310C2×D61—41no—
311C2×D61—42no—
312C2×D61—41no—
313C2×D61—84no—
314C2×D61—88no—
315C4×C61—22no—
316C4×C61—21no—
317C4×C61—22no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—22no—
321C4×C61#40712yes—
322C4×C61C22×M1611yescharacteristic
323C4×C61C22×M1611yescharacteristic
324C4×C61#57611yescharacteristic
325C4×C61#57711yescharacteristic
326Dic61—22no—
327Dic61—42no—
328Dic61—41no—
329Dic61—41no—
330C4×H(5,𝔽2)—611no—
331#2989—611no—
332#2965D6111yescharacteristic, complemented
333#1490—612no—
334#1489—612no—
335#1488—612no—
336#1486—612no—
337#1478—611no—
338#1477—611no—
339#1476—611no—
340C2×D4⋊C8—611no—
341C2×C61—22no—
342C2×C61—42no—
343C2×C61C2×#40711yescharacteristic
344C2×C61#289711yescharacteristic
345C2×C61#290111yescharacteristic
346D61—44no—
347D61—42no—
348C2×H(5,𝔽2)—611no—
349#520—611no—
350#520—611no—
351#520—612no—
352#520—1221no—
353#518—611no—
354#518—612no—
355#518—1222no—
356#518—1221no—
357C4×#41—611no—
358C4×#41—612no—
359C4×#41—1221no—
360C4×#41C2×D6111yescharacteristic
361C2×C4×D4—611no—
362C2×C4×D4—612no—
363C2×C4×D4—1222no—
364C2×C4×D4—1221no—
365#446—21no—
366#446—1221no—
367#436—21no—
368#436—1221no—
369#435—22no—
370#435—1222no—
371#432—612no—
372#431—611no—
373#431C2×D6111yescharacteristic, complemented
374#431C2×D6112yescomplemented
375#423—612no—
376C2×C4⋊C8—611no—
377C4×M16C2×D6112yescomplemented
378C2×C4×C8C2×D6111yescharacteristic, complemented
379Q8⋊C8—611no—
380Q8⋊C8—611no—
381Q8⋊C8—612no—
382D4⋊C8—612no—
383D4⋊C8—612no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386D4⋊C8—612no—
387D4⋊C8—612no—
388D4⋊C8—612no—
389C61#1745311yescharacteristic, complemented, Sylow
390H(5,𝔽2)—1224no—
391H(5,𝔽2)—1224no—
392C2×#41—611no—
393C2×#41—612no—
394C2×#41—1221no—
395C2×#41C4×D6111yescharacteristic
396C22×D4—611no—
397C22×D4—612no—
398C22×D4—1222no—
399C22×D4—1221no—
400C23×C4—1221no—
401C23×C4—1222no—
402C2×M16—22no—
403C2×M16—1222no—
404C22×C8—21no—
405C22×C8—1221no—
406C4×Q8—611no—
407C4×Q8—612no—
408C4×Q8—611no—
409C4×Q8—1221no—
410C4×Q8D24411yescharacteristic
411C4×Q8C22⋊D6111yescharacteristic
412C4×D4—612no—
413C4×D4—612no—
414C4×D4—612no—
415C4×D4—611no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—1222no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1222no—
436C4×D4D24412yes—
437C4×D4D24411yescharacteristic
438C4×D4C22⋊D6112yes—
439C4×D4C22⋊D6111yescharacteristic
440#117—612no—
441#117—611no—
442#117—1222no—
443#117—1221no—
444#117—1221no—
445#117—1222no—
446#117C4×D6112yes—
447#117C4×D6111yescharacteristic
448C2×C4⋊C4—612no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1222no—
456C2×C22⋊C4—1222no—
457C2×C42—1221no—
458C2×C42—1222no—
459C2×C42C22×D6111yescharacteristic
460C2×C42C22×D6112yes—
461C4⋊C8—22no—
462C4⋊C8—21no—
463C4⋊C8—612no—
464C4⋊C8—611no—
465C4⋊C8—611no—
466C4⋊C8—1221no—
467C4⋊C8—1222no—
468C22⋊C8—21no—
469C22⋊C8—22no—
470C22⋊C8—1221no—
471C22⋊C8—1222no—
472#97—22no—
473#97—1222no—
474C4×C8—21no—
475C4×C8—1221no—
476C4×C8C22×D6111yescharacteristic
477C4×C8C22×D6111yescharacteristic
478C4×C8C22×D6112yes—
479C24—1222no—
480C24—1221no—
481#41—22no—
482#41—22no—
483#41—614no—
484#41—1224no—
485#41—1224no—
486#41—1228no—
487#41—1222no—
488C2×Q8—611no—
489C2×Q8#1175819811yescharacteristic
490C2×D4—612no—
491C2×D4—611no—
492C2×D4—614no—
493C2×D4—1224no—
494C2×D4—1224no—
495C2×D4—12216no—
496C2×D4—1228no—
497C2×D4—1222no—
498C2×D4—1221no—
499C2×D4—1222no—
500C2×D4—1221no—
501C2×D4—1222no—
502C2×D4#1175819812yes—
503C2×D4#1175819811yescharacteristic
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1221no—
514C22×C4—1222no—
515C22×C4—4881no—
516C22×C4—4882no—
517C22×C4—4881no—
518C22×C4—4882no—
519C22×C4C2×C4×D6112yes—
520C22×C4C2×C4×D6111yescharacteristic
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24412yes—
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6112yes—
525M16—44no—
526M16—4882no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C2×C8—42no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1222no—
534C2×C8—4881no—
535C4⋊C4—611no—
536C4⋊C4—612no—
537C4⋊C4—611no—
538C4⋊C4—1221no—
539C4⋊C4—1222no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1222no—
543C4⋊C4#1175819812yes—
544C4⋊C4#1175819811yescharacteristic
545C4⋊C4#1175819812yes—
546C4⋊C4#1175819811yescharacteristic
547C22⋊C4—1222no—
548C22⋊C4—1221no—
549C22⋊C4—1222no—
550C22⋊C4—1221no—
551C22⋊C4—1221no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4#1175819812yes—
557C22⋊C4#1175819812yes—
558C22⋊C4#1175819812yes—
559C42—1221no—
560C42—1222no—
561C42—1221no—
562C42—1221no—
563C42—1222no—
564C42C2×C4×D6111yescharacteristic
565C42C2×C4×D6112yes—
566C42C2×D24411yescharacteristic
567C42C2×D24412yes—
568C42C2×C22⋊D6111yescharacteristic
569C42C2×C22⋊D6112yes—
570C42C23×D6111yescharacteristic
571C23—1228no—
572C23—1224no—
573C23—1222no—
574C23—1221no—
575C23—4881no—
576C23—4882no—
577C23—4881no—
578C23—4882no—
579C23C2×#1175819812yes—
580C23C2×#1175819811yescharacteristic
581Q8—22no—
582Q8—612no—
583Q8—1221no—
584D4—24no—
585D4—22no—
586D4—614no—
587D4—612no—
588D4—1228no—
589D4—1228no—
590D4—1224no—
591D4—1224no—
592D4—1221no—
593D4—1222no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—1228no—
603C2×C4—1221no—
604C2×C4—1224no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—4881no—
609C2×C4—4882no—
610C2×C4—4882no—
611C2×C4—4882no—
612C2×C4—4882no—
613C2×C4—4881no—
614C2×C4—4882no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2441no—
619C2×C4C2×#1175819811yescharacteristic
620C2×C4C2×#1175819812yes—
621C2×C4C2×#1175819812yes—
622C2×C4C2×#1175819811yescharacteristic
623C2×C4C2×#1175819812yes—
624C2×C4C2×#1175819811yescharacteristic
625C2×C4M16×D6112yes—
626C2×C4M16×D6112yes—
627C2×C4C22×C4×D6111yescharacteristic
628C2×C4C22×C22⋊D6111yescharacteristic
629C2×C4C22×D24411yescharacteristic
630C8—22no—
631C8—41no—
632C8—42no—
633C8—2442no—
634C8—2441no—
635C8—2441no—
636C22—21no—
637C22—21no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—1224no—
642C22—1222no—
643C22—4884no—
644C22—4888no—
645C22—2442no—
646C22—2441no—
647C22—2442no—
648C22—2441no—
649C22#1683972512yes—
650C22C22×#1175819811yescharacteristic
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—22no—
656C4—22no—
657C4—1222no—
658C4—2442no—
659C4—2441no—
660C4—2441no—
661C4#1683972512yes—
662C4C2×M16×D6111yescharacteristic
663C4C2×M16×D6111yescharacteristic
664C4#1684056711yescharacteristic
665C4#1684057411yescharacteristic
666C2—22no—
667C2—42no—
668C2—2444no—
669C2—2442no—
670C2C2×#1683972511yescharacteristic
671C2#2145434611yescharacteristic
672C2#2145436711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2965C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519823C4
cosocleC42×C61C24
hypercenter#2965D61
hypocenterC61#17453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2965

Complements

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

NH ≅ G/Ntype
2. #21454936C2semidirect
3. #21444263C2semidirect
4. #21444259C2semidirect
5. #21444254C2semidirect
6. #21444245C2semidirect
7. #21444211C2semidirect
8. #21444207C2semidirect
9. #21444202C2semidirect
10. C2×#16839491C2semidirect
11. C61×#2965C2semidirect
31. #16839819C22semidirect
32. #16839815C22semidirect
33. #16839786C22semidirect
34. C2×C4⋊#92418C22semidirect
35. #16839494C22semidirect
36. #16839494C22semidirect
37. #16839494C22semidirect
38. #16839491C22semidirect
39. #16839491C22semidirect
40. #16839491C22semidirect
41. #16839491C22semidirect
42. #16839491C22semidirect
43. #16839491C22semidirect
44. #16839491C22semidirect
49. C61×#431C22semidirect
50. C61×#431C22semidirect
51. C4×M16×C61C22semidirect
52. C2×C4×C8×C61C22semidirect
165. #11758222M16semidirect
171. C2×D244M16semidirect
332. #2965D61semidirect
373. #431C2×D61semidirect
374. #431C2×D61semidirect
377. C4×M16C2×D61semidirect
378. C2×C4×C8C2×D61semidirect
389. C61#17453semidirect

Maximal subgroups

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

subgroupindex
C4×#145198232
#214549362
#214442632
#214442592
#214442542
#214442452
#214442112
#214442072
#214442022
C2×#168394912
C61×#29652
#1745361