Finite Group Databasebeta

← #30565077

Subgroups — #30565077

Order 15616.

2086 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459668C211yesmaximal, characteristic
2C2×#16840427C211yesmaximal, characteristic, complemented
3#21458116C214yesmaximal, complemented
4#21458101C214yesmaximal, complemented
5#21457484C214yesmaximal, complemented
6#21457468C214yesmaximal, complemented
7#21456767C211yesmaximal, characteristic, complemented
8#21456646C214yesmaximal, complemented
9#21455859C211yesmaximal, characteristic, complemented
10#21455846C211yesmaximal, characteristic, complemented
11#21455050C214yesmaximal, complemented
12#21454859C211yesmaximal, characteristic, complemented
13C61×#3206C211yesmaximal, characteristic, complemented
14C22×Q8×D61C2212yes—
15C2×#14519806C2218yescomplemented
16#16840494—22no—
17#16840486C2212yes—
18#16840477C2212yes—
19#16840458—22no—
20#16840432C2212yes—
21#16840427—22no—
22#16840427—22no—
23#16840427C2214yes—
24#16840427C2214yes—
25#16840416C2212yes—
26#16840349—22no—
27#16840326—21no—
28#16840319C2211yescharacteristic
29C2×#14519776—21no—
30#16840272C2214yes—
31C2×#14519755—22no—
32C2×#14519753C2212yes—
33#16840245—21no—
34C2×C4×Dic122C2211yescharacteristic
35C2×C4×D244C2211yescharacteristic
36C2×#14519739—21no—
37#16840141C2212yescomplemented
38#16840112C2212yescomplemented
39#16840102C2218yescomplemented
40#16840098C2212yes—
41#16840078C2212yes—
42#16840058C2212yescomplemented
43#16840044C2212yescomplemented
44#16840019C2218yescomplemented
45#16840005C2218yescomplemented
46#16839973C2212yescomplemented
47#16839969C2214yescomplemented
48#16839959C2212yescomplemented
49#16839921C2218yescomplemented
50#16839920C2212yes—
51#16839919C2218yescomplemented
52#16839918C2214yes—
53#16839900C2212yes—
54#16839899C2218yescomplemented
55C4×Dic244C2214yescomplemented
56C4×#11758200C2214yescomplemented
57#16839841C2214yes—
58#16839839C2214yes—
59#16839815C2211yescharacteristic
60#16839769C2218yescomplemented
61#16839724C2214yes—
62C61×#532C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#492C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#461C2214yescomplemented
67C61×#448C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758224C2314yes—
70C22×Dic122—22no—
71C22×Dic122C2311yescharacteristic
72C2×Q8×D61—24no—
73C2×Q8×D61—48no—
74C2×Q8×D61C2318yes—
75C22×D244D411yescharacteristic
76C22×C4×D61—22no—
77C22×C4×D61C2312yes—
78#14519806—416no—
79C2×Dic244C2314yes—
80C2×#11758202C2318yes—
81C2×#11758200C2314yes—
82C2×#11758189C2318yes—
83#14519777C2311yescharacteristic
84#14519777C2311yescharacteristic
85#14519776—24no—
86#14519776—24no—
87#14519773—22no—
88#14519772—24no—
89#14519769—21no—
90#14519766—22no—
91#14519763—24no—
92#14519761—24no—
93#14519759—24no—
94#14519757C2314yes—
95#14519756—21no—
96#14519756C2312yes—
97#14519755—28no—
98#14519755—28no—
99#14519753—24no—
100#14519753C2318yes—
101#14519751—22no—
102#14519751—22no—
103#14519749—24no—
104#14519748C2314yes—
105#14519745—24no—
106Q8×Dic61—22no—
107C4×Dic122—21no—
108C4×Dic122—21no—
109C4×Dic122—21no—
110C4×Dic122C2312yes—
111C4×Dic122C2312yes—
112C4×Dic122C2312yes—
113C4×D244D412yes—
114C4×D244D412yes—
115C4×D244C2312yes—
116C4×D244C2312yes—
117D4×Dic61—24no—
118D4×Dic61—22no—
119#14519739—24no—
120#14519739—24no—
121#14519737—24no—
122#14519736—21no—
123#14519736—22no—
124#14519735C2314yes—
125C2×C4⋊Dic61—22no—
126C2×C4⋊Dic61D411yescharacteristic
127C2×C4⋊Dic61C2311yescharacteristic
128C2×#11758196—21no—
129C2×#11758196—22no—
130C2×#11758198—22no—
131C2×#11758198D412yes—
132C2×C4×Dic61—21no—
133#14519711C2314yes—
134#14519708C2314yes—
135C4⋊#92418C2312yes—
136#14519702C2314yes—
137#14519702C2314yes—
138#14519701C2318yes—
139#14519700C2314yes—
140#14519698C2314yes—
141C22⋊#92418C2312yes—
142#14519689C2318yes—
143C4×#92418C2312yes—
144C2×#41×C61—21no—
145C22×Q8×C61C2312yes—
146C2×SD16×C61C2318yes—
147C4⋊Q8×C61—21no—
148C4⋊Q8×C61—21no—
149C4⋊Q8×C61C2312yes—
150C4⋊Q8×C61C2312yes—
151C4⋊Q8×C61C2311yescharacteristic
152#127×C61C2311yescharacteristic
153#124×C61—21no—
154#121×C61—21no—
155#121×C61C2312yes—
156C4×D4×C61—21no—
157C4×D4×C61C2312yes—
158C2×C4⋊C4×C61—21no—
159C2×C4⋊C4×C61—21no—
160C2×C42×C61C2311yescharacteristic
161C4⋊C8×C61C2312yes—
162#103×C61C2318yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61C2×D412yes—
166#41×C61—44no—
167C2×Q8×C61—22no—
168C2×Q8×C61—21no—
169C2×Q8×C61—44no—
170C2×Q8×C61C2414yes—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C2412yes—
175C22×C4×C61—22no—
176C22×C4×C61—22no—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C2×C22⋊D61—24no—
181C2×C22⋊D61—24no—
182C22×Dic61—21no—
183C22×Dic61—21no—
184C22×Dic61—24no—
185C22×Dic61C2×D412yes—
186Q8×D61—216no—
187Q8×D61—48no—
188Q8×D61—416no—
189#11758224—48no—
190C2×D244—24no—
191C2×D244C2×D412yes—
192C2×D244C2×D412yes—
193C2×C4×D61—28no—
194C2×C4×D61—44no—
195C2×C4×D61—48no—
196C2×C4×D61C2×D414yes—
197C2×C4×D61C2414yes—
198C2×Dic122—21no—
199C2×Dic122—24no—
200C2×Dic122—28no—
201C2×Dic122—21no—
202C2×Dic122—48no—
203C2×Dic122C2412yes—
204C2×Dic122C2412yes—
205SD16×C61—48no—
206C2×C8×C61C2414yes—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—22no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61C2412yes—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2411yescharacteristic
218C42×C61C2411yescharacteristic
219C22⋊Dic61—22no—
220C22⋊Dic61—24no—
221#11758202—48no—
222#11758200—28no—
223#11758198—28no—
224#11758198—24no—
225#11758198—24no—
226#11758198C2×D414yes—
227#11758198C2×D414yes—
228C4⋊Dic61—21no—
229C4⋊Dic61—21no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232C4⋊Dic61—24no—
233C4⋊Dic61C2×D412yes—
234C4⋊Dic61C2×D412yes—
235C4⋊Dic61C2412yes—
236C4⋊Dic61C2412yes—
237#11758196—22no—
238#11758196—22no—
239#11758196—22no—
240#11758196—22no—
241#11758196—28no—
242C4×Dic61—22no—
243C4×Dic61—22no—
244C4×Dic61—22no—
245C4×Dic61—24no—
246C2×#92418C2414yes—
247Dic244—28no—
248#11758189—48no—
249C23×C61—22no—
250C23×C61C22×D411yescharacteristic
251C22×D61—24no—
252C22×D61—42no—
253C22×D61—42no—
254C22×D61C22×D414yes—
255Q8×C61—28no—
256Q8×C61—44no—
257Q8×C61—42no—
258D4×C61—24no—
259D4×C61—44no—
260D4×C61—42no—
261D4×C61—81no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—24no—
265C2×C4×C61—41no—
266C2×C4×C61—44no—
267C2×C4×C61—44no—
268C2×C4×C61#13714yescomplemented
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2511yescharacteristic
275C22⋊D61—84no—
276C22⋊D61—84no—
277C2×Dic61—24no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—84no—
285C2×Dic61C22×D414yes—
286D244—24no—
287D244—24no—
288C4×D61—28no—
289C4×D61—44no—
290C4×D61—416no—
291Dic122—24no—
292Dic122—22no—
293Dic122—24no—
294Dic122—416no—
295Dic122—81no—
296Dic122H(5,𝔽2)14yes—
297C8×C61—24no—
298#92418—24no—
299#61584—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61C2×#13712yes—
306C22×C61C23×D411yescharacteristic
307C2×D61—24no—
308C2×D61—24no—
309C2×D61—24no—
310C2×D61—48no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—44no—
315C4×C61—42no—
316C4×C61C2×#13712yes—
317C4×C61C2×#13712yes—
318C4×C61C2×H(5,𝔽2)12yes—
319Dic61—24no—
320Dic61—24no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324#3506—611no—
325C2×D4×Q8—611no—
326#3329—614no—
327#3328—614no—
328#3278—614no—
329#3276—614no—
330#3209—611no—
331#3206D6111yescharacteristic, complemented
332#3196—614no—
333#3104—611no—
334#3103—611no—
335#3004—614no—
336#2975—611no—
337C2×C61—22no—
338C2×C61—82no—
339C2×C61#348411yescharacteristic
340C2×C61C22×#13711yescharacteristic
341C2×C61#358611yescharacteristic
342D61—44no—
343C23×Q8—612no—
344C2×#137—618no—
345#559—1222no—
346#556—612no—
347#555—612no—
348#553—1222no—
349#550—612no—
350D4×Q8—614no—
351D4×Q8—614no—
352D4×Q8—1222no—
353D4×Q8—1222no—
354#547—612no—
355#538—1222no—
356#535—1221no—
357#533—611no—
358#532C2×D6111yescharacteristic
359C2×C4⋊Q8—1221no—
360C2×C4⋊Q8C2×D6111yescharacteristic
361#525—614no—
362C2×C22⋊Q8—612no—
363C2×C22⋊Q8—1222no—
364#520—1221no—
365C2×C4×Q8—611no—
366C2×C4×D4—611no—
367C2×C4×D4—1221no—
368#494—612no—
369#492C2×D6112yescomplemented
370#487—612no—
371#487C2×D6112yescomplemented
372#485—618no—
373#484—612no—
374#481—612no—
375#477—612no—
376#475—612no—
377#470—618no—
378#468—618no—
379#462—614no—
380#462—612no—
381#461C2×D6114yescomplemented
382#460—612no—
383#452—618no—
384#451—614no—
385#451—618no—
386#451—612no—
387#448—618no—
388#448—612no—
389#448C2×D6114yes—
390C4×Q16—614no—
391C4×SD16—614no—
392#436—614no—
393#436—614no—
394#431—611no—
395#431C2×D6111yescharacteristic
396#419—618no—
397#407—614no—
398C61#6158411yescharacteristic, complemented, Sylow
399C2×#41—21no—
400C2×#41—614no—
401C22×Q8—618no—
402C22×Q8—611no—
403C22×Q8—1222no—
404C22×Q8—1224no—
405C22×Q8—2448no—
406C22×Q8C22×D6112yes—
407C22×D4—611no—
408C23×C4—612no—
409C23×C4—1222no—
410#137—24416no—
411C2×Q16—618no—
412C2×Q16—614no—
413C2×SD16—614no—
414C2×SD16C22×D6118yes—
415C2×M16—618no—
416C4⋊Q8—21no—
417C4⋊Q8—21no—
418C4⋊Q8—611no—
419C4⋊Q8—611no—
420C4⋊Q8—1224no—
421C4⋊Q8—1224no—
422C4⋊Q8C22×D6112yes—
423C4⋊Q8C22×D6112yes—
424C4⋊Q8C22×D6111yescharacteristic
425#127—1222no—
426#127C22×D6111yescharacteristic
427#126—1224no—
428#125—1221no—
429#124—21no—
430#124—1224no—
431#124—1222no—
432#123—1224no—
433#123—1224no—
434C22⋊Q8—618no—
435C22⋊Q8—612no—
436C22⋊Q8—614no—
437C22⋊Q8—1228no—
438C22⋊Q8—1228no—
439C22⋊Q8—1224no—
440C22⋊Q8—1221no—
441#121—21no—
442#121—614no—
443#121—1224no—
444#121—1222no—
445#121—1222no—
446#121C22×D6112yes—
447#120—1224no—
448C4×Q8—612no—
449C4×Q8—612no—
450C4×Q8—612no—
451C4×Q8—1222no—
452C4×Q8—1221no—
453C4×Q8—1221no—
454C4×Q8—1221no—
455C4×D4—21no—
456C4×D4—612no—
457C4×D4—612no—
458C4×D4—612no—
459C4×D4—612no—
460C4×D4—1224no—
461C4×D4—1224no—
462C4×D4—1224no—
463C4×D4—1222no—
464C4×D4—1224no—
465C4×D4C22×D6112yes—
466#117—614no—
467#117—1222no—
468#117—1221no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—611no—
472C2×C4⋊C4—611no—
473C2×C4⋊C4—1222no—
474C2×C4⋊C4—1222no—
475C2×C4⋊C4—1221no—
476C2×C22⋊C4—612no—
477C2×C22⋊C4—1222no—
478C2×C42—1221no—
479C2×C42C22×D6111yescharacteristic
480#107—614no—
481#106—614no—
482C4⋊C8—612no—
483C4⋊C8C22×D6112yes—
484#103—618no—
485#103—614no—
486#103—614no—
487#103—614no—
488#103C22×D6118yes—
489#102—614no—
490C22⋊C8—612no—
491C22⋊C8—618no—
492C22⋊C8C22×D6112yes—
493C4×C8—612no—
494C4×C8C22×D6112yes—
495C24—612no—
496#41—44no—
497#41—2448no—
498C2×Q8—21no—
499C2×Q8—22no—
500C2×Q8—44no—
501C2×Q8—612no—
502C2×Q8—612no—
503C2×Q8—1228no—
504C2×Q8—1224no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—12216no—
508C2×Q8—2448no—
509C2×Q8—2448no—
510C2×Q8—24416no—
511C2×Q8C23×D6114yes—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—612no—
516C2×D4—612no—
517C2×D4—1224no—
518C2×D4—1224no—
519C2×D4—1224no—
520C2×D4C23×D6112yes—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—612no—
524C22×C4—614no—
525C22×C4—614no—
526C22×C4—1228no—
527C22×C4—1224no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—2444no—
531C22×C4—2448no—
532C22×C4D4×D6111yescharacteristic
533C22×C4D4×D6111yescharacteristic
534C22×C4C23×D6111yescharacteristic
535Q16—1228no—
536Q16—2448no—
537SD16—48no—
538SD16—1228no—
539M16—2448no—
540C2×C8—614no—
541C2×C8C23×D6114yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—612no—
550C4⋊C4—612no—
551C4⋊C4—1228no—
552C4⋊C4—1224no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4C23×D6112yes—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—614no—
565C22⋊C4—614no—
566C22⋊C4—1228no—
567C22⋊C4—1224no—
568C22⋊C4—1224no—
569C22⋊C4—1224no—
570C22⋊C4—1222no—
571C42—1224no—
572C42—1222no—
573C42—1222no—
574C42—1222no—
575C42D4×D6111yescharacteristic
576C42D4×D6111yescharacteristic
577C42C23×D6111yescharacteristic
578C42C23×D6111yescharacteristic
579C23—22no—
580C23—614no—
581C23—1224no—
582C23—2442no—
583C23—2442no—
584C23C2×D4×D6111yescharacteristic
585Q8—28no—
586Q8—44no—
587Q8—42no—
588Q8—614no—
589Q8—1224no—
590Q8—1222no—
591Q8—1224no—
592Q8—4881no—
593Q8—24416no—
594D4—24no—
595D4—44no—
596D4—42no—
597D4—81no—
598D4—1224no—
599D4—1224no—
600D4—4884no—
601D4—4884no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—614no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1228no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—4884no—
616C2×C4—2444no—
617C2×C4—24416no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4#1451980614yescomplemented
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C24×D6111yescharacteristic
627C8—24no—
628C8—1224no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—1224no—
635C22—1224no—
636C22—1224no—
637C22—2448no—
638C22C2×#1451980612yes—
639C22C22×D4×D6111yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—42no—
644C4—44no—
645C4—1224no—
646C4—1224no—
647C4—2442no—
648C4—2442no—
649C4—2442no—
650C4C2×#1451980612yes—
651C4C2×#1451980612yes—
652C4H(5,𝔽2)×D6112yes—
653C2—22no—
654C2—82no—
655C2—2444no—
656C2#2145945311yescharacteristic
657C2C22×#1451980611yescharacteristic
658C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3206C2
socleC22×C61C23×D4
prime radical#21459668C2
hypercenter#3206D61
hypocenterC61#61584
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. C2×#16840427C2semidirect
3. #21458116C2semidirect
4. #21458101C2semidirect
5. #21457484C2semidirect
6. #21457468C2semidirect
7. #21456767C2semidirect
8. #21456646C2semidirect
9. #21455859C2semidirect
10. #21455846C2semidirect
11. #21455050C2semidirect
12. #21454859C2semidirect
13. C61×#3206C2semidirect
15. C2×#14519806C22semidirect
37. #16840141C22semidirect
38. #16840112C22semidirect
39. #16840102C22semidirect
42. #16840058C22semidirect
43. #16840044C22semidirect
44. #16840019C22semidirect
45. #16840005C22semidirect
46. #16839973C22semidirect
47. #16839969C22semidirect
48. #16839959C22semidirect
49. #16839921C22semidirect
51. #16839919C22semidirect
54. #16839899C22semidirect
55. C4×Dic244C22semidirect
56. C4×#11758200C22semidirect
60. #16839769C22semidirect
64. C61×#492C22semidirect
65. C61×#487C22semidirect
66. C61×#461C22semidirect
268. C2×C4×C61#137semidirect
331. #3206D61semidirect
369. #492C2×D61semidirect
371. #487C2×D61semidirect
381. #461C2×D61semidirect
398. C61#61584semidirect
620. C2×C4#14519806semidirect

Maximal subgroups

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

subgroupindex
#214596682
C2×#168404272
#214581162
#214581012
#214574842
#214574682
#214567672
#214566462
#214558592
#214558462
#214550502
#214548592
C61×#32062
#6158461