Finite Group Databasebeta

← #30565076

Subgroups — #30565076

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×Q8×D244C211yesmaximal, characteristic, complemented
3#21458115C214yesmaximal, complemented
4#21458099C214yesmaximal, complemented
5#21457488C214yesmaximal, complemented
6#21457466C214yesmaximal, complemented
7#21456769C211yesmaximal, characteristic, complemented
8#21456741C211yesmaximal, characteristic, complemented
9#21456648C214yesmaximal, complemented
10#21455858C211yesmaximal, characteristic, complemented
11#21455050C214yesmaximal, complemented
12#21454859C211yesmaximal, characteristic, complemented
13C61×#3103C211yesmaximal, characteristic, complemented
14C22×Q8×D61C2212yes—
15C2×#14519806C2218yescomplemented
16#16840494—22no—
17#16840486C2212yes—
18#16840477C2212yes—
19#16840458—22no—
20#16840432C2212yes—
21Q8×D244—22no—
22Q8×D244—22no—
23Q8×D244C2214yes—
24Q8×D244C2214yes—
25#16840416C2212yes—
26#16840349—22no—
27#16840326—21no—
28#16840319C2211yescharacteristic
29C2×#14519777—21no—
30C2×#14519777C2211yescharacteristic
31#16840272C2214yes—
32C2×#14519757—22no—
33C2×#14519757C2212yes—
34#16840245—21no—
35C2×C4×D244—21no—
36C2×C4×D244C2211yescharacteristic
37#16840142C2212yescomplemented
38#16840134C2212yescomplemented
39#16840114C2212yescomplemented
40#16840113C2212yescomplemented
41#16840106C2218yescomplemented
42#16840097C2212yes—
43#16840057C2212yescomplemented
44#16840021C2218yescomplemented
45#16840007C2218yescomplemented
46#16839973C2214yescomplemented
47#16839972C2212yescomplemented
48#16839967C2214yescomplemented
49#16839925C2218yescomplemented
50#16839920C2214yes—
51#16839919C2218yescomplemented
52#16839918C2212yes—
53#16839900C2214yes—
54#16839899C2218yescomplemented
55C4×#11758202C2214yescomplemented
56#16839841C2214yes—
57#16839839C2214yes—
58#16839815C2211yescharacteristic
59#16839769C2218yescomplemented
60#16839724C2214yes—
61C61×#532C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#481C2212yes—
64C61×#475C2212yescomplemented
65C61×#460C2212yescomplemented
66C61×#448C2212yes—
67C4×SD16×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758224C2314yes—
70C22×Dic122—22no—
71C22×Dic122C2312yes—
72C2×Q8×D61—24no—
73C2×Q8×D61—48no—
74C2×Q8×D61C2318yes—
75C22×D244D411yescharacteristic
76C22×C4×D61—22no—
77C22×C4×D61C2312yes—
78#14519806—416no—
79C2×Dic244C2318yes—
80C2×#11758202C2314yes—
81C2×#11758200C2318yes—
82C2×#11758189C2318yes—
83#14519777—21no—
84#14519777—21no—
85#14519777—24no—
86#14519777—24no—
87#14519777C2312yes—
88#14519777C2311yescharacteristic
89#14519777C2312yes—
90#14519777C2311yescharacteristic
91#14519773—22no—
92#14519772—24no—
93#14519769—21no—
94#14519766—22no—
95#14519763—24no—
96#14519761—24no—
97#14519759—24no—
98#14519757—24no—
99#14519757—28no—
100#14519757—28no—
101#14519757C2314yes—
102#14519757C2318yes—
103#14519756—21no—
104#14519756C2312yes—
105#14519751—22no—
106#14519751—22no—
107#14519749—24no—
108#14519748C2314yes—
109#14519745—24no—
110Q8×Dic61—22no—
111C4×Dic122—21no—
112C4×Dic122C2312yes—
113C4×D244—24no—
114C4×D244—24no—
115C4×D244D412yes—
116C4×D244D412yes—
117C4×D244C2312yes—
118C4×D244C2312yes—
119D4×Dic61—24no—
120D4×Dic61—22no—
121#14519737—24no—
122#14519736—21no—
123#14519736—22no—
124#14519735C2314yes—
125C2×C4⋊Dic61—21no—
126C2×C4⋊Dic61—21no—
127C2×C4⋊Dic61—22no—
128C2×C4⋊Dic61—22no—
129C2×C4⋊Dic61D411yescharacteristic
130C2×#11758198—22no—
131C2×#11758198D412yes—
132#14519710C2314yes—
133C4⋊#92418C2312yes—
134#14519702C2318yes—
135#14519701C2314yes—
136#14519700C2318yes—
137#14519700C2314yes—
138C22⋊#92418C2312yes—
139#14519689C2318yes—
140C4×#92418C2312yes—
141C2×#41×C61—21no—
142C22×Q8×C61C2311yescharacteristic
143C2×SD16×C61C2314yes—
144C4⋊Q8×C61C2311yescharacteristic
145#127×C61C2311yescharacteristic
146#124×C61—21no—
147#121×C61—21no—
148#121×C61C2312yes—
149C4×Q8×C61—21no—
150C4×Q8×C61—21no—
151C4×Q8×C61C2312yes—
152C4×Q8×C61C2312yes—
153C4×D4×C61—21no—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C42×C61—21no—
158C2×C42×C61C2311yescharacteristic
159#106×C61C2314yes—
160C4⋊C8×C61C2312yes—
161#103×C61C2314yes—
162#102×C61C2314yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61C2×D412yes—
166#41×C61—44no—
167C2×Q8×C61—24no—
168C2×Q8×C61—21no—
169C2×Q8×C61—21no—
170C2×Q8×C61C2412yes—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C2412yes—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—22no—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2411yescharacteristic
181C2×C22⋊D61—24no—
182C2×C22⋊D61—24no—
183C22×Dic61—22no—
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—22no—
199C2×Dic122—28no—
200C2×Dic122—21no—
201C2×Dic122—44no—
202C2×Dic122—48no—
203C2×Dic122C2414yes—
204C2×Dic122C2412yes—
205SD16×C61—28no—
206C2×C8×C61C2414yes—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61C2412yes—
212C4⋊C4×C61C2412yes—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C42×C61—22no—
216C42×C61—22no—
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219C42×C61C2411yescharacteristic
220C42×C61C2411yescharacteristic
221C22⋊Dic61—22no—
222C22⋊Dic61—24no—
223#11758202—28no—
224#11758200—48no—
225#11758198—28no—
226#11758198—24no—
227#11758198—24no—
228#11758198C2×D414yes—
229#11758198C2×D414yes—
230C4⋊Dic61—21no—
231C4⋊Dic61—22no—
232C4⋊Dic61—22no—
233C4⋊Dic61—22no—
234C4⋊Dic61—22no—
235C4⋊Dic61—21no—
236C4⋊Dic61—21no—
237C4⋊Dic61—24no—
238C4⋊Dic61—28no—
239C4⋊Dic61C2×D412yes—
240C4⋊Dic61C2×D412yes—
241C4⋊Dic61C2412yes—
242#11758196—22no—
243#11758196—22no—
244C4×Dic61—22no—
245C4×Dic61—24no—
246C2×#92418C2414yes—
247Dic244—48no—
248#11758189—48no—
249C23×C61—22no—
250C23×C61C22×D411yescharacteristic
251C22×D61—24no—
252C22×D61—42no—
253C22×D61—42no—
254C22×D61C22×D414yes—
255Q8×C61—24no—
256Q8×C61—22no—
257Q8×C61—42no—
258Q8×C61H(5,𝔽2)14yes—
259D4×C61—24no—
260D4×C61—44no—
261D4×C61—42no—
262D4×C61—81no—
263C2×C4×C61—24no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61—44no—
271C2×C4×C61#13714yescomplemented
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C2511yescharacteristic
278C22⋊D61—84no—
279C22⋊D61—84no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—24no—
283C2×Dic61—42no—
284C2×Dic61—44no—
285C2×Dic61—84no—
286C2×Dic61C22×D414yes—
287D244—24no—
288D244—24no—
289C4×D61—28no—
290C4×D61—44no—
291C4×D61—416no—
292Dic122—24no—
293Dic122—28no—
294Dic122—416no—
295Dic122—44no—
296Dic122—81no—
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—24no—
314C4×C61—21no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61C2×#13712yes—
318C4×C61C2×#13712yes—
319C4×C61C2×H(5,𝔽2)12yes—
320Dic61—24no—
321Dic61—42no—
322Dic61—44no—
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#3206—611no—
332#3196—614no—
333#3104—611no—
334#3103D6111yescharacteristic, complemented
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—611no—
360C2×C4⋊Q8—1221no—
361#525—614no—
362C2×C22⋊Q8—612no—
363C2×C22⋊Q8—1222no—
364#520—1221no—
365C2×C4×Q8C2×D6111yescharacteristic
366C2×C4×D4—611no—
367C2×C4×D4—1221no—
368#494—612no—
369#492—612no—
370#487—612no—
371#487—612no—
372#485—618no—
373#484—612no—
374#481C2×D6112yes—
375#477—612no—
376#475C2×D6112yescomplemented
377#470—618no—
378#468—618no—
379#462—612no—
380#462—614no—
381#461—614no—
382#460C2×D6112yescomplemented
383#452—618no—
384#451—612no—
385#451—618no—
386#451—614no—
387#448—618no—
388#448—614no—
389#448C2×D6112yes—
390C4×Q16—614no—
391C4×SD16C2×D6114yescomplemented
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—612no—
403C22×Q8—1222no—
404C22×Q8—1224no—
405C22×Q8—2448no—
406C22×Q8C22×D6111yescharacteristic
407C22×D4—611no—
408C23×C4—612no—
409C23×C4—1222no—
410#137—24416no—
411C2×Q16—614no—
412C2×Q16—618no—
413C2×SD16—618no—
414C2×SD16C22×D6114yes—
415C2×M16—618no—
416C4⋊Q8—611no—
417C4⋊Q8—612no—
418C4⋊Q8—612no—
419C4⋊Q8—611no—
420C4⋊Q8—1224no—
421C4⋊Q8—1224no—
422C4⋊Q8—1221no—
423C4⋊Q8—1221no—
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—612no—
435C22⋊Q8—618no—
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—21no—
449C4×Q8—21no—
450C4×Q8—612no—
451C4×Q8—1222no—
452C4×Q8—1221no—
453C4×Q8C22×D6112yes—
454C4×Q8C22×D6112yes—
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—611no—
471C2×C4⋊C4—1222no—
472C2×C4⋊C4—1222no—
473C2×C4⋊C4—1221no—
474C2×C4⋊C4—1221no—
475C2×C4⋊C4C22×D6111yescharacteristic
476C2×C22⋊C4—612no—
477C2×C22⋊C4—1222no—
478C2×C42—21no—
479C2×C42C22×D6111yescharacteristic
480#107—614no—
481#106C22×D6114yes—
482C4⋊C8—612no—
483C4⋊C8C22×D6112yes—
484#103—614no—
485#103—618no—
486#103—618no—
487#103—614no—
488#103C22×D6114yes—
489#102C22×D6114yes—
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—24no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—614no—
502C2×Q8—612no—
503C2×Q8—1228no—
504C2×Q8—1221no—
505C2×Q8—1222no—
506C2×Q8—12216no—
507C2×Q8—2444no—
508C2×Q8—2448no—
509C2×Q8—2448no—
510C2×Q8—24416no—
511C2×Q8C23×D6112yes—
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—21no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—612no—
525C22×C4—614no—
526C22×C4—614no—
527C22×C4—1228no—
528C22×C4—1224no—
529C22×C4—1222no—
530C22×C4—2444no—
531C22×C4—2448no—
532C22×C4D4×D6111yescharacteristic
533C22×C4D4×D6111yescharacteristic
534C22×C4C23×D6111yescharacteristic
535Q16—1228no—
536Q16—2448no—
537SD16—28no—
538SD16—2448no—
539M16—2448no—
540C2×C8—614no—
541C2×C8C23×D6114yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—1224no—
550C4⋊C4—1228no—
551C4⋊C4—1221no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4C23×D6112yes—
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—22no—
572C42—22no—
573C42—1224no—
574C42—1222no—
575C42D4×D6111yescharacteristic
576C42D4×D6111yescharacteristic
577C42C23×D6111yescharacteristic
578C42C23×D6111yescharacteristic
579C23—22no—
580C23—614no—
581C23—1224no—
582C23—2442no—
583C23—2442no—
584C23C2×D4×D6111yescharacteristic
585Q8—24no—
586Q8—22no—
587Q8—42no—
588Q8—1228no—
589Q8—1224no—
590Q8—4881no—
591Q8—24416no—
592Q8—2444no—
593Q8#1451982314yes—
594D4—24no—
595D4—44no—
596D4—42no—
597D4—81no—
598D4—1224no—
599D4—1224no—
600D4—4884no—
601D4—4884no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—614no—
611C2×C4—1228no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1224no—
615C2×C4—4884no—
616C2×C4—2444no—
617C2×C4—24416no—
618C2×C4—2442no—
619C2×C4—2444no—
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—24no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—1224no—
647C4—2442no—
648C4—2442no—
649C4—2444no—
650C4C2×#1451980612yes—
651C4C2×#1451980612yes—
652C4C2×#1451982312yes—
653C2—22no—
654C2—82no—
655C2—2444no—
656C2#2145945211yescharacteristic
657C2C22×#1451980611yescharacteristic
658C2#2146060311yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3103C2
socleC22×C61C23×D4
prime radical#21459668C2
hypercenter#3103D61
hypocenterC61#61584
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×Q8×D244C2semidirect
3. #21458115C2semidirect
4. #21458099C2semidirect
5. #21457488C2semidirect
6. #21457466C2semidirect
7. #21456769C2semidirect
8. #21456741C2semidirect
9. #21456648C2semidirect
10. #21455858C2semidirect
11. #21455050C2semidirect
12. #21454859C2semidirect
13. C61×#3103C2semidirect
15. C2×#14519806C22semidirect
37. #16840142C22semidirect
38. #16840134C22semidirect
39. #16840114C22semidirect
40. #16840113C22semidirect
41. #16840106C22semidirect
43. #16840057C22semidirect
44. #16840021C22semidirect
45. #16840007C22semidirect
46. #16839973C22semidirect
47. #16839972C22semidirect
48. #16839967C22semidirect
49. #16839925C22semidirect
51. #16839919C22semidirect
54. #16839899C22semidirect
55. C4×#11758202C22semidirect
59. #16839769C22semidirect
64. C61×#475C22semidirect
65. C61×#460C22semidirect
67. C4×SD16×C61C22semidirect
271. C2×C4×C61#137semidirect
334. #3103D61semidirect
376. #475C2×D61semidirect
382. #460C2×D61semidirect
391. C4×SD16C2×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×Q8×D2442
#214581152
#214580992
#214574882
#214574662
#214567692
#214567412
#214566482
#214558582
#214550502
#214548592
C61×#31032
#6158461