Finite Group Databasebeta

← #30565078

Subgroups — #30565078

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#21459671C211yesmaximal, characteristic
2#21458113C214yesmaximal, complemented
3#21458102C214yesmaximal, complemented
4#21457481C214yesmaximal, complemented
5#21457465C214yesmaximal, complemented
6#21456767C211yesmaximal, characteristic, complemented
7#21456744C211yesmaximal, characteristic, complemented
8#21456649C214yesmaximal, complemented
9#21455858C211yesmaximal, characteristic, complemented
10#21455850C211yesmaximal, characteristic, complemented
11#21455053C214yesmaximal, complemented
12#21454860C211yesmaximal, characteristic, complemented
13C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
14C2×#14519808C2218yescomplemented
15#16840495—22no—
16#16840482C2212yes—
17#16840477C2212yes—
18#16840462—22no—
19#16840433C2212yes—
20#16840417C2212yes—
21#16840350—22no—
22#16840329—21no—
23#16840319C2211yescharacteristic
24#16840312C2211yescharacteristic
25#16840273C2214yes—
26#16840243—21no—
27#16840141C2212yescomplemented
28#16840136C2212yescomplemented
29#16840112C2212yescomplemented
30#16840112C2212yescomplemented
31#16840100C2218yescomplemented
32#16840097C2212yes—
33#16840080C2212yes—
34#16840057C2212yescomplemented
35#16840047C2212yescomplemented
36#16840018C2218yescomplemented
37#16840010C2218yescomplemented
38#16839972C2212yescomplemented
39#16839969C2214yescomplemented
40#16839964C2214yescomplemented
41#16839962C2212yescomplemented
42#16839922C2218yescomplemented
43#16839918C2214yes—
44#16839918C2212yes—
45#16839918C2218yescomplemented
46#16839901C2214yes—
47#16839901C2212yes—
48#16839901C2218yescomplemented
49C4×#11758202C2214yescomplemented
50C4×#11758201C2214yescomplemented
51#16839840C2214yes—
52#16839840C2214yes—
53#16839815C2211yescharacteristic
54#16839815C2211yescharacteristic
55#16839771C2218yescomplemented
56#16839726C2214yes—
57C23×Q8×C61C2212yes—
58D4×Q8×C61—22no—
59D4×Q8×C61—22no—
60D4×Q8×C61C2214yes—
61D4×Q8×C61C2214yes—
62C2×C4⋊Q8×C61—21no—
63C2×C4⋊Q8×C61C2211yescharacteristic
64C2×C22⋊Q8×C61—22no—
65C2×C22⋊Q8×C61C2212yes—
66C2×C4×Q8×C61C2211yescharacteristic
67C2×C4×D4×C61—21no—
68C2×C4×D4×C61C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758222C2314yes—
71#14519808—416no—
72C2×#11758202C2314yes—
73C2×#11758202C2318yes—
74C2×#11758201C2314yes—
75C2×#11758201C2318yes—
76C2×#11758194C2318yes—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519773—22no—
82#14519770—24no—
83#14519769—21no—
84#14519765—21no—
85#14519764—22no—
86#14519763—24no—
87#14519760—24no—
88#14519759—24no—
89#14519756—21no—
90#14519756C2314yes—
91#14519756C2312yes—
92#14519751—21no—
93#14519751C2314yes—
94#14519751C2312yes—
95#14519750—24no—
96#14519748—22no—
97#14519748—22no—
98#14519746—24no—
99C4×Dic122—21no—
100C4×Dic122C2312yes—
101#14519742—22no—
102C4×D244—21no—
103C4×D244C2312yes—
104#14519739—22no—
105C4×C22⋊D61—24no—
106C4×C22⋊D61—24no—
107#14519736—21no—
108#14519736C2314yes—
109#14519735—22no—
110C4⋊C4×D61—22no—
111#14519710C2314yes—
112#14519708C2314yes—
113C4⋊#92418C2312yes—
114C4⋊#92418C2312yes—
115#14519701C2318yes—
116#14519701C2318yes—
117#14519701C2314yes—
118#14519701C2314yes—
119#14519700C2314yes—
120#14519697C2314yes—
121C22⋊#92418C2312yes—
122C22⋊#92418C2312yes—
123C22⋊#92418C2318yes—
124C4×#92418C2312yes—
125C4×#92418C2312yes—
126C22×Q8×C61—22no—
127C22×Q8×C61—24no—
128C22×Q8×C61—48no—
129C22×Q8×C61C2311yescharacteristic
130C22×Q8×C61C2318yes—
131C22×Q8×C61C2312yes—
132C22×D4×C61D411yescharacteristic
133C23×C4×C61—22no—
134C23×C4×C61C2312yes—
135C4⋊Q8×C61—24no—
136C4⋊Q8×C61—24no—
137C4⋊Q8×C61—21no—
138C4⋊Q8×C61—21no—
139C4⋊Q8×C61C2312yes—
140C4⋊Q8×C61C2312yes—
141C22⋊Q8×C61—24no—
142C22⋊Q8×C61—28no—
143C22⋊Q8×C61—28no—
144C22⋊Q8×C61C2318yes—
145C4×Q8×C61—21no—
146C4×Q8×C61—21no—
147C4×Q8×C61C2312yes—
148C4×Q8×C61C2312yes—
149C4×D4×C61—24no—
150C4×D4×C61—24no—
151C4×D4×C61D412yes—
152C4×D4×C61D412yes—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61—22no—
157C2×C4⋊C4×C61—21no—
158C2×C4⋊C4×C61—21no—
159C2×C4⋊C4×C61D411yescharacteristic
160C2×C4⋊C4×C61C2311yescharacteristic
161C2×C22⋊C4×C61—22no—
162C2×C22⋊C4×C61D412yes—
163C2×C42×C61—21no—
164C2×C42×C61C2311yescharacteristic
165C24×C61C2×D412yes—
166C2×Q8×C61—24no—
167C2×Q8×C61—21no—
168C2×Q8×C61—28no—
169C2×Q8×C61—22no—
170C2×Q8×C61—216no—
171C2×Q8×C61—48no—
172C2×Q8×C61—44no—
173C2×Q8×C61—48no—
174C2×Q8×C61—416no—
175C2×Q8×C61C2414yes—
176C2×Q8×C61C2412yes—
177C2×D4×C61—24no—
178C2×D4×C61C2×D412yes—
179C2×D4×C61C2×D412yes—
180C22×C4×C61—21no—
181C22×C4×C61—21no—
182C22×C4×C61—22no—
183C22×C4×C61—28no—
184C22×C4×C61—44no—
185C22×C4×C61—48no—
186C22×C4×C61C2×D411yescharacteristic
187C22×C4×C61C2×D412yes—
188C22×C4×C61C2×D414yes—
189C22×C4×C61C2×D411yescharacteristic
190C22×C4×C61C2414yes—
191C22×C4×C61C2411yescharacteristic
192C2×C22⋊D61—22no—
193C2×C22⋊D61—24no—
194C2×C22⋊D61—24no—
195#11758222—48no—
196#11758222—44no—
197C2×D244—21no—
198C2×D244—21no—
199C2×D244C2412yes—
200C2×C4×D61—22no—
201C2×C4×D61—24no—
202C2×Dic122—21no—
203C2×Dic122—21no—
204C2×Dic122C2412yes—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—22no—
211C4⋊C4×C61—28no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61C2×D412yes—
214C4⋊C4×C61C2×D412yes—
215C4⋊C4×C61C2412yes—
216C22⋊C4×C61—28no—
217C22⋊C4×C61C2×D414yes—
218C22⋊C4×C61C2×D414yes—
219C42×C61—22no—
220C42×C61—22no—
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2411yescharacteristic
224C42×C61C2411yescharacteristic
225C22⋊Dic61—22no—
226C22⋊Dic61—24no—
227C22⋊Dic61—24no—
228#11758202—28no—
229#11758202—48no—
230#11758201—28no—
231#11758201—48no—
232#11758198—24no—
233#11758198—22no—
234#11758198—22no—
235C4⋊Dic61—21no—
236C4⋊Dic61—21no—
237C4⋊Dic61—21no—
238C4⋊Dic61—21no—
239C4⋊Dic61C2412yes—
240C4⋊Dic61C2412yes—
241#11758196—24no—
242#11758196—22no—
243#11758196—22no—
244C4×Dic61—24no—
245C4×Dic61—22no—
246#11758194—48no—
247C2×#92418C2414yes—
248C2×#92418C2414yes—
249C23×C61—24no—
250C23×C61—42no—
251C23×C61—42no—
252C23×C61C22×D414yes—
253C23×C61C22×D411yescharacteristic
254C22×D61—22no—
255Q8×C61—24no—
256Q8×C61—22no—
257Q8×C61—28no—
258Q8×C61—416no—
259Q8×C61—44no—
260Q8×C61H(5,𝔽2)14yes—
261D4×C61—24no—
262D4×C61—24no—
263C2×C4×C61—24no—
264C2×C4×C61—22no—
265C2×C4×C61—28no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—24no—
269C2×C4×C61—42no—
270C2×C4×C61—44no—
271C2×C4×C61—41no—
272C2×C4×C61—416no—
273C2×C4×C61—42no—
274C2×C4×C61—44no—
275C2×C4×C61#13714yescomplemented
276C2×C4×C61C22×D414yes—
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C2511yescharacteristic
283C22⋊D61—44no—
284C22⋊D61—84no—
285C22⋊D61—84no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289D244—24no—
290D244—42no—
291D244—81no—
292C4×D61—44no—
293C4×D61—84no—
294Dic122—24no—
295Dic122—42no—
296Dic122—81no—
297#92418—24no—
298#92418—24no—
299#61584—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—24no—
303C22×C61—24no—
304C22×C61—24no—
305C22×C61—48no—
306C22×C61C2×#13712yes—
307C22×C61C23×D411yescharacteristic
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—24no—
314C4×C61—24no—
315C4×C61—21no—
316C4×C61—42no—
317C4×C61—44no—
318C4×C61C2×#13712yes—
319C4×C61C2×#13712yes—
320C4×C61C2×H(5,𝔽2)12yes—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324#3506—611no—
325C2×D4×Q8D6111yescharacteristic, complemented
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#3103—611no—
335#3004—614no—
336#2975—611no—
337C2×C61—22no—
338C2×C61—44no—
339C2×C61#348411yescharacteristic
340C2×C61C22×#13711yescharacteristic
341C2×C61#358611yescharacteristic
342D61—82no—
343C23×Q8C2×D6112yes—
344C2×#137—618no—
345#559—1222no—
346#556—612no—
347#555—612no—
348#553—1222no—
349#550—612no—
350D4×Q8—22no—
351D4×Q8—22no—
352D4×Q8C2×D6114yes—
353D4×Q8C2×D6114yes—
354#547—612no—
355#538—1222no—
356#535—1221no—
357#533—611no—
358#532—611no—
359C2×C4⋊Q8—21no—
360C2×C4⋊Q8C2×D6111yescharacteristic
361#525—614no—
362C2×C22⋊Q8—22no—
363C2×C22⋊Q8C2×D6112yes—
364#520—1221no—
365C2×C4×Q8C2×D6111yescharacteristic
366C2×C4×D4—21no—
367C2×C4×D4C2×D6111yescharacteristic
368#494—612no—
369#492—612no—
370#487—612no—
371#487—612no—
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#461—614no—
382#460—612no—
383#452—618no—
384#451—618no—
385#451—612no—
386#451—614no—
387#448—618no—
388#448—612no—
389#448—614no—
390C4×Q16—614no—
391C4×SD16—614no—
392#436—614no—
393#436—614no—
394#431—611no—
395#431—611no—
396#419—618no—
397#407—614no—
398C61#6158411yescharacteristic, complemented, Sylow
399C2×#41—614no—
400C2×#41—1221no—
401C22×Q8—22no—
402C22×Q8—24no—
403C22×Q8—48no—
404C22×Q8C22×D6111yescharacteristic
405C22×Q8C22×D6118yes—
406C22×Q8C22×D6112yes—
407C22×D4C22⋊D6111yescharacteristic
408C23×C4—22no—
409C23×C4C22×D6112yes—
410#137—24416no—
411C2×Q16—614no—
412C2×Q16—618no—
413C2×SD16—614no—
414C2×SD16—618no—
415C2×M16—618no—
416C4⋊Q8—24no—
417C4⋊Q8—24no—
418C4⋊Q8—21no—
419C4⋊Q8—21no—
420C4⋊Q8—611no—
421C4⋊Q8—611no—
422C4⋊Q8—611no—
423C4⋊Q8C22×D6112yes—
424C4⋊Q8C22×D6112yes—
425#127—611no—
426#127—1222no—
427#126—1224no—
428#125—1221no—
429#124—1222no—
430#124—1224no—
431#124—1221no—
432#123—1224no—
433#123—1224no—
434C22⋊Q8—24no—
435C22⋊Q8—28no—
436C22⋊Q8—28no—
437C22⋊Q8—612no—
438C22⋊Q8—614no—
439C22⋊Q8—1221no—
440C22⋊Q8C22×D6118yes—
441#121—612no—
442#121—614no—
443#121—1222no—
444#121—1222no—
445#121—1221no—
446#121—1224no—
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—24no—
456C4×D4—24no—
457C4×D4—612no—
458C4×D4—1224no—
459C4×D4—1224no—
460C4×D4—1222no—
461C4×D4—1221no—
462C4×D4C22⋊D6112yes—
463C4×D4C22⋊D6112yes—
464C4×D4C22×D6112yes—
465C4×D4C22×D6112yes—
466#117—614no—
467#117—1221no—
468#117—1222no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—22no—
473C2×C4⋊C4—1222no—
474C2×C4⋊C4C22⋊D6111yescharacteristic
475C2×C4⋊C4C22×D6111yescharacteristic
476C2×C22⋊C4—22no—
477C2×C22⋊C4C22⋊D6112yes—
478C2×C42—21no—
479C2×C42C22×D6111yescharacteristic
480#107—614no—
481#106—614no—
482C4⋊C8—612no—
483C4⋊C8—612no—
484#103—614no—
485#103—614no—
486#103—618no—
487#103—618no—
488#103—614no—
489#102—614no—
490C22⋊C8—612no—
491C22⋊C8—612no—
492C22⋊C8—618no—
493C4×C8—612no—
494C4×C8—612no—
495C24C2×C22⋊D6112yes—
496#41—2448no—
497#41—2444no—
498C2×Q8—24no—
499C2×Q8—21no—
500C2×Q8—28no—
501C2×Q8—22no—
502C2×Q8—216no—
503C2×Q8—48no—
504C2×Q8—44no—
505C2×Q8—48no—
506C2×Q8—416no—
507C2×Q8—612no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8C23×D6114yes—
511C2×Q8C23×D6112yes—
512C2×D4—24no—
513C2×D4—612no—
514C2×D4—1224no—
515C2×D4—1224no—
516C2×D4—1222no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4C2×C22⋊D6112yes—
520C2×D4C2×C22⋊D6112yes—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—28no—
525C22×C4—44no—
526C22×C4—48no—
527C22×C4—1224no—
528C22×C4—1222no—
529C22×C4C2×C22⋊D6111yescharacteristic
530C22×C4C2×C22⋊D6112yes—
531C22×C4C2×C22⋊D6114yes—
532C22×C4C2×C22⋊D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534C22×C4C23×D6114yes—
535Q16—1228no—
536Q16—2448no—
537SD16—1228no—
538SD16—2448no—
539M16—2448no—
540C2×C8—614no—
541C2×C8—614no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—28no—
549C4⋊C4—21no—
550C4⋊C4—612no—
551C4⋊C4—612no—
552C4⋊C4—1221no—
553C4⋊C4—1224no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4C2×C22⋊D6112yes—
560C4⋊C4C2×C22⋊D6112yes—
561C4⋊C4C23×D6112yes—
562C22⋊C4—28no—
563C22⋊C4—1224no—
564C22⋊C4—1222no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4C2×C22⋊D6114yes—
570C22⋊C4C2×C22⋊D6114yes—
571C42—22no—
572C42—22no—
573C42—1224no—
574C42—1222no—
575C42C2×C22⋊D6111yescharacteristic
576C42C2×C22⋊D6111yescharacteristic
577C42C23×D6111yescharacteristic
578C42C23×D6111yescharacteristic
579C23—24no—
580C23—42no—
581C23—42no—
582C23—1222no—
583C23C22×C22⋊D6114yes—
584C23C22×C22⋊D6111yescharacteristic
585Q8—24no—
586Q8—22no—
587Q8—28no—
588Q8—416no—
589Q8—44no—
590Q8—1224no—
591Q8—4881no—
592Q8—2442no—
593Q8#1451982214yes—
594D4—24no—
595D4—24no—
596D4—1224no—
597D4—4881no—
598D4—4884no—
599D4—4884no—
600D4—2444no—
601D4—2442no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—4884no—
618C2×C4—2444no—
619C2×C4#1451980814yescomplemented
620C2×C4C22×C22⋊D6114yes—
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4C22×C22⋊D6111yescharacteristic
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4C22×C22⋊D6111yescharacteristic
625C2×C4C22×C22⋊D6111yescharacteristic
626C2×C4C24×D6111yescharacteristic
627C8—1224no—
628C8—1224no—
629C22—21no—
630C22—21no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—48no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22C2×#1451980812yes—
639C22C23×C22⋊D6111yescharacteristic
640C4—21no—
641C4—21no—
642C4—24no—
643C4—21no—
644C4—24no—
645C4—42no—
646C4—44no—
647C4—2442no—
648C4—2442no—
649C4—2442no—
650C4C2×#1451980812yes—
651C4C2×#1451980812yes—
652C4C2×#1451982212yes—
653C2—22no—
654C2—44no—
655C2—4882no—
656C2#2145945411yescharacteristic
657C2C22×#1451980811yescharacteristic
658C2#2146060611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×D4×Q8×C61C2
socleC22×C61C23×D4
prime radical#21459671C2
hypercenterC2×D4×Q8D61
hypocenterC61#61584
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. #21458113C2semidirect
3. #21458102C2semidirect
4. #21457481C2semidirect
5. #21457465C2semidirect
6. #21456767C2semidirect
7. #21456744C2semidirect
8. #21456649C2semidirect
9. #21455858C2semidirect
10. #21455850C2semidirect
11. #21455053C2semidirect
12. #21454860C2semidirect
13. C2×D4×Q8×C61C2semidirect
14. C2×#14519808C22semidirect
27. #16840141C22semidirect
28. #16840136C22semidirect
29. #16840112C22semidirect
30. #16840112C22semidirect
31. #16840100C22semidirect
34. #16840057C22semidirect
35. #16840047C22semidirect
36. #16840018C22semidirect
37. #16840010C22semidirect
38. #16839972C22semidirect
39. #16839969C22semidirect
40. #16839964C22semidirect
41. #16839962C22semidirect
42. #16839922C22semidirect
45. #16839918C22semidirect
48. #16839901C22semidirect
49. C4×#11758202C22semidirect
50. C4×#11758201C22semidirect
55. #16839771C22semidirect
275. C2×C4×C61#137semidirect
325. C2×D4×Q8D61semidirect
398. C61#61584semidirect
619. C2×C4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214581132
#214581022
#214574812
#214574652
#214567672
#214567442
#214566492
#214558582
#214558502
#214550532
#214548602
C2×D4×Q8×C612
#6158461