Finite Group Databasebeta

← #29648229

Subgroups — #29648229

Order 15616.

1152 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459609C211yesmaximal, characteristic
2#21454834C211yesmaximal, characteristic, complemented
3#21454831C211yesmaximal, characteristic, complemented
4#21444453C211yesmaximal, characteristic, complemented
5#21444441C212yesmaximal, complemented
6#21444428C212yesmaximal, complemented
7#21444428C212yesmaximal, complemented
8#21444417C211yesmaximal, characteristic, complemented
9#21444410C211yesmaximal, characteristic, complemented
10#329×D61C212yesmaximal, complemented
11C61×#1527C211yesmaximal, characteristic, complemented
12C2×#14519825C411yescharacteristic
13C2×#14519822C411yescharacteristic
14#16840420—21no—
15#16840420—21no—
16#16840405—21no—
17#16840405—21no—
18#16840402—22no—
19#16840402—22no—
20#16840336C411yescharacteristic
21#16840332C412yes—
22#16840330C412yes—
23#16840329C411yescharacteristic
24#16840312C2211yescharacteristic
25#16840311C2212yes—
26#124×D61C2212yes—
27C4×#11758222C2211yescharacteristic
28#16839892—21no—
29#16839889—21no—
30#16839836—22no—
31#16839831—22no—
32C8×D244—21no—
33#16839827—21no—
34#16839819C2211yescharacteristic, complemented
35#16839817C2212yescomplemented
36#16839817C2212yescomplemented
37C4×#11758194C2211yescharacteristic, complemented
38#97×D61C2212yescomplemented
39#97×D61C2212yescomplemented
40#16839502C2212yescomplemented
41#16839502C2212yescomplemented
42#16839501C2211yescharacteristic, complemented
43#16839501C2212yescomplemented
44#16839501C2211yescharacteristic, complemented
45#16839501C2211yescharacteristic, complemented
46#16839501C2211yescharacteristic, complemented
47#16839501C2212yescomplemented
48C2×#124×C61C2211yescharacteristic
49C61×#432C2211yescharacteristic, complemented
50C4×M16×C61C2211yescharacteristic, complemented
51C61×#329C2212yescomplemented
52C61×#329C2212yescomplemented
53#14519825—44no—
54#14519822—44no—
55C2×#11758226—21no—
56C2×#11758226C2×C412yes—
57C2×#11758222—22no—
58C2×#11758222C2×C411yescharacteristic
59C2×#11758222C2×C411yescharacteristic
60C2×#11758222C2×C411yescharacteristic
61C2×#11758224—21no—
62C2×#11758224C2×C412yes—
63C2×Q8×D61—21no—
64C2×Q8×D61C2×C412yes—
65C22×C22⋊D61—22no—
66C2×D4×D61—21no—
67C2×D4×D61C2×C412yes—
68C2×#11758194—21no—
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71#14519777C2×C411yescharacteristic
72#14519775C2×C412yes—
73#14519774C2×C411yescharacteristic
74#14519773C2×C412yes—
75#14519766C2×C412yes—
76#14519765C2311yescharacteristic
77#14519765C2311yescharacteristic
78#14519764C2×C412yes—
79#14519763—22no—
80#14519763—22no—
81#14519760—22no—
82#14519759—22no—
83#14519759—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753—21no—
88#14519751—21no—
89#14519751—21no—
90#14519750—22no—
91#14519750—22no—
92#14519750—22no—
93#14519750—22no—
94#14519748—21no—
95#14519748—21no—
96#14519747—22no—
97#14519747—22no—
98#14519746—22no—
99C4×Dic122D411yescharacteristic
100C4×Dic122D411yescharacteristic
101C4×D244—21no—
102C4×D244—21no—
103C4×D244D411yescharacteristic
104C4×D244D411yescharacteristic
105C4×C22⋊D61D412yes—
106C4×C22⋊D61D412yes—
107#14519737—22no—
108#14519737—22no—
109#14519736C2×C411yescharacteristic
110#14519734C2×C412yes—
111C22⋊C4×D61—22no—
112C22⋊C4×D61—22no—
113C42×D61C2312yes—
114C4⋊#92418—21no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118#14519689—22no—
119#14519689—22no—
120#14519688C2311yescharacteristic
121#14519688C2312yes—
122#14519688C2311yescharacteristic
123#14519687—22no—
124#14519687—22no—
125C4×#92418—21no—
126C22×Q8×C61C2×C411yescharacteristic
127C22×D4×C61C2×C411yescharacteristic
128C2×M16×C61—21no—
129#124×C61—21no—
130#124×C61—21no—
131#124×C61C2×C412yes—
132#124×C61C2312yes—
133C2×C22⋊C4×C61—21no—
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61C2311yescharacteristic
139#97×C61C2312yes—
140#97×C61C2311yescharacteristic
141C4×C8×C61—21no—
142C24×C61—21no—
143C23×D61—22no—
144C2×Q8×C61—21no—
145C2×Q8×C61—42no—
146C2×Q8×C61C22×C412yes—
147C2×D4×C61—21no—
148C2×D4×C61—42no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—48no—
161C2×C22⋊D61C22⋊C412yes—
162C22×Dic61—22no—
163#11758226—44no—
164#11758226—44no—
165Q8×D61—44no—
166Q8×D61—44no—
167#11758224—44no—
168#11758224—44no—
169D4×D61—44no—
170D4×D61—44no—
171#11758222—24no—
172#11758222—48no—
173#11758222—42no—
174#11758222—42no—
175C2×D244—22no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244C22⋊C411yescharacteristic
179C2×D244C22×C411yescharacteristic
180C2×D244C22×C411yescharacteristic
181C2×C4×D61—22no—
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—22no—
185C2×C4×D61C22×C412yes—
186C2×C4×D61C2×D412yes—
187C2×C4×D61C2×D412yes—
188C2×Dic122—22no—
189C2×Dic122C22⋊C411yescharacteristic
190C2×Dic122C22×C411yescharacteristic
191C2×Dic122C22×C411yescharacteristic
192M16×C61—42no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C42×C61C22×C411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C42×C61C2411yescharacteristic
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206C22⋊Dic61C22⋊C412yes—
207#11758198—21no—
208#11758198—21no—
209#11758198—21no—
210#11758198—21no—
211#11758198—22no—
212#11758198—22no—
213#11758198C22⋊C412yes—
214#11758198C22⋊C412yes—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61C22⋊C411yescharacteristic
218C4⋊Dic61C22⋊C411yescharacteristic
219#11758196—22no—
220#11758196—22no—
221#11758196C22⋊C412yes—
222#11758196C22⋊C412yes—
223C4×Dic61C22×C412yes—
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D412yes—
226#11758194—42no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—22no—
230C8×D61—44no—
231C8×D61—44no—
232C23×C61—22no—
233C23×C61—42no—
234C23×C61—81no—
235C23×C61—81no—
236C23×C61C2×C22⋊C411yescharacteristic
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—44no—
240C22×D61—82no—
241C22×D61—82no—
242C22×D61C2×C22⋊C412yes—
243Q8×C61—42no—
244Q8×C61—42no—
245D4×C61—42no—
246D4×C61—42no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—44no—
260C22⋊D61—42no—
261C22⋊D61—42no—
262C22⋊D61—44no—
263C22⋊D61—82no—
264C22⋊D61—82no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—44no—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271D244—41no—
272D244—41no—
273D244—44no—
274D244—81no—
275D244—81no—
276D244#10014yes—
277C4×D61—24no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—44no—
281C4×D61—44no—
282C4×D61—42no—
283C4×D61—42no—
284Dic122—22no—
285Dic122—44no—
286Dic122—41no—
287Dic122—41no—
288C8×C61—22no—
289C8×C61—42no—
290C8×C61—41no—
291#92418—22no—
292#92418—42no—
293#92418—41no—
294#17685—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—162no—
302C22×C61C22×C22⋊C411yescharacteristic
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—164no—
312C2×D61C2×#10014yes—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—42no—
319C4×C61C2×#10012yes—
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—41no—
324Dic61—41no—
325#3500—611no—
326#2970—611no—
327#2970—611no—
328#1536—611no—
329#1533—612no—
330#1530—612no—
331#1530—612no—
332#1527—611no—
333#1527D6111yescharacteristic, complemented
334#1525—611no—
335C2×#329—612no—
336C2×C61—41no—
337C2×C61—82no—
338C2×C61C22×#10011yescharacteristic
339C2×C61#290111yescharacteristic
340C2×C61#290111yescharacteristic
341D61—24no—
342D61—81no—
343D61—81no—
344C2×2−1+4—611no—
345C2×H(5,𝔽2)—611no—
346#548—1221no—
347#548—1221no—
348#546—1222no—
349#546—1222no—
350#546—1221no—
351#546—1221no—
352#536—611no—
353#536—612no—
354#535—611no—
355#535—612no—
356#532—612no—
357#532—611no—
358C2×#124—612no—
359C2×#124C2×D6111yescharacteristic
360C4×#41—611no—
361#446—1221no—
362#446—1221no—
363#435—1222no—
364#435—1222no—
365C8×D4—1221no—
366C8×D4—1221no—
367#432—611no—
368#432—612no—
369#432—612no—
370#432C2×D6111yescharacteristic, complemented
371C4×M16—611no—
372C4×M16C2×D6111yescharacteristic, complemented
373C2×#97—612no—
374C2×#97—612no—
375#329—612no—
376#329—612no—
377#329—612no—
378#329—612no—
379#329—611no—
380#329—611no—
381#329—611no—
382#329—611no—
383#329C2×D6112yescomplemented
384#329C2×D6112yescomplemented
385C61#1768511yescharacteristic, complemented, Sylow
3862−1+4—2444no—
387H(5,𝔽2)—2444no—
388C2×#41—611no—
389C2×#41—612no—
390C2×#41—612no—
391C2×#41—611no—
392C2×#41—611no—
393C2×#41—1222no—
394C2×#41—1221no—
395C2×#41—1221no—
396C22×Q8—612no—
397C22×Q8—1221no—
398C22×Q8C4×D6111yescharacteristic
399C22×D4—612no—
400C22×D4—1221no—
401C22×D4—1222no—
402C22×D4C4×D6111yescharacteristic
403C2×M16—21no—
404C2×M16—1221no—
405C22×C8—1222no—
406C22×C8—1222no—
407C4⋊Q8—612no—
408C4⋊Q8—611no—
409#127—612no—
410#127—611no—
411#124—21no—
412#124—21no—
413#124—612no—
414#124—612no—
415#124—611no—
416#124—611no—
417#124—1222no—
418#124—1222no—
419#124C4×D6112yes—
420#124C22×D6112yes—
421#123—1222no—
422#123—1222no—
423#123—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426C22⋊Q8—1221no—
427C22⋊Q8—1221no—
428#121—1222no—
429#121—1222no—
430#121—1222no—
431#121—1222no—
432#121—1221no—
433#121—1221no—
434#121—1221no—
435#121—1221no—
436#120—1222no—
437#120—1222no—
438#120—1222no—
439C4×Q8—611no—
440C4×Q8—611no—
441C4×D4—612no—
442C4×D4—612no—
443C4×D4—611no—
444C4×D4—611no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—1221no—
448C4×D4—1221no—
449#117—611no—
450#117—612no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C42—612no—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—21no—
458C4⋊C8—1222no—
459C4⋊C8—1222no—
460C4⋊C8—1221no—
461C22⋊C8—21no—
462C22⋊C8—1222no—
463C22⋊C8—1222no—
464C22⋊C8—1221no—
465#97—611no—
466#97—612no—
467#97—611no—
468#97—1222no—
469#97—1222no—
470#97C22×D6111yescharacteristic
471#97C22×D6112yes—
472#97C22×D6111yescharacteristic
473C4×C8—21no—
474C4×C8—1221no—
475C24—21no—
476C24—1222no—
477#41—1224no—
478#41—2442no—
479#41—2442no—
480#41—2448no—
481#41—2444no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485C2×Q8—21no—
486C2×Q8—42no—
487C2×Q8—611no—
488C2×Q8—611no—
489C2×Q8—611no—
490C2×Q8—1222no—
491C2×Q8—2444no—
492C2×Q8—2444no—
493C2×Q8C2×C4×D6112yes—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—612no—
497C2×D4—611no—
498C2×D4—611no—
499C2×D4—611no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—2444no—
510C2×D4—2444no—
511C2×D4—2448no—
512C2×D4C2×C4×D6112yes—
513C22×C4—21no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—612no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4C2×C4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525M16—42no—
526M16—2442no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1222no—
533C2×C8—2444no—
534C2×C8—2444no—
535C4⋊C4—612no—
536C4⋊C4—612no—
537C4⋊C4—611no—
538C4⋊C4—611no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—612no—
548C22⋊C4—612no—
549C22⋊C4—612no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1221no—
555C22⋊C4—1221no—
556C22⋊C4—1221no—
557C22⋊C4—1221no—
558C42—612no—
559C42—612no—
560C42—612no—
561C42C2×C4×D6111yescharacteristic
562C42D4×D6111yescharacteristic
563C42D4×D6111yescharacteristic
564C42C23×D6111yescharacteristic
565C23—22no—
566C23—42no—
567C23—81no—
568C23—81no—
569C23—612no—
570C23—1221no—
571C23—1221no—
572C23—4882no—
573C23—4882no—
574C23—2444no—
575C23C22⋊C4×D6111yescharacteristic
576Q8—42no—
577Q8—42no—
578Q8—1222no—
579Q8—2441no—
580Q8—2441no—
581Q8—2444no—
582D4—42no—
583D4—42no—
584D4—614no—
585D4—1224no—
586D4—4881no—
587D4—4881no—
588D4—4882no—
589D4—4882no—
590D4—2441no—
591D4—2441no—
592D4—2444no—
593D4—2444no—
594D4—2442no—
595D4—2442no—
596D4—2444no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1224no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2444no—
611C2×C4—2444no—
612C2×C4—2444no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4C22⋊C4×D6111yescharacteristic
616C2×C4C22⋊C4×D6111yescharacteristic
617C2×C4C22⋊C4×D6111yescharacteristic
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C8—22no—
622C8—42no—
623C8—41no—
624C8—1222no—
625C8—2442no—
626C8—2441no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—162no—
634C22—614no—
635C22—1222no—
636C22—1222no—
637C22—9764no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22C2×C22⋊C4×D6111yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—1222no—
652C4—1222no—
653C4—1222no—
654C4—2441no—
655C4—2441no—
656C4#100×D6112yes—
657C2—41no—
658C2—82no—
659C2—1224no—
660C2—4881no—
661C2—4881no—
662C2C2×#100×D6111yescharacteristic
663C2#2145436311yescharacteristic
664C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1527C2
socleC22×C61C22×C22⋊C4
prime radical#21459609C2
cosocleC42×C61C24
hypercenter#1527D61
hypocenterC61#17685
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #21454834C2semidirect
3. #21454831C2semidirect
4. #21444453C2semidirect
5. #21444441C2semidirect
6. #21444428C2semidirect
7. #21444428C2semidirect
8. #21444417C2semidirect
9. #21444410C2semidirect
10. #329×D61C2semidirect
11. C61×#1527C2semidirect
34. #16839819C22semidirect
35. #16839817C22semidirect
36. #16839817C22semidirect
37. C4×#11758194C22semidirect
38. #97×D61C22semidirect
39. #97×D61C22semidirect
40. #16839502C22semidirect
41. #16839502C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
45. #16839501C22semidirect
46. #16839501C22semidirect
47. #16839501C22semidirect
49. C61×#432C22semidirect
50. C4×M16×C61C22semidirect
51. C61×#329C22semidirect
52. C61×#329C22semidirect
333. #1527D61semidirect
370. #432C2×D61semidirect
372. C4×M16C2×D61semidirect
383. #329C2×D61semidirect
384. #329C2×D61semidirect
385. C61#17685semidirect

Maximal subgroups

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

subgroupindex
#214596092
#214548342
#214548312
#214444532
#214444412
#214444282
#214444172
#214444102
#329×D612
C61×#15272
#1768561