Finite Group Databasebeta

← #29648227

Subgroups — #29648227

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#21459612C211yesmaximal, characteristic
2#21454833C211yesmaximal, characteristic, complemented
3#21444453C211yesmaximal, characteristic, complemented
4#21444442C212yesmaximal, complemented
5#21444431C212yesmaximal, complemented
6#21444429C212yesmaximal, complemented
7#21444418C211yesmaximal, characteristic, complemented
8#21444416C211yesmaximal, characteristic, complemented
9#21444410C211yesmaximal, characteristic, complemented
10C2×#16839501C212yesmaximal, complemented
11C61×#2970C211yesmaximal, characteristic, complemented
12C2×#14519824C411yescharacteristic
13C2×#14519823C411yescharacteristic
14#16840424—21no—
15#16840424—21no—
16#16840407—21no—
17#16840407—22no—
18#16840407—21no—
19#16840407—22no—
20#16840338C412yes—
21#16840338C411yescharacteristic
22#16840331C412yes—
23#16840331C411yescharacteristic
24#16840312C2211yescharacteristic
25#16840312C2212yes—
26C2×#14519765C2212yes—
27C2×#14519765C2211yescharacteristic
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic, complemented
32#16839819C2212yescomplemented
33C4×#11758194C2211yescharacteristic, complemented
34C2×#14519688C2212yescomplemented
35#16839501C2211yescharacteristic, complemented
36#16839501C2212yescomplemented
37#16839501C2212yescomplemented
38#16839501C2212yescomplemented
39#16839501C2211yescharacteristic, complemented
40#16839501C2211yescharacteristic, complemented
41#16839501C2211yescharacteristic, complemented
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic, complemented
50C61×#432C2212yescomplemented
51C4×M16×C61C2211yescharacteristic, complemented
52C2×#97×C61C2212yescomplemented
53#14519824—44no—
54#14519823—44no—
55C2×#11758226—21no—
56C2×#11758222C2×C412yes—
57C2×#11758222C2×C411yescharacteristic
58C2×#11758222C2×C412yes—
59C2×#11758222C2×C411yescharacteristic
60C2×#11758224—22no—
61C2×#11758224—21no—
62C22×Dic122C2×C412yes—
63C22×Dic122C2×C411yescharacteristic
64C2×Q8×D61—21no—
65C22×D244C2×C412yes—
66C22×D244C2×C411yescharacteristic
67C2×D4×D61—22no—
68C2×D4×D61—21no—
69C2×#11758194—21no—
70C22×#92418—22no—
71#14519777C2×C411yescharacteristic
72#14519777C2×C412yes—
73#14519774C2×C411yescharacteristic
74#14519774C2×C412yes—
75#14519765—21no—
76#14519765—22no—
77#14519765—21no—
78#14519765—22no—
79#14519765C2×C412yes—
80#14519765C2×C412yes—
81#14519765C2×C412yes—
82#14519765C2311yescharacteristic
83#14519765C2312yes—
84#14519765C2311yescharacteristic
85#14519761—22no—
86#14519761—22no—
87#14519761—22no—
88#14519757—22no—
89#14519757—21no—
90#14519757—22no—
91#14519757—21no—
92#14519752—22no—
93#14519752—21no—
94#14519752—22no—
95#14519752—21no—
96#14519751—22no—
97#14519751—21no—
98#14519751—22no—
99#14519751—21no—
100#14519747—22no—
101#14519747—22no—
102#14519747—22no—
103C4×D244—21no—
104C4×D244—22no—
105C4×D244—21no—
106C4×D244—22no—
107C2×#11758198—22no—
108C2×#11758198—21no—
109C2×#11758198—22no—
110C2×#11758198—21no—
111C4⋊#92418—21no—
112C4⋊#92418—22no—
113C22⋊#92418—22no—
114C22⋊#92418—21no—
115#14519688—22no—
116#14519688C2311yescharacteristic
117#14519688C2312yes—
118#14519688C2311yescharacteristic
119C4×#92418—21no—
120C2×#41×C61C2×C411yescharacteristic
121C2×M16×C61—21no—
122C22×C8×C61—22no—
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61D411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127C4×D4×C61D411yescharacteristic
128C4×D4×C61D412yes—
129#117×C61C2×C411yescharacteristic
130#117×C61C2×C412yes—
131C2×C42×C61C2312yes—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—22no—
135C22⋊C8×C61—22no—
136C22⋊C8×C61—21no—
137#97×C61—22no—
138#97×C61C2311yescharacteristic
139#97×C61C2312yes—
140#97×C61C2311yescharacteristic
141C4×C8×C61—21no—
142C23×D61—21no—
143C23×D61—22no—
144#41×C61—24no—
145C2×Q8×C61C22⋊C411yescharacteristic
146C2×D4×C61C22⋊C412yes—
147C2×D4×C61C22⋊C411yescharacteristic
148C22×C4×C61C22×C412yes—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C2×C22⋊D61—21no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—22no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160#11758226—44no—
161Q8×D61—44no—
162#11758224—48no—
163#11758224—44no—
164D4×D61—48no—
165D4×D61—44no—
166#11758222—44no—
167#11758222—44no—
168#11758222—42no—
169#11758222—42no—
170C2×D244—22no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—22no—
174C2×D244—21no—
175C2×D244—22no—
176C2×D244—44no—
177C2×D244—42no—
178C2×D244C22×C411yescharacteristic
179C2×D244C22×C412yes—
180C2×D244C22×C411yescharacteristic
181C2×C4×D61—22no—
182C2×C4×D61—21no—
183C2×C4×D61—22no—
184C2×C4×D61—21no—
185C2×Dic122—22no—
186C2×Dic122—21no—
187C2×Dic122—44no—
188C2×Dic122—42no—
189C2×Dic122C22×C411yescharacteristic
190C2×Dic122C22×C412yes—
191C2×Dic122C22×C411yescharacteristic
192M16×C61—42no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—22no—
196C2×C8×C61—44no—
197C4⋊C4×C61C22⋊C412yes—
198C4⋊C4×C61C22⋊C412yes—
199C4⋊C4×C61C22⋊C411yescharacteristic
200C4⋊C4×C61C22⋊C411yescharacteristic
201C22⋊C4×C61C22⋊C412yes—
202C22⋊C4×C61C22⋊C412yes—
203C22⋊C4×C61C22⋊C412yes—
204C42×C61C22×C412yes—
205C42×C61C22×C411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D412yes—
208C42×C61C2×D411yescharacteristic
209C42×C61C2×D412yes—
210C42×C61C2411yescharacteristic
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—21no—
218#11758198—22no—
219#11758198—21no—
220#11758198—22no—
221#11758198—21no—
222#11758198—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—22no—
225C4⋊Dic61—21no—
226C4⋊Dic61—22no—
227#11758194—42no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—22no—
231C2×#92418—44no—
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61C2×C22⋊C412yes—
234C22×D61—22no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—42no—
238C22×D61—44no—
239C22×D61—82no—
240C22×D61—82no—
241C22×D61—81no—
242C22×D61—81no—
243Q8×C61—22no—
244D4×C61—24no—
245D4×C61#10014yes—
246C2×C4×C61—24no—
247C2×C4×C61—22no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—42no—
262C22⋊D61—42no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—42no—
269C2×Dic61—44no—
270D244—41no—
271D244—42no—
272D244—41no—
273D244—44no—
274D244—42no—
275D244—82no—
276D244—81no—
277D244—82no—
278D244—81no—
279C4×D61—44no—
280C4×D61—44no—
281C4×D61—42no—
282C4×D61—42no—
283Dic122—44no—
284Dic122—42no—
285Dic122—41no—
286Dic122—42no—
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—22no—
296C22×C61—22no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61C2×#10014yes—
301C22×C61C22×C22⋊C411yescharacteristic
302C2×D61—41no—
303C2×D61—42no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—42no—
309C2×D61—41no—
310C2×D61—42no—
311C2×D61—164no—
312C2×D61—162no—
313C4×C61—21no—
314C4×C61—22no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—22no—
318C4×C61—21no—
319C4×C61—22no—
320C4×C61—21no—
321C4×C61C2×#10012yes—
322Dic61—41no—
323Dic61—42no—
324Dic61—41no—
325#3500—611no—
326#2970—611no—
327#2970D6111yescharacteristic, complemented
328#1536—611no—
329#1533—612no—
330#1530—612no—
331#1530—612no—
332#1527—611no—
333#1527—611no—
334#1525—611no—
335C2×#329—612no—
336C2×C61—24no—
337C2×C61—41no—
338C2×C61C22×#10011yescharacteristic
339C2×C61#290111yescharacteristic
340C2×C61#290111yescharacteristic
341D61—81no—
342D61—81no—
343D61—82no—
344C2×2−1+4—611no—
345C2×H(5,𝔽2)—611no—
346#548—1221no—
347#548—1221no—
348#546—1221no—
349#546—1222no—
350#546—1221no—
351#546—1222no—
352#536—611no—
353#536—612no—
354#535—611no—
355#535—612no—
356#532—611no—
357#532—612no—
358C2×#124—612no—
359C2×#124—611no—
360C4×#41C2×D6111yescharacteristic
361#446—21no—
362#446—1221no—
363#435—22no—
364#435—1222no—
365C8×D4—21no—
366C8×D4—1221no—
367#432—611no—
368#432—612no—
369#432C2×D6111yescharacteristic, complemented
370#432C2×D6112yescomplemented
371C4×M16—611no—
372C4×M16C2×D6111yescharacteristic, complemented
373C2×#97—612no—
374C2×#97C2×D6112yescomplemented
375#329—612no—
376#329—612no—
377#329—611no—
378#329—612no—
379#329—612no—
380#329—611no—
381#329—611no—
382#329—611no—
383#329—612no—
384#329—612no—
385C61#1768511yescharacteristic, complemented, Sylow
3862−1+4—2444no—
387H(5,𝔽2)—2444no—
388C2×#41—612no—
389C2×#41—611no—
390C2×#41—612no—
391C2×#41—611no—
392C2×#41—1221no—
393C2×#41—1222no—
394C2×#41—1221no—
395C2×#41C4×D6111yescharacteristic
396C22×Q8—611no—
397C22×Q8—612no—
398C22×Q8—1221no—
399C22×D4—612no—
400C22×D4—611no—
401C22×D4—1222no—
402C22×D4—1221no—
403C2×M16—21no—
404C2×M16—1221no—
405C22×C8—22no—
406C22×C8—1222no—
407C4⋊Q8—611no—
408C4⋊Q8—612no—
409#127—611no—
410#127—612no—
411#124—612no—
412#124—612no—
413#124—612no—
414#124—611no—
415#124—612no—
416#124—611no—
417#124—1221no—
418#124—1222no—
419#124—1221no—
420#124—1222no—
421#123—1222no—
422#123—1222no—
423#123—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8—1221no—
426C22⋊Q8—1222no—
427C22⋊Q8—1221no—
428#121—1222no—
429#121—1221no—
430#121—1222no—
431#121—1221no—
432#121—1222no—
433#121—1221no—
434#121—1222no—
435#121—1221no—
436#120—1222no—
437#120—1222no—
438#120—1222no—
439C4×Q8D24411yescharacteristic
440C4×Q8C22⋊D6111yescharacteristic
441C4×D4—1221no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4—1222no—
445C4×D4D24411yescharacteristic
446C4×D4D24412yes—
447C4×D4C22⋊D6111yescharacteristic
448C4×D4C22⋊D6112yes—
449#117C4×D6111yescharacteristic
450#117C4×D6112yes—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1221no—
455C2×C42C22×D6112yes—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—21no—
458C4⋊C8—22no—
459C4⋊C8—1221no—
460C4⋊C8—1222no—
461C22⋊C8—22no—
462C22⋊C8—21no—
463C22⋊C8—1222no—
464C22⋊C8—1221no—
465#97—22no—
466#97—611no—
467#97—612no—
468#97—611no—
469#97—1222no—
470#97C22×D6111yescharacteristic
471#97C22×D6112yes—
472#97C22×D6111yescharacteristic
473C4×C8—21no—
474C4×C8—1221no—
475C24—1221no—
476C24—1222no—
477#41—24no—
478#41—2444no—
479#41—2444no—
480#41—2442no—
481#41—2442no—
482#41—2448no—
483#41—2444no—
484#41—2444no—
485C2×Q8—612no—
486C2×Q8—611no—
487C2×Q8—611no—
488C2×Q8—1222no—
489C2×Q8—1221no—
490C2×Q8—2442no—
491C2×Q8—2444no—
492C2×Q8—2444no—
493C2×Q8#1175819811yescharacteristic
494C2×D4—611no—
495C2×D4—612no—
496C2×D4—611no—
497C2×D4—1221no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—1222no—
501C2×D4—1221no—
502C2×D4—1222no—
503C2×D4—1221no—
504C2×D4—1222no—
505C2×D4—1222no—
506C2×D4—1221no—
507C2×D4—2444no—
508C2×D4—2442no—
509C2×D4—2448no—
510C2×D4—2444no—
511C2×D4#1175819812yes—
512C2×D4#1175819811yescharacteristic
513C22×C4—1221no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1222no—
518C22×C4—1221no—
519C22×C4C2×C4×D6112yes—
520C22×C4C2×C4×D6111yescharacteristic
521C22×C4C2×D24412yes—
522C22×C4C2×D24411yescharacteristic
523C22×C4C2×C22⋊D6112yes—
524C22×C4C2×C22⋊D6111yescharacteristic
525M16—42no—
526M16—2442no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C2×C8—44no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1222no—
534C2×C8—2444no—
535C4⋊C4—1221no—
536C4⋊C4—1222no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4#1175819812yes—
540C4⋊C4#1175819811yescharacteristic
541C4⋊C4#1175819812yes—
542C4⋊C4#1175819811yescharacteristic
543C22⋊C4—1221no—
544C22⋊C4—1222no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—1222no—
548C22⋊C4—1221no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4#1175819812yes—
556C22⋊C4#1175819812yes—
557C22⋊C4#1175819812yes—
558C42C2×C4×D6112yes—
559C42C2×C4×D6111yescharacteristic
560C42C2×D24411yescharacteristic
561C42C2×D24412yes—
562C42C2×C22⋊D6111yescharacteristic
563C42C2×C22⋊D6112yes—
564C42C23×D6111yescharacteristic
565C23—1222no—
566C23—1221no—
567C23—1221no—
568C23—4882no—
569C23—4882no—
570C23—4881no—
571C23—4881no—
572C23—2442no—
573C23—2444no—
574C23C2×#1175819811yescharacteristic
575C23C2×#1175819812yes—
576Q8—22no—
577Q8—2442no—
578Q8—2441no—
579Q8—2441no—
580Q8—2444no—
581Q8—2442no—
582D4—24no—
583D4—4882no—
584D4—4881no—
585D4—4882no—
586D4—4881no—
587D4—2441no—
588D4—2442no—
589D4—2441no—
590D4—2444no—
591D4—2442no—
592D4—2442no—
593D4—2444no—
594D4—2444no—
595D4—2442no—
596D4#1451969314yes—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2442no—
609C2×C4—2444no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4C2×#1175819811yescharacteristic
613C2×C4C2×#1175819812yes—
614C2×C4C2×#1175819811yescharacteristic
615C2×C4C2×#1175819812yes—
616C2×C4C2×#1175819812yes—
617C2×C4C2×#1175819811yescharacteristic
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C22×C22⋊D6111yescharacteristic
620C2×C4C22×D24411yescharacteristic
621C8—22no—
622C8—42no—
623C8—41no—
624C8—1222no—
625C8—2442no—
626C8—2441no—
627C22—22no—
628C22—22no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—9764no—
633C22—9762no—
634C22—2441no—
635C22—2441no—
636C22—2442no—
637C22—2441no—
638C22—2441no—
639C22—2442no—
640C22—2441no—
641C22—2442no—
642C22—2441no—
643C22C2×#1451969314yes—
644C22C22×#1175819811yescharacteristic
645C4—21no—
646C4—22no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—2441no—
654C4—2442no—
655C4—2441no—
656C4C2×#1451969312yes—
657C2—24no—
658C2—41no—
659C2—4881no—
660C2—4882no—
661C2—4881no—
662C2C22×#1451969311yescharacteristic
663C2#2145436411yescharacteristic
664C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2970C2
socleC22×C61C22×C22⋊C4
prime radical#21459612C2
cosocleC42×C61C24
hypercenter#2970D61
hypocenterC61#17685
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21444453C2semidirect
4. #21444442C2semidirect
5. #21444431C2semidirect
6. #21444429C2semidirect
7. #21444418C2semidirect
8. #21444416C2semidirect
9. #21444410C2semidirect
10. C2×#16839501C2semidirect
11. C61×#2970C2semidirect
31. #16839819C22semidirect
32. #16839819C22semidirect
33. C4×#11758194C22semidirect
34. C2×#14519688C22semidirect
35. #16839501C22semidirect
36. #16839501C22semidirect
37. #16839501C22semidirect
38. #16839501C22semidirect
39. #16839501C22semidirect
40. #16839501C22semidirect
41. #16839501C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
49. C61×#432C22semidirect
50. C61×#432C22semidirect
51. C4×M16×C61C22semidirect
52. C2×#97×C61C22semidirect
327. #2970D61semidirect
369. #432C2×D61semidirect
370. #432C2×D61semidirect
372. C4×M16C2×D61semidirect
374. C2×#97C2×D61semidirect
385. C61#17685semidirect

Maximal subgroups

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

subgroupindex
#214596122
#214548332
#214444532
#214444422
#214444312
#214444292
#214444182
#214444162
#214444102
C2×#168395012
C61×#29702
#1768561