Finite Group Databasebeta

← #29646491

Subgroups — #29646491

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519822C211yesmaximal, characteristic
2#21454932C211yesmaximal, characteristic, complemented
3#21454780C211yesmaximal, characteristic, complemented
4#21444260C212yesmaximal, complemented
5#21444255C212yesmaximal, complemented
6#21444250C212yesmaximal, complemented
7#21444242C212yesmaximal, complemented
8#21444207C211yesmaximal, characteristic, complemented
9#21444199C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1478C211yesmaximal, characteristic, complemented
12C2×#14519822C411yescharacteristic
13#16840243C411yescharacteristic
14#16840243C411yescharacteristic
15#16840241—21no—
16#16840241C412yes—
17#16840237C411yescharacteristic
18#16840235—22no—
19#16840234—21no—
20#16840234C412yes—
21C4×#11758222C2211yescharacteristic
22C4×#11758222C2211yescharacteristic
23C4×#11758224—21no—
24C4×#11758224C2212yes—
25C2×C4×C22⋊D61—22no—
26C4×D4×D61—21no—
27C4×D4×D61C2212yes—
28#16839892—21no—
29#16839889—21no—
30#16839841—21no—
31#16839839—21no—
32#16839830—22no—
33#16839830—22no—
34#16839818C2212yescomplemented
35#16839813C2212yescomplemented
36#16839784C2212yescomplemented
37C2×C4⋊#92418C2211yescharacteristic, complemented
38C4×#11758189C2212yescomplemented
39C2×C4×#92418C2211yescharacteristic, complemented
40#16839494C2212yescomplemented
41#16839492C2211yescharacteristic, complemented
42#16839492C2211yescharacteristic, complemented
43#16839491C2212yescomplemented
44D4⋊#92418C2212yescomplemented
45D4⋊#92418C2212yescomplemented
46#16839489C2211yescharacteristic, complemented
47#16839489C2211yescharacteristic, complemented
48C2×C4×D4×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic, complemented
50C61×#431C2211yescharacteristic, complemented
51C61×D4⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yescomplemented
53#14519822—24no—
54#14519822—24no—
55C2×#11758222C2×C411yescharacteristic
56C2×#11758222C2×C411yescharacteristic
57C2×#11758224—21no—
58C2×#11758224C2×C412yes—
59C22×C22⋊D61—22no—
60C2×D4×D61—21no—
61C2×D4×D61C2×C412yes—
62C22×C4×D61—22no—
63C2×#11758189—22no—
64C2×#11758189—22no—
65C22×#92418—21no—
66C22×#92418—21no—
67C4×Dic122C2×C411yescharacteristic
68C4×Dic122D411yescharacteristic
69C4×Dic122D411yescharacteristic
70C4×Dic122C2311yescharacteristic
71#14519742—21no—
72#14519742C2×C412yes—
73C4×D244C2×C411yescharacteristic
74C4×D244D411yescharacteristic
75C4×D244D411yescharacteristic
76C4×D244C2311yescharacteristic
77D4×Dic61—21no—
78D4×Dic61—21no—
79D4×Dic61C2×C412yes—
80D4×Dic61C2×C412yes—
81#14519739—21no—
82#14519739C2×C412yes—
83C4×C22⋊D61—21no—
84C4×C22⋊D61—21no—
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61D412yes—
92C4×C22⋊D61D412yes—
93#14519737—22no—
94#14519737—22no—
95#14519737—22no—
96#14519737—22no—
97#14519736C2×C411yescharacteristic
98#14519736C2×C411yescharacteristic
99#14519735—21no—
100#14519735C2×C412yes—
101#14519734—21no—
102#14519734C2×C412yes—
103#14519733—22no—
104#14519733—22no—
105C2×#11758196—22no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61C2×C412yes—
108C2×C22⋊Dic61—22no—
109C2×#11758198—22no—
110C22⋊C4×D61—22no—
111C22⋊C4×D61—22no—
112C2×C4×Dic61—22no—
113C42×D61—21no—
114C42×D61C2312yes—
115C4⋊#92418—21no—
116C4⋊#92418C2312yes—
117#14519706—22no—
118#14519706—22no—
119#14519689—22no—
120#14519689—22no—
121#14519687—22no—
122#14519687—22no—
123C4×#92418—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C22×D4×C61C2×C411yescharacteristic
127C23×C4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C412yes—
131C4×D4×C61C2312yes—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61—21no—
139C22⋊C8×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2312yes—
142C24×C61—21no—
143C23×D61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61M1614yescomplemented
147C2×D4×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—81no—
152C22×C4×C61—81no—
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C2×C22⋊D61—216no—
157C2×C22⋊D61—21no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61C22⋊C412yes—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164#11758224—28no—
165#11758224—24no—
166#11758224—24no—
167D4×D61—28no—
168D4×D61—24no—
169D4×D61—24no—
170#11758222—22no—
171#11758222—22no—
172#11758222—22no—
173#11758222M1614yescomplemented
174C2×D244C22⋊C411yescharacteristic
175C2×D244C22×C411yescharacteristic
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—21no—
180C2×C4×D61—22no—
181C2×C4×D61—82no—
182C2×C4×D61—82no—
183C2×C4×D61C22×C412yes—
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×Dic122C22⋊C411yescharacteristic
187C2×Dic122C22×C411yescharacteristic
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—22no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199C22⋊Dic61—21no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61C22⋊C412yes—
203#11758198—22no—
204#11758198—21no—
205#11758198—21no—
206#11758198—22no—
207#11758198C22⋊C412yes—
208#11758198C22⋊C412yes—
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22×C411yescharacteristic
212C4⋊Dic61C22×C411yescharacteristic
213#11758196—21no—
214#11758196—21no—
215#11758196—22no—
216#11758196—22no—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61C22×C412yes—
225C4×Dic61C2×D412yes—
226C4×Dic61C2×D412yes—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—22no—
230C2×#92418—42no—
231C2×#92418—81no—
232#11758189—44no—
233#11758189—82no—
234C23×C61—22no—
235C23×C61—24no—
236C23×C61—81no—
237C23×C61—81no—
238C23×C61C2×C22⋊C411yescharacteristic
239C22×D61—21no—
240C22×D61—28no—
241C22×D61—82no—
242C22×D61—82no—
243C22×D61C2×C22⋊C412yes—
244D4×C61—22no—
245D4×C61—24no—
246D4×C61C2×M1614yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—82no—
254C2×C4×C61—81no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×M1612yes—
259C2×C4×C61C2×M1612yes—
260C2×C4×C61C23×C411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C22⋊D61—28no—
264C22⋊D61—28no—
265C22⋊D61—24no—
266C22⋊D61—24no—
267C2×Dic61—28no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—82no—
272C2×Dic61—82no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276D244—21no—
277D244—22no—
278D244C2×M1612yes—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—24no—
284C4×D61—42no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—82no—
288C4×D61—81no—
289C4×D61—82no—
290Dic122—21no—
291Dic122—22no—
292Dic122C2×M1612yes—
293C8×C61—42no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—22no—
297#92418—41no—
298#92418—42no—
299#17453—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—24no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—84no—
306C22×C61#40712yes—
307C22×C61C22×C22⋊C411yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—88no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—42no—
319C4×C61#40712yes—
320C4×C61C22×M1611yescharacteristic
321C4×C61C22×M1611yescharacteristic
322C4×C61#57611yescharacteristic
323C4×C61#57711yescharacteristic
324Dic61—22no—
325Dic61—22no—
326Dic61—22no—
327Dic61—22no—
328Dic61—41no—
329Dic61—41no—
330C4×H(5,𝔽2)—611no—
331#2989—611no—
332#2965—611no—
333#1490—612no—
334#1489—612no—
335#1488—612no—
336#1486—612no—
337#1478D6111yescharacteristic, complemented
338#1477—611no—
339#1476—611no—
340C2×D4⋊C8—611no—
341C2×C61—22no—
342C2×C61—44no—
343C2×C61C2×#40711yescharacteristic
344C2×C61#289711yescharacteristic
345C2×C61#290111yescharacteristic
346D61—42no—
347D61—42no—
348C2×H(5,𝔽2)—611no—
349#520—611no—
350#520—612no—
351#520—611no—
352#520—1221no—
353#518—611no—
354#518—612no—
355#518—1222no—
356#518—1221no—
357C4×#41—611no—
358C4×#41—612no—
359C4×#41—611no—
360C4×#41—1221no—
361C2×C4×D4—612no—
362C2×C4×D4—1221no—
363C2×C4×D4—1222no—
364C2×C4×D4C2×D6111yescharacteristic
365#446—1221no—
366#446—1221no—
367#436—1221no—
368#436—1221no—
369#435—1222no—
370#435—1222no—
371#432—612no—
372#431—612no—
373#431C2×D6111yescharacteristic, complemented
374#431C2×D6111yescharacteristic, complemented
375#423—612no—
376C2×C4⋊C8—611no—
377C4×M16—612no—
378C2×C4×C8—611no—
379Q8⋊C8—611no—
380Q8⋊C8—611no—
381Q8⋊C8—612no—
382D4⋊C8—612no—
383D4⋊C8—612no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386D4⋊C8—612no—
387D4⋊C8C2×D6112yescomplemented
388D4⋊C8C2×D6112yescomplemented
389C61#1745311yescharacteristic, complemented, Sylow
390H(5,𝔽2)—1224no—
391H(5,𝔽2)—1224no—
392C2×#41—611no—
393C2×#41—612no—
394C2×#41—611no—
395C2×#41—1221no—
396C22×D4—612no—
397C22×D4—1221no—
398C22×D4—1222no—
399C22×D4C4×D6111yescharacteristic
400C23×C4—21no—
401C23×C4—1222no—
402C2×M16—1222no—
403C2×M16—1222no—
404C22×C8—1221no—
405C22×C8—1221no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—612no—
409C4×Q8—611no—
410C4×Q8—611no—
411C4×Q8—1221no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—611no—
420C4×D4—611no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1221no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1221no—
437C4×D4—1221no—
438C4×D4C4×D6112yes—
439C4×D4C22×D6112yes—
440#117—611no—
441#117—612no—
442#117—611no—
443#117—612no—
444#117—1222no—
445#117—1222no—
446#117—1221no—
447#117—1221no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4C4×D6111yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1222no—
456C2×C22⋊C4—1222no—
457C2×C42—612no—
458C2×C42—1222no—
459C2×C42—1221no—
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—21no—
462C4⋊C8—612no—
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465C4⋊C8—1221no—
466C4⋊C8C22×D6111yescharacteristic
467C4⋊C8C22×D6111yescharacteristic
468C22⋊C8—21no—
469C22⋊C8—21no—
470C22⋊C8—1222no—
471C22⋊C8—1222no—
472#97—1222no—
473#97—1222no—
474C4×C8—21no—
475C4×C8—611no—
476C4×C8—611no—
477C4×C8—1221no—
478C4×C8C22×D6112yes—
479C24—21no—
480C24—1222no—
481#41—614no—
482#41—1224no—
483#41—1228no—
484#41—1224no—
485#41—1222no—
486#41—1222no—
487#41—1222no—
488C2×Q8—611no—
489C2×Q8—611no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—612no—
493C2×D4—611no—
494C2×D4—611no—
495C2×D4—1222no—
496C2×D4—1224no—
497C2×D4—1228no—
498C2×D4—1222no—
499C2×D4—12216no—
500C2×D4—1221no—
501C2×D4—1224no—
502C2×D4#1175818914yescomplemented
503C2×D4C2×C4×D6112yes—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—4882no—
521C22×C4—4882no—
522C22×C4C2×C4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525M16—4882no—
526M16—2444no—
527C2×C8—22no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1222no—
533C2×C8—4881no—
534C2×C8—2442no—
535C4⋊C4—21no—
536C4⋊C4—612no—
537C4⋊C4—611no—
538C4⋊C4—612no—
539C4⋊C4—611no—
540C4⋊C4—611no—
541C4⋊C4—611no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4C2×C4×D6112yes—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—612no—
550C22⋊C4—612no—
551C22⋊C4—612no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1221no—
557C22⋊C4—1221no—
558C22⋊C4—1221no—
559C42—612no—
560C42—612no—
561C42—612no—
562C42—1222no—
563C42—1222no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C42C2×C4×D6111yescharacteristic
568C42D4×D6111yescharacteristic
569C42D4×D6111yescharacteristic
570C42C23×D6111yescharacteristic
571C23—22no—
572C23—24no—
573C23—81no—
574C23—81no—
575C23—612no—
576C23—1228no—
577C23—1221no—
578C23—4882no—
579C23—4882no—
580C23C22⋊C4×D6111yescharacteristic
581Q8—612no—
582Q8—1222no—
583Q8—1221no—
584D4—24no—
585D4—22no—
586D4—612no—
587D4—1228no—
588D4—1228no—
589D4—1224no—
590D4—1224no—
591D4—1222no—
592D4—1221no—
593D4C2×#1175818914yes—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1228no—
606C2×C4—1221no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—4881no—
615C2×C4—4882no—
616C2×C4—4882no—
617C2×C4—4882no—
618C2×C4—4882no—
619C2×C4—2442no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4C22⋊C4×D6111yescharacteristic
623C2×C4C22⋊C4×D6111yescharacteristic
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4M16×D6112yes—
626C2×C4C2×#1175818912yes—
627C2×C4C22×C4×D6111yescharacteristic
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C8—42no—
631C8—41no—
632C8—41no—
633C8—1222no—
634C8—2441no—
635C8—2442no—
636C22—21no—
637C22—21no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—84no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—4888no—
647C22—2441no—
648C22—2441no—
649C22#1683972412yes—
650C22C2×C22⋊C4×D6111yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—42no—
655C4—1222no—
656C4—1222no—
657C4—1222no—
658C4—1222no—
659C4—2441no—
660C4—2441no—
661C4#1683972412yes—
662C4C2×M16×D6111yescharacteristic
663C4C22×#1175818911yescharacteristic
664C4#1684056611yescharacteristic
665C4#1684057211yescharacteristic
666C2—22no—
667C2—44no—
668C2—2442no—
669C2—2442no—
670C2C2×#1683972411yescharacteristic
671C2#2145434411yescharacteristic
672C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1478C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519822C4
cosocleC42×C61C24
hypercenter#1478D61
hypocenterC61#17453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #21454932C2semidirect
3. #21454780C2semidirect
4. #21444260C2semidirect
5. #21444255C2semidirect
6. #21444250C2semidirect
7. #21444242C2semidirect
8. #21444207C2semidirect
9. #21444199C2semidirect
10. C2×D4⋊#92418C2semidirect
11. C61×#1478C2semidirect
34. #16839818C22semidirect
35. #16839813C22semidirect
36. #16839784C22semidirect
37. C2×C4⋊#92418C22semidirect
38. C4×#11758189C22semidirect
39. C2×C4×#92418C22semidirect
40. #16839494C22semidirect
41. #16839492C22semidirect
42. #16839492C22semidirect
43. #16839491C22semidirect
44. D4⋊#92418C22semidirect
45. D4⋊#92418C22semidirect
46. #16839489C22semidirect
47. #16839489C22semidirect
49. C61×#431C22semidirect
50. C61×#431C22semidirect
51. C61×D4⋊C8C22semidirect
52. C61×D4⋊C8C22semidirect
146. C2×D4×C61M16semidirect
173. #11758222M16semidirect
337. #1478D61semidirect
373. #431C2×D61semidirect
374. #431C2×D61semidirect
387. D4⋊C8C2×D61semidirect
388. D4⋊C8C2×D61semidirect
389. C61#17453semidirect
502. C2×D4#11758189semidirect

Maximal subgroups

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

subgroupindex
C4×#145198222
#214549322
#214547802
#214442602
#214442552
#214442502
#214442422
#214442072
#214441992
C2×D4⋊#924182
C61×#14782
#1745361