Finite Group Databasebeta

← #30611983

Subgroups — #30611983

Order 15616.

1112 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462748C212yesmaximal, complemented
2#21461357C211yesmaximal, characteristic, complemented
3#21461353C211yesmaximal, characteristic, complemented
4#21461352C211yesmaximal, characteristic, complemented
5#21461339C212yesmaximal, complemented
6#21461338C212yesmaximal, complemented
7#21461333C212yesmaximal, complemented
8#21461323C211yesmaximal, characteristic, complemented
9D61⋊#333C211yesmaximal, characteristic, complemented
10C2×#16840622C211yesmaximal, characteristic, complemented
11#21459601C211yesmaximal, characteristic
12#16840814—22no—
13#16840792—22no—
14#16840791—22no—
15#16840789—22no—
16#16840785C2212yes—
17#16840782C2212yes—
18#16840780C2212yes—
19#16840767C2212yes—
20C2×#14519844C2212yes—
21D61⋊(C4⋊C8)C2212yes—
22#16840622C2211yescharacteristic
23#16840622C2212yes—
24#16840622C2211yescharacteristic
25#16840622C2212yes—
26#16840622C2211yescharacteristic
27#16840622C2211yescharacteristic
28#16840622C2212yes—
29#16840622C2212yes—
30#16840620C2212yes—
31#16840620C2212yes—
32C2×#14519825C411yescharacteristic
33C2×#14519824C411yescharacteristic
34#16840425—22no—
35#16840414—22no—
36#16840410—22no—
37#16840410—22no—
38#16840338C411yescharacteristic
39#16840337C411yescharacteristic
40#16840337C411yescharacteristic
41#16840332C411yescharacteristic
42#16840332C411yescharacteristic
43#16840311C2211yescharacteristic
44#16840311C2211yescharacteristic
45#16840308C2211yescharacteristic
46C2×#14519775C2211yescharacteristic
47C4⋊Q8×D61C2211yescharacteristic
48C4⋊Q8×D61C2211yescharacteristic
49#16840230C2211yescharacteristic
50C61×#536C411yescharacteristic
51C2×#11758213—22no—
52C2×D61⋊C8—22no—
53C22×#92420—22no—
54#14519844—22no—
55#14519844—22no—
56#14519844C2312yes—
57#14519844C2312yes—
58#14519844C2312yes—
59#14519844C2312yes—
60C4⋊#92420—22no—
61#14519834—22no—
62#14519834—22no—
63C22⋊#92420—22no—
64#14519832—22no—
65#14519832—22no—
66C4×#92420—22no—
67#14519825—44no—
68#14519824—44no—
69C2×#11758226—21no—
70C2×#11758226C2×C411yescharacteristic
71C2×#11758226C2×C411yescharacteristic
72C2×#11758222—21no—
73C2×#11758222—21no—
74C2×#11758222—21no—
75C2×#11758222C2×C411yescharacteristic
76C2×#11758222C2×C411yescharacteristic
77C2×#11758224—21no—
78C2×#11758224—21no—
79C2×#11758224C2×C411yescharacteristic
80C2×#11758224C2×C411yescharacteristic
81C22×Dic122—21no—
82C22×Dic122C2×C411yescharacteristic
83C2×Q8×D61—21no—
84C2×Q8×D61C2×C411yescharacteristic
85C2×Q8×D61C2×C411yescharacteristic
86C2×Q8×D61C2×C411yescharacteristic
87C2×Q8×D61C2×C411yescharacteristic
88#14519777C2×C411yescharacteristic
89#14519777C2×C411yescharacteristic
90#14519776—22no—
91#14519776—22no—
92#14519775—22no—
93#14519775C2×C411yescharacteristic
94#14519775C2×C411yescharacteristic
95#14519775C2311yescharacteristic
96#14519775C2311yescharacteristic
97#14519775C2311yescharacteristic
98#14519773C2311yescharacteristic
99#14519766C2×C411yescharacteristic
100#14519765C2×C411yescharacteristic
101#14519765C2×C411yescharacteristic
102#14519764C2×C411yescharacteristic
103#14519762—22no—
104#14519762—22no—
105#14519761—22no—
106#14519759—22no—
107#14519759—22no—
108#14519757—22no—
109#14519755—22no—
110#14519755—22no—
111#14519754—22no—
112#14519754—22no—
113#14519750—22no—
114#14519750—22no—
115#14519749—22no—
116#14519742D412yes—
117#14519739D412yes—
118C4×C22⋊D61—22no—
119#14519737—22no—
120#14519737—22no—
121#14519737D412yes—
122#14519737D412yes—
123#14519734C2×C411yescharacteristic
124#14519734C2×C411yescharacteristic
125C2×#11758196—22no—
126C4⋊C4×D61—22no—
127C4⋊C4×D61—22no—
128C2×C4×Dic61C2311yescharacteristic
129C42×D61C2311yescharacteristic
130C42×D61C2311yescharacteristic
131C2×#41×C61C2×C411yescharacteristic
132C22×Q8×C61C2×C411yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic
134C4⋊Q8×C61C2×C411yescharacteristic
135#124×C61C2×C411yescharacteristic
136#124×C61C2×C411yescharacteristic
137#123×C61—22no—
138C22⋊Q8×C61—22no—
139#117×C61C2×C411yescharacteristic
140#41×C61—42no—
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61—21no—
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C411yescharacteristic
156C2×C22⋊D61C22⋊C411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D412yes—
161#11758226—42no—
162#11758226—42no—
163#11758226—44no—
164Q8×D61—44no—
165Q8×D61—42no—
166Q8×D61—42no—
167Q8×D61—42no—
168Q8×D61—42no—
169#11758224—42no—
170#11758224—42no—
171#11758224—44no—
172#11758224—44no—
173#11758222—24no—
174#11758222—44no—
175#11758222—44no—
176#11758222—44no—
177#11758222—42no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244C22⋊C411yescharacteristic
181C2×D244C22×C411yescharacteristic
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×C4×D61—21no—
187C2×C4×D61—21no—
188C2×C4×D61—21no—
189C2×C4×D61C22×C411yescharacteristic
190C2×C4×D61C22×C411yescharacteristic
191C2×C4×D61C2×D412yes—
192C2×C4×D61C2×D412yes—
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122—21no—
196C2×Dic122—21no—
197C2×Dic122—21no—
198C2×Dic122—44no—
199C2×Dic122—42no—
200C2×Dic122C22⋊C411yescharacteristic
201C2×Dic122C22×C411yescharacteristic
202C2×Dic122C22×C411yescharacteristic
203C2×Dic122C22×C411yescharacteristic
204C2×#92420—22no—
205C2×#92420—22no—
206C2×#92420—22no—
207C2×#92420—22no—
208C2×#92420—82no—
209#11758213—82no—
210D61⋊C8—82no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61C22⋊C412yes—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61C22⋊C412yes—
217C42×C61C22×C411yescharacteristic
218C42×C61C22×C411yescharacteristic
219C22⋊Dic61—22no—
220#11758198—22no—
221#11758198—22no—
222#11758198—22no—
223#11758198C22⋊C412yes—
224#11758198C22⋊C412yes—
225C4⋊Dic61—22no—
226C4⋊Dic61—22no—
227#11758196—22no—
228#11758196—22no—
229#11758196—22no—
230#11758196—22no—
231#11758196—22no—
232#11758196C22⋊C412yes—
233#11758196C22⋊C412yes—
234C4×Dic61C22×C411yescharacteristic
235C4×Dic61C2×D412yes—
236C4×Dic61C2×D412yes—
237C4×Dic61C2×D412yes—
238C4×Dic61C2411yescharacteristic
239C23×C61—21no—
240C23×C61C2×C22⋊C411yescharacteristic
241C22×D61—21no—
242C22×D61C2×C22⋊C411yescharacteristic
243C22×D61C2×C22⋊C411yescharacteristic
244Q8×C61—42no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248D4×C61—41no—
249D4×C61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C23×C411yescharacteristic
260C22⋊D61—22no—
261C22⋊D61—42no—
262C22⋊D61—42no—
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—42no—
266C22⋊D61#10014yes—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C22×D412yes—
279D244—22no—
280D244—41no—
281D244—42no—
282D244—42no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—42no—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—42no—
293C4×D61—42no—
294C4×D61—42no—
295C4×D61—42no—
296Dic122—22no—
297Dic122—42no—
298Dic122—42no—
299Dic122—42no—
300Dic122—42no—
301Dic122—42no—
302Dic122—41no—
303Dic122—41no—
304Dic122—41no—
305#92420—42no—
306#92420—42no—
307#92420—42no—
308#92420—42no—
309#17779—611nomaximal, Sylow
310C22×C61—22no—
311C22×C61—41no—
312C22×C61—42no—
313C22×C61C2×#10012yes—
314C22×C61C22×C22⋊C411yescharacteristic
315C2×D61—22no—
316C2×D61—21no—
317C2×D61—22no—
318C2×D61—41no—
319C2×D61—42no—
320C2×D61C2×#10012yes—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—22no—
324C4×C61—22no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328Dic61—21no—
329Dic61—22no—
330Dic61—41no—
331Dic61—41no—
332Dic61—41no—
333Dic61C2×#10012yes—
334Dic61#57712yes—
335Dic61#57812yes—
336#3499—611no—
337#2990—612no—
338#1560—611no—
339#1558—611no—
340#1558—611no—
341#1551—612no—
342#1551—612no—
343#1548—612no—
344#1543—611no—
345C2×#333—611no—
346C2×#333—611no—
347C2×C61—22no—
348C2×C61—81no—
349C2×C61C22×#10011yescharacteristic
350C2×C61#289812yes—
351D61—22no—
352D61—41no—
353D61—81no—
354C2×2−1+4—611no—
355C2×2−1+4—611no—
356D4×Q8—1222no—
357#547—1222no—
358#547—1222no—
359#547—1222no—
360#536—611no—
361#536—611no—
362#536—611no—
363#536—611no—
364#536—611no—
365#536#692311yescharacteristic
366#532—611no—
367#532—611no—
368#532—611no—
369C2×C4⋊Q8—611no—
370C2×C4⋊Q8—611no—
371C2×C4⋊Q8—611no—
372C4×#41—611no—
373#446—1222no—
374#435—1222no—
375#435—1222no—
376C8×D4—1222no—
377#432—612no—
378#432—612no—
379#431—612no—
380#423—612no—
381C2×C4⋊C8—612no—
382C2×C4⋊C8—612no—
383#333—611no—
384#333—612no—
385#333—612no—
386#333—611no—
387#333—611no—
388#333—611no—
389#333—612no—
390#333—612no—
391#331—612no—
392#331—612no—
393C61#1777911yescharacteristic, complemented, Sylow
3942−1+4—2444no—
3952−1+4—2444no—
396C2×#41—611no—
397C2×#41—611no—
398C2×#41—611no—
399C2×#41—611no—
400C2×#41—611no—
401C2×#41—611no—
402C2×#41—1221no—
403C2×#41—1221no—
404C2×#41—1221no—
405C2×#41—1221no—
406C2×#41—1221no—
407C2×#41—1221no—
408C2×#41C2×#692311yescharacteristic
409C22×Q8—611no—
410C22×Q8—611no—
411C22×Q8—611no—
412C22×Q8—611no—
413C22×Q8—611no—
414C22×Q8—1221no—
415C22×Q8—1221no—
416C22×Q8C2×#692311yescharacteristic
417C2×M16—1222no—
418C22×C8—1222no—
419C22×C8—1222no—
420C4⋊Q8—611no—
421C4⋊Q8—611no—
422C4⋊Q8—611no—
423C4⋊Q8—611no—
424C4⋊Q8—611no—
425C4⋊Q8—611no—
426C4⋊Q8—611no—
427C4⋊Q8—1222no—
428C4⋊Q8—1222no—
429C4⋊Q8—1222no—
430C4⋊Q8C2×#692311yescharacteristic
431C4⋊Q8C2×#692311yescharacteristic
432#127—611no—
433#124—611no—
434#124—611no—
435#124—611no—
436#124—611no—
437#124C2×#692311yescharacteristic
438#124C2×#692311yescharacteristic
439#123—22no—
440#123—1222no—
441#123—1222no—
442#123—1222no—
443#123—1222no—
444#123—1222no—
445C22⋊Q8—22no—
446C22⋊Q8—1222no—
447C22⋊Q8—1222no—
448C22⋊Q8—1222no—
449C22⋊Q8—1222no—
450C22⋊Q8—1222no—
451#121—1222no—
452#121—1222no—
453#121—1222no—
454C4×Q8—612no—
455C4×D4—612no—
456C4×D4—612no—
457C4×D4—612no—
458C4×D4—1222no—
459C4×D4—1222no—
460C4×D4—1222no—
461#117—611no—
462#117—611no—
463#117C2×#692311yescharacteristic
464C2×C4⋊C4—1222no—
465C2×C4⋊C4—1222no—
466C2×C4⋊C4—1222no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C4⋊C8—612no—
471C4⋊C8—612no—
472C4⋊C8—612no—
473C4⋊C8—612no—
474C4⋊C8—1222no—
475C4⋊C8—1222no—
476C4⋊C8—1222no—
477C22⋊C8—1222no—
478C22⋊C8—1222no—
479C22⋊C8—1222no—
480#97—1222no—
481#97—1222no—
482C4×C8—1222no—
483#41—42no—
484#41—1224no—
485#41—2442no—
486#41—2444no—
487#41—2444no—
488#41—2444no—
489#41—2442no—
490#41—2442no—
491#41—2442no—
492#41—2442no—
493#41—2444no—
494#41—2444no—
495#41—2444no—
496C2×Q8—21no—
497C2×Q8—42no—
498C2×Q8—611no—
499C2×Q8—611no—
500C2×Q8—611no—
501C2×Q8—611no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—2442no—
508C2×Q8—2444no—
509C2×Q8—2442no—
510C2×Q8—2442no—
511C2×Q8—2442no—
512C2×Q8—2442no—
513C2×Q8—2444no—
514C2×Q8C22×#692311yescharacteristic
515C2×Q8C22×#692311yescharacteristic
516C2×Q8C22×#692311yescharacteristic
517C2×D4—21no—
518C2×D4—611no—
519C2×D4—611no—
520C2×D4—611no—
521C2×D4—611no—
522C2×D4—1221no—
523C2×D4—1221no—
524C2×D4—1222no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4C22×#692311yescharacteristic
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—612no—
532C22×C4—612no—
533C22×C4—612no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—1221no—
539C22×C4—1221no—
540C22×C4—1221no—
541C22×C4—1221no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—1221no—
546C22×C4C22×#692311yescharacteristic
547M16—4882no—
548C2×C8—1222no—
549C2×C8—1222no—
550C2×C8—1222no—
551C2×C8—1222no—
552C2×C8—4882no—
553C2×C8—4882no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—612no—
557C4⋊C4—612no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—1222no—
562C4⋊C4—1222no—
563C4⋊C4—1222no—
564C4⋊C4—1222no—
565C4⋊C4C22⋊#692312yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—612no—
569C22⋊C4—612no—
570C22⋊C4—1222no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4C22⋊#692312yes—
575C42—612no—
576C42—612no—
577C42—612no—
578C42—611no—
579C42—611no—
580C42C22×#692311yescharacteristic
581C42C22×#692311yescharacteristic
582C23—21no—
583C23—611no—
584C23—611no—
585C23—1221no—
586C23C2×C22⋊#692311yescharacteristic
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—42no—
591Q8—1222no—
592Q8—2441no—
593Q8—2441no—
594Q8—2441no—
595Q8—2442no—
596Q8—2442no—
597Q8—2442no—
598Q8—2442no—
599Q8—2442no—
600D4—41no—
601D4—42no—
602D4—614no—
603D4—1222no—
604D4—1222no—
605D4—2441no—
606D4—2442no—
607D4—2442no—
608D4—2442no—
609D4—2442no—
610D4—2442no—
611D4—2442no—
612D4—2442no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—611no—
620C2×C4—612no—
621C2×C4—612no—
622C2×C4—611no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4C2×C22⋊#692311yescharacteristic
645C2×C4C2×C22⋊#692312yes—
646C2×C4C2×C22⋊#692312yes—
647C2×C4C23×#692311yescharacteristic
648C8—2442no—
649C8—2442no—
650C8—2442no—
651C8—2442no—
652C22—22no—
653C22—41no—
654C22—42no—
655C22—612no—
656C22—1222no—
657C22—1221no—
658C22—1222no—
659C22—2441no—
660C22—2442no—
661C22D61⋊#10012yes—
662C22C22×C22⋊#692311yescharacteristic
663C4—21no—
664C4—21no—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—612no—
671C4—612no—
672C4—612no—
673C4—1221no—
674C4—1222no—
675C4—2441no—
676C4—2441no—
677C4—2441no—
678C2—22no—
679C2—81no—
680C2—1222no—
681C2—4881no—
682C2—2441no—
683C2C2×D61⋊#10011yescharacteristic
684C2#2146244112yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC22×C61C22×C22⋊C4
prime radical#21459601C2
cosocleC4×Dic61C24
hypercenter#536#6923
hypocenterC61#17779
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #21462748C2semidirect
2. #21461357C2semidirect
3. #21461353C2semidirect
4. #21461352C2semidirect
5. #21461339C2semidirect
6. #21461338C2semidirect
7. #21461333C2semidirect
8. #21461323C2semidirect
9. D61⋊#333C2semidirect
10. C2×#16840622C2semidirect
393. C61#17779semidirect

Maximal subgroups

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

subgroupindex
#214627482
#214613572
#214613532
#214613522
#214613392
#214613382
#214613332
#214613232
D61⋊#3332
C2×#168406222
#214596012
#1777961