Finite Group Databasebeta

← #29648231

Subgroups — #29648231

Order 15616.

1152 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459609C211yesmaximal, characteristic
2#21454828C212yesmaximal, complemented
3#21444452C211yesmaximal, characteristic, complemented
4#21444441C211yesmaximal, characteristic, complemented
5#21444431C212yesmaximal, complemented
6#21444428C212yesmaximal, complemented
7#21444415C212yesmaximal, complemented
8#21444409C211yesmaximal, characteristic, complemented
9C2×#16839502C211yesmaximal, characteristic, complemented
10#329×D61C211yesmaximal, characteristic, complemented
11C61×#1533C211yesmaximal, characteristic, complemented
12C2×#14519825C411yescharacteristic
13C2×#14519822C411yescharacteristic
14#16840420—22no—
15#16840405—22no—
16#16840402—22no—
17#16840402—22no—
18#16840336C411yescharacteristic
19#16840332C411yescharacteristic
20#16840332C411yescharacteristic
21#16840330C411yescharacteristic
22#16840330C411yescharacteristic
23#16840329C411yescharacteristic
24#16840312C2211yescharacteristic
25#16840311C2211yescharacteristic
26#16840311C2211yescharacteristic
27#124×D61C2211yescharacteristic
28#124×D61C2211yescharacteristic
29C4×#11758222C2211yescharacteristic
30#16839891—22no—
31#16839832—22no—
32#16839830—22no—
33#16839825—22no—
34#16839817C2212yescomplemented
35#16839817C2212yescomplemented
36#16839714C2212yescomplemented
37C2×#14519688C2212yescomplemented
38#97×D61C2212yescomplemented
39#16839502C2211yescharacteristic, complemented
40#16839502C2212yescomplemented
41#16839502C2211yescharacteristic, complemented
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45#16839501C2212yescomplemented
46C2×#124×C61C2211yescharacteristic
47C61×#432C2212yescomplemented
48C61×#329C2211yescharacteristic, complemented
49C61×#329C2211yescharacteristic, complemented
50C61×#329C2212yescomplemented
51#14519825—44no—
52#14519822—44no—
53C2×#11758226—21no—
54C2×#11758226C2×C411yescharacteristic
55C2×#11758226C2×C411yescharacteristic
56C2×#11758222—21no—
57C2×#11758222—21no—
58C2×#11758222C2×C411yescharacteristic
59C2×#11758222C2×C411yescharacteristic
60C2×#11758222C2×C411yescharacteristic
61C2×#11758224—21no—
62C2×#11758224C2×C411yescharacteristic
63C2×#11758224C2×C411yescharacteristic
64C2×Q8×D61—21no—
65C2×Q8×D61C2×C411yescharacteristic
66C2×Q8×D61C2×C411yescharacteristic
67C22×C22⋊D61—21no—
68C22×C22⋊D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61C2×C411yescharacteristic
71C2×D4×D61C2×C411yescharacteristic
72C2×#11758189—22no—
73C22×#92418—22no—
74C2×C8×D61—22no—
75#14519777C2×C411yescharacteristic
76#14519775C2×C411yescharacteristic
77#14519775C2×C411yescharacteristic
78#14519774C2×C411yescharacteristic
79#14519773C2×C411yescharacteristic
80#14519773C2×C411yescharacteristic
81#14519766C2×C411yescharacteristic
82#14519766C2×C411yescharacteristic
83#14519765C2311yescharacteristic
84#14519765C2311yescharacteristic
85#14519764C2×C411yescharacteristic
86#14519764C2×C411yescharacteristic
87#14519763—22no—
88#14519763—22no—
89#14519760—22no—
90#14519759—22no—
91#14519759—22no—
92#14519754—22no—
93#14519754—22no—
94#14519753—22no—
95#14519751—22no—
96#14519750—22no—
97#14519750—22no—
98#14519750—22no—
99#14519750—22no—
100#14519748—22no—
101#14519747—22no—
102#14519747—22no—
103#14519746—22no—
104C4×Dic122D412yes—
105C4×D244—22no—
106C4×D244D412yes—
107C4×C22⋊D61D412yes—
108C4×C22⋊D61D412yes—
109#14519737—22no—
110#14519737—22no—
111#14519736C2×C411yescharacteristic
112#14519734C2×C411yescharacteristic
113#14519734C2×C411yescharacteristic
114C22⋊C4×D61—22no—
115C22⋊C4×D61—22no—
116C42×D61C2311yescharacteristic
117C42×D61C2311yescharacteristic
118#14519706—22no—
119#14519706—22no—
120#14519689—22no—
121#14519689—22no—
122#14519688—22no—
123#14519688C2312yes—
124#14519688C2312yes—
125#14519687—22no—
126C8×Dic61—22no—
127C22×Q8×C61C2×C411yescharacteristic
128C22×D4×C61C2×C411yescharacteristic
129#124×C61—22no—
130#124×C61C2×C411yescharacteristic
131#124×C61C2×C411yescharacteristic
132#124×C61C2311yescharacteristic
133#124×C61C2311yescharacteristic
134C2×C22⋊C4×C61—22no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—22no—
137C22⋊C8×C61—22no—
138#97×C61C2312yes—
139#97×C61C2312yes—
140C24×C61—21no—
141C23×D61—21no—
142C23×D61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—42no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—42no—
149C2×D4×C61C22×C411yescharacteristic
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D412yes—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—21no—
158C2×C22⋊D61—21no—
159C2×C22⋊D61—21no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—44no—
163C2×C22⋊D61—44no—
164C2×C22⋊D61C22⋊C411yescharacteristic
165C2×C22⋊D61C22⋊C411yescharacteristic
166C22×Dic61—21no—
167C22×Dic61—21no—
168#11758226—42no—
169#11758226—42no—
170#11758226—44no—
171Q8×D61—44no—
172Q8×D61—42no—
173Q8×D61—42no—
174#11758224—44no—
175#11758224—42no—
176#11758224—42no—
177D4×D61—42no—
178D4×D61—42no—
179D4×D61—44no—
180#11758222—24no—
181#11758222—44no—
182#11758222—44no—
183#11758222—42no—
184#11758222—42no—
185C2×D244—21no—
186C2×D244—21no—
187C2×D244—22no—
188C2×D244C22⋊C411yescharacteristic
189C2×D244C22×C411yescharacteristic
190C2×D244C22×C411yescharacteristic
191C2×C4×D61—21no—
192C2×C4×D61—21no—
193C2×C4×D61—21no—
194C2×C4×D61—21no—
195C2×C4×D61—21no—
196C2×C4×D61—21no—
197C2×C4×D61C22×C411yescharacteristic
198C2×C4×D61C22×C411yescharacteristic
199C2×C4×D61C2×D412yes—
200C2×C4×D61C2×D412yes—
201C2×Dic122—21no—
202C2×Dic122—21no—
203C2×Dic122C22⋊C411yescharacteristic
204C2×Dic122C22×C411yescharacteristic
205C2×Dic122C22×C411yescharacteristic
206C2×C8×C61—22no—
207C2×C8×C61—22no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C42×C61C22×C411yescharacteristic
213C42×C61C2×D412yes—
214C42×C61C2411yescharacteristic
215C22⋊Dic61—22no—
216C22⋊Dic61—22no—
217C22⋊Dic61C22⋊C412yes—
218#11758198—22no—
219#11758198—22no—
220#11758198—22no—
221#11758198—22no—
222#11758198C22⋊C412yes—
223#11758198C22⋊C412yes—
224C4⋊Dic61—22no—
225C4⋊Dic61C22⋊C412yes—
226#11758196—22no—
227#11758196—22no—
228#11758196C22⋊C412yes—
229#11758196C22⋊C412yes—
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C2×D412yes—
233C4×Dic61C2×D412yes—
234C2×#92418—22no—
235C2×#92418—22no—
236C2×#92418—44no—
237#11758189—44no—
238C8×D61—44no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—42no—
242C23×C61—82no—
243C23×C61C2×C22⋊C411yescharacteristic
244C22×D61—21no—
245C22×D61—21no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61—82no—
249C22×D61—82no—
250C22×D61C2×C22⋊C411yescharacteristic
251C22×D61C2×C22⋊C411yescharacteristic
252Q8×C61—42no—
253Q8×C61—41no—
254Q8×C61—41no—
255D4×C61—41no—
256D4×C61—41no—
257D4×C61—42no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61C2×C22⋊C412yes—
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D412yes—
267C22⋊D61—22no—
268C22⋊D61—42no—
269C22⋊D61—42no—
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—42no—
273C22⋊D61—42no—
274C22⋊D61—82no—
275C22⋊D61—82no—
276C22⋊D61#10014yes—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285D244—22no—
286D244—41no—
287D244—41no—
288D244—42no—
289D244—42no—
290D244—82no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—42no—
294C4×D61—42no—
295C4×D61—42no—
296C4×D61—42no—
297C4×D61—42no—
298C4×D61—42no—
299C4×D61—42no—
300C4×D61—42no—
301Dic122—22no—
302Dic122—42no—
303Dic122—42no—
304Dic122—41no—
305Dic122—41no—
306C8×C61—42no—
307C8×C61—42no—
308#92418—24no—
309#92418—42no—
310#17685—611nomaximal, Sylow
311C22×C61—22no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—42no—
315C22×C61—42no—
316C22×C61—162no—
317C22×C61C2×#10012yes—
318C22×C61C22×C22⋊C411yescharacteristic
319C2×D61—22no—
320C2×D61—21no—
321C2×D61—22no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—162no—
327C2×D61—162no—
328C2×D61C2×#10012yes—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—22no—
332C4×C61—22no—
333C4×C61—41no—
334C4×C61—41no—
335Dic61—21no—
336Dic61—22no—
337Dic61—22no—
338Dic61—41no—
339Dic61—41no—
340Dic61C2×#10012yes—
341#3500—611no—
342#2970—612no—
343#1536—611no—
344#1533—611no—
345#1533D6111yescharacteristic, complemented
346#1530—612no—
347#1530—612no—
348#1527—612no—
349#1525—611no—
350C2×#329—611no—
351C2×#329—611no—
352C2×C61—22no—
353C2×C61—81no—
354C2×C61—81no—
355C2×C61C22×#10011yescharacteristic
356C2×C61#290112yes—
357D61—22no—
358D61—41no—
359D61—81no—
360D61—81no—
361C2×2−1+4—611no—
362C2×H(5,𝔽2)—611no—
363#548—1222no—
364#546—1222no—
365#546—1222no—
366#546—1222no—
367#536—611no—
368#536—611no—
369#536—611no—
370#535—611no—
371#535—611no—
372#535—611no—
373#532—611no—
374#532—611no—
375#532—611no—
376C2×#124—611no—
377C2×#124—611no—
378C2×#124C2×D6111yescharacteristic
379C4×#41—611no—
380#446—1222no—
381#435—1222no—
382#435—1222no—
383C8×D4—1222no—
384#432—612no—
385#432—612no—
386#432C2×D6112yescomplemented
387C4×M16—612no—
388C2×#97—612no—
389C2×#97—612no—
390#329—611no—
391#329—611no—
392#329—612no—
393#329—612no—
394#329—612no—
395#329—612no—
396#329—612no—
397#329C2×D6111yescharacteristic, complemented
398#329C2×D6111yescharacteristic, complemented
399#329C2×D6112yescomplemented
400C61#1768511yescharacteristic, complemented, Sylow
4012−1+4—2444no—
402H(5,𝔽2)—2444no—
403C2×#41—611no—
404C2×#41—611no—
405C2×#41—611no—
406C2×#41—611no—
407C2×#41—611no—
408C2×#41—611no—
409C2×#41—611no—
410C2×#41—1221no—
411C2×#41—1221no—
412C2×#41—1221no—
413C2×#41—1221no—
414C22×Q8—611no—
415C22×Q8—611no—
416C22×Q8—1221no—
417C22×Q8C4×D6111yescharacteristic
418C22×D4—611no—
419C22×D4—611no—
420C22×D4—1221no—
421C22×D4—1221no—
422C22×D4—1221no—
423C22×D4C4×D6111yescharacteristic
424C2×M16—1222no—
425C22×C8—1222no—
426C22×C8—1222no—
427C4⋊Q8—611no—
428C4⋊Q8—611no—
429C4⋊Q8—611no—
430#127—611no—
431#127—611no—
432#127—611no—
433#124—22no—
434#124—611no—
435#124—611no—
436#124—611no—
437#124—611no—
438#124—611no—
439#124—611no—
440#124—1222no—
441#124—1222no—
442#124C4×D6111yescharacteristic
443#124C4×D6111yescharacteristic
444#124C22×D6111yescharacteristic
445#124C22×D6111yescharacteristic
446#123—1222no—
447#123—1222no—
448#123—1222no—
449C22⋊Q8—1222no—
450C22⋊Q8—1222no—
451C22⋊Q8—1222no—
452#121—1222no—
453#121—1222no—
454#121—1222no—
455#121—1222no—
456#121—1222no—
457#121—1222no—
458#120—1222no—
459#120—1222no—
460#120—1222no—
461C4×Q8—612no—
462C4×D4—612no—
463C4×D4—612no—
464C4×D4—612no—
465C4×D4—1222no—
466C4×D4—1222no—
467C4×D4—1222no—
468#117—611no—
469#117—611no—
470#117—611no—
471C2×C22⋊C4—22no—
472C2×C22⋊C4—1222no—
473C2×C22⋊C4—1222no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42C22×D6111yescharacteristic
477C4⋊C8—22no—
478C4⋊C8—1222no—
479C4⋊C8—1222no—
480C22⋊C8—22no—
481C22⋊C8—1222no—
482C22⋊C8—1222no—
483#97—612no—
484#97—612no—
485#97—1222no—
486#97—1222no—
487#97C22×D6112yes—
488#97C22×D6112yes—
489C4×C8—1222no—
490C24—21no—
491C24—1221no—
492C24—1221no—
493#41—1224no—
494#41—2442no—
495#41—2442no—
496#41—2444no—
497#41—2444no—
498#41—2442no—
499#41—2442no—
500#41—2442no—
501#41—2442no—
502#41—2444no—
503#41—2444no—
504C2×Q8—21no—
505C2×Q8—42no—
506C2×Q8—611no—
507C2×Q8—611no—
508C2×Q8—611no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—2442no—
512C2×Q8—2442no—
513C2×Q8—2444no—
514C2×Q8C2×C4×D6111yescharacteristic
515C2×Q8C2×C4×D6111yescharacteristic
516C2×D4—21no—
517C2×D4—42no—
518C2×D4—611no—
519C2×D4—611no—
520C2×D4—611no—
521C2×D4—611no—
522C2×D4—611no—
523C2×D4—1221no—
524C2×D4—1221no—
525C2×D4—1222no—
526C2×D4—1222no—
527C2×D4—1221no—
528C2×D4—1221no—
529C2×D4—1221no—
530C2×D4—1221no—
531C2×D4—1221no—
532C2×D4—1221no—
533C2×D4—1222no—
534C2×D4—2442no—
535C2×D4—2442no—
536C2×D4—2444no—
537C2×D4—2444no—
538C2×D4—2444no—
539C2×D4C2×C4×D6111yescharacteristic
540C2×D4C2×C4×D6111yescharacteristic
541C22×C4—21no—
542C22×C4—611no—
543C22×C4—611no—
544C22×C4—612no—
545C22×C4—612no—
546C22×C4—1221no—
547C22×C4—1221no—
548C22×C4—1221no—
549C22×C4—1221no—
550C22×C4—1221no—
551C22×C4—1221no—
552C22×C4—1221no—
553C22×C4—1221no—
554C22×C4C2×C4×D6111yescharacteristic
555C22×C4D4×D6112yes—
556M16—2444no—
557C2×C8—22no—
558C2×C8—22no—
559C2×C8—1222no—
560C2×C8—1222no—
561C2×C8—2444no—
562C2×C8—2444no—
563C4⋊C4—612no—
564C4⋊C4—612no—
565C4⋊C4—612no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1222no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—612no—
574C22⋊C4—612no—
575C22⋊C4—612no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C42—611no—
583C42—611no—
584C42—612no—
585C42—612no—
586C42C2×C4×D6111yescharacteristic
587C42D4×D6112yes—
588C42C23×D6111yescharacteristic
589C23—21no—
590C23—21no—
591C23—42no—
592C23—82no—
593C23—611no—
594C23—611no—
595C23—1221no—
596C23—1221no—
597C23—4882no—
598C23—4882no—
599C23—2442no—
600C23—2442no—
601C23C22⋊C4×D6111yescharacteristic
602Q8—41no—
603Q8—41no—
604Q8—42no—
605Q8—1222no—
606Q8—2441no—
607Q8—2441no—
608Q8—2442no—
609Q8—2442no—
610D4—41no—
611D4—41no—
612D4—42no—
613D4—614no—
614D4—1222no—
615D4—1222no—
616D4—4882no—
617D4—4882no—
618D4—4882no—
619D4—2441no—
620D4—2441no—
621D4—2442no—
622D4—2442no—
623D4—2442no—
624D4—2442no—
625D4—2442no—
626D4—2442no—
627D4—2442no—
628D4—2442no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—611no—
635C2×C4—612no—
636C2×C4—612no—
637C2×C4—611no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1221no—
641C2×C4—1221no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4—2442no—
650C2×C4—2442no—
651C2×C4—2442no—
652C2×C4C22⋊C4×D6112yes—
653C2×C4C22⋊C4×D6111yescharacteristic
654C2×C4C22×C4×D6111yescharacteristic
655C2×C4C2×D4×D6112yes—
656C8—42no—
657C8—42no—
658C8—1224no—
659C8—2442no—
660C22—22no—
661C22—42no—
662C22—42no—
663C22—41no—
664C22—41no—
665C22—162no—
666C22—612no—
667C22—1222no—
668C22—1222no—
669C22—1221no—
670C22—9762no—
671C22—9762no—
672C22—2442no—
673C22—2442no—
674C22—2441no—
675C22—2441no—
676C22#100×D6112yes—
677C22C2×C22⋊C4×D6111yescharacteristic
678C4—22no—
679C4—22no—
680C4—21no—
681C4—21no—
682C4—41no—
683C4—41no—
684C4—612no—
685C4—1222no—
686C4—1222no—
687C4—1221no—
688C4—2441no—
689C4—2441no—
690C2—22no—
691C2—81no—
692C2—81no—
693C2—1222no—
694C2—4881no—
695C2—4881no—
696C2—2441no—
697C2C2×#100×D6111yescharacteristic
698C2#2145436512yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1533C2
socleC22×C61C22×C22⋊C4
prime radical#21459609C2
cosocleC42×C61C24
hypercenter#1533D61
hypocenterC61#17685
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #21454828C2semidirect
3. #21444452C2semidirect
4. #21444441C2semidirect
5. #21444431C2semidirect
6. #21444428C2semidirect
7. #21444415C2semidirect
8. #21444409C2semidirect
9. C2×#16839502C2semidirect
10. #329×D61C2semidirect
11. C61×#1533C2semidirect
34. #16839817C22semidirect
35. #16839817C22semidirect
36. #16839714C22semidirect
37. C2×#14519688C22semidirect
38. #97×D61C22semidirect
39. #16839502C22semidirect
40. #16839502C22semidirect
41. #16839502C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
45. #16839501C22semidirect
47. C61×#432C22semidirect
48. C61×#329C22semidirect
49. C61×#329C22semidirect
50. C61×#329C22semidirect
345. #1533D61semidirect
386. #432C2×D61semidirect
397. #329C2×D61semidirect
398. #329C2×D61semidirect
399. #329C2×D61semidirect
400. C61#17685semidirect

Maximal subgroups

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

subgroupindex
#214596092
#214548282
#214444522
#214444412
#214444312
#214444282
#214444152
#214444092
C2×#168395022
#329×D612
C61×#15332
#1768561