Finite Group Databasebeta

← #29688608

Subgroups — #29688608

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840148C211yesmaximal, characteristic
2C2×#16839872C211yesmaximal, characteristic, complemented
3C2×#16839860C211yesmaximal, characteristic, complemented
4C2×C4×D488C211yesmaximal, characteristic, complemented
5#21448541C211yesmaximal, characteristic, complemented
6#21448526C211yesmaximal, characteristic, complemented
7#21446702C211yesmaximal, characteristic, complemented
8#21446693C211yesmaximal, characteristic, complemented
9#21446682C211yesmaximal, characteristic, complemented
10#21446670C211yesmaximal, characteristic, complemented
11#21446627C211yesmaximal, characteristic, complemented
12#21446623C211yesmaximal, characteristic, complemented
13#21445984C211yesmaximal, characteristic, complemented
14#21445505C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×D488C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19C2×#14519773C2211yescharacteristic
20C2×#14519766C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840148C418yescomplemented
24#16839872—24no—
25#16839860—24no—
26C4×D488—24no—
27C2×#14519711C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37C2×#14519687C2211yescharacteristic
38C2×#14519687C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839663C2211yescharacteristic
44#16839663C2211yescharacteristic
45#16839653—22no—
46#16839641—22no—
47#16839638—22no—
48#16839633—22no—
49#16839631C2211yescharacteristic
50#16839627C2211yescharacteristic
51#16839608C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C4×SD16×C61—24no—
56C2×#106×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201C2×C414yescomplemented
68C2×D488—22no—
69C2×D488—22no—
70C2×D488C2×C414yescomplemented
71C2×#11758199—22no—
72C2×#11758199—22no—
73C2×#11758199C2×C414yescomplemented
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519773—22no—
77#14519773C2×C414yescomplemented
78#14519766—22no—
79#14519766C2×C414yescomplemented
80C4×D244—22no—
81C4×D244—22no—
82C4×D244—22no—
83C4×D244—22no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—22no—
87C2×C22⋊Dic61—22no—
88C2×#11758198—22no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519711—22no—
96#14519710—22no—
97#14519708—22no—
98#14519701—22no—
99#14519699—22no—
100#14519699—22no—
101#14519698—22no—
102#14519697—22no—
103#14519697—22no—
104#14519688D414yescomplemented
105#14519687—22no—
106#14519687C2×C414yescomplemented
107C8×Dic61—22no—
108C8×Dic61—22no—
109#14519684—22no—
110#14519684—22no—
111#14519684D412yes—
112#14519684D412yes—
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61C2×C414yescomplemented
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—22no—
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—22no—
132#103×C61—22no—
133#102×C61—22no—
134C4×C8×C61D414yescomplemented
135C24×C61—21no—
136C23×D61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—42no—
154C22×C4×C61—82no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—28no—
159C2×C22⋊D61—82no—
160C2×C22⋊D61—82no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244C22×C412yes—
176C2×D244C22×C412yes—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61—82no—
180C2×C4×D61—82no—
181SD16×C61—28no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C22×C412yes—
185C2×C8×C61C2×D412yes—
186C2×C8×C61C2×D412yes—
187C2×C8×C61#4112yes—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—42no—
193C22⋊C4×C61—42no—
194C42×C61—42no—
195C42×C61C2×D412yes—
196C42×C61C2×D412yes—
197C22⋊Dic61—82no—
198#11758201—28no—
199#11758199—28no—
200#11758198—28no—
201#11758198—42no—
202#11758198—42no—
203#11758198—82no—
204#11758198—82no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209#11758196—82no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61C22×C412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C22×C412yes—
217C2×#92418C2×D412yes—
218C2×#92418C2×D412yes—
219C2×#92418#4112yes—
220D488—28no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—42no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—41no—
240Q8×C61—41no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—41no—
244D4×C61—41no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—84no—
255C2×C4×C61#12712yes—
256C2×C4×C61#12712yes—
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C22⋊D61—88no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61C4×D414yes—
274D244—22no—
275D244—22no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282C4×D61—84no—
283C4×D61—84no—
284C8×C61—22no—
285C8×C61C4×D414yes—
286#92418—22no—
287#92418C4×D414yes—
288#21704—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—44no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—82no—
294C22×C61C2×C4×D411yescharacteristic
295C22×C61C2×#12711yescharacteristic
296C22×C61#53211yescharacteristic
297C22×C61#57612yes—
298C22×C61#57712yes—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—44no—
302C2×D61—44no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—82no—
308C2×D61—82no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C2×C4×D411yescharacteristic
316C4×C61#53211yescharacteristic
317C4×C61#53211yescharacteristic
318Dic61—24no—
319Dic61—42no—
320Dic61—81no—
321Dic61—81no—
322C2×#496—611no—
323C2×#442—611no—
324C2×#440—611no—
325C2×C4×SD16D6111yescharacteristic, complemented
326C2×C4×D8—611no—
327#2328—611no—
328#2326—611no—
329#1973—611no—
330#1972—611no—
331#1970—611no—
332#1968—611no—
333#1958—611no—
334#1957—611no—
335#1847—611no—
336#1726—611no—
337C2×C61—82no—
338C2×C61C4×#12711yescharacteristic
339C2×C61#294612yes—
340C2×C61#315411yescharacteristic
341C2×C61#315511yescharacteristic
342C2×C61#317912yes—
343D61—82no—
344D61—82no—
345C22×SD16—611no—
346C22×SD16C2×D6111yescharacteristic
347C22×D8—611no—
348C22×D8—611no—
349C2×#127—611no—
350C2×#124—611no—
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4C2×D6111yescharacteristic
355#496—618no—
356#442—1224no—
357#440—1224no—
358C4×SD16—24no—
359C4×D8—1224no—
360C2×#107—611no—
361C2×#107—611no—
362C2×#106—611no—
363C2×#106C2×D6111yescharacteristic
364C2×#103—611no—
365C2×#103C2×D6111yescharacteristic
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102—611no—
371C2×#102C2×D6111yescharacteristic
372C2×#97—611no—
373C2×#97—611no—
374C2×#97—611no—
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8C2×D6111yescharacteristic
378#391—611no—
379#390—611no—
380#390—611no—
381#390—611no—
382#388—1222no—
383#386—1222no—
384#386—1222no—
385#385—1222no—
386#384—611no—
387#382—611no—
388#376—611no—
389C61#2170411yescharacteristic, complemented, Sylow
390C22×Q8C22×D6111yescharacteristic
391C22×D4—611no—
392C22×D4—611no—
393C22×D4—1222no—
394C22×D4C22×D6111yescharacteristic
395C23×C4—21no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×SD16—22no—
399C2×SD16—22no—
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C2×SD16C4×D6114yescomplemented
404C2×D8—614no—
405C2×D8—614no—
406C2×D8—1222no—
407C2×D8—1222no—
408C2×D8—1222no—
409C2×D8—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#127—614no—
415#127—1222no—
416#124—614no—
417#124—1222no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1222no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C42—21no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42C22×D6111yescharacteristic
443#107—1222no—
444#107—1222no—
445#106—22no—
446#106—1222no—
447#103—22no—
448#103—1222no—
449#102—22no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#102—1222no—
455#97—614no—
456#97—614no—
457#97—1222no—
458C4×C8—1222no—
459C4×C8—1222no—
460C4×C8C22⋊D6114yescomplemented
461#95—612no—
462#95—612no—
463#95—1222no—
464#95—1222no—
465C24—21no—
466C24—1221no—
467C24—1221no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C2×C4×D6112yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—612no—
480C2×D4—612no—
481C2×D4—1228no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—4882no—
493C2×D4—4882no—
494C2×D4C2×C4×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—82no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—4882no—
505C22×C4—4882no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—28no—
512SD16—1228no—
513D8—1228no—
514D8—1228no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—612no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×C4×D6112yes—
524C2×C8#1175822212yes—
525C2×C8C2×C22⋊D6112yes—
526C2×C8C2×C22⋊D6112yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—4882no—
537C22⋊C4—42no—
538C22⋊C4—1228no—
539C22⋊C4—4882no—
540C22⋊C4—4882no—
541C22⋊C4—4882no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—42no—
545C42—612no—
546C42—612no—
547C42—1221no—
548C42—1221no—
549C42D4×D6112yes—
550C42C2×C22⋊D6112yes—
551C23—22no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—1222no—
557C23—1222no—
558C23—2441no—
559C23—2441no—
560C23—2442no—
561C23—2442no—
562C23—2441no—
563C23—2441no—
564C23—2442no—
565C23—2442no—
566C23C2×D4×D6111yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—1222no—
576D4—1222no—
577D4—1222no—
578D4—1222no—
579D4—4888no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4—614no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—4884no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4#1451977312yes—
607C2×C4#1451977312yes—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C22×C22⋊D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4C2×#1175822211yescharacteristic
614C8—22no—
615C8—614no—
616C8—1222no—
617C8C4×C22⋊D6114yes—
618C22—44no—
619C22—42no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—4882no—
624C22—4882no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—2444no—
630C22—2444no—
631C22—2442no—
632C22—2442no—
633C22C4×D4×D6111yescharacteristic
634C22C2×#1451977311yescharacteristic
635C22#1684030911yescharacteristic
636C22#1684056412yes—
637C22#1684056812yes—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—1224no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4C4×D4×D6111yescharacteristic
648C4C2×C4×C22⋊D6111yescharacteristic
649C4#1684030811yescharacteristic
650C4#1684030911yescharacteristic
651C2—82no—
652C2—4882no—
653C2—4882no—
654C2C4×#1451977311yescharacteristic
655C2#2145464612yes—
656C2#2145628911yescharacteristic
657C2#2145630611yescharacteristic
658C2#2145649812yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840148C2
cosocleC22×C4×C61C24
hypercenterC2×C4×SD16D61
hypocenterC61#21704
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#16839872C2semidirect
3. C2×#16839860C2semidirect
4. C2×C4×D488C2semidirect
5. #21448541C2semidirect
6. #21448526C2semidirect
7. #21446702C2semidirect
8. #21446693C2semidirect
9. #21446682C2semidirect
10. #21446670C2semidirect
11. #21446627C2semidirect
12. #21446623C2semidirect
13. #21445984C2semidirect
14. #21445505C2semidirect
15. C2×C4×SD16×C61C2semidirect
23. #16840148C4semidirect
67. C2×#11758201C2×C4semidirect
70. C2×D488C2×C4semidirect
73. C2×#11758199C2×C4semidirect
77. #14519773C2×C4semidirect
79. #14519766C2×C4semidirect
104. #14519688D4semidirect
106. #14519687C2×C4semidirect
118. C2×SD16×C61C2×C4semidirect
134. C4×C8×C61D4semidirect
325. C2×C4×SD16D61semidirect
389. C61#21704semidirect
403. C2×SD16C4×D61semidirect
460. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401482
C2×#168398722
C2×#168398602
C2×C4×D4882
#214485412
#214485262
#214467022
#214466932
#214466822
#214466702
#214466272
#214466232
#214459842
#214455052
C2×C4×SD16×C612
#2170461