Finite Group Databasebeta

← #29699173

Subgroups — #29699173

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839847C211yesmaximal, characteristic
2#21448764C211yesmaximal, characteristic
3Q8×#11758196C211yesmaximal, characteristic, complemented
4#21446516C211yesmaximal, characteristic
5#21446514C211yesmaximal, characteristic
6#21446397C211yesmaximal, characteristic, complemented
7#21446395C211yesmaximal, characteristic, complemented
8#21446174C211yesmaximal, characteristic
9#21446172C211yesmaximal, characteristic
10#21445945C211yesmaximal, characteristic, complemented
11#21445822C211yesmaximal, characteristic, complemented
12#21445818C211yesmaximal, characteristic
13#21445687C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519734—21no—
21#16839847C418yescomplemented
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839660—21no—
33#16839660C2211yescharacteristic
34#16839658—21no—
35#16839658C2211yescharacteristic
36#16839653—21no—
37#16839651—21no—
38#16839631—21no—
39#16839631C2211yescharacteristic
40#16839631C2211yescharacteristic
41#16839621—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70Q8×Dic61C2×C414yes—
71#14519739—22no—
72#14519739C2×C414yes—
73#14519734—44no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196D411yescharacteristic
86C2×#11758196D411yescharacteristic
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519708C2×C414yes—
98#14519706—22no—
99#14519701—22no—
100#14519699—41no—
101#14519699C2×C414yes—
102#14519697—41no—
103#14519687—41no—
104C8×Dic61C2×C414yes—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×Q8×C61—22no—
114C4×Q8×C61—41no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#103×C61—22no—
123#103×C61C2×C414yes—
124#97×C61—41no—
125#95×C61—21no—
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C22×C412yes—
131C2×Q8×C61#4112yes—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61C22×C412yes—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—42no—
172C42×C61—42no—
173#11758201—28no—
174#11758198—24no—
175#11758198—42no—
176#11758198—42no—
177#11758198—81no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188#11758196—22no—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192#11758196—81no—
193#11758196—81no—
194#11758196—81no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—24no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61C22×C412yes—
201C4×Dic61#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61#11714yescomplemented
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61C4×D412yes—
262C2×Dic61C4×D412yes—
263C2×Dic61AGL(1,ℤ/8)12yes—
264C2×Dic61#13712yes—
265D244—22no—
266D244—22no—
267D244—41no—
268D244—41no—
269C4×D61—48no—
270C8×C61—22no—
271C8×C61—41no—
272#92418—22no—
273#92418—41no—
274#22759—611nomaximal, Sylow
275C22×C61C2×C4×D411yescharacteristic
276C22×C61#54611yescharacteristic
277C22×C61#54811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57711yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—24no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61C2×#11711yescharacteristic
296C4×C61#52011yescharacteristic
297C4×C61#55211yescharacteristic
298C4×C61#55311yescharacteristic
299Dic61—22no—
300Dic61—41no—
301Dic61—41no—
302Dic61—42no—
303Dic61—81no—
304Dic61—81no—
305Dic61#44014yes—
306C2×C4×SD16—611no—
307#2354—611no—
308Q8×C4⋊C4—611no—
309#1936—611no—
310#1936—611no—
311#1919—611no—
312#1919—611no—
313#1880—611no—
314#1880—611no—
315#1834—611no—
316#1803—611no—
317#1802—611no—
318#1770—611no—
319#1763—611no—
320#1763D6111yescharacteristic, complemented
321C2×C61#235211yescharacteristic
322C2×C61C2×#44011yescharacteristic
323C2×C61#295211yescharacteristic
324C2×C61#332911yescharacteristic
325C2×C61#333211yescharacteristic
326C2×C61#336811yescharacteristic
327C2×C61#336911yescharacteristic
328D61—82no—
329C22×SD16—611no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4—611no—
334C2×#117—1221no—
335C4×SD16—618no—
336C2×#106—611no—
337C2×C4⋊C8—611no—
338C2×#103—611no—
339C2×#103C2×D6111yescharacteristic
340C2×#103C2×D6111yescharacteristic
341C2×#102—611no—
342C2×#102—611no—
343C2×#102—611no—
344C2×#97—611no—
345C2×#97C2×D6111yescharacteristic
346C2×C4×C8—611no—
347#390—611no—
348#390—611no—
349#389—611no—
350#389—611no—
351#389—1221no—
352#389—1221no—
353#388—1221no—
354#388—1221no—
355#384—611no—
356#384—611no—
357#384—1221no—
358#380—1221no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C4⋊C4C2×D6111yescharacteristic
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340C2×D6111yescharacteristic
370#340C2×D6111yescharacteristic
371#336—611no—
372#336—611no—
373C61#2275911yescharacteristic, complemented, Sylow
374C22×Q8C22×D6111yescharacteristic
375C22×D4—611no—
376C23×C4—1221no—
377C2×SD16—614no—
378C2×SD16—1222no—
379C2×SD16—1222no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C4×Q8—22no—
385C4×Q8—41no—
386C4×Q8—614no—
387C4×Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—2441no—
390C4×D4—614no—
391C4×D4—1222no—
392#117—2444no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#106—614no—
424C4⋊C8—1222no—
425#103—22no—
426#103—1222no—
427#103C4×D6114yes—
428#102—614no—
429#102—2441no—
430#102—2441no—
431#97—41no—
432#97—2441no—
433C4×C8—614no—
434#95—21no—
435#95—611no—
436#95—611no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C24—1221no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D6112yes—
445C2×Q8#1175822212yes—
446C2×Q8#1175822212yes—
447C2×D4—612no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—1224no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470SD16—1228no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—612no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8C2×C4×D6112yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—612no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—4881no—
499C4⋊C4—4881no—
500C4⋊C4—4881no—
501C4⋊C4—4881no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—1224no—
507C22⋊C4—4881no—
508C22⋊C4—4881no—
509C22⋊C4—2442no—
510C22⋊C4—2442no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—612no—
516C42—612no—
517C42—1224no—
518C42—1221no—
519C42—1221no—
520C42—2442no—
521C42—2442no—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23C2×D4×D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1451973414yescomplemented
534D4—1222no—
535D4—1222no—
536D4—2441no—
537D4—2441no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—2448no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4C2×#1175822211yescharacteristic
582C2×C4#1451982211yescharacteristic
583C2×C4#1451982411yescharacteristic
584C8—22no—
585C8—41no—
586C8—1222no—
587C8—2441no—
588C22—4881no—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—2442no—
593C22—2442no—
594C22—2442no—
595C22C4×D4×D6111yescharacteristic
596C22#1684040211yescharacteristic
597C22#1684042111yescharacteristic
598C22AGL(1,ℤ/8)×D6111yescharacteristic
599C22#137×D6111yescharacteristic
600C22#1684057111yescharacteristic
601C22#1684057111yescharacteristic
602C4—24no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—614no—
609C4—1222no—
610C4—4881no—
611C4—4881no—
612C4—2442no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451973411yescharacteristic
616C4#1684024111yescharacteristic
617C4#1684044511yescharacteristic
618C4#1684046111yescharacteristic
619C2—4882no—
620C2#2144874211yescharacteristic
621C2#440×D6111yescharacteristic
622C2#2145469911yescharacteristic
623C2#2145810911yescharacteristic
624C2#2145814211yescharacteristic
625C2#2145854911yescharacteristic
626C2#2145856011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#22759
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
3. Q8×#11758196C2semidirect
6. #21446397C2semidirect
7. #21446395C2semidirect
10. #21445945C2semidirect
11. #21445822C2semidirect
13. #21445687C2semidirect
14. #21445651C2semidirect
15. C61×#1763C2semidirect
21. #16839847C4semidirect
219. Q8×C61#117semidirect
320. #1763D61semidirect
373. C61#22759semidirect
533. Q8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
#214487642
Q8×#117581962
#214465162
#214465142
#214463972
#214463952
#214461742
#214461722
#214459452
#214458222
#214458182
#214456872
#214456512
C61×#17632
#2275961