Finite Group Databasebeta

← #29699843

Subgroups — #29699843

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839875C211yesmaximal, characteristic
2#21448790C211yesmaximal, characteristic, complemented
3#21448766C211yesmaximal, characteristic, complemented
4#21446695C211yesmaximal, characteristic
5#21446685C211yesmaximal, characteristic
6#21446447C211yesmaximal, characteristic, complemented
7#21446447C211yesmaximal, characteristic, complemented
8#21446440C211yesmaximal, characteristic, complemented
9#21446440C211yesmaximal, characteristic, complemented
10#21445987C211yesmaximal, characteristic, complemented
11#21445830C211yesmaximal, characteristic, complemented
12#21445820C211yesmaximal, characteristic, complemented
13#21445716C211yesmaximal, characteristic, complemented
14C4×#14519709C211yesmaximal, characteristic, complemented
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519735—21no—
20#16839875C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839654—21no—
37#16839654—21no—
38#16839652—21no—
39#16839652—21no—
40#16839632C2211yescharacteristic
41#16839628C2211yescharacteristic
42#16839624—21no—
43#16839619—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C2×D488—22no—
65C2×D488—22no—
66C2×D488C2×C414yes—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244C2×C414yes—
70C4×D244C2×C414yes—
71#14519735—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94#14519711—22no—
95#14519711—41no—
96#14519709—22no—
97#14519709—41no—
98#14519709C2×C414yes—
99#14519699—41no—
100#14519699—41no—
101#14519699—41no—
102#14519699—41no—
103#14519699C2×C414yes—
104#14519699C2×C414yes—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121#97×C61—22no—
122#97×C61C2×C414yes—
123C4×C8×C61—22no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C23×D61—21no—
127C23×D61—21no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C22×C412yes—
168C2×C8×C61C22×C412yes—
169C2×C8×C61#4112yes—
170C2×C8×C61#4112yes—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—41no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C22×C412yes—
178C42×C61#4112yes—
179#11758198—24no—
180#11758198—24no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—81no—
186#11758198—81no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—81no—
201C4⋊Dic61C22×C412yes—
202C4⋊Dic61C22×C412yes—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C4×Dic61—81no—
207D488—28no—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
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—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
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×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265D244—22no—
266D244—22no—
267D244—22no—
268D244—22no—
269D244—41no—
270D244—41no—
271D244—41no—
272D244—41no—
273C4×D61—48no—
274C4×D61—84no—
275C8×C61—22no—
276C8×C61—41no—
277C8×C61—41no—
278C8×C61#11714yescomplemented
279#22817—611nomaximal, Sylow
280C22×C61C2×C4×D411yescharacteristic
281C22×C61#54711yescharacteristic
282C22×C61#55011yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57611yescharacteristic
286C22×C61#57711yescharacteristic
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61C2×#11711yescharacteristic
308C4×C61#52011yescharacteristic
309C4×C61#55511yescharacteristic
310C4×C61#55611yescharacteristic
311Dic61—42no—
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#442—611no—
318#2357—611no—
319#2354—611no—
320#1972—611no—
321#1970—611no—
322#1926—611no—
323#1926—611no—
324#1925—611no—
325#1925—611no—
326#1847—611no—
327#1804—611no—
328#1802—611no—
329#1776—611no—
330C4×#106—611no—
331#1771D6111yescharacteristic, complemented
332C2×C61#235411yescharacteristic
333C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
334C2×C61#294611yescharacteristic
335C2×C61#334511yescharacteristic
336C2×C61#334711yescharacteristic
337C2×C61#339911yescharacteristic
338C2×C61#340111yescharacteristic
339D61—82no—
340D61—82no—
341C22×D8—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×#117—1221no—
345#442—618no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×#102—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#97C2×D6111yescharacteristic
359C2×#97C2×D6111yescharacteristic
360C2×C4×C8C2×D6111yescharacteristic
361#390—611no—
362#390—611no—
363#390—611no—
364#390—611no—
365#388—1221no—
366#388—1221no—
367#388—1221no—
368#388—1221no—
369#384—611no—
370#382—611no—
371#381—1221no—
372#380—1221no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#336C2×D6111yescharacteristic
384#336C2×D6111yescharacteristic
385C61#2281711yescharacteristic, complemented, Sylow
386C22×D4—611no—
387C22×D4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C2×D8—614no—
391C2×D8—1222no—
392C2×D8—1222no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4×D4—614no—
398C4×D4—614no—
399C4×D4—1222no—
400C4×D4—1222no—
401#117—2444no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4D24411yescharacteristic
411C2×C4⋊C4D24411yescharacteristic
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—2441no—
431#106—614no—
432#106—1222no—
433#106—2441no—
434C4⋊C8—22no—
435#102—614no—
436#102—614no—
437#102—2441no—
438#102—2441no—
439#102—2441no—
440#102—2441no—
441#97—22no—
442#97C4×D6114yes—
443C4×C8—22no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95D24411yescharacteristic
451#95D24411yescharacteristic
452C24—1221no—
453C24—1221no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—4881no—
475C22×C4—4881no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4C2×D24411yescharacteristic
479C22×C4C2×D24411yescharacteristic
480C22×C4C2×D24411yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C23×D6111yescharacteristic
485D8—1228no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8C2×C4×D6112yes—
495C2×C8C2×C4×D6112yes—
496C2×C8#1175822412yes—
497C2×C8#1175822612yes—
498C4⋊C4—22no—
499C4⋊C4—41no—
500C4⋊C4—612no—
501C4⋊C4—612no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—4881no—
513C4⋊C4—2442no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—4881no—
517C22⋊C4—4881no—
518C22⋊C4—4881no—
519C22⋊C4—4881no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—4881no—
529C42—4881no—
530C42—2442no—
531C42—2442no—
532C42C2×C4×D6112yes—
533C42#1175822212yes—
534C23—1222no—
535C23—1222no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23C22×D24411yescharacteristic
549D4—1222no—
550D4—1222no—
551D4—1222no—
552D4—1222no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—4884no—
575C2×C4—2448no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4C4×D24412yes—
593C2×C4C4×D24412yes—
594C2×C4#1451980512yes—
595C2×C4C22×C4×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822611yescharacteristic
599C2×C4#1451982311yescharacteristic
600C2×C4#1451982411yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8#1451973514yescomplemented
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22C2×C4×D24411yescharacteristic
620C22#1684041611yescharacteristic
621C22#1684043411yescharacteristic
622C22C2×#1451980511yescharacteristic
623C22#1684056211yescharacteristic
624C22#1684056711yescharacteristic
625C22#1684057411yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—4881no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4—2442no—
638C4C2×#1451973511yescharacteristic
639C4#1684024211yescharacteristic
640C4#1684047711yescharacteristic
641C4#1684048511yescharacteristic
642C2—4882no—
643C2—4882no—
644C2#2144876611yescharacteristic
645C2C4×#1451980511yescharacteristic
646C2#2145465311yescharacteristic
647C2#2145829111yescharacteristic
648C2#2145832111yescharacteristic
649C2#2145886311yescharacteristic
650C2#2145888711yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#22817
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21448790C2semidirect
3. #21448766C2semidirect
6. #21446447C2semidirect
7. #21446447C2semidirect
8. #21446440C2semidirect
9. #21446440C2semidirect
10. #21445987C2semidirect
11. #21445830C2semidirect
12. #21445820C2semidirect
13. #21445716C2semidirect
14. C4×#14519709C2semidirect
15. C61×#1771C2semidirect
20. #16839875C4semidirect
278. C8×C61#117semidirect
331. #1771D61semidirect
385. C61#22817semidirect
604. C8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×#168398752
#214487902
#214487662
#214466952
#214466852
#214464472
#214464402
#214459872
#214458302
#214458202
#214457162
C4×#145197092
C61×#17712
#2281761