Finite Group Databasebeta

← #29683031

Subgroups — #29683031

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839864C211yesmaximal, characteristic
2#21448296C211yesmaximal, characteristic, complemented
3C42×D244C211yesmaximal, characteristic
4#21446461C211yesmaximal, characteristic, complemented
5#21446454C211yesmaximal, characteristic, complemented
6#21446412C211yesmaximal, characteristic, complemented
7#21446391C211yesmaximal, characteristic, complemented
8#21446152C211yesmaximal, characteristic
9#21446152C211yesmaximal, characteristic
10#21446083C211yesmaximal, characteristic
11#21446069C211yesmaximal, characteristic
12#21445716C211yesmaximal, characteristic, complemented
13#21445650C211yesmaximal, characteristic, complemented
14#21445645C211yesmaximal, characteristic
15C61×#1745C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C42×D61—21no—
22#16839864C418yescomplemented
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839646—21no—
37#16839646—21no—
38C4×#11758196—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198—21no—
44C4×#11758198—21no—
45C4×#11758198—21no—
46#16839608—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C43×C61C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190C2×C414yes—
68C4×Dic122—22no—
69C4×Dic122C2×C414yescomplemented
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244C2×C414yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C42×D61—44no—
94#14519711—22no—
95#14519711C2×C414yescomplemented
96#14519709—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702C2×C414yescomplemented
100#14519699—22no—
101#14519699—22no—
102#14519699C2×C414yes—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#97×C61—22no—
125#97×C61C2×C414yescomplemented
126C23×D61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
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×D244C22×C412yes—
145C2×D244#4112yes—
146C2×D244#4112yes—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22×C412yes—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C22×C412yes—
173C2×C8×C61C22×C412yes—
174C42×C61—22no—
175C42×C61—22no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61—41no—
182C42×C61C22×C412yes—
183C42×C61#4112yes—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61#4112yes—
198C4⋊Dic61#4112yes—
199#11758196—24no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—24no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207#11758190—28no—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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×C61C4×D412yes—
231C2×C4×C61C4×D412yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61#13712yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
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—
261D244—22no—
262D244—22no—
263D244—22no—
264D244#11714yes—
265C4×D61—48no—
266C4×D61—48no—
267C4×D61—48no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272C8×C61—22no—
273C8×C61—22no—
274C8×C61—41no—
275C8×C61—41no—
276#21121—611nomaximal, Sylow
277C22×C61C2×C4×D411yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61#52511yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C61C2×#13711yescharacteristic
282C22×C61#57511yescharacteristic
283C22×C61#57511yescharacteristic
284C2×D61—24no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61C2×#11711yescharacteristic
298C4×C61C4×#4111yescharacteristic
299C4×C61#52911yescharacteristic
300C4×C61#52911yescharacteristic
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304Dic61—42no—
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308C2×#440—611no—
309#2295—611no—
310C42×D4—611no—
311#1928—611no—
312#1927—611no—
313#1921—611no—
314#1918—611no—
315#1877—611no—
316#1877—611no—
317#1864—611no—
318#1862—611no—
319#1776—611no—
320#1763—611no—
321#1762—611no—
322#1745D6111yescharacteristic, complemented
323C2×C61#229011yescharacteristic
324C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
325C2×C61C4×#13711yescharacteristic
326C2×C61#311011yescharacteristic
327C2×C61#311111yescharacteristic
328C2×C61#311411yescharacteristic
329C2×C61#311611yescharacteristic
330D61—44no—
331C22×SD16—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×C4×D4—611no—
335C2×C4×D4—611no—
336C22×C42—1221no—
337#440—618no—
338C2×#107—611no—
339C2×#107—611no—
340C2×#106—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×#97C2×D6111yescharacteristic
348C2×#97C2×D6111yescharacteristic
349#391—611no—
350#391—611no—
351#389—611no—
352#389—611no—
353#387—1221no—
354#387—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C22⋊C4—1221no—
361C4×C22⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#376—1221no—
364C43C2×D6111yescharacteristic
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#336C2×D6111yescharacteristic
372#336C2×D6111yescharacteristic
373#336C2×D6111yescharacteristic
374#336C2×D6111yescharacteristic
375C61#2112111yescharacteristic, complemented, Sylow
376C22×Q8—611no—
377C22×D4—611no—
378C23×C4—1221no—
379C23×C4—1221no—
380C23×C4—1221no—
381C2×SD16—614no—
382C2×SD16—1222no—
383C2×SD16—1222no—
384C22×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4×Q8—614no—
389C4×Q8—1222no—
390C4×D4—614no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—2444no—
414C2×C42D24411yescharacteristic
415C2×C42D24411yescharacteristic
416C2×C42D24411yescharacteristic
417C2×C42D24411yescharacteristic
418C2×C42C22×D6111yescharacteristic
419C2×C42C22×D6111yescharacteristic
420C2×C42C22×D6111yescharacteristic
421#107—614no—
422#107—1222no—
423#106—1222no—
424#103—614no—
425#103—1222no—
426#103—1222no—
427#102—614no—
428#102—1222no—
429#102—1222no—
430#97—22no—
431#97C4×D6114yescomplemented
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C24—1221no—
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—612no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1224no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4C2×D24411yescharacteristic
472C22×C4C2×D24411yescharacteristic
473C22×C4C2×D24411yescharacteristic
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×D24411yescharacteristic
476C22×C4C2×D24411yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—1228no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8C2×C4×D6112yes—
490C2×C8C2×C4×D6112yes—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—612no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C42—22no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—1224no—
518C42—2442no—
519C42—2442no—
520C42—2442no—
521C42—2442no—
522C42C2×C4×D6112yes—
523C42#1175822212yes—
524C23—1222no—
525C23—1222no—
526C23—1222no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23C22×D24411yescharacteristic
532Q8—1222no—
533Q8—1222no—
534Q8—2441no—
535Q8—2441no—
536D4—614no—
537D4—1222no—
538D4—1222no—
539D4—1222no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—2448no—
560C2×C4—2448no—
561C2×C4—2448no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
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×C4C4×D24412yes—
579C2×C4C4×D24412yes—
580C2×C4#1451980512yes—
581C2×C4#1451981012yes—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#41×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C2×C4C2×#1175822211yescharacteristic
588C8—22no—
589C8—22no—
590C8—41no—
591C8—41no—
592C22—1224no—
593C22—2442no—
594C22—2442no—
595C22—2442no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22C2×C4×D24411yescharacteristic
600C22#1684027411yescharacteristic
601C22#1684027411yescharacteristic
602C22C2×#1451980511yescharacteristic
603C22C2×#1451981011yescharacteristic
604C22#1684056211yescharacteristic
605C22#1684056211yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—4881no—
613C4—4881no—
614C4—2442no—
615C4—2442no—
616C4—2442no—
617C4—2442no—
618C4—2442no—
619C4#117×D6111yescharacteristic
620C4#1684023011yescharacteristic
621C4#1684029011yescharacteristic
622C4#1684029011yescharacteristic
623C2—2444no—
624C2#2144826311yescharacteristic
625C2C4×#1451980511yescharacteristic
626C2C4×#1451981011yescharacteristic
627C2#2145591911yescharacteristic
628C2#2145593011yescharacteristic
629C2#2145595911yescharacteristic
630C2#2145598211yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1745C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1745D61
hypocenterC61#21121
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #21448296C2semidirect
4. #21446461C2semidirect
5. #21446454C2semidirect
6. #21446412C2semidirect
7. #21446391C2semidirect
12. #21445716C2semidirect
13. #21445650C2semidirect
15. C61×#1745C2semidirect
22. #16839864C4semidirect
69. C4×Dic122C2×C4semidirect
95. #14519711C2×C4semidirect
99. #14519702C2×C4semidirect
125. #97×C61C2×C4semidirect
322. #1745D61semidirect
375. C61#21121semidirect
431. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214482962
C42×D2442
#214464612
#214464542
#214464122
#214463912
#214461522
#214460832
#214460692
#214457162
#214456502
#214456452
C61×#17452
#2112161