Finite Group Databasebeta

← #29683046

Subgroups — #29683046

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839875C211yesmaximal, characteristic
2#21448283C211yesmaximal, characteristic, complemented
3C42×D244C211yesmaximal, characteristic, complemented
4#21446447C211yesmaximal, characteristic, complemented
5#21446440C211yesmaximal, characteristic, complemented
6#21446405C211yesmaximal, characteristic, complemented
7#21446398C211yesmaximal, characteristic, complemented
8#21446166C211yesmaximal, characteristic
9#21446166C211yesmaximal, characteristic
10#21446083C211yesmaximal, characteristic, complemented
11#21446069C211yesmaximal, characteristic, complemented
12#21445716C211yesmaximal, characteristic, complemented
13#21445645C211yesmaximal, characteristic, complemented
14#21445645C211yesmaximal, characteristic, complemented
15C61×#1745C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519734—21no—
22C2×C42×D61—21no—
23#16839875C418yescomplemented
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839653—21no—
38#16839653—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198—21no—
44C4×#11758198—21no—
45C4×#11758198—21no—
46C4×#11758198—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×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C2×D488—22no—
67C2×D488—22no—
68C2×D488C2×C414yes—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244C2×C414yes—
76C4×D244C2×C414yescomplemented
77#14519734—44no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C42×D61—44no—
99#14519711—22no—
100#14519709—22no—
101#14519709C2×C414yescomplemented
102#14519699—22no—
103#14519699—22no—
104#14519699—22no—
105#14519699—22no—
106#14519699C2×C414yes—
107#14519699C2×C414yescomplemented
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125#97×C61—22no—
126#97×C61C2×C414yescomplemented
127C23×D61—21no—
128C23×D61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—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×D244C22×C412yes—
151C2×D244C22×C412yes—
152C2×D244#4112yes—
153C2×D244#4112yes—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
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×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C22×C412yes—
177C2×C8×C61C22×C412yes—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61—41no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C22×C412yes—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61#4112yes—
207C4⋊Dic61#4112yes—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214D488—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
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×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61AGL(1,ℤ/8)12yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272D244—22no—
273D244—22no—
274D244—22no—
275D244—22no—
276D244—22no—
277D244—41no—
278D244—41no—
279D244#11714yescomplemented
280C4×D61—48no—
281C4×D61—48no—
282C4×D61—48no—
283C4×D61—48no—
284C8×C61—22no—
285C8×C61—22no—
286C8×C61—41no—
287C8×C61—41no—
288#21122—611nomaximal, Sylow
289C22×C61C2×C4×D411yescharacteristic
290C22×C61#52511yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61C2×#11711yescharacteristic
317C4×C61C4×#4111yescharacteristic
318C4×C61#52911yescharacteristic
319C4×C61#52911yescharacteristic
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326C2×#442—611no—
327#2293—611no—
328C42×D4—611no—
329#1926—611no—
330#1925—611no—
331#1920—611no—
332#1919—611no—
333#1879—611no—
334#1879—611no—
335#1864—611no—
336#1862—611no—
337#1776—611no—
338#1762—611no—
339#1762—611no—
340#1745D6111yescharacteristic, complemented
341C2×C61#229011yescharacteristic
342C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C61#310911yescharacteristic
345C2×C61#311111yescharacteristic
346C2×C61#311411yescharacteristic
347C2×C61#311511yescharacteristic
348D61—44no—
349D61—82no—
350C22×D8—611no—
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×#117—1221no—
356C22×C42—1221no—
357#442—618no—
358C2×#107—611no—
359C2×#106—611no—
360C2×#106—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#97C2×D6111yescharacteristic
368C2×#97C2×D6111yescharacteristic
369#390—611no—
370#390—611no—
371#389—611no—
372#389—611no—
373#388—1221no—
374#388—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C22⋊C4—1221no—
380C4×C22⋊C4—1221no—
381C4×C22⋊C4—1221no—
382C4×C22⋊C4—1221no—
383C43C2×D6111yescharacteristic
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336C2×D6111yescharacteristic
391#336C2×D6111yescharacteristic
392#336C2×D6111yescharacteristic
393#336C2×D6111yescharacteristic
394C61#2112211yescharacteristic, complemented, Sylow
395C22×D4—611no—
396C22×D4—611no—
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×D8—614no—
402C2×D8—1222no—
403C2×D8—1222no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4×D4—614no—
409C4×D4—614no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416#117—2444no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—2444no—
438C2×C42D24411yescharacteristic
439C2×C42D24411yescharacteristic
440C2×C42D24411yescharacteristic
441C2×C42D24411yescharacteristic
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#106—614no—
447#106—1222no—
448#102—614no—
449#102—614no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#97—22no—
455#97C4×D6114yescomplemented
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—1221no—
463C24—1221no—
464C2×D4—612no—
465C2×D4—612no—
466C2×D4—612no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×D24411yescharacteristic
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C2×D24411yescharacteristic
500C22×C4C23×D6111yescharacteristic
501D8—1228no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8C2×C4×D6112yes—
513C2×C8C2×C4×D6112yes—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C22⋊C4—1224no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—2442no—
545C42—2442no—
546C42—2442no—
547C42—2442no—
548C42C2×C4×D6112yes—
549C42#1175822212yes—
550C23—1222no—
551C23—1222no—
552C23—1222no—
553C23—1222no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C22×D24411yescharacteristic
565D4—614no—
566D4—1222no—
567D4—1222no—
568D4—1222no—
569D4—1222no—
570D4—1222no—
571D4—2441no—
572D4—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—2448no—
592C2×C4—2448no—
593C2×C4—2448no—
594C2×C4—2448no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4C4×D24412yes—
610C2×C4C4×D24412yes—
611C2×C4#1451980512yes—
612C2×C4#1451980512yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4C2×#1175822211yescharacteristic
619C8—22no—
620C8—22no—
621C8—41no—
622C8—41no—
623C22—1224no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C2×C4×D24411yescharacteristic
638C22#1684027411yescharacteristic
639C22#1684027411yescharacteristic
640C22C2×#1451980511yescharacteristic
641C22C2×#1451980511yescharacteristic
642C22#1684056211yescharacteristic
643C22#1684056211yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—4881no—
651C4—4881no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4—2442no—
656C4#117×D6111yescharacteristic
657C4#1684023011yescharacteristic
658C4#1684029011yescharacteristic
659C4#1684029011yescharacteristic
660C2—4882no—
661C2—2444no—
662C2#2144826311yescharacteristic
663C2C4×#1451980511yescharacteristic
664C2C4×#1451980511yescharacteristic
665C2#2145591111yescharacteristic
666C2#2145593011yescharacteristic
667C2#2145595911yescharacteristic
668C2#2145597111yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1745C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1745D61
hypocenterC61#21122
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. #21448283C2semidirect
3. C42×D244C2semidirect
4. #21446447C2semidirect
5. #21446440C2semidirect
6. #21446405C2semidirect
7. #21446398C2semidirect
10. #21446083C2semidirect
11. #21446069C2semidirect
12. #21445716C2semidirect
13. #21445645C2semidirect
14. #21445645C2semidirect
15. C61×#1745C2semidirect
23. #16839875C4semidirect
76. C4×D244C2×C4semidirect
101. #14519709C2×C4semidirect
107. #14519699C2×C4semidirect
126. #97×C61C2×C4semidirect
279. D244#117semidirect
340. #1745D61semidirect
394. C61#21122semidirect
455. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398752
#214482832
C42×D2442
#214464472
#214464402
#214464052
#214463982
#214461662
#214460832
#214460692
#214457162
#214456452
C61×#17452
#2112261