Finite Group Databasebeta

← #29699437

Subgroups — #29699437

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839873C211yesmaximal, characteristic
2#21448764C211yesmaximal, characteristic, complemented
3#21448728C211yesmaximal, characteristic, complemented
4#21446515C211yesmaximal, characteristic, complemented
5#21446503C211yesmaximal, characteristic, complemented
6#21446406C211yesmaximal, characteristic, complemented
7#21446168C211yesmaximal, characteristic
8#21446166C211yesmaximal, characteristic
9#21445951C211yesmaximal, characteristic, complemented
10#21445828C211yesmaximal, characteristic, complemented
11#21445818C211yesmaximal, characteristic, complemented
12#21445692C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1919C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519734—21no—
22#16839873C418yescomplemented
23C2×#14519706C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839653—21no—
37#16839651—21no—
38#16839636—21no—
39#16839633—21no—
40#16839631C2211yescharacteristic
41#16839627C2211yescharacteristic
42#16839621—21no—
43#16839619—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
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—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×D244—22no—
72C4×D244—41no—
73#14519739—22no—
74#14519739—22no—
75#14519739—22no—
76#14519739—41no—
77#14519739C2×C414yes—
78#14519739C2×C414yescomplemented
79#14519734—44no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
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×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519706—22no—
99#14519699—22no—
100#14519699—41no—
101#14519697—22no—
102#14519697—41no—
103#14519697C2×C414yes—
104#14519697C2×C414yescomplemented
105#14519688—41no—
106#14519687C2×C414yescomplemented
107C8×Dic61—41no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#106×C61C2×C414yescomplemented
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
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×D244C22×C412yes—
150C2×D244C22×C412yes—
151C2×D244#4112yes—
152C2×D244#4112yes—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—81no—
178C4⋊C4×C61C22×C412yes—
179C4⋊C4×C61C22×C412yes—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—42no—
183#11758198—24no—
184#11758198—24no—
185#11758198—42no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—81no—
190#11758198—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—81no—
196#11758196—22no—
197#11758196—41no—
198#11758196—42no—
199#11758196—81no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—42no—
203C4×Dic61C22×C412yes—
204C4×Dic61#4112yes—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C22×C412yes—
211D488—28no—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
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—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61C4×D412yes—
267C2×Dic61C4×D412yes—
268C2×Dic61AGL(1,ℤ/8)12yes—
269D244—22no—
270D244—22no—
271D244—22no—
272D244—22no—
273D244—22no—
274D244—41no—
275D244—41no—
276D244#11714yescomplemented
277C4×D61—48no—
278C4×D61—84no—
279C4×D61—84no—
280C4×D61—84no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#22780—611nomaximal, Sylow
286C22×C61C2×C4×D411yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61#54811yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57711yescharacteristic
293C2×D61—24no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61C2×#11711yescharacteristic
314C4×C61#52011yescharacteristic
315C4×C61#55211yescharacteristic
316C4×C61#55311yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C2×#442—611no—
324#2354—611no—
325#2349—611no—
326#1936—611no—
327#1934—611no—
328#1920—611no—
329#1919D6111yescharacteristic, complemented
330#1879—611no—
331#1879—611no—
332#1836—611no—
333#1804—611no—
334#1802—611no—
335#1771—611no—
336#1762—611no—
337C4×#102—611no—
338C2×C61#235211yescharacteristic
339C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
340C2×C61#294611yescharacteristic
341C2×C61#330611yescharacteristic
342C2×C61#331911yescharacteristic
343C2×C61#336611yescharacteristic
344C2×C61#337611yescharacteristic
345D61—44no—
346D61—82no—
347C22×D8—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—611no—
351C2×C4×D4—611no—
352C2×#117—1221no—
353#442—618no—
354C2×#106C2×D6111yescharacteristic
355C2×C4⋊C8—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#97—611no—
363C2×#97—611no—
364C2×C4×C8—611no—
365#390—611no—
366#390—611no—
367#389—611no—
368#389C2×D6111yescharacteristic
369#388—1221no—
370#388—1221no—
371#385—1221no—
372#385—1221no—
373#384—611no—
374#382—611no—
375#380—1221no—
376#380—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4C2×D6111yescharacteristic
380C2×#95—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389#336—611no—
390#336C2×D6111yescharacteristic
391C61#2278011yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4—611no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×D8—614no—
399C2×D8—1222no—
400C2×D8—1222no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4×D4—614no—
406C4×D4—614no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—2441no—
412C4×D4—2441no—
413#117—2444no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
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×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#106C4×D6114yescomplemented
440C4⋊C8—1222no—
441#102—614no—
442#102—614no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446#102—2441no—
447#97—614no—
448#97—2441no—
449C4×C8—2441no—
450#95—21no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—1221no—
459C24—1221no—
460C2×D4—612no—
461C2×D4—612no—
462C2×D4—612no—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—4881no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497D8—1228no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—612no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8C2×C4×D6112yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—81no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—4881no—
524C4⋊C4—2441no—
525C4⋊C4—2442no—
526C4⋊C4C2×C4×D6112yes—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—4881no—
531C22⋊C4—4881no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—612no—
540C42—612no—
541C42—1221no—
542C42—1221no—
543C42—2442no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—1222no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23C2×D4×D6111yescharacteristic
559D4—614no—
560D4—1222no—
561D4—1222no—
562D4—1222no—
563D4—1222no—
564D4—1222no—
565D4—2441no—
566D4—2441no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—4884no—
598C2×C4—4884no—
599C2×C4—4884no—
600C2×C4—2448no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4#1451982311yescharacteristic
612C2×C4#1451982511yescharacteristic
613C8—22no—
614C8—41no—
615C8—1222no—
616C8—2441no—
617C22—1224no—
618C22—4881no—
619C22—4881no—
620C22—4881no—
621C22—4881no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C4×D4×D6111yescharacteristic
632C22#1684040411yescharacteristic
633C22#1684042211yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22#1684056111yescharacteristic
636C22#1684056411yescharacteristic
637C22#1684056811yescharacteristic
638C4—42no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—1222no—
645C4—1222no—
646C4—1222no—
647C4—4881no—
648C4—4881no—
649C4—2441no—
650C4#117×D6111yescharacteristic
651C4#1684024411yescharacteristic
652C4#1684044411yescharacteristic
653C4#1684046311yescharacteristic
654C2—4882no—
655C2—2444no—
656C2#2144874311yescharacteristic
657C2#2145462411yescharacteristic
658C2#2145464611yescharacteristic
659C2#2145778311yescharacteristic
660C2#2145797811yescharacteristic
661C2#2145852911yescharacteristic
662C2#2145865311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1919C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1919D61
hypocenterC61#22780
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
2. #21448764C2semidirect
3. #21448728C2semidirect
4. #21446515C2semidirect
5. #21446503C2semidirect
6. #21446406C2semidirect
9. #21445951C2semidirect
10. #21445828C2semidirect
11. #21445818C2semidirect
12. #21445692C2semidirect
13. #21445647C2semidirect
14. #21445633C2semidirect
15. C61×#1919C2semidirect
22. #16839873C4semidirect
78. #14519739C2×C4semidirect
104. #14519697C2×C4semidirect
106. #14519687C2×C4semidirect
123. #106×C61C2×C4semidirect
276. D244#117semidirect
329. #1919D61semidirect
391. C61#22780semidirect
439. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398732
#214487642
#214487282
#214465152
#214465032
#214464062
#214461682
#214461662
#214459512
#214458282
#214458182
#214456922
#214456472
#214456332
C61×#19192
#2278061