Finite Group Databasebeta

← #29699345

Subgroups — #29699345

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×#16839872C211yesmaximal, characteristic
2C4⋊C4×D244C211yesmaximal, characteristic, complemented
3#21446519C211yesmaximal, characteristic, complemented
4#21446505C211yesmaximal, characteristic, complemented
5#21446404C211yesmaximal, characteristic, complemented
6#21446397C211yesmaximal, characteristic, complemented
7#21446170C211yesmaximal, characteristic
8#21446170C211yesmaximal, characteristic
9#21445952C211yesmaximal, characteristic, complemented
10#21445826C211yesmaximal, characteristic, complemented
11#21445819C211yesmaximal, characteristic, complemented
12#21445694C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic, complemented
15C61×#2357C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16839872C418yescomplemented
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839636—21no—
36#16839636—21no—
37#16839632C2211yescharacteristic
38#16839619—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C61×#390C2211yescharacteristic
51C61×#390C2211yescharacteristic
52C61×#388—21no—
53C61×#388—21no—
54C61×#382C2211yescharacteristic
55C61×#381—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C22⋊C4×C61—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C2×#11758199—22no—
65C2×#11758199—22no—
66C2×#11758199C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—41no—
75C4×D244—41no—
76C4×D244C2×C414yes—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C4⋊C4×D61—44no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88#14519708C2×C414yescomplemented
89C4⋊#92418—22no—
90#14519698—41no—
91#14519698—41no—
92#14519698C2×C414yescomplemented
93#14519697—22no—
94#14519697—22no—
95#14519697C2×C414yes—
96#14519688—41no—
97#14519688C2×C414yescomplemented
98C4×#92418—41no—
99#14519684—21no—
100#14519684—21no—
101C22×D4×C61C2311yescharacteristic
102C23×C4×C61—21no—
103C4×D4×C61—22no—
104C4×D4×C61C2×C414yescomplemented
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61—21no—
127C23×D61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C22×C412yes—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—81no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244#4112yes—
155C2×D244#4112yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C22×C412yes—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—81no—
177C22⋊C4×C61—81no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—81no—
183C42×C61—81no—
184C42×C61C22×C412yes—
185C42×C61#4112yes—
186#11758199—28no—
187#11758198—24no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197C4⋊Dic61C22×C412yes—
198#11758196—42no—
199#11758196—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C22×C412yes—
211C2×#92418C22×C412yes—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
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—22no—
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×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—84no—
256C2×C4×C61C4×D412yes—
257C2×C4×C61C4×D412yes—
258C2×C4×C61AGL(1,ℤ/8)12yes—
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274D244—22no—
275D244—22no—
276D244—22no—
277D244#11714yes—
278C4×D61—48no—
279C4×D61—84no—
280C4×D61—84no—
281#92418—22no—
282#92418—22no—
283#92418—41no—
284#92418—41no—
285#22774—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57711yescharacteristic
300C2×D61—24no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×#11711yescharacteristic
317C4×C61#52011yescharacteristic
318C4×C61#55211yescharacteristic
319C4×C61#55311yescharacteristic
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#442—611no—
324#2357D6111yescharacteristic, complemented
325D4×C4⋊C4—611no—
326#1936—611no—
327#1934—611no—
328#1920—611no—
329#1919—611no—
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—82no—
339C2×C61#235211yescharacteristic
340C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C61#294611yescharacteristic
342C2×C61#330711yescharacteristic
343C2×C61#331811yescharacteristic
344C2×C61#337011yescharacteristic
345C2×C61#337311yescharacteristic
346D61—44no—
347C22×D8—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—611no—
351C2×C4×D4C2×D6111yescharacteristic
352C22×C4⋊C4—1221no—
353#442—618no—
354C2×#106—611no—
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#390C2×D6111yescharacteristic
366#390C2×D6111yescharacteristic
367#389—611no—
368#389—611no—
369#388—21no—
370#388—21no—
371#385—1221no—
372#385—1221no—
373#384—611no—
374#382C2×D6111yescharacteristic
375#381—21no—
376#380—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C22⋊C4—21no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#336—611no—
390#336—611no—
391C61#2277411yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
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×C8—611no—
404C22×C8—611no—
405C4×D4—22no—
406C4×D4—614no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—2441no—
411C4×D4—2441no—
412C4×D4C4×D6114yescomplemented
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—2444no—
421C2×C4⋊C4C22⋊D6111yescharacteristic
422C2×C4⋊C4C22⋊D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#106—614no—
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—21no—
453#95—21no—
454#95—1221no—
455#95—1221no—
456#95C22⋊D6111yescharacteristic
457#95C22⋊D6111yescharacteristic
458C24—21no—
459C24—1221no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—612no—
466C2×D4—612no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4C2×C4×D6112yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—1221no—
479C22×C4—1224no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—4881no—
483C22×C4—4881no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497D8—1228no—
498C2×C8—612no—
499C2×C8—612no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—41no—
517C4⋊C4—612no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C22⋊C4—1224no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—81no—
541C42—81no—
542C42C2×C4×D6112yes—
543C42#1175822212yes—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1222no—
552C23—1222no—
553C23—1222no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23C22×C22⋊D6111yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—614no—
564D4—1222no—
565D4—1222no—
566D4—1222no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
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—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—1222no—
593C2×C4—4884no—
594C2×C4—4884no—
595C2×C4—2448no—
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×C4C4×C22⋊D6112yes—
605C2×C4C4×C22⋊D6112yes—
606C2×C4#1451980212yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4#1451982311yescharacteristic
612C2×C4#1451982411yescharacteristic
613C8—1222no—
614C8—1222no—
615C8—2441no—
616C8—2441no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—1224no—
625C22—2442no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C2×C4×C22⋊D6111yescharacteristic
632C22#1684040011yescharacteristic
633C22#1684041911yescharacteristic
634C22C2×#1451980211yescharacteristic
635C22#1684056011yescharacteristic
636C22#1684056511yescharacteristic
637C22#1684057311yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—4881no—
648C4—4881no—
649C4—2442no—
650C4#117×D6111yescharacteristic
651C4#1684024211yescharacteristic
652C4#1684045111yescharacteristic
653C4#1684045611yescharacteristic
654C2—82no—
655C2—2444no—
656C2#2144874911yescharacteristic
657C2C4×#1451980211yescharacteristic
658C2#2145465111yescharacteristic
659C2#2145778911yescharacteristic
660C2#2145795411yescharacteristic
661C2#2145856611yescharacteristic
662C2#2145860711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2357C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#2357D61
hypocenterC61#22774
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×D244C2semidirect
3. #21446519C2semidirect
4. #21446505C2semidirect
5. #21446404C2semidirect
6. #21446397C2semidirect
9. #21445952C2semidirect
10. #21445826C2semidirect
11. #21445819C2semidirect
12. #21445694C2semidirect
13. #21445647C2semidirect
14. C4×#14519697C2semidirect
15. C61×#2357C2semidirect
21. #16839872C4semidirect
88. #14519708C2×C4semidirect
92. #14519698C2×C4semidirect
97. #14519688C2×C4semidirect
104. C4×D4×C61C2×C4semidirect
324. #2357D61semidirect
391. C61#22774semidirect
412. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398722
C4⋊C4×D2442
#214465192
#214465052
#214464042
#214463972
#214461702
#214459522
#214458262
#214458192
#214456942
#214456472
C4×#145196972
C61×#23572
#2277461