Finite Group Databasebeta

← #29687240

Subgroups — #29687240

Order 15616.

960 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×#16840086C211yesmaximal, characteristic
2C4×#14519751C211yesmaximal, characteristic
3#21446446C211yesmaximal, characteristic, complemented
4#21446439C211yesmaximal, characteristic, complemented
5#21446397C211yesmaximal, characteristic, complemented
6#21446390C211yesmaximal, characteristic, complemented
7#21446141C211yesmaximal, characteristic
8#21446018C211yesmaximal, characteristic, complemented
9#21445914C211yesmaximal, characteristic, complemented
10#21445717C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445647C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic, complemented
14#21445581C211yesmaximal, characteristic, complemented
15C4×#117×C61C211yesmaximal, characteristic, complemented
16C2×#14519751C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×C22⋊D61—21no—
19#16840086C418yescomplemented
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839653—21no—
35#16839641—21no—
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38C4×#11758198—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×#117×C61C2211yescharacteristic
50C2×#117×C61C2211yescharacteristic
51C22×C42×C61C2211yescharacteristic
52C4×C4⋊C4×C61C2211yescharacteristic
53C4×C4⋊C4×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C4×C22⋊C4×C61—21no—
57C4×C22⋊C4×C61—21no—
58C61×#376—21no—
59C43×C61—21no—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519751—22no—
68#14519751C2×C414yes—
69C4×D244—22no—
70C4×D244—22no—
71C4×C22⋊D61—44no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519710C2×C414yescomplemented
84#14519708—22no—
85#14519708C2×C414yescomplemented
86#14519697—22no—
87#14519697—22no—
88#14519697C2×C414yescomplemented
89#14519697C2×C414yescomplemented
90C22⋊#92418—22no—
91C22⋊#92418C2×C414yescomplemented
92#14519688—22no—
93#14519688—22no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61D411yescharacteristic
98C23×C4×C61D411yescharacteristic
99C23×C4×C61C2311yescharacteristic
100#117×C61—22no—
101#117×C61—22no—
102#117×C61—22no—
103#117×C61C2×C414yescomplemented
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61—21no—
112C2×C42×C61—22no—
113C2×C42×C61—22no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61C2×D411yescharacteristic
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C412yes—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C412yes—
170C4⋊C4×C61C22×C412yes—
171C4⋊C4×C61#4112yes—
172C4⋊C4×C61#4112yes—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C42×C61—24no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—42no—
182C42×C61—42no—
183C42×C61—42no—
184C42×C61—42no—
185C42×C61#4112yes—
186C42×C61#4112yes—
187C22⋊Dic61—42no—
188#11758198—24no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C22×C412yes—
198#11758196—42no—
199C4×Dic61—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—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61C4×D412yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
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—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—44no—
247C2×C4×C61—44no—
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—41no—
256C2×C4×C61C4×D412yes—
257C2×C4×C61#12312yes—
258C2×C4×C61#12312yes—
259C2×C4×C61AGL(1,ℤ/8)12yes—
260C2×C4×C61C23×C411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C22⋊D61—48no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—84no—
281#92418—22no—
282#92418—22no—
283#92418—41no—
284#92418—41no—
285#21582—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C2×#12311yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57511yescharacteristic
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61C4×#4111yescharacteristic
317C4×C61C4×#4111yescharacteristic
318C4×C61#52911yescharacteristic
319C4×C61#52911yescharacteristic
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#482—611no—
324C4×#121—611no—
325C4×#117D6111yescharacteristic, complemented
326#1926—611no—
327#1925—611no—
328#1919—611no—
329#1918—611no—
330#1875—611no—
331#1854—611no—
332#1825—611no—
333#1776—611no—
334#1776—611no—
335#1762—611no—
336#1762—611no—
337#1749—611no—
338C2×C61—42no—
339C2×C61C4×#12311yescharacteristic
340C2×C61#294311yescharacteristic
341C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
342C2×C61#309311yescharacteristic
343C2×C61#309411yescharacteristic
344C2×C61#311111yescharacteristic
345C2×C61#311511yescharacteristic
346D61—82no—
347C2×#121—611no—
348C2×C4×D4—611no—
349C2×C4×D4—1221no—
350C2×#117C2×D6111yescharacteristic
351C2×#117C2×D6111yescharacteristic
352C22×C42C2×D6111yescharacteristic
353#482—618no—
354C2×#107—611no—
355C2×#107—611no—
356C2×#106—611no—
357C2×#106—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8—611no—
364C2×#97—611no—
365C2×#97—611no—
366#390—611no—
367#389—611no—
368#388—1221no—
369#386—1221no—
370#380—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4C2×D6111yescharacteristic
373C4×C4⋊C4C2×D6111yescharacteristic
374C4×C4⋊C4C2×D6111yescharacteristic
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—21no—
377C4×C22⋊C4—21no—
378C4×C22⋊C4—1221no—
379#376—21no—
380C43—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#2158211yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4—1221no—
394C23×C4—1221no—
395C23×C4C22⋊D6111yescharacteristic
396C23×C4C22⋊D6111yescharacteristic
397C23×C4C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—611no—
402#121—614no—
403#121—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2444no—
407#117—22no—
408#117—22no—
409#117—22no—
410#117C4×D6114yescomplemented
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—22no—
429C2×C42—22no—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—1221no—
435C2×C42C22⋊D6111yescharacteristic
436C2×C42C22⋊D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#107—614no—
440#107—1222no—
441#106—614no—
442#106—1222no—
443#102—614no—
444#102—614no—
445#102—1222no—
446#102—1222no—
447C22⋊C8—614no—
448C22⋊C8—1222no—
449#97—1222no—
450#97—1222no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—1221no—
459C24C2×C22⋊D6111yescharacteristic
460C2×D4—612no—
461C2×D4—1224no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—2442no—
468C2×D4—2442no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4C2×C4×D6112yes—
490C22×C4#1175822212yes—
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497C2×C8—612no—
498C2×C8—612no—
499C2×C8—1221no—
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—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—612no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—2442no—
520C4⋊C4C2×C4×D6112yes—
521C4⋊C4C2×C4×D6112yes—
522C4⋊C4#1175822212yes—
523C4⋊C4#1175822212yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—1224no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—24no—
532C42—21no—
533C42—21no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—2442no—
542C42#1175822212yes—
543C42#1175822212yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—1222no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23C4×C22⋊D6112yes—
558C23C22×C22⋊D6111yescharacteristic
559D4—1222no—
560D4—1222no—
561D4—2448no—
562D4—2441no—
563D4—2441no—
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—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—1222no—
594C2×C4—4884no—
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×C4C4×C22⋊D6112yes—
604C2×C4#1451976012yes—
605C2×C4#1451976012yes—
606C2×C4#1451980212yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4C2×#1175822211yescharacteristic
612C2×C4C2×#1175822211yescharacteristic
613C8—1222no—
614C8—1222no—
615C8—2441no—
616C8—2441no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C2×C4×C22⋊D6111yescharacteristic
632C22C2×#1451976011yescharacteristic
633C22#1684027311yescharacteristic
634C22C2×#1451980211yescharacteristic
635C22#1684056011yescharacteristic
636C22#1684056011yescharacteristic
637C22#1684056011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—41no—
647C4—4881no—
648C4—4881no—
649C4—2442no—
650C4C4×#1175822211yescharacteristic
651C4C4×#1175822211yescharacteristic
652C4#1684029611yescharacteristic
653C4#1684029611yescharacteristic
654C2—42no—
655C2—4882no—
656C2C4×#1451976011yescharacteristic
657C2#2145461711yescharacteristic
658C2C4×#1451980211yescharacteristic
659C2#2145575011yescharacteristic
660C2#2145575911yescharacteristic
661C2#2145592811yescharacteristic
662C2#2145597011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#117×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenterC4×#117D61
hypocenterC61#21582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #21446446C2semidirect
4. #21446439C2semidirect
5. #21446397C2semidirect
6. #21446390C2semidirect
8. #21446018C2semidirect
9. #21445914C2semidirect
10. #21445717C2semidirect
11. #21445717C2semidirect
12. #21445647C2semidirect
13. #21445647C2semidirect
14. #21445581C2semidirect
15. C4×#117×C61C2semidirect
19. #16840086C4semidirect
83. #14519710C2×C4semidirect
85. #14519708C2×C4semidirect
88. #14519697C2×C4semidirect
89. #14519697C2×C4semidirect
91. C22⋊#92418C2×C4semidirect
103. #117×C61C2×C4semidirect
325. C4×#117D61semidirect
391. C61#21582semidirect
410. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400862
C4×#145197512
#214464462
#214464392
#214463972
#214463902
#214461412
#214460182
#214459142
#214457172
#214456472
#214455812
C4×#117×C612
#2158261