Finite Group Databasebeta

← #29697182

Subgroups — #29697182

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839755C211yesmaximal, characteristic
2#21448700C211yesmaximal, characteristic, complemented
3#21448658C211yesmaximal, characteristic
4#21446505C211yesmaximal, characteristic, complemented
5#21446499C211yesmaximal, characteristic, complemented
6#21446441C211yesmaximal, characteristic, complemented
7#21446076C211yesmaximal, characteristic, complemented
8#21446070C211yesmaximal, characteristic, complemented
9#21445912C211yesmaximal, characteristic, complemented
10#21445845C211yesmaximal, characteristic, complemented
11#21445756C211yesmaximal, characteristic
12C4×#14519700C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#21445589C211yesmaximal, characteristic, complemented
15C61×#1925C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22#16839755C418yescomplemented
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839662—21no—
34#16839662—21no—
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839653—21no—
38#16839651—21no—
39#16839626—21no—
40#16839620—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45C4×#11758198C2211yescharacteristic
46#16839608—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C2×#11758224—22no—
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—41no—
64C2×#11758224—41no—
65C2×#11758224C2×C414yes—
66C22×Dic122C2311yescharacteristic
67C22×C22⋊D61—21no—
68C23×Dic61—21no—
69C22×C4×D61C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C2×C8×D61—41no—
73C2×C8×D61—41no—
74C2×C8×D61C2×C414yes—
75C4×Dic122—22no—
76C4×Dic122—41no—
77#14519742—22no—
78#14519742—41no—
79#14519735—22no—
80#14519735C2×C414yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100#14519711—41no—
101#14519702—22no—
102#14519702C2×C414yes—
103#14519700—22no—
104#14519700C2×C414yes—
105#14519698—41no—
106#14519698C2×C414yes—
107#14519689—22no—
108C4×#92418—41no—
109C8×Dic61—41no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#107×C61—41no—
128#102×C61—41no—
129#102×C61C2×C414yes—
130#95×C61—21no—
131C24×C61—21no—
132C23×D61C2×D411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—24no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158#11758224—28no—
159#11758224—44no—
160#11758224—44no—
161#11758224—44no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C22×C412yes—
167C2×C4×D61#4112yes—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C22×C412yes—
172C2×Dic122#4112yes—
173C2×Dic122#4112yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—42no—
187C22⋊C4×C61—81no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—81no—
193#11758198—24no—
194#11758198—22no—
195#11758198—22no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C22×C412yes—
204#11758196—42no—
205#11758196—42no—
206#11758196—81no—
207#11758196—81no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—24no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220C8×D61—44no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61C4×D412yes—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—41no—
239D4×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
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#13512yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C22⋊D61—48no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—48no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61C4×D412yes—
283C2×Dic61#13512yes—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122#11714yes—
292C8×C61—22no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#22573—611nomaximal, Sylow
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61C2×C4×D411yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61#57611yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61C2×#11711yescharacteristic
324C4×C61#51811yescharacteristic
325C4×C61#55111yescharacteristic
326C4×C61#55111yescharacteristic
327Dic61—24no—
328Dic61—22no—
329Dic61—22no—
330Dic61—42no—
331Dic61—41no—
332Dic61—81no—
333Dic61—81no—
334C2×#416—611no—
335#2346—611no—
336#2341—611no—
337#1934—611no—
338#1934—611no—
339#1925—611no—
340#1925D6111yescharacteristic, complemented
341#1863—611no—
342#1863—611no—
343#1825—611no—
344#1808—611no—
345#1788—611no—
346C4×#103—611no—
347C4×#103—611no—
348C8×C22⋊C4—611no—
349C2×C61—82no—
350C2×C61#234211yescharacteristic
351C2×C61C4×#13511yescharacteristic
352C2×C61#295111yescharacteristic
353C2×C61#330011yescharacteristic
354C2×C61#330011yescharacteristic
355C2×C61#330511yescharacteristic
356C2×C61#330511yescharacteristic
357D61—42no—
358C22×#41—611no—
359C23×C8—611no—
360C2×C4×Q8—611no—
361C2×C4×Q8—611no—
362C2×#117—611no—
363C2×#107—611no—
364C2×#107C2×D6111yescharacteristic
365#416—618no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102C2×D6111yescharacteristic
373C2×#102C2×D6111yescharacteristic
374C2×C22⋊C8—611no—
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377#390—611no—
378#390—1221no—
379#390—1221no—
380#390C2×D6111yescharacteristic
381#389—611no—
382#389—611no—
383#388—1221no—
384#388—1221no—
385#382—1221no—
386#380—611no—
387#380—611no—
388#380—1221no—
389C4×C4⋊C4—611no—
390C4×C4⋊C4C2×D6111yescharacteristic
391C4×C22⋊C4—611no—
392C4×C22⋊C4—1221no—
393#376—1221no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400#336—611no—
401#336C2×D6111yescharacteristic
402C61#2257311yescharacteristic, complemented, Sylow
403C2×#41—614no—
404C2×#41—1222no—
405C2×#41—1222no—
406C2×#41—1222no—
407C2×#41—2441no—
408C2×#41—2441no—
409C22×Q8—611no—
410C22×D4—1221no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—611no—
413C23×C4—1221no—
414C22×C8—611no—
415C22×C8—614no—
416C22×C8—611no—
417C22×C8—2441no—
418C22×C8—2441no—
419C22×C8C22×D6111yescharacteristic
420C22×C8C22×D6111yescharacteristic
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—2441no—
424C4×Q8—2441no—
425#117—614no—
426#117—1222no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4—611no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—21no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#107—41no—
454#107—2441no—
455#103—614no—
456#103—614no—
457#103—1222no—
458#103—1222no—
459#102—41no—
460#102—614no—
461#102—2441no—
462#102C4×D6114yes—
463C22⋊C8—1222no—
464C4×C8—2441no—
465C4×C8—2441no—
466#95—21no—
467#95—611no—
468#95—611no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—21no—
475C24—611no—
476#41—1228no—
477#41—2444no—
478#41—2444no—
479#41—2444no—
480C2×Q8—612no—
481C2×Q8—612no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—1224no—
492C2×D4—2442no—
493C2×D4—2442no—
494C2×D4C2×C4×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—611no—
503C22×C4—1224no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—612no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—2444no—
529C2×C8C2×C4×D6112yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—612no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—4881no—
542C4⋊C4—4881no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4C2×C4×D6112yes—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—1222no—
549C22⋊C4—1224no—
550C22⋊C4—1222no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C22⋊C4—2442no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—1224no—
559C42—1221no—
560C42—1221no—
561C42—2442no—
562C42—2442no—
563C42—2442no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—612no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23C2×D4×D6111yescharacteristic
579Q8—614no—
580Q8—1222no—
581Q8—1222no—
582Q8—1222no—
583D4—22no—
584D4—22no—
585D4—41no—
586D4—41no—
587D4—2448no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—2448no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4#1451979612yes—
629C2×C4C22×C4×D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#1451982211yescharacteristic
634C2×C4#1451982311yescharacteristic
635C8—22no—
636C8—41no—
637C8—1222no—
638C8—2441no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22C4×D4×D6111yescharacteristic
654C22#1684040411yescharacteristic
655C22#1684040911yescharacteristic
656C22#135×D6111yescharacteristic
657C22C2×#1451979611yescharacteristic
658C22#1684056611yescharacteristic
659C22#1684057811yescharacteristic
660C4—22no—
661C4—41no—
662C4—42no—
663C4—81no—
664C4—81no—
665C4—1222no—
666C4—1222no—
667C4—1224no—
668C4—4881no—
669C4—4881no—
670C4—2442no—
671C4—2441no—
672C4#117×D6111yescharacteristic
673C4#1684023411yescharacteristic
674C4#1684043711yescharacteristic
675C4#1684044211yescharacteristic
676C2—82no—
677C2—2442no—
678C2#2144866311yescharacteristic
679C2C4×#1451979611yescharacteristic
680C2#2145469111yescharacteristic
681C2#2145770411yescharacteristic
682C2#2145771311yescharacteristic
683C2#2145776711yescharacteristic
684C2#2145777211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1925D61
hypocenterC61#22573
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. #21448700C2semidirect
4. #21446505C2semidirect
5. #21446499C2semidirect
6. #21446441C2semidirect
7. #21446076C2semidirect
8. #21446070C2semidirect
9. #21445912C2semidirect
10. #21445845C2semidirect
12. C4×#14519700C2semidirect
13. #21445638C2semidirect
14. #21445589C2semidirect
15. C61×#1925C2semidirect
22. #16839755C4semidirect
340. #1925D61semidirect
402. C61#22573semidirect

Maximal subgroups

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

subgroupindex
C2×#168397552
#214487002
#214486582
#214465052
#214464992
#214464412
#214460762
#214460702
#214459122
#214458452
#214457562
C4×#145197002
#214456382
#214455892
C61×#19252
#2257361