Finite Group Databasebeta

← #29691328

Subgroups — #29691328

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448503C211yesmaximal, characteristic
2#21448405C211yesmaximal, characteristic
3#21446691C211yesmaximal, characteristic, complemented
4#21446681C211yesmaximal, characteristic, complemented
5#21446511C211yesmaximal, characteristic, complemented
6#21446497C211yesmaximal, characteristic, complemented
7#21446462C211yesmaximal, characteristic, complemented
8#21446455C211yesmaximal, characteristic, complemented
9#21446162C211yesmaximal, characteristic, complemented
10#21446162C211yesmaximal, characteristic, complemented
11#21446132C211yesmaximal, characteristic
12#21446020C211yesmaximal, characteristic
13#21445754C211yesmaximal, characteristic
14#21445587C211yesmaximal, characteristic
15C2×C61×#451C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839669—21no—
31#16839669—21no—
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839669C2211yescharacteristic
37#16839661—21no—
38#16839661C2211yescharacteristic
39#16839655—21no—
40#16839655—21no—
41#16839648—21no—
42#16839625—21no—
43#16839625—21no—
44#16839622C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×Q8×C61C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C22⋊Q8×C61C2211yescharacteristic
56C61×#451C418yescomplemented
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62Q8×Dic61—41no—
63Q8×Dic61—41no—
64#14519736—41no—
65#14519736—41no—
66#14519736—41no—
67#14519736—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61D411yescharacteristic
79C2×C22⋊Dic61D411yescharacteristic
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519711—41no—
86#14519709—41no—
87#14519701—41no—
88#14519701—41no—
89#14519701—41no—
90#14519701—41no—
91#14519701—41no—
92#14519701—41no—
93C22⋊#92418—22no—
94#14519687—41no—
95C8×Dic61—41no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×Q8×C61—21no—
110C22×Q8×C61—22no—
111C22×Q8×C61—41no—
112C22×Q8×C61—41no—
113C22×Q8×C61—41no—
114C22×Q8×C61—41no—
115C22×Q8×C61C2×C414yes—
116C22×Q8×C61C2311yescharacteristic
117C22×Q8×C61C2311yescharacteristic
118C22×Q8×C61C2311yescharacteristic
119C23×C4×C61—21no—
120C23×C4×C61C2311yescharacteristic
121C2×Q16×C61—41no—
122C2×Q16×C61—41no—
123C2×Q16×C61—41no—
124C2×Q16×C61—41no—
125C2×Q16×C61C2×C414yes—
126C2×Q16×C61C2×C414yes—
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22⋊Q8×C61—22no—
130C22⋊Q8×C61C2×C414yes—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C22⋊C4×C61—21no—
134#103×C61C2×C414yes—
135#103×C61C2×C414yes—
136C22⋊C8×C61C2×C414yes—
137C24×C61C2×D411yescharacteristic
138C2×Q8×C61—24no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—44no—
155C2×Q8×C61—44no—
156C2×Q8×C61—42no—
157C2×Q8×C61—42no—
158C2×Q8×C61—44no—
159C2×Q8×C61—44no—
160C2×Q8×C61C22×C412yes—
161C2×Q8×C61C22×C412yes—
162C2×Q8×C61C22×C412yes—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—24no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—42no—
173C22×C4×C61—42no—
174C22×C4×C61C22×C412yes—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61#4112yes—
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—21no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61C2×D411yescharacteristic
183C22×Dic61C2×D411yescharacteristic
184C22×Dic61C2×D411yescharacteristic
185C22×Dic61C2×D411yescharacteristic
186Q16×C61—44no—
187Q16×C61—44no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61C22×C412yes—
193C2×C8×C61C22×C412yes—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C22×C412yes—
197C22⋊C4×C61—24no—
198C22⋊Dic61—41no—
199C22⋊Dic61—41no—
200C22⋊Dic61—42no—
201C22⋊Dic61—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—41no—
211C4⋊Dic61—41no—
212C4⋊Dic61—42no—
213C4⋊Dic61—81no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C4×Dic61—42no—
220C4×Dic61—81no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C4×D412yes—
234C23×C61C22×D411yescharacteristic
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—48no—
242Q8×C61—41no—
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—48no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61C4×D412yes—
268C2×C4×C61C23×C411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×C4×C61H(5,𝔽2)11yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C8×C61—22no—
299C8×C61—22no—
300#92418—41no—
301#92418—41no—
302#22054—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C22×C61#53411yescharacteristic
312C22×C61#53511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C22×C61#57811yescharacteristic
317C4×C61—22no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—42no—
322C4×C61#51811yescharacteristic
323C4×C61#51811yescharacteristic
324C4×C61#53811yescharacteristic
325C4×C61#53811yescharacteristic
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#451D6111yescharacteristic
335#2323—611no—
336#2310—611no—
337#1971—611no—
338#1969—611no—
339#1935—611no—
340#1933—611no—
341#1928—611no—
342#1927—611no—
343#1878—611no—
344#1878—611no—
345#1873—611no—
346#1854—611no—
347#1787—611no—
348#1750—611no—
349C2×C61—42no—
350C2×C61#232111yescharacteristic
351C2×C61#294711yescharacteristic
352C2×C61#295011yescharacteristic
353C2×C61#320111yescharacteristic
354C2×C61#320211yescharacteristic
355C2×C61#322311yescharacteristic
356C2×C61#322511yescharacteristic
357C23×Q8C2×D6111yescharacteristic
358C22×Q16C2×D6111yescharacteristic
359C22×Q16C2×D6111yescharacteristic
360C2×C22⋊Q8C2×D6111yescharacteristic
361C2×C4×Q8—611no—
362C2×#117—611no—
363C2×#117—611no—
364#451Dic6118yescomplemented
365C2×#107—611no—
366C2×#106—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×C22⋊C8—611no—
376C2×C22⋊C8C2×D6111yescharacteristic
377C2×#97—611no—
378C2×C4×C8—611no—
379#391—611no—
380#391—611no—
381#391—611no—
382#391—611no—
383#391—611no—
384#391—1221no—
385#391—1221no—
386#389—611no—
387#389—1221no—
388#388—1221no—
389#388—1221no—
390#387—1221no—
391#381—1221no—
392#381—1221no—
393#380—611no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#336—611no—
400#336—611no—
401C61#2205411yescharacteristic, complemented, Sylow
402C22×Q8—21no—
403C22×Q8—22no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8—41no—
408C22×Q8C2×Dic6114yes—
409C22×Q8C22×D6111yescharacteristic
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—21no—
413C23×C4C22×D6111yescharacteristic
414C2×Q16—41no—
415C2×Q16—41no—
416C2×Q16—41no—
417C2×Q16—41no—
418C2×Q16C2×Dic6114yes—
419C2×Q16C2×Dic6114yes—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—22no—
425C22⋊Q8C2×Dic6114yes—
426C4×Q8—2441no—
427C4×Q8—2441no—
428#117—2441no—
429#117—2441no—
430#117—2441no—
431#117—2441no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—611no—
444C2×C22⋊C4—611no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452#107—2441no—
453#106—2441no—
454#103—2441no—
455#103—2441no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460#103C2×Dic6114yes—
461#103C2×Dic6114yes—
462C22⋊C8—1222no—
463C22⋊C8C2×Dic6114yes—
464#97—2441no—
465C4×C8—2441no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24D4×D6111yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—24no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—44no—
497C2×Q8—44no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8C22×Dic6112yes—
503C2×Q8C22×Dic6112yes—
504C2×Q8C22×Dic6112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—24no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4D4×D6111yescharacteristic
525C22×C4#1175822412yes—
526C22×C4C22×Dic6112yes—
527C22×C4C23×D6111yescharacteristic
528Q16—44no—
529Q16—44no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8C22×Dic6112yes—
541C2×C8C22×Dic6112yes—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4881no—
553C4⋊C4—2441no—
554C4⋊C4—2441no—
555C4⋊C4—2442no—
556C4⋊C4C22×Dic6112yes—
557C22⋊C4—24no—
558C22⋊C4—4881no—
559C22⋊C4—2441no—
560C22⋊C4—2441no—
561C22⋊C4—2442no—
562C42—1221no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—4881no—
567C42—2442no—
568C42—2442no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23D4×Dic6112yes—
576C23C2×D4×D6111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4D4×Dic6112yes—
634C2×C4C23×Dic6111yescharacteristic
635C2×C4C2×#1175822411yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982211yescharacteristic
640C8—22no—
641C8—22no—
642C8—2441no—
643C8—2441no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×D4×Dic6111yescharacteristic
652C22#1684032211yescharacteristic
653C22#1684032711yescharacteristic
654C22#1684057111yescharacteristic
655C22#1684057511yescharacteristic
656C22#1684057511yescharacteristic
657C22#1684057811yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—2441no—
668C4—2441no—
669C4—2441no—
670C4—2441no—
671C4#1684023711yescharacteristic
672C4#1684023711yescharacteristic
673C4#1684034711yescharacteristic
674C4#1684034711yescharacteristic
675C2—42no—
676C2#2144848011yescharacteristic
677C2#2145466311yescharacteristic
678C2#2145468711yescharacteristic
679C2#2145670511yescharacteristic
680C2#2145671211yescharacteristic
681C2#2145689811yescharacteristic
682C2#2145693411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#451C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenterC2×#451D61
hypocenterC61#22054
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #21446691C2semidirect
4. #21446681C2semidirect
5. #21446511C2semidirect
6. #21446497C2semidirect
7. #21446462C2semidirect
8. #21446455C2semidirect
9. #21446162C2semidirect
10. #21446162C2semidirect
56. C61×#451C4semidirect
364. #451Dic61semidirect
401. C61#22054semidirect

Maximal subgroups

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

subgroupindex
#214485032
#214484052
#214466912
#214466812
#214465112
#214464972
#214464622
#214464552
#214461622
#214461322
#214460202
#214457542
#214455872
C2×C61×#4512
#2205461