Finite Group Databasebeta

← #29697246

Subgroups — #29697246

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839763C211yesmaximal, characteristic
2#21448698C211yesmaximal, characteristic, complemented
3#21448649C211yesmaximal, characteristic
4#21446506C211yesmaximal, characteristic, complemented
5#21446492C211yesmaximal, characteristic, complemented
6#21446463C211yesmaximal, characteristic, complemented
7#21446456C211yesmaximal, characteristic, complemented
8#21446078C211yesmaximal, characteristic, complemented
9#21446064C211yesmaximal, characteristic, complemented
10#21445906C211yesmaximal, characteristic, complemented
11#21445850C211yesmaximal, characteristic, complemented
12#21445646C211yesmaximal, characteristic, complemented
13#102×Dic61C211yesmaximal, characteristic, complemented
14#21445605C211yesmaximal, characteristic, complemented
15C61×#1788C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22#16839763C418yescomplemented
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839669C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839647—21no—
34#16839647—21no—
35#16839642—21no—
36#16839642—21no—
37#16839625—21no—
38#16839622C2211yescharacteristic
39#16839620—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C22⋊C4×Dic61—21no—
43C4×C22⋊Dic61C2211yescharacteristic
44C2×#14519684—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×#41×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C2×#11758194—41no—
64C2×#11758194—41no—
65C2×#11758194C2×C414yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71D4×Dic61—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61C2×C414yescomplemented
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—41no—
91#14519708—41no—
92#14519701C2×C414yescomplemented
93#14519701C2×C414yescomplemented
94#14519698C2×C414yes—
95#14519698C2×C414yes—
96#14519687—41no—
97C8×Dic61—41no—
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#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C2×#41×C61—22no—
110C2×#41×C61—22no—
111C2×#41×C61—22no—
112C2×#41×C61—41no—
113C2×#41×C61—41no—
114C2×#41×C61C2×C414yes—
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125#103×C61—41no—
126#103×C61—41no—
127#102×C61—22no—
128#102×C61—22no—
129C22⋊C8×C61—22no—
130#95×C61—21no—
131C24×C61—21no—
132C24×C61C2×D411yescharacteristic
133#41×C61—28no—
134#41×C61—44no—
135#41×C61—44no—
136#41×C61—44no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—24no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61C22×C412yes—
150C2×D4×C61#4112yes—
151C2×D4×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—24no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—42no—
161C22×C4×C61—42no—
162C22×C4×C61C22×C412yes—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—24no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—42no—
172C22×Dic61—42no—
173C22×Dic61—42no—
174C22×Dic61—42no—
175C22×Dic61—81no—
176C22×Dic61—81no—
177C22×Dic61—81no—
178C22×Dic61—81no—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊C4×C61—42no—
194C22⋊Dic61—22no—
195C22⋊Dic61—41no—
196C22⋊Dic61—42no—
197C22⋊Dic61—42no—
198C22⋊Dic61—81no—
199C4⋊Dic61—24no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—81no—
209#11758196—81no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—81no—
215C4×Dic61—81no—
216#11758194—44no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C22×C412yes—
222C2×#92418C22×C412yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61C4×D412yes—
237C23×C61C22×D411yescharacteristic
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—48no—
246D4×C61#11714yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—48no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61C4×D412yes—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—22no—
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—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—84no—
294C2×Dic61—84no—
295C2×Dic61—84no—
296C2×Dic61#13712yes—
297C8×C61—41no—
298C8×C61—41no—
299#92418—22no—
300#92418—22no—
301#22578—611nomaximal, Sylow
302C22×C61—22no—
303C22×C61—24no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—41no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—41no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—41no—
314C22×C61—42no—
315C22×C61—41no—
316C22×C61C2×C4×D411yescharacteristic
317C22×C61#54611yescharacteristic
318C22×C61#54611yescharacteristic
319C22×C61C2×#13711yescharacteristic
320C22×C61#57511yescharacteristic
321C22×C61#57611yescharacteristic
322C22×C61#57711yescharacteristic
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61C2×#11711yescharacteristic
328C4×C61#51811yescharacteristic
329C4×C61#55111yescharacteristic
330C4×C61#55111yescharacteristic
331Dic61—22no—
332Dic61—41no—
333Dic61—42no—
334Dic61—42no—
335Dic61—41no—
336Dic61—41no—
337Dic61—81no—
338Dic61—81no—
339C2×#417—611no—
340#2345—611no—
341#2339—611no—
342#1935—611no—
343#1933—611no—
344#1928—611no—
345#1927—611no—
346#1864—611no—
347#1862—611no—
348#1823—611no—
349#1809—611no—
350#1788D6111yescharacteristic
351#1762—611no—
352C4×#102—611no—
353#1753—611no—
354C2×C61—42no—
355C2×C61—44no—
356C2×C61#234211yescharacteristic
357C2×C61C4×#13711yescharacteristic
358C2×C61#294611yescharacteristic
359C2×C61#330711yescharacteristic
360C2×C61#331411yescharacteristic
361C2×C61#332211yescharacteristic
362C2×C61#333111yescharacteristic
363C22×#41C2×D6111yescharacteristic
364C22×M16—611no—
365C2×C4×D4—611no—
366C2×C4×D4—611no—
367C22×C4⋊C4—611no—
368C2×#107—611no—
369C2×#106—611no—
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×#102—611no—
376C2×#102—611no—
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×C22⋊C8C2×D6111yescharacteristic
380C2×#97—611no—
381C2×C4×C8—611no—
382#391—611no—
383#391—611no—
384#389—611no—
385#389—611no—
386#387—1221no—
387#387—1221no—
388#386—1221no—
389#386—1221no—
390#381—1221no—
391#380—611no—
392#380—1221no—
393#380C2×D6111yescharacteristic
394C4×C4⋊C4—611no—
395C4×C4⋊C4—611no—
396C4×C22⋊C4—611no—
397C4×C22⋊C4—1221no—
398C2×#95—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#336—611no—
406#336—611no—
407C61#2257811yescharacteristic, complemented, Sylow
408C2×#41—22no—
409C2×#41—22no—
410C2×#41—22no—
411C2×#41—41no—
412C2×#41—41no—
413C2×#41C4×D6114yes—
414C22×Q8C22×D6111yescharacteristic
415C22×D4—21no—
416C22×D4C22×D6111yescharacteristic
417C23×C4—21no—
418C23×C4—1221no—
419C23×C4—1221no—
420C23×C4—1221no—
421C23×C4C22×D6111yescharacteristic
422C2×M16—614no—
423C2×M16—2441no—
424C2×M16—2441no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—2441no—
432C4×D4—2441no—
433C2×C4⋊C4—614no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—611no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42—1221no—
454#107—2441no—
455#106—2441no—
456#103—41no—
457#103—41no—
458#103—614no—
459#103—614no—
460#102—22no—
461#102—22no—
462#102—614no—
463#102—614no—
464C22⋊C8—22no—
465#97—2441no—
466C4×C8—2441no—
467#95—21no—
468#95—611no—
469#95—611no—
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—
479C24—21no—
480C24D4×D6111yescharacteristic
481#41—28no—
482#41—44no—
483#41—44no—
484#41—44no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8C2×C4×D6112yes—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D6112yes—
498C2×D4#1175822412yes—
499C2×D4#1175822412yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1224no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—4881no—
520C22×C4—4881no—
521C22×C4—4881no—
522C22×C4—4881no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4C2×C4×D6112yes—
528C22×C4D4×D6111yescharacteristic
529C22×C4#1175822412yes—
530C22×C4C23×D6111yescharacteristic
531M16—2444no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—612no—
539C2×C8—612no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—612no—
549C4⋊C4—612no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1224no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—4881no—
558C4⋊C4—4881no—
559C22⋊C4—42no—
560C22⋊C4—1222no—
561C22⋊C4—4881no—
562C22⋊C4—2441no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—1221no—
566C42—1221no—
567C42—1221no—
568C42—1221no—
569C42—4881no—
570C42—4881no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23#1451973712yes—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4#1451973314yes—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—4884no—
626C2×C4—4884no—
627C2×C4—4884no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4#1451973712yes—
639C2×C4C22×C4×D6111yescharacteristic
640C2×C4C2×#1175822411yescharacteristic
641C2×C4C2×#1175822411yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C2×C4#1451982211yescharacteristic
644C2×C4#1451982211yescharacteristic
645C8—41no—
646C8—41no—
647C8—1222no—
648C8—1222no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—41no—
663C22C2×#1451973711yescharacteristic
664C22#1684039911yescharacteristic
665C22#1684039911yescharacteristic
666C22#137×D6111yescharacteristic
667C22#1684055811yescharacteristic
668C22#1684056411yescharacteristic
669C22#1684056911yescharacteristic
670C4—22no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—1222no—
675C4—4881no—
676C4—4881no—
677C4—2442no—
678C4—2442no—
679C4—2441no—
680C4—2441no—
681C4—2441no—
682C4C2×#1451973311yescharacteristic
683C4#1684023511yescharacteristic
684C4#1684044311yescharacteristic
685C4#1684044311yescharacteristic
686C2—42no—
687C2—44no—
688C2#2144867011yescharacteristic
689C2#2145464311yescharacteristic
690C2#2145465411yescharacteristic
691C2#2145778811yescharacteristic
692C2#2145789311yescharacteristic
693C2#2145801311yescharacteristic
694C2#2145812811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1788C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#1788D61
hypocenterC61#22578
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #21448698C2semidirect
4. #21446506C2semidirect
5. #21446492C2semidirect
6. #21446463C2semidirect
7. #21446456C2semidirect
8. #21446078C2semidirect
9. #21446064C2semidirect
10. #21445906C2semidirect
11. #21445850C2semidirect
12. #21445646C2semidirect
13. #102×Dic61C2semidirect
14. #21445605C2semidirect
22. #16839763C4semidirect
65. C2×#11758194C2×C4semidirect
74. C2×C4⋊Dic61C2×C4semidirect
92. #14519701C2×C4semidirect
93. #14519701C2×C4semidirect
407. C61#22578semidirect

Maximal subgroups

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

subgroupindex
C2×#168397632
#214486982
#214486492
#214465062
#214464922
#214464632
#214464562
#214460782
#214460642
#214459062
#214458502
#214456462
#102×Dic612
#214456052
C61×#17882
#2257861