Finite Group Databasebeta

← #29697175

Subgroups — #29697175

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839757C211yesmaximal, characteristic
2#21448703C211yesmaximal, characteristic, complemented
3#21448648C211yesmaximal, characteristic
4#21446497C211yesmaximal, characteristic, complemented
5#21446456C211yesmaximal, characteristic, complemented
6#21446450C211yesmaximal, characteristic, complemented
7#21446081C211yesmaximal, characteristic, complemented
8#21446079C211yesmaximal, characteristic, complemented
9#21445913C211yesmaximal, characteristic, complemented
10#21445843C211yesmaximal, characteristic, complemented
11#21445758C211yesmaximal, characteristic
12C4×#14519699C211yesmaximal, characteristic, complemented
13#21445633C211yesmaximal, characteristic, complemented
14C8×#11758198C211yesmaximal, characteristic, complemented
15C61×#1933C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23#16839757C418yescomplemented
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839654—21no—
35#16839652—21no—
36#16839640—21no—
37#16839640—21no—
38#16839627—21no—
39#16839624—21no—
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×#11758198C2211yescharacteristic
45#16839607—21no—
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—41no—
64C2×#11758226—41no—
65C2×#11758226C2×C414yes—
66C22×D244—21no—
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C2×C8×D61—41no—
75C2×C8×D61—41no—
76C2×C8×D61C2×C414yescomplemented
77C4×D244—22no—
78C4×D244—41no—
79#14519739—22no—
80#14519739—41no—
81#14519735—22no—
82#14519735C2×C414yescomplemented
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198D411yescharacteristic
93C2×#11758198D411yescharacteristic
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519708—41no—
99#14519708—41no—
100#14519701—41no—
101#14519701C2×C414yescomplemented
102#14519699—22no—
103#14519699C2×C414yes—
104#14519697—22no—
105#14519697C2×C414yes—
106#14519689—22no—
107C8×Dic61—41no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×Q8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125#103×C61—41no—
126#103×C61C2×C414yescomplemented
127C4×C8×C61—41no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131C23×D61—21no—
132C23×D61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758226—28no—
153#11758226—44no—
154#11758226—44no—
155#11758226—44no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—24no—
160C2×D244—42no—
161C2×D244—42no—
162C2×D244C22×C412yes—
163C2×D244#4112yes—
164C2×D244#4112yes—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—24no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—81no—
177C2×C4×D61—81no—
178C2×C4×D61C22×C412yes—
179C2×C4×D61#4112yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—81no—
192C4⋊C4×C61C22×C412yes—
193C42×C61—21no—
194C42×C61—21no—
195#11758198—24no—
196#11758198—22no—
197#11758198—22no—
198#11758198—42no—
199#11758198—42no—
200#11758198—81no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—81no—
208C4⋊Dic61C22×C412yes—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—24no—
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×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61C4×D412yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
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—41no—
256C2×C4×C61—41no—
257C2×C4×C61#13512yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61C4×D412yes—
281C2×Dic61#13512yes—
282D244—22no—
283D244—22no—
284D244—22no—
285D244—48no—
286D244#11714yes—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—48no—
292C4×D61—84no—
293C4×D61—84no—
294C8×C61—22no—
295C8×C61—41no—
296#92418—22no—
297#92418—41no—
298#22572—611nomaximal, Sylow
299C22×C61C2×C4×D411yescharacteristic
300C22×C61#54611yescharacteristic
301C22×C61#54611yescharacteristic
302C22×C61C2×#13511yescharacteristic
303C22×C61C2×#13511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57711yescharacteristic
306C2×D61—22no—
307C2×D61—24no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61C2×#11711yescharacteristic
327C4×C61#51811yescharacteristic
328C4×C61#55111yescharacteristic
329C4×C61#55111yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—41no—
334Dic61—81no—
335Dic61—81no—
336C2×#416—611no—
337#2346—611no—
338#2339—611no—
339#1933—611no—
340#1933D6111yescharacteristic, complemented
341#1927—611no—
342#1927—611no—
343#1864—611no—
344#1864—611no—
345#1825—611no—
346#1807—611no—
347#1788—611no—
348C4×#102—611no—
349C4×#102—611no—
350C8×C22⋊C4—611no—
351C2×C61#234211yescharacteristic
352C2×C61C4×#13511yescharacteristic
353C2×C61#295211yescharacteristic
354C2×C61#329611yescharacteristic
355C2×C61#329611yescharacteristic
356C2×C61#330311yescharacteristic
357C2×C61#330311yescharacteristic
358D61—42no—
359D61—44no—
360C22×#41—611no—
361C23×C8—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×#117—611no—
365C2×#106—611no—
366C2×#106—611no—
367#416—618no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103C2×D6111yescharacteristic
371C2×#103C2×D6111yescharacteristic
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×C22⋊C8—611no—
377C2×C4×C8—611no—
378C2×C4×C8C2×D6111yescharacteristic
379#391—611no—
380#391C2×D6111yescharacteristic
381#389—611no—
382#389C2×D6111yescharacteristic
383#388—1221no—
384#388—1221no—
385#386—1221no—
386#386—1221no—
387#382—1221no—
388#381—1221no—
389#380—611no—
390#380—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C22⋊C4—611no—
394#376—1221no—
395C2×#95—1221no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340C2×D6111yescharacteristic
401#340C2×D6111yescharacteristic
402#336—611no—
403#336—611no—
404C61#2257211yescharacteristic, complemented, Sylow
405C2×#41—614no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—2441no—
410C2×#41—2441no—
411C22×Q8C22×D6111yescharacteristic
412C22×D4—611no—
413C22×D4—1221no—
414C23×C4—611no—
415C23×C4—1221no—
416C23×C4—1221no—
417C23×C4—1221no—
418C22×C8—611no—
419C22×C8—614no—
420C22×C8—611no—
421C22×C8—2441no—
422C22×C8—2441no—
423C22×C8C22×D6111yescharacteristic
424C22×C8C22×D6111yescharacteristic
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—2441no—
428C4×D4—2441no—
429#117—614no—
430#117—1222no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
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—611no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#106—2441no—
452#106—2441no—
453#103—41no—
454#103—614no—
455#103—2441no—
456#103C4×D6114yescomplemented
457#102—614no—
458#102—614no—
459#102—1222no—
460#102—1222no—
461C22⋊C8—1222no—
462C4×C8—41no—
463C4×C8—2441no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—611no—
468#95—611no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—611no—
477C24—1221no—
478#41—1228no—
479#41—2444no—
480#41—2444no—
481#41—2444no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8C2×C4×D6112yes—
488C2×D4—612no—
489C2×D4—612no—
490C2×D4—612no—
491C2×D4—1224no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—2442no—
496C2×D4—2442no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—611no—
506C22×C4—1224no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—4881no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—2444no—
537C2×C8C2×C4×D6112yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—81no—
544C4⋊C4—612no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4C2×C4×D6112yes—
552C22⋊C4—1222no—
553C22⋊C4—1224no—
554C22⋊C4—1222no—
555C22⋊C4—4881no—
556C22⋊C4—4881no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—1224no—
562C42—1221no—
563C42—1221no—
564C42—2442no—
565C42—2442no—
566C23—612no—
567C23—1222no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1222no—
573C23—1221no—
574C23—1222no—
575C23—1221no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23C2×D4×D6111yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585D4—614no—
586D4—1222no—
587D4—1222no—
588D4—1222no—
589D4—2448no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—4884no—
622C2×C4—4884no—
623C2×C4—2448no—
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#1451980012yes—
633C2×C4C22×C4×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4#41×D6111yescharacteristic
636C2×C4#41×D6111yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982311yescharacteristic
639C8—22no—
640C8—41no—
641C8—1222no—
642C8—2441no—
643C22—1222no—
644C22—1222no—
645C22—1224no—
646C22—1222no—
647C22—2441no—
648C22—2442no—
649C22—2441no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22—2441no—
654C22—2442no—
655C22—2441no—
656C22—2442no—
657C22C4×D4×D6111yescharacteristic
658C22#1684040111yescharacteristic
659C22#1684040511yescharacteristic
660C22#135×D6111yescharacteristic
661C22C2×#1451980011yescharacteristic
662C22#1684056811yescharacteristic
663C22#1684056911yescharacteristic
664C4—22no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—81no—
669C4—81no—
670C4—1222no—
671C4—1222no—
672C4—4881no—
673C4—4881no—
674C4—2441no—
675C4—2442no—
676C4#117×D6111yescharacteristic
677C4#1684023411yescharacteristic
678C4#1684043711yescharacteristic
679C4#1684044011yescharacteristic
680C2—2442no—
681C2—2444no—
682C2#2144866711yescharacteristic
683C2C4×#1451980011yescharacteristic
684C2#2145469711yescharacteristic
685C2#2145765311yescharacteristic
686C2#2145766011yescharacteristic
687C2#2145774211yescharacteristic
688C2#2145774611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1933D61
hypocenterC61#22572
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. #21448703C2semidirect
4. #21446497C2semidirect
5. #21446456C2semidirect
6. #21446450C2semidirect
7. #21446081C2semidirect
8. #21446079C2semidirect
9. #21445913C2semidirect
10. #21445843C2semidirect
12. C4×#14519699C2semidirect
13. #21445633C2semidirect
14. C8×#11758198C2semidirect
15. C61×#1933C2semidirect
23. #16839757C4semidirect
76. C2×C8×D61C2×C4semidirect
82. #14519735C2×C4semidirect
101. #14519701C2×C4semidirect
126. #103×C61C2×C4semidirect
340. #1933D61semidirect
404. C61#22572semidirect
456. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397572
#214487032
#214486482
#214464972
#214464562
#214464502
#214460812
#214460792
#214459132
#214458432
#214457582
C4×#145196992
#214456332
C8×#117581982
C61×#19332
#2257261