Finite Group Databasebeta

← #29712234

Subgroups — #29712234

Order 15616.

1436 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839912C211yesmaximal, characteristic, complemented
2C22×#14519697C211yesmaximal, characteristic, complemented
3#21449175C211yesmaximal, characteristic
4#21446962C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446174C212yesmaximal, complemented
8#21446141C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445905C211yesmaximal, characteristic, complemented
11C2×C61×#552C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201C2212yes—
14C23×#92418C2211yescharacteristic
15C2×#14519751C2212yes—
16C2×#14519745—21no—
17#16839912—24no—
18C2×#14519701C2212yescomplemented
19C2×#14519697—24no—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2212yescomplemented
25#16839679—21no—
26#16839671—21no—
27#16839664C2212yes—
28#16839622C2211yescharacteristic
29#16839617—21no—
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32#16839524C2212yescomplemented
33#16839517C2212yescomplemented
34C61×#552—24no—
35C61×#552—24no—
36C2×#126×C61—21no—
37C2×#125×C61—21no—
38C2×#125×C61C2211yescharacteristic, complemented
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61C2211yescharacteristic
41C2×C22⋊Q8×C61C2211yescharacteristic, complemented
42C2×C4×Q8×C61C2211yescharacteristic, complemented
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C2×#117×C61C2211yescharacteristic, complemented
46C22×C4⋊C4×C61C2211yescharacteristic
47C24×D61—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—42no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C2×#11758201—42no—
58C22×#92418—22no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519751—24no—
63#14519751—42no—
64#14519745—44no—
65C2×C4⋊Dic61C2312yes—
66C2×C22⋊Dic61—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70#14519701—42no—
71#14519697—24no—
72#14519697—24no—
73#14519697—24no—
74C22⋊#92418—22no—
75C22⋊#92418—42no—
76#14519684—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—21no—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82#126×C61—28no—
83#125×C61—28no—
84#125×C61—22no—
85#125×C61—22no—
86#123×C61—28no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61D414yescomplemented
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95#117×C61—28no—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D414yescomplemented
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C23×D61—22no—
121C23×D61—44no—
122C23×D61—42no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61—82no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D412yes—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—44no—
154C2×C22⋊D61—44no—
155C2×C22⋊D61—82no—
156C22×Dic61—22no—
157C2×D244—28no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—48no—
164C2×D244—44no—
165C2×D244—44no—
166C2×D244—42no—
167C2×D244#4114yes—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61C2×D412yes—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61—24no—
187C42×C61—24no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—42no—
191#11758201—48no—
192#11758198—44no—
193#11758198—44no—
194#11758198—82no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—44no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61#12012yes—
212C23×C61#13512yes—
213C23×C61AGL(1,ℤ/8)12yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—24no—
216C22×D61—48no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—88no—
224C22×D61—88no—
225C22×D61—84no—
226C22×D61—84no—
227Q8×C61—24no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—44no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—48no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—42no—
256C2×C4×C61#12012yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4112yes—
260C2×C4×C61H(5,𝔽2)12yes—
261C22⋊D61—88no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265D244—24no—
266D244—24no—
267D244—48no—
268D244—42no—
269D244—42no—
270#92418—42no—
271#92418—42no—
272#23966—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61#44914yes—
283C22×C61C2×#12011yescharacteristic
284C22×C61#54612yes—
285C22×C61C2×#13511yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61#57712yes—
288C2×D61—44no—
289C2×D61—44no—
290C2×D61—44no—
291C2×D61—816no—
292C2×D61—82no—
293C2×D61—82no—
294C2×D61—82no—
295C2×D61—82no—
296C4×C61—22no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61#52511yescharacteristic
305C4×C61#53511yescharacteristic
306C4×C61#54612yes—
307Dic61—82no—
308C2×#552D6111yescharacteristic, complemented
309C2×#450—611no—
310C22×#102—611no—
311#2408—611no—
312#2025—612no—
313#2000—612no—
314#1892—611no—
315#1880—612no—
316#1875—612no—
317#1869—611no—
318#1823—611no—
319C2×C61—24no—
320C2×C61—81no—
321C2×C61#240411yescharacteristic
322C2×C61C2×#44911yescharacteristic
323C2×C61#301011yescharacteristic
324C2×C61#329612yes—
325C2×C61#331012yes—
326D61—84no—
327C23×D4—611no—
328C22×SD16—612no—
329C23×C8—611no—
330#552—24no—
331#552—24no—
332C2×#126—21no—
333C2×#125—21no—
334C2×#125C2×D6111yescharacteristic, complemented
335C2×#123—21no—
336C2×C22⋊Q8C2×D6111yescharacteristic
337C2×C22⋊Q8C2×D6111yescharacteristic, complemented
338C2×#121—612no—
339C2×#120—1221no—
340C2×C4×Q8C2×D6111yescharacteristic, complemented
341C2×C4×D4C2×D6111yescharacteristic
342C2×#117—21no—
343C2×#117C2×D6111yescharacteristic, complemented
344C22×C4⋊C4C2×D6111yescharacteristic
345#450—1224no—
346C2×#103—612no—
347C2×#102—612no—
348C2×#102—612no—
349C2×#102—612no—
350C2×#102—1224no—
351C2×C22⋊C8—611no—
352C2×C22⋊C8—612no—
353#394—1221no—
354#392—1221no—
355#390—612no—
356#380—611no—
357#379—1221no—
358#340—612no—
359#340—612no—
360#340—612no—
361#336—612no—
362C61#2396611yescharacteristic, complemented, Sylow
363C25—1221no—
364C22×Q8C22×D6111yescharacteristic
365C22×D4—612no—
366C22×D4—1224no—
367C22×D4—1222no—
368C22×D4—1222no—
369C22×D4—1222no—
370C22×D4—1221no—
371C22×D4—2442no—
372C22×D4C22⋊D6111yescharacteristic
373C23×C4—21no—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C2×SD16—1224no—
377C2×SD16—1224no—
378C2×SD16—2442no—
379C22×C8—612no—
380C22×C8—612no—
381C22×C8—1222no—
382C22×C8—2442no—
383#126—28no—
384#125—28no—
385#125—22no—
386#125—22no—
387#123—28no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8C22⋊D6114yescomplemented
392#121—1224no—
393#121—2442no—
394#120—2444no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—28no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4C22⋊D6111yescharacteristic
411C2×C4⋊C4C22⋊D6114yescomplemented
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D6111yescharacteristic
427#103—2442no—
428#102—1224no—
429#102—1224no—
430#102—1224no—
431C22⋊C8—1222no—
432C22⋊C8—2442no—
433#95—1221no—
434C24—1222no—
435C24—4882no—
436C24—2444no—
437C24—2442no—
438C24—2442no—
439C24—2442no—
440C24C2×C22⋊D6111yescharacteristic
441C24C2×C22⋊D6111yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8C2×C22⋊D6112yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—614no—
450C2×D4—1228no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—4882no—
457C2×D4—2448no—
458C2×D4—2444no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2442no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—1222no—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6112yes—
481C22×C4C2×C22⋊D6112yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—2448no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—2444no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4C2×C22⋊D6112yes—
505C4⋊C4C2×C22⋊D6112yes—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—4882no—
513C22⋊C4—2444no—
514C22⋊C4—2444no—
515C22⋊C4—2442no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—1224no—
528C23—4884no—
529C23—4888no—
530C23—4888no—
531C23—4884no—
532C23—2448no—
533C23—2442no—
534C23—2442no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23#1451974612yes—
540C23#1451979712yes—
541C23#1451980212yes—
542C23C22×C22⋊D6111yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546D4—44no—
547D4—1224no—
548D4—1224no—
549D4—4888no—
550D4—2448no—
551D4—2442no—
552D4—2442no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4#1451974612yes—
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4#41×D6112yes—
585C2×C4#1451982312yes—
586C8—2442no—
587C8—2442no—
588C22—22no—
589C22—24no—
590C22—22no—
591C22—24no—
592C22—41no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—48816no—
598C22—4882no—
599C22—4882no—
600C22—4882no—
601C22—4882no—
602C22—2444no—
603C22—2444no—
604C22—2444no—
605C22#1683990714yes—
606C22C2×#1451974611yescharacteristic
607C22#1684040012yes—
608C22C2×#1451979711yescharacteristic
609C22C2×#1451980211yescharacteristic
610C22#1684057012yes—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—4882no—
620C4#1684026911yescharacteristic
621C4#1684032611yescharacteristic
622C4#1684040012yes—
623C2—24no—
624C2—81no—
625C2—4884no—
626C2#2144913411yescharacteristic
627C2C2×#1683990711yescharacteristic
628C2#2145510511yescharacteristic
629C2#2145764812yes—
630C2#2145783412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#552C2
socleC23×C61C22×D4
prime radical#21449175C2
hypercenterC2×#552D61
hypocenterC61#23966
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16839912C2semidirect
2. C22×#14519697C2semidirect
4. #21446962C2semidirect
5. #21446820C2semidirect
6. #21446243C2semidirect
7. #21446174C2semidirect
8. #21446141C2semidirect
9. #21446112C2semidirect
10. #21445905C2semidirect
11. C2×C61×#552C2semidirect
18. C2×#14519701C22semidirect
24. C2×C22⋊#92418C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
32. #16839524C22semidirect
33. #16839517C22semidirect
38. C2×#125×C61C22semidirect
41. C2×C22⋊Q8×C61C22semidirect
42. C2×C4×Q8×C61C22semidirect
45. C2×#117×C61C22semidirect
90. C22⋊Q8×C61D4semidirect
106. C2×C4⋊C4×C61D4semidirect
308. C2×#552D61semidirect
334. C2×#125C2×D61semidirect
337. C2×C22⋊Q8C2×D61semidirect
340. C2×C4×Q8C2×D61semidirect
343. C2×#117C2×D61semidirect
362. C61#23966semidirect
391. C22⋊Q8C22⋊D61semidirect
411. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399122
C22×#145196972
#214491752
#214469622
#214468202
#214462432
#214461742
#214461412
#214461122
#214459052
C2×C61×#5522
#2396661