Finite Group Databasebeta

← #29733361

Subgroups — #29733361

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840128C211yesmaximal, characteristic, complemented
2C2×#16839962C211yesmaximal, characteristic, complemented
3#21450765C211yesmaximal, characteristic
4#21446820C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446136C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#552C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773—21no—
15C2×#14519751C2212yes—
16C2×#14519751C2212yes—
17#16840128—24no—
18#16839962—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2212yes—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839671—21no—
28#16839671—21no—
29#16839663—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34C61×#552—24no—
35C61×#552—24no—
36C2×#126×C61—21no—
37C2×#125×C61—21no—
38C2×#125×C61C2211yescharacteristic
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61C2211yescharacteristic
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C2×#117×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51C22×C22⋊D61—22no—
52C2×#11758201—24no—
53C2×#11758201—42no—
54C2×#11758201—42no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519774—22no—
58#14519774—22no—
59#14519773—44no—
60#14519751—24no—
61#14519751—24no—
62#14519751—42no—
63#14519751—42no—
64C2×C4⋊Dic61C2312yes—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68C4⋊#92418—22no—
69#14519701—24no—
70#14519697—24no—
71#14519697—24no—
72#14519697—24no—
73C22⋊#92418—42no—
74C22⋊#92418—42no—
75#14519688—22no—
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×C61D414yescomplemented
86#123×C61—28no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61D414yescomplemented
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95#117×C61—28no—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
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—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
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—
121C2×Q8×C61—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—41no—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—44no—
149C2×C22⋊D61—44no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C2×C22⋊D61—82no—
153C2×C22⋊D61—82no—
154C22×Dic61—22no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—42no—
162C2×D244—42no—
163C2×D244—42no—
164C2×D244—42no—
165C2×D244—82no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D412yes—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C42×C61—24no—
185C42×C61—24no—
186C42×C61C2×D412yes—
187C42×C61C2×D412yes—
188#11758201—48no—
189#11758198—44no—
190#11758198—44no—
191C4⋊Dic61—22no—
192C4⋊Dic61#4114yes—
193C4×Dic61—42no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61AGL(1,ℤ/8)12yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—42no—
221D4×C61—44no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61#12012yes—
247C2×C4×C61#12012yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4112yes—
251C2×C4×C61H(5,𝔽2)12yes—
252C22⋊D61—88no—
253C22⋊D61—88no—
254C2×Dic61—24no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257D244—42no—
258D244—42no—
259D244—42no—
260D244—42no—
261D244—84no—
262D244—84no—
263#92418—42no—
264#92418—42no—
265#25583—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61C2×#12011yescharacteristic
277C22×C61#54612yes—
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57712yes—
281C2×D61—82no—
282C2×D61—82no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61#53211yescharacteristic
297C4×C61#53511yescharacteristic
298C4×C61#55012yes—
299Dic61—44no—
300C2×#552D6111yescharacteristic, complemented
301C2×#490—611no—
302C2×#460—611no—
303#2541—611no—
304#2000—612no—
305#2000—612no—
306#1989—611no—
307#1955—611no—
308#1875—612no—
309#1874—612no—
310#1763—611no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61#243111yescharacteristic
314C2×C61#300611yescharacteristic
315C2×C61#301011yescharacteristic
316C2×C61#331012yes—
317C2×C61#331712yes—
318D61—162no—
319C22×SD16—612no—
320#552—24no—
321#552—24no—
322C2×#127—611no—
323C2×#127—1221no—
324C2×#126—21no—
325C2×#125—21no—
326C2×#125C2×D6111yescharacteristic
327C2×#123—21no—
328C2×C22⋊Q8C2×D6111yescharacteristic
329C2×C22⋊Q8C2×D6111yescharacteristic
330C2×#121—612no—
331C2×#121—612no—
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4C2×D6111yescharacteristic
334C2×#117—21no—
335C2×#117C2×D6111yescharacteristic
336C22×C4⋊C4C2×D6111yescharacteristic
337#490—1224no—
338#460—1224no—
339C2×C4⋊C8—611no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343C2×#102—612no—
344C2×C22⋊C8—612no—
345C2×C22⋊C8—612no—
346C2×#97—611no—
347#392—1221no—
348#392—1221no—
349#390—1221no—
350C4×C4⋊C4—611no—
351#340—612no—
352#340—612no—
353#340—612no—
354C61#2558311yescharacteristic, complemented, Sylow
355C22×Q8C22×D6111yescharacteristic
356C22×D4—612no—
357C22×D4—1222no—
358C22×D4—1222no—
359C22×D4—1221no—
360C22×D4—1221no—
361C22×D4C22⋊D6111yescharacteristic
362C23×C4—21no—
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C2×SD16—1224no—
366C2×SD16—2442no—
367C2×SD16—2442no—
368C22×C8—612no—
369C22×C8—612no—
370#127—1222no—
371#127—1222no—
372#127—2444no—
373#126—28no—
374#125—28no—
375#125—22no—
376#125C22⋊D6114yescomplemented
377#123—28no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382#121—1224no—
383#121—1224no—
384#121—2442no—
385#121—2442no—
386C4×Q8—22no—
387C4×Q8C22⋊D6114yescomplemented
388C4×D4—22no—
389C4×D4—22no—
390#117—28no—
391#117—22no—
392#117—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4C22⋊D6111yescharacteristic
412C2×C22⋊C4C22⋊D6111yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—1222no—
418#103—1224no—
419#102—1224no—
420#102—1224no—
421#102—1224no—
422C22⋊C8—2442no—
423C22⋊C8—2442no—
424#97—1222no—
425#95—1221no—
426C24—1222no—
427C24C2×C22⋊D6111yescharacteristic
428C24C2×C22⋊D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C22⋊D6112yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—4882no—
443C2×D4—4882no—
444C2×D4—4882no—
445C2×D4—2442no—
446C2×D4—2442no—
447C2×D4—2444no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4—2442no—
452C2×D4—2442no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—1222no—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—2448no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—614no—
492C4⋊C4—1222no—
493C4⋊C4C2×C22⋊D6112yes—
494C4⋊C4C2×C22⋊D6112yes—
495C4⋊C4C2×C22⋊D6112yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—2444no—
502C22⋊C4—2444no—
503C42—24no—
504C42—24no—
505C42—2442no—
506C42C2×C22⋊D6112yes—
507C42C2×C22⋊D6112yes—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23#1451980212yes—
525C23C22×C22⋊D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529D4—44no—
530D4—4884no—
531D4—4884no—
532D4—4888no—
533D4—4888no—
534D4—2442no—
535D4—2442no—
536D4—2442no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—1224no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4#1451974612yes—
566C2×C4#1451974612yes—
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#41×D6112yes—
570C2×C4#1451982312yes—
571C8—2442no—
572C8—2442no—
573C22—24no—
574C22—22no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—4882no—
584C22—4882no—
585C22—4882no—
586C22—4882no—
587C22—4882no—
588C22—4882no—
589C22—4882no—
590C22C2×#1451974611yescharacteristic
591C22#1684040012yes—
592C22C2×#1451980211yescharacteristic
593C22#1684056011yescharacteristic
594C22#1684057012yes—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—2444no—
604C4#1684030811yescharacteristic
605C4#1684032611yescharacteristic
606C4#1684043112yes—
607C2—42no—
608C2—81no—
609C2—9762no—
610C2#2144943411yescharacteristic
611C2#2145507111yescharacteristic
612C2#2145510511yescharacteristic
613C2#2145783412yes—
614C2#2145793912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#552C2
socleC23×C61C22×D4
prime radical#21450765C2
hypercenterC2×#552D61
hypocenterC61#25583
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×#16840128C2semidirect
2. C2×#16839962C2semidirect
4. #21446820C2semidirect
5. #21446820C2semidirect
6. #21446777C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446136C2semidirect
10. #21445649C2semidirect
11. C2×C61×#552C2semidirect
85. #125×C61D4semidirect
92. C4×Q8×C61D4semidirect
300. C2×#552D61semidirect
354. C61#25583semidirect
376. #125C22⋊D61semidirect
387. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
C2×#168399622
#214507652
#214468202
#214467772
#214466182
#214461412
#214461362
#214456492
C2×C61×#5522
#2558361