Finite Group Databasebeta

← #29733686

Subgroups — #29733686

Order 15616.

1216 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840152C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21451569C211yesmaximal, characteristic, complemented
4#21446820C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446631C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal, complemented
9#21446167C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C2×C61×#542C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758199C2212yes—
14C2×#14519774C2211yescharacteristic
15C2×#14519751C2212yes—
16#16840152—24no—
17#16840110—24no—
18C2×#14519701C2212yes—
19C2×#14519698C2212yes—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×C22⋊#92418C2212yes—
23C2×#14519688C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839679—21no—
26#16839679—21no—
27#16839671—21no—
28#16839671—21no—
29#16839664C2212yes—
30#16839628C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#542—24no—
35C61×#542—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—21no—
38C2×#124×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×#123×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C2×#117×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C2×#11758201—42no—
54C2×#11758199—24no—
55C2×#11758199—24no—
56C2×#11758199—42no—
57C22×#92418C2312yes—
58C22×#92418C2312yes—
59#14519774—22no—
60#14519774—22no—
61#14519751—42no—
62#14519751—42no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67#14519701—42no—
68#14519698—42no—
69#14519697—24no—
70#14519697—24no—
71C22⋊#92418—24no—
72#14519688—22no—
73C4×#92418—22no—
74#14519684—21no—
75#14519684—21no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yescomplemented
84#126×C61—28no—
85#124×C61—28no—
86#124×C61—22no—
87#124×C61D414yescomplemented
88#123×C61—28no—
89C22⋊Q8×C61—28no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—22no—
92#121×C61—22no—
93#121×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—22no—
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—82no—
125C2×Q8×C61C2×D412yes—
126C2×D4×C61—24no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—82no—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—82no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—44no—
148C2×C22⋊D61—82no—
149C2×C22⋊D61—82no—
150C22×Dic61—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244—82no—
160C2×D244—82no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—24no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182#11758201—48no—
183#11758199—48no—
184#11758198—44no—
185#11758198—82no—
186#11758198—82no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418#4114yes—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61C22×D411yescharacteristic
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213Q8×C61—24no—
214Q8×C61—42no—
215Q8×C61—84no—
216D4×C61—24no—
217D4×C61—42no—
218D4×C61—84no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—84no—
238C2×C4×C61#12012yes—
239C2×C4×C61#12012yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4112yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C22⋊D61—88no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248D244—42no—
249D244—42no—
250D244—42no—
251D244—42no—
252D244—84no—
253D244—84no—
254#92418—24no—
255#92418—42no—
256#25614—611nomaximal, Sylow
257C22×C61—42no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61C2×#12011yescharacteristic
268C22×C61#53211yescharacteristic
269C22×C61#53511yescharacteristic
270C22×C61#57612yes—
271C22×C61#57712yes—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61—82no—
278C2×D61—82no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61#54612yes—
288C4×C61#55012yes—
289Dic61—82no—
290C2×#542D6111yescharacteristic, complemented
291C2×#496—611no—
292C2×#486—611no—
293#2619—611no—
294#2000—612no—
295#1999—612no—
296#1993—611no—
297#1958—611no—
298#1880—612no—
299#1879—612no—
300#1750—611no—
301C2×C61—81no—
302C2×C61—82no—
303C2×C61#243111yescharacteristic
304C2×C61#300811yescharacteristic
305C2×C61#301011yescharacteristic
306C2×C61#317912yes—
307C2×C61#322412yes—
308D61—162no—
309C22×SD16—612no—
310C22×D8—612no—
311#542—24no—
312#542—24no—
313C2×C4⋊Q8C2×D6111yescharacteristic
314C2×#127—611no—
315C2×#126—21no—
316C2×#124—21no—
317C2×#124C2×D6111yescharacteristic
318C2×#123—21no—
319C2×C22⋊Q8—21no—
320C2×C22⋊Q8C2×D6111yescharacteristic
321C2×#121—612no—
322C2×#121C2×D6111yescharacteristic
323C2×C4×Q8C2×D6111yescharacteristic
324C2×C4×D4C2×D6111yescharacteristic
325C2×#117C2×D6111yescharacteristic
326#496—1224no—
327#486—1224no—
328C2×#103—612no—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102—612no—
332C2×C22⋊C8—612no—
333C2×#97—611no—
334C2×C4×C8—611no—
335#394—1221no—
336#394—1221no—
337#392—1221no—
338#392—1221no—
339#390—612no—
340#382—611no—
341#340—612no—
342#340—612no—
343#336—612no—
344C61#2561411yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8C22×D6111yescharacteristic
347C22×D4—21no—
348C22×D4—612no—
349C22×D4—1222no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4C22×D6111yescharacteristic
353C23×C4—21no—
354C23×C4C22×D6111yescharacteristic
355C2×SD16—1224no—
356C2×SD16—1224no—
357C2×SD16—2442no—
358C2×D8—1224no—
359C2×D8—1224no—
360C2×D8—2442no—
361C22×C8—612no—
362C22×C8—612no—
363C4⋊Q8—22no—
364C4⋊Q8C22⋊D6114yescomplemented
365#127—1222no—
366#127—1222no—
367#126—28no—
368#124—28no—
369#124—22no—
370#124C22⋊D6114yescomplemented
371#123—28no—
372C22⋊Q8—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375#121—22no—
376#121—22no—
377#121—2442no—
378#121—2442no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—22no—
382C4×D4—22no—
383#117—22no—
384#117—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C4⋊C4C22⋊D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4C22⋊D6111yescharacteristic
402C2×C22⋊C4C22⋊D6111yescharacteristic
403C2×C42—21no—
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406#103—2442no—
407#102—1224no—
408#102—1224no—
409#102—2442no—
410C22⋊C8—1224no—
411#97—1222no—
412C4×C8—1222no—
413#95—1221no—
414#95—1221no—
415C24—21no—
416C24—1222no—
417C24C2×C22⋊D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—82no—
423C2×Q8C2×C22⋊D6112yes—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—82no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—4882no—
436C2×D4—4882no—
437C2×D4—4882no—
438C2×D4—4882no—
439C2×D4—2444no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4C2×C22⋊D6112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—82no—
452C22×C4—1222no—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459SD16—2448no—
460D8—2448no—
461C2×C8—614no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4C2×C22⋊D6112yes—
477C4⋊C4C2×C22⋊D6112yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—4882no—
486C22⋊C4—4882no—
487C22⋊C4—2444no—
488C42—24no—
489C42—21no—
490C42—21no—
491C42C2×C22⋊D6112yes—
492C42C2×C22⋊D6112yes—
493C23—21no—
494C23—22no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23C22×C22⋊D6111yescharacteristic
511Q8—24no—
512Q8—42no—
513Q8—84no—
514D4—24no—
515D4—42no—
516D4—84no—
517D4—4884no—
518D4—4884no—
519D4—4888no—
520D4—2442no—
521D4—2442no—
522D4—2442no—
523D4—2442no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—84no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4#1451974612yes—
547C2×C4#1451974612yes—
548C2×C4C22×C22⋊D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4#41×D6112yes—
551C2×C4#1451982312yes—
552C8—1224no—
553C8—2442no—
554C22—41no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—82no—
563C22—82no—
564C22—4882no—
565C22—4882no—
566C22—4882no—
567C22—4882no—
568C22—4882no—
569C22—4882no—
570C22—4882no—
571C22C2×#1451974611yescharacteristic
572C22#1684030811yescharacteristic
573C22#1684032611yescharacteristic
574C22#1684056512yes—
575C22#1684057012yes—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—42no—
584C4—4882no—
585C4#1684040012yes—
586C4#1684043112yes—
587C2—81no—
588C2—82no—
589C2—9762no—
590C2#2144943411yescharacteristic
591C2#2145508711yescharacteristic
592C2#2145510511yescharacteristic
593C2#2145649612yes—
594C2#2145690712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#542C2
socleC23×C61C22×D4
hypercenterC2×#542D61
hypocenterC61#25614
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#16840152C2semidirect
2. C2×#16840110C2semidirect
3. #21451569C2semidirect
4. #21446820C2semidirect
5. #21446815C2semidirect
6. #21446793C2semidirect
7. #21446631C2semidirect
8. #21446174C2semidirect
9. #21446167C2semidirect
10. #21445585C2semidirect
11. C2×C61×#542C2semidirect
83. C4⋊Q8×C61D4semidirect
87. #124×C61D4semidirect
290. C2×#542D61semidirect
344. C61#25614semidirect
364. C4⋊Q8C22⋊D61semidirect
370. #124C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401522
C2×#168401102
#214515692
#214468202
#214468152
#214467932
#214466312
#214461742
#214461672
#214455852
C2×C61×#5422
#2561461