Finite Group Databasebeta

← #29691217

Subgroups — #29691217

Order 15616.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839898C211yesmaximal, characteristic
2#21448325C211yesmaximal, characteristic, complemented
3C2×#16839663C211yesmaximal, characteristic, complemented
4#21446693C212yesmaximal, complemented
5#21446514C212yesmaximal, complemented
6#21446445C212yesmaximal, complemented
7#21446168C212yesmaximal, complemented
8#21446136C211yesmaximal, characteristic, complemented
9#21445749C211yesmaximal, characteristic, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1853C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×D488C2212yes—
14C2×#14519751C2211yescharacteristic
15C2×#14519739C2211yescharacteristic, complemented
16C2×#14519737—21no—
17C22×#11758198—21no—
18C22×#11758198—21no—
19C22×C4×Dic61C2211yescharacteristic
20#16839898C418yescomplemented
21C2×#14519699C2212yes—
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yescomplemented
25C2×C22⋊#92418C2211yescharacteristic
26C2×#14519687C2212yescomplemented
27#16839663—21no—
28#16839663—21no—
29#16839663C2212yes—
30#16839663C2212yes—
31#16839663C2211yescharacteristic, complemented
32#16839658C2211yescharacteristic, complemented
33#16839651—21no—
34#16839639—21no—
35C4⋊C4×Dic61C2211yescharacteristic, complemented
36#16839609—21no—
37#16839524C2212yescomplemented
38#16839516C2212yescomplemented
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#106×C61C2212yescomplemented
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C61×#340C2212yescomplemented
44C24×D61—21no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—42no—
48C22×D244—42no—
49C22×D244C2×C414yescomplemented
50C22×D244C2311yescharacteristic
51C22×D244C2312yes—
52C22×C22⋊D61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C2×D488—42no—
57C2×D488—42no—
58C2×D488C2×C418yescomplemented
59C22×#92418C2312yes—
60#14519751—22no—
61#14519751C2×C414yes—
62#14519739—42no—
63#14519737—44no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×#11758198—44no—
75C2×#11758198—44no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—42no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82#14519699C2×C418yescomplemented
83#14519697—42no—
84#14519697—42no—
85#14519697—42no—
86C22⋊#92418—22no—
87#14519687—42no—
88#14519684—21no—
89#14519684—21no—
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—42no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99#106×C61—42no—
100C22⋊C8×C61C2×C414yescomplemented
101#95×C61—21no—
102C24×C61C2×D411yescharacteristic
103C23×D61—24no—
104C23×D61—21no—
105C23×D61—22no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61—42no—
110C23×D61—42no—
111C23×D61—42no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C22×Dic61—41no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244—48no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—48no—
151C2×D244C22×C414yes—
152C2×D244C22×C412yes—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61C22×C414yes—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165#11758198—24no—
166#11758198—42no—
167#11758198—44no—
168#11758198—44no—
169#11758198—44no—
170#11758198—42no—
171#11758198—42no—
172#11758198—44no—
173C4⋊Dic61—21no—
174C4⋊Dic61C22×C412yes—
175#11758196—42no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—41no—
179C4×Dic61—41no—
180C4×Dic61—42no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184D488—48no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61C4×D412yes—
190C23×C61AGL(1,ℤ/8)12yes—
191C23×C61AGL(1,ℤ/8)12yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—48no—
196C22×D61—48no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—41no—
208C22×D61—42no—
209C22×D61—48no—
210C22×D61—84no—
211C22×D61—84no—
212C22×D61—84no—
213C22×D61—84no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—84no—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)12yes—
232C2×C4×C61H(5,𝔽2)12yes—
233C22⋊D61—48no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61AGL(1,ℤ/8)12yes—
251D244—24no—
252D244—24no—
253D244—24no—
254D244—48no—
255D244—42no—
256D244—42no—
257D244—42no—
258C4×D61—84no—
259C8×C61—24no—
260#92418—42no—
261#22046—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61#44214yes—
269C22×C61C2×C4×D411yescharacteristic
270C22×C61#53411yescharacteristic
271C22×C61#53511yescharacteristic
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57511yescharacteristic
276C2×D61—44no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—44no—
280C2×D61—42no—
281C2×D61—44no—
282C2×D61—816no—
283C2×D61—81no—
284C2×D61—82no—
285C2×D61—81no—
286C2×D61—82no—
287C2×D61—81no—
288C2×D61—82no—
289C2×D61—81no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61—82no—
295C4×C61#51812yes—
296C4×C61#53812yes—
297Dic61—22no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—42no—
302C2×#447—611no—
303C4×#121—611no—
304C2×#390—611no—
305#1972—612no—
306#1936—612no—
307#1926—612no—
308#1879—612no—
309#1874—611no—
310#1853D6111yescharacteristic, complemented
311#1786—611no—
312#1749—611no—
313C2×C61—24no—
314C2×C61#232111yescharacteristic
315C2×C61C2×#44211yescharacteristic
316C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C61#319211yescharacteristic
318C2×C61#319411yescharacteristic
319C2×C61#321311yescharacteristic
320C2×C61#321711yescharacteristic
321D61—82no—
322D61—84no—
323C23×D4—611no—
324C22×D8—612no—
325C2×#121—611no—
326C2×C4×D4—611no—
327C2×C4×D4—1221no—
328C22×C4⋊C4C2×D6111yescharacteristic
329C22×C22⋊C4—1221no—
330C22×C22⋊C4—1221no—
331C22×C42—611no—
332#447—618no—
333C2×#106C2×D6112yescomplemented
334C2×#102—612no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8C2×D6111yescharacteristic
340C2×#97—612no—
341#390—612no—
342#390—612no—
343#390—611no—
344#390—1221no—
345#390—1221no—
346#389—611no—
347#388—1221no—
348#386—1221no—
349#380C2×D6111yescharacteristic
350C4×C4⋊C4—611no—
351C4×C22⋊C4—1221no—
352#340—612no—
353#340C2×D6112yescomplemented
354#336—612no—
355C61#2204611yescharacteristic, complemented, Sylow
356C25—1221no—
357C22×D4—614no—
358C22×D4—611no—
359C22×D4—612no—
360C22×D4—1221no—
361C22×D4—1221no—
362C22×D4—1222no—
363C22×D4—2442no—
364C22×D4—2442no—
365C23×C4—21no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370C2×D8—618no—
371C2×D8—2442no—
372C2×D8—2442no—
373C22×C8—612no—
374C22×C8C22×D6112yes—
375#121—614no—
376#121—1222no—
377C4×D4—2442no—
378C4×D4—2444no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6112yes—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—2444no—
396C2×C22⋊C4—2444no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—1221no—
402C2×C42—2442no—
403#106—42no—
404#102—618no—
405#102—2442no—
406#102—2442no—
407#102—2442no—
408C22⋊C8—1222no—
409C22⋊C8C4×D6114yescomplemented
410#97—2442no—
411#95—21no—
412#95—1221no—
413#95—1221no—
414C24—1221no—
415C24—1222no—
416C24—1224no—
417C24—2442no—
418C24—2442no—
419C24—2442no—
420C24—2442no—
421C24—2442no—
422C24—2442no—
423C24D4×D6111yescharacteristic
424C2×D4—614no—
425C2×D4—612no—
426C2×D4—1224no—
427C2×D4—1221no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1224no—
436C2×D4—2448no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4—2448no—
440C2×D4—2442no—
441C2×D4—2442no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4—2441no—
464C22×C4C2×C4×D6112yes—
465C22×C4D4×D6111yescharacteristic
466C22×C4#1175822612yes—
467C22×C4C23×D6111yescharacteristic
468D8—2448no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8C2×C4×D6114yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—2442no—
483C22⋊C4—42no—
484C22⋊C4—1224no—
485C22⋊C4—2444no—
486C22⋊C4—2444no—
487C22⋊C4—2444no—
488C22⋊C4—2444no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C42—1222no—
493C42—1222no—
494C42—2441no—
495C42—2441no—
496C42—2442no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—1222no—
502C23—1224no—
503C23—4884no—
504C23—4884no—
505C23—4884no—
506C23—4884no—
507C23—2448no—
508C23—2448no—
509C23—2442no—
510C23—2441no—
511C23—2442no—
512C23—2441no—
513C23—2441no—
514C23—2442no—
515C23—2441no—
516C23—2442no—
517C23—2441no—
518C23—2442no—
519C23—2442no—
520C23—2441no—
521C23—2448no—
522C23#1451973912yes—
523C23#1451980412yes—
524C23#1451980412yes—
525C23C2×D4×D6111yescharacteristic
526D4—1224no—
527D4—1224no—
528D4—1224no—
529D4—2448no—
530D4—2448no—
531D4—2442no—
532D4—2442no—
533D4—2442no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—612no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—4884no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4#1451973912yes—
567C2×C4C22×C4×D6111yescharacteristic
568C2×C4C2×#1175822611yescharacteristic
569C2×C4#1451982212yes—
570C2×C4#1451982312yes—
571C8—24no—
572C8—2442no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—48816no—
580C22—4882no—
581C22—4881no—
582C22—4882no—
583C22—4881no—
584C22—4881no—
585C22—4882no—
586C22—4882no—
587C22—4881no—
588C22—2442no—
589C22—2444no—
590C22—2444no—
591C22—2442no—
592C22—2442no—
593C22—2444no—
594C22#1683987314yes—
595C22C2×#1451973911yescharacteristic
596C22#1684032311yescharacteristic
597C22#1684032711yescharacteristic
598C22AGL(1,ℤ/8)×D6111yescharacteristic
599C22C2×#1451980411yescharacteristic
600C22C2×#1451980411yescharacteristic
601C22#1684056111yescharacteristic
602C4—22no—
603C4—81no—
604C4—82no—
605C4—81no—
606C4—1222no—
607C4—2442no—
608C4—2441no—
609C4—2441no—
610C4—2441no—
611C4#1684023412yes—
612C4#1684035212yes—
613C2—24no—
614C2—4884no—
615C2—4882no—
616C2#2144847611yescharacteristic
617C2C2×#1683987311yescharacteristic
618C2#2145462611yescharacteristic
619C2#2145661811yescharacteristic
620C2#2145663811yescharacteristic
621C2#2145680511yescharacteristic
622C2#2145684011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1853C2
socleC23×C61C22×D4
prime radicalC2×#16839898C2
cosocleC22×C4×C61C24
hypercenter#1853D61
hypocenterC61#22046
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. #21448325C2semidirect
3. C2×#16839663C2semidirect
4. #21446693C2semidirect
5. #21446514C2semidirect
6. #21446445C2semidirect
7. #21446168C2semidirect
8. #21446136C2semidirect
9. #21445749C2semidirect
10. #21445583C2semidirect
11. C61×#1853C2semidirect
15. C2×#14519739C22semidirect
20. #16839898C4semidirect
24. C2×#14519697C22semidirect
26. C2×#14519687C22semidirect
31. #16839663C22semidirect
32. #16839658C22semidirect
35. C4⋊C4×Dic61C22semidirect
37. #16839524C22semidirect
38. #16839516C22semidirect
40. C2×#106×C61C22semidirect
43. C61×#340C22semidirect
49. C22×D244C2×C4semidirect
58. C2×D488C2×C4semidirect
82. #14519699C2×C4semidirect
100. C22⋊C8×C61C2×C4semidirect
310. #1853D61semidirect
333. C2×#106C2×D61semidirect
353. #340C2×D61semidirect
355. C61#22046semidirect
409. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398982
#214483252
C2×#168396632
#214466932
#214465142
#214464452
#214461682
#214461362
#214457492
#214455832
C61×#18532
#2204661