Finite Group Databasebeta

← #29713908

Subgroups — #29713908

Order 15616.

1434 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic, complemented
4#21447003C212yesmaximal, complemented
5#21446816C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758199C2211yescharacteristic
14#16840415—24no—
15#16840415—24no—
16C2×#14519762—22no—
17C2×#14519753—21no—
18C2×#14519752C2212yes—
19C2×#14519751—21no—
20C2×#14519748C2212yes—
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24C4×#11758199—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519689C2212yes—
30C2×#14519689C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839678—22no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#474—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C2×#11758226—28no—
46C2×#11758226—44no—
47C2×#11758226—44no—
48C22×D244—22no—
49C22×D244D411yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—21no—
54C22×C4×D61C2312yes—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C2×#11758199—22no—
58C22×#92418C2311yescharacteristic
59C22×#92418C2311yescharacteristic
60#14519762—216no—
61#14519753—28no—
62#14519752—24no—
63#14519752—24no—
64#14519751—44no—
65#14519748—24no—
66#14519748—42no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D414yescomplemented
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—22no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84#14519710—22no—
85#14519699—42no—
86#14519698—22no—
87#14519697—22no—
88#14519689—24no—
89#14519689—42no—
90C4×#92418—22no—
91#14519684—22no—
92C22×Q8×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C22×C8×C61C2312yes—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61D414yescomplemented
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#107×C61—24no—
106C4⋊C8×C61—22no—
107#102×C61—24no—
108C24×C61—21no—
109C23×D61—21no—
110C23×D61C2×D412yes—
111C2×Q8×C61—24no—
112C2×Q8×C61—42no—
113C2×Q8×C61—42no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61#4112yescomplemented
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—82no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61C2×D412yes—
134#11758226—432no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—28no—
142C2×D244—21no—
143C2×D244—44no—
144C2×D244—44no—
145C2×D244D814yescomplemented
146C2×D244C2×D412yes—
147C2×C4×D61—28no—
148C2×C4×D61—24no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C2×D414yes—
167C22⋊C4×C61—42no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—44no—
171#11758199—28no—
172#11758198—28no—
173#11758198—24no—
174#11758198—24no—
175#11758198—44no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—42no—
181C4⋊Dic61—81no—
182C4⋊Dic61C2×D412yes—
183#11758196—28no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418#4112yescomplemented
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61C22×D411yescharacteristic
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61C2×D814yes—
212Q8×C61—48no—
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—24no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—84no—
226C2×C4×C61#12012yes—
227C2×C4×C61#12012yes—
228C2×C4×C61C2×D812yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C22⋊D61—88no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246D244—22no—
247D244—22no—
248D244—28no—
249D244—416no—
250D244—41no—
251D244—41no—
252C4×D61—416no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—48no—
258C8×C61—42no—
259#92418—22no—
260#92418—41no—
261#24116—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12011yescharacteristic
270C22×C61#54611yescharacteristic
271C22×C61#54711yescharacteristic
272C22×C61C22×D811yescharacteristic
273C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57811yescharacteristic
276C2×D61—24no—
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—22no—
291C4×C61—44no—
292C4×C61—81no—
293C4×C61#44714yes—
294C4×C61#52511yescharacteristic
295C4×C61#53711yescharacteristic
296C4×C61#54811yescharacteristic
297C4×C61#55011yescharacteristic
298Dic61—42no—
299Dic61—42no—
300Dic61—82no—
301C2×#547—611no—
302C2×#474D6111yescharacteristic, complemented
303C2×C4×D8—611no—
304#2537—611no—
305#2033—612no—
306#1999—612no—
307#1990—611no—
308#1954—611no—
309#1874—612no—
310#1852—612no—
311#1802—611no—
312C2×C61—82no—
313C2×C61#240711yescharacteristic
314C2×C61C2×#44711yescharacteristic
315C2×C61#301311yescharacteristic
316C2×C61#329411yescharacteristic
317C2×C61#333011yescharacteristic
318C2×C61#333411yescharacteristic
319C2×C61#335411yescharacteristic
320D61—44no—
321D61—82no—
322C22×#41—1221no—
323C22×D8—611no—
324#547—1224no—
325#547—1224no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#123—1222no—
328C2×C22⋊Q8—1221no—
329C2×#121—612no—
330C2×#121—612no—
331C2×#121—1221no—
332C2×C4×D4—611no—
333C2×C4×D4—611no—
334C2×C4×D4C2×D6111yescharacteristic
335C22×C4⋊C4—612no—
336#474—24no—
337C4×D8—1224no—
338C2×#107—611no—
339C2×#107C2×D6112yes—
340C2×C4⋊C8C2×D6111yescharacteristic
341C2×#102—611no—
342C2×#102—611no—
343C2×#102—612no—
344C2×#102C2×D6112yes—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—612no—
347C2×C4×C8—611no—
348#394—1222no—
349#384—611no—
350#380—612no—
351#380—1221no—
352#340—612no—
353#340—612no—
354#340—612no—
355C61#2411611yescharacteristic, complemented, Sylow
356C2×#41—1228no—
357C2×#41—2444no—
358C2×#41—2444no—
359C22×Q8—21no—
360C22×D4—611no—
361C22×D4—611no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4C22×D6111yescharacteristic
365C23×C4—21no—
366C23×C4—612no—
367C23×C4—1222no—
368C23×C4—1221no—
369C2×D8—1222no—
370C2×D8—1222no—
371C2×D8—1222no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6112yes—
375C4⋊Q8—22no—
376C4⋊Q8C22⋊D6114yescomplemented
377#123—12216no—
378C22⋊Q8—1228no—
379#121—1224no—
380#121—1224no—
381#121—1224no—
382#121—2442no—
383#121—2444no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—614no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6112yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C42C22×D6111yescharacteristic
408#107—24no—
409#107—1222no—
410C4⋊C8—22no—
411#102—24no—
412#102—1222no—
413#102—1222no—
414#102—2442no—
415C22⋊C8—1224no—
416C22⋊C8—2442no—
417C4×C8—1222no—
418#95—1222no—
419C24—21no—
420C24—612no—
421C24—1221no—
422#41—24432no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—612no—
432C2×D4—614no—
433C2×D4—1228no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—4882no—
442C2×D4—2444no—
443C2×D4—2444no—
444C2×D4—2444no—
445C2×D4—2444no—
446C2×D4#1175822612yescomplemented
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—612no—
453C22×C4—1228no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—1222no—
463C22×C4—2444no—
464C22×C4—2444no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470D8—1228no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—612no—
485C4⋊C4—1228no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—4881no—
491C4⋊C4—2442no—
492C4⋊C4C2×C22⋊D6114yes—
493C22⋊C4—42no—
494C22⋊C4—1228no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—2444no—
498C22⋊C4—2444no—
499C42D4×D6112yes—
500C42C2×C22⋊D6112yes—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—614no—
509C23—1222no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1222no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23C2×D4×D6111yescharacteristic
523Q8—48no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528D4—1222no—
529D4—1222no—
530D4—1228no—
531D4—4888no—
532D4—24416no—
533D4—2441no—
534D4—2441no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—24416no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2448no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4#1451974512yes—
560C2×C4#1451974512yes—
561C2×C4D8×D6112yes—
562C2×C4#1451980112yes—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822611yescharacteristic
567C2×C4#1451982311yescharacteristic
568C2×C4#1451982511yescharacteristic
569C8—42no—
570C8—1222no—
571C8—2441no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—1224no—
580C22—1224no—
581C22—1224no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22—2442no—
593C22C2×#1451974511yescharacteristic
594C22#1684040311yescharacteristic
595C22#1684041411yescharacteristic
596C22C2×D8×D6111yescharacteristic
597C22C2×#1451980111yescharacteristic
598C22#1684057111yescharacteristic
599C22#1684057811yescharacteristic
600C4—22no—
601C4—44no—
602C4—81no—
603C4—4882no—
604C4—2442no—
605C4—2442no—
606C4#1683989414yes—
607C4#1684027511yescharacteristic
608C4#1684034211yescharacteristic
609C4#1684042311yescharacteristic
610C4#1684043111yescharacteristic
611C2—82no—
612C2—4882no—
613C2—2444no—
614C2#2144916611yescharacteristic
615C2C2×#1683989411yescharacteristic
616C2#2145513311yescharacteristic
617C2#2145762911yescharacteristic
618C2#2145812111yescharacteristic
619C2#2145816511yescharacteristic
620C2#2145841711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
hypercenterC2×#474D61
hypocenterC61#24116
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840415C2semidirect
2. C2×C4×#11758199C2semidirect
3. #21450732C2semidirect
4. #21447003C2semidirect
5. #21446816C2semidirect
6. #21446781C2semidirect
7. #21446612C2semidirect
8. #21446133C2semidirect
9. #21446005C2semidirect
10. #21445820C2semidirect
11. C2×C61×#474C2semidirect
70. C4×D244D4semidirect
97. C4⋊Q8×C61D4semidirect
119. C2×D4×C61#41semidirect
145. C2×D244D8semidirect
189. C2×#92418#41semidirect
302. C2×#474D61semidirect
355. C61#24116semidirect
376. C4⋊Q8C22⋊D61semidirect
446. C2×D4#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×C4×#117581992
#214507322
#214470032
#214468162
#214467812
#214466122
#214461332
#214460052
#214458202
C2×C61×#4742
#2411661