Finite Group Databasebeta

← #29730462

Subgroups — #29730462

Order 15616.

1290 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840138C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic, complemented
3C2×#16839954C212yesmaximal, complemented
4#21448883C211yesmaximal, characteristic, complemented
5#21447129C212yesmaximal, complemented
6#21446890C212yesmaximal, complemented
7#21446747C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446170C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#2541C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C22×#11758199C2212yes—
14C22×#11758199C2212yes—
15C2×#14519774C2211yescharacteristic
16C2×C4×D244C2212yes—
17C2×#14519739—21no—
18#16840138—24no—
19#16839957—24no—
20#16839954—28no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×C4⋊#92418C2212yes—
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×#14519698C2211yescharacteristic
27C2×#14519697C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839664C2212yes—
30#16839663—21no—
31#16839640—22no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839517C2212yes—
36C2×#127×C61C2211yescharacteristic
37C2×#127×C61C2212yes—
38C2×#121×C61—21no—
39C2×#121×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#392—22no—
42C61×#390C2212yes—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C4×D61—22no—
48C2×#11758199—24no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199—24no—
52C2×#11758199—24no—
53C2×#11758199—24no—
54C2×#11758199—42no—
55C2×#11758199—42no—
56C2×#11758199—42no—
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59C22×#92418C2311yescharacteristic
60#14519774—22no—
61#14519774D414yescomplemented
62C4×D244—24no—
63C4×D244—42no—
64#14519739—44no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61—21no—
71C4⋊#92418—22no—
72C4⋊#92418—24no—
73#14519698—24no—
74#14519698—22no—
75#14519698—42no—
76#14519698D414yescomplemented
77#14519697—24no—
78C4×#92418—22no—
79#14519684—21no—
80C22×D4×C61—21no—
81C22×D4×C61—22no—
82C22×D4×C61—22no—
83C22×D4×C61—21no—
84C22×D4×C61C2312yes—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87#127×C61—24no—
88#127×C61—22no—
89#127×C61—22no—
90#127×C61—42no—
91#121×C61—44no—
92#121×C61—44no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2312yes—
102C2×C42×C61C2311yescharacteristic
103#95×C61D412yes—
104C24×C61—21no—
105C24×C61—22no—
106C23×D61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—44no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—44no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—82no—
127C2×D4×C61—81no—
128C2×D4×C61—81no—
129C2×D4×C61—82no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—21no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—22no—
148C2×D244—24no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—81no—
152C2×D244—81no—
153C2×D244C2×D414yes—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—82no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61D814yescomplemented
161C22⋊C4×C61—44no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—44no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61C2×D412yes—
167C42×C61#4112yes—
168#11758199—48no—
169#11758199—48no—
170#11758199—48no—
171#11758198—44no—
172#11758198—44no—
173#11758198—82no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61C2×D412yes—
178C4×Dic61—42no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C2×D412yes—
187C2×#92418#4112yes—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61C22×D411yescharacteristic
203C22×D61—24no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—42no—
213D4×C61—41no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—41no—
217D4×C61—42no—
218D4×C61—88no—
219D4×C61—84no—
220D4×C61—88no—
221D4×C61—84no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—84no—
233C2×C4×C61#12112yes—
234C2×C4×C61#12112yes—
235C2×C4×C61C2×D814yes—
236C2×C4×C61C2×D812yes—
237C2×C4×C61AGL(1,ℤ/8)12yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250D244—24no—
251D244—24no—
252D244—42no—
253D244—42no—
254D244—48no—
255D244—84no—
256C4×D61—88no—
257#92418—22no—
258#92418—22no—
259#92418—42no—
260#25364—611nomaximal, Sylow
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61C2×#12111yescharacteristic
276C22×C61D4211yescharacteristic
277C22×C61#55011yescharacteristic
278C22×C61C22×D811yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57611yescharacteristic
281C22×C61#57611yescharacteristic
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C2×D61—82no—
289C4×C61—24no—
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61#45914yescomplemented
294C4×C61#53211yescharacteristic
295C4×C61#54011yescharacteristic
296C4×C61D4211yescharacteristic
297C4×C61#54611yescharacteristic
298Dic61—42no—
299Dic61—81no—
300C2×#493—611no—
301C2×#459—612no—
302C2×#459—611no—
303#2541D6111yescharacteristic, complemented
304#2366—611no—
305#2054—612no—
306#2011—612no—
307#1982—611no—
308#1939—611no—
309#1879—612no—
310C4×#102—611no—
311C2×C61—82no—
312C2×C61—162no—
313C2×C61#242811yescharacteristic
314C2×C61C2×#45911yescharacteristic
315C2×C61#304111yescharacteristic
316C2×C61#327411yescharacteristic
317C2×C61D4×D811yescharacteristic
318C2×C61#339111yescharacteristic
319C2×C61#339911yescharacteristic
320D61—84no—
321C22×D8—612no—
322C22×D8—612no—
323C22×D8—612no—
324C2×#127—611no—
325C2×#127C2×D6111yescharacteristic
326C2×#127C2×D6112yes—
327C2×#121—21no—
328C2×#121—21no—
329C2×C4×D4—612no—
330C2×C4×D4—1221no—
331C2×C4×D4C2×D6111yescharacteristic
332#493—1224no—
333#459—1228no—
334#459—1224no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—611no—
337C2×#102—611no—
338C2×#102—612no—
339C2×#102—611no—
340C2×#102—612no—
341C2×#102—612no—
342C2×C4×C8—611no—
343#392—22no—
344#390—612no—
345#390—1221no—
346#390C2×D6112yes—
347#386—1222no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4C2×D6111yescharacteristic
350#340—611no—
351#340—611no—
352#336—612no—
353C61#2536411yescharacteristic, complemented, Sylow
354C22×D4—21no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—21no—
358C22×D4—612no—
359C22×D4—1221no—
360C22×D4—1221no—
361C22×D4C22×D6112yes—
362C22×D4C22×D6111yescharacteristic
363C23×C4—21no—
364C23×C4—1222no—
365C2×D8—1224no—
366C2×D8—1224no—
367C2×D8—1224no—
368C2×D8—1224no—
369C2×D8—1224no—
370C2×D8—1224no—
371C2×D8—2442no—
372C2×D8—2442no—
373C2×D8—2442no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8—612no—
377#127—24no—
378#127—22no—
379#127—22no—
380#127—42no—
381#127—614no—
382#127—1222no—
383#121—44no—
384#121—44no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—1224no—
388C4×D4—2442no—
389C4×D4—2444no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4C22⋊D6111yescharacteristic
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C42—1221no—
402C2×C42C22×D6112yes—
403C2×C42C22×D6111yescharacteristic
404C4⋊C8—1224no—
405C4⋊C8—1222no—
406#102—614no—
407#102—1224no—
408#102—1224no—
409#102—1222no—
410#102—2442no—
411C4×C8—1222no—
412#95—1221no—
413#95C22⋊D6112yes—
414C24—21no—
415C24—22no—
416C24—1222no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—82no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—82no—
440C2×D4—614no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—4881no—
449C2×D4—4881no—
450C2×D4—2442no—
451C2×D4—2442no—
452C2×D4D4×D6112yes—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—4882no—
460C22×C4—2444no—
461C22×C4—2444no—
462C22×C4C2×C22⋊D6112yes—
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6112yes—
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C23×D6111yescharacteristic
467D8—2448no—
468D8—2448no—
469D8—2448no—
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—612no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4#1175819914yescomplemented
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—4882no—
491C22⋊C4—2444no—
492C22⋊C4—2444no—
493C42—22no—
494C42—22no—
495C42—2442no—
496C42D4×D6112yes—
497C42#1175822412yes—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—1224no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23C22×C22⋊D6111yescharacteristic
520D4—22no—
521D4—22no—
522D4—42no—
523D4—41no—
524D4—42no—
525D4—42no—
526D4—41no—
527D4—42no—
528D4—88no—
529D4—84no—
530D4—88no—
531D4—84no—
532D4—1224no—
533D4—1224no—
534D4—4884no—
535D4—2448no—
536D4—2442no—
537D4—2442no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4—1222no—
550C2×C4—4888no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451974812yes—
557C2×C4#1451974812yes—
558C2×C4C2×#1175819914yes—
559C2×C4C2×#1175819912yes—
560C2×C4#1451980312yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4#1451982211yescharacteristic
566C2×C4#1451982311yescharacteristic
567C8—1222no—
568C8—1222no—
569C8—2442no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—4882no—
585C22—4882no—
586C22—4882no—
587C22—4882no—
588C22—2444no—
589C22—2444no—
590C22—2444no—
591C22C2×#1451974811yescharacteristic
592C22D4×C22⋊D6111yescharacteristic
593C22#1684043111yescharacteristic
594C22C22×#1175819911yescharacteristic
595C22C2×#1451980311yescharacteristic
596C22#1684056511yescharacteristic
597C22#1684056511yescharacteristic
598C4—24no—
599C4—22no—
600C4—42no—
601C4—81no—
602C4—4881no—
603C4—2442no—
604C4#1683995414yescomplemented
605C4#1684030711yescharacteristic
606C4#1684036511yescharacteristic
607C4#1684039611yescharacteristic
608C4#1684040911yescharacteristic
609C2—82no—
610C2—162no—
611C2—4884no—
612C2#2144940311yescharacteristic
613C2C2×#1683995411yescharacteristic
614C2#2145531811yescharacteristic
615C2#2145743511yescharacteristic
616C2D4×#1175819911yescharacteristic
617C2#2145879111yescharacteristic
618C2#2145886111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2541C2
socleC23×C61C22×D4
hypercenter#2541D61
hypocenterC61#25364
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16840138C2semidirect
2. C2×#16839957C2semidirect
3. C2×#16839954C2semidirect
4. #21448883C2semidirect
5. #21447129C2semidirect
6. #21446890C2semidirect
7. #21446747C2semidirect
8. #21446538C2semidirect
9. #21446170C2semidirect
10. C4×#14519698C2semidirect
11. C61×#2541C2semidirect
61. #14519774D4semidirect
76. #14519698D4semidirect
160. C4⋊C4×C61D8semidirect
293. C4×C61#459semidirect
303. #2541D61semidirect
353. C61#25364semidirect
486. C4⋊C4#11758199semidirect
604. C4#16839954semidirect

Maximal subgroups

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

subgroupindex
C2×#168401382
C2×#168399572
C2×#168399542
#214488832
#214471292
#214468902
#214467472
#214465382
#214461702
C4×#145196982
C61×#25412
#2536461