Finite Group Databasebeta

← #29713902

Subgroups — #29713902

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×#16840416C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic, complemented
4#21447001C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446137C212yesmaximal, complemented
9#21446006C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2537C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758199C2211yescharacteristic
14#16840416—24no—
15#16840416—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519761—22no—
18C2×#14519757—21no—
19C2×#14519751C2212yes—
20C2×C4×D244C2211yescharacteristic
21C22×C4⋊Dic61C2212yes—
22#16840042—24no—
23C4×#11758199—24no—
24C2×#14519710C2212yes—
25C2×#14519710C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2212yes—
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#121×C61—21no—
38C2×#121×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#394—22no—
42C61×#384C2211yescharacteristic
43C61×#380—21no—
44C61×#380C2212yes—
45C2×#11758224—28no—
46C2×#11758224—44no—
47C2×#11758224—44no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—22no—
51C23×Dic61—22no—
52C22×C4×D61—21no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519777—22no—
60#14519777D414yescomplemented
61#14519761—216no—
62#14519757—28no—
63#14519751—24no—
64#14519751—24no—
65C4×D244—22no—
66C4×D244D414yescomplemented
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—21no—
74C2×#11758198—22no—
75#14519710—24no—
76#14519710—22no—
77C4⋊#92418—22no—
78#14519698—24no—
79#14519698—22no—
80#14519697—22no—
81#14519697—42no—
82C22⋊#92418—24no—
83C22⋊#92418—42no—
84C4×#92418—22no—
85C22×D4×C61—22no—
86C22×D4×C61D411yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61C2312yes—
90#121×C61—24no—
91#121×C61—42no—
92#121×C61—44no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61D412yes—
106C2×C42×C61C2311yescharacteristic
107#95×C61—22no—
108C24×C61—21no—
109C24×C61C2×D412yes—
110C23×D61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—44no—
119C2×D4×C61—44no—
120C2×D4×C61—82no—
121C2×D4×C61D814yescomplemented
122C2×D4×C61#4112yescomplemented
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—28no—
139C2×C22⋊D61—44no—
140C2×C22⋊D61—44no—
141C22×Dic61—28no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—44no—
145C22×Dic61—44no—
146#11758224—432no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C2×D412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×Dic122—24no—
157C2×Dic122—42no—
158C2×Dic122—42no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—81no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—44no—
168C22⋊C4×C61—44no—
169C22⋊C4×C61—42no—
170C42×C61C2×D412yes—
171C42×C61C2×D412yes—
172#11758199—28no—
173#11758198—28no—
174#11758198—24no—
175C4⋊Dic61—28no—
176C4⋊Dic61—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61C2×D412yes—
180C4⋊Dic61C2×D414yes—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—21no—
187C2×#92418—22no—
188C2×#92418—21no—
189C2×#92418#4112yescomplemented
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61C2×D814yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—28no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—88no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—84no—
234C2×C4×C61#12012yes—
235C2×C4×C61#12012yes—
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×C612−1+411yescharacteristic
244C22⋊D61—416no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—416no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—48no—
257Dic122—48no—
258#92418—22no—
259#92418—41no—
260#92418—42no—
261#24116—611nomaximal, Sylow
262C22×C61—24no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×#12011yescharacteristic
277C22×C61#54611yescharacteristic
278C22×C61#54711yescharacteristic
279C22×C61C22×D811yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57811yescharacteristic
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—42no—
292C4×C61—81no—
293C4×C61—82no—
294C4×C61#44714yes—
295C4×C61#52511yescharacteristic
296C4×C61#53711yescharacteristic
297C4×C61#54811yescharacteristic
298C4×C61#55011yescharacteristic
299Dic61—44no—
300Dic61—42no—
301C2×#547—611no—
302C2×#474—611no—
303C2×C4×D8—611no—
304#2537D6111yescharacteristic, complemented
305#2033—612no—
306#1999—612no—
307#1990—611no—
308#1954—611no—
309#1874—612no—
310#1852—612no—
311#1802—611no—
312C2×C61—44no—
313C2×C61—82no—
314C2×C61#240711yescharacteristic
315C2×C61C2×#44711yescharacteristic
316C2×C61#301311yescharacteristic
317C2×C61#329411yescharacteristic
318C2×C61#333011yescharacteristic
319C2×C61#333411yescharacteristic
320C2×C61#335411yescharacteristic
321D61—82no—
322C22×#41—1221no—
323C22×D8—611no—
324#547—1224no—
325#547—1224no—
326C2×C4⋊Q8—611no—
327C2×#123—1222no—
328C2×C22⋊Q8—1221no—
329C2×#121—21no—
330C2×#121—612no—
331C2×#121C2×D6112yes—
332C2×C4×D4—611no—
333C2×C4×D4C2×D6111yescharacteristic
334C2×C4×D4C2×D6111yescharacteristic
335C22×C4⋊C4—612no—
336#474—1224no—
337C4×D8—1224no—
338C2×#107—611no—
339C2×#107—612no—
340C2×C4⋊C8—611no—
341C2×#102—611no—
342C2×#102—612no—
343C2×#102—611no—
344C2×#102—612no—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—612no—
347C2×C4×C8—611no—
348#394—22no—
349#384C2×D6111yescharacteristic
350#380—21no—
351#380C2×D6112yes—
352#340—612no—
353#340—612no—
354#340—612no—
355C61#2411611yescharacteristic, complemented, Sylow
356C2×#41—1228no—
357C2×#41—2444no—
358C2×#41—2444no—
359C22×Q8—1221no—
360C22×D4—22no—
361C22×D4—611no—
362C22×D4—1222no—
363C22×D4C22⋊D6111yescharacteristic
364C22×D4C22×D6111yescharacteristic
365C23×C4—21no—
366C23×C4—1222no—
367C23×C4—1221no—
368C23×C4C22×D6112yes—
369C2×D8—1222no—
370C2×D8—1222no—
371C2×D8—1222no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—612no—
375C4⋊Q8—614no—
376C4⋊Q8—1222no—
377#123—12216no—
378C22⋊Q8—1228no—
379#121—24no—
380#121—42no—
381#121—44no—
382#121—1224no—
383#121—1224no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—614no—
389C4×D4—1222no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22⋊D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6112yes—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4C22⋊D6112yes—
407C2×C42C22×D6111yescharacteristic
408#107—1222no—
409#107—1224no—
410C4⋊C8—1222no—
411#102—1222no—
412#102—1222no—
413#102—1224no—
414#102—2442no—
415C22⋊C8—1224no—
416C22⋊C8—2442no—
417C4×C8—1222no—
418#95—22no—
419C24—21no—
420C24—1221no—
421C24C2×C22⋊D6112yes—
422#41—24432no—
423C2×Q8—1224no—
424C2×Q8—2442no—
425C2×Q8—2442no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—82no—
436C2×D4—612no—
437C2×D4—1228no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2444no—
444C2×D4—2444no—
445C2×D4#1175819914yescomplemented
446C2×D4#1175822412yescomplemented
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—1228no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—2444no—
463C22×C4—2444no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4C2×C22⋊D6112yes—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470D8—1228no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1221no—
478C2×C8—1222no—
479C2×C8—1221no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—612no—
488C4⋊C4—614no—
489C4⋊C4—1228no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—1228no—
498C22⋊C4—1224no—
499C42D4×D6112yes—
500C42D4×D6112yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—1222no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23C2×#1175819914yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—2448no—
524D4—22no—
525D4—28no—
526D4—22no—
527D4—41no—
528D4—41no—
529D4—88no—
530D4—1222no—
531D4—1222no—
532D4—24416no—
533D4—2441no—
534D4—2441no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—1222no—
552C2×C4—1224no—
553C2×C4—24416no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2442no—
558C2×C4—2448no—
559C2×C4#1451974512yes—
560C2×C4#1451974512yes—
561C2×C4C2×#1175819912yes—
562C2×C4#1451980212yes—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822411yescharacteristic
567C2×C4#1451982211yescharacteristic
568C2×C4#1451982411yescharacteristic
569C8—1222no—
570C8—2441no—
571C8—2442no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22—2442no—
591C22—2442no—
592C22—2442no—
593C22C2×#1451974511yescharacteristic
594C22#1684040611yescharacteristic
595C22#1684041111yescharacteristic
596C22C22×#1175819911yescharacteristic
597C22C2×#1451980211yescharacteristic
598C22#1684057311yescharacteristic
599C22#1684057711yescharacteristic
600C4—22no—
601C4—42no—
602C4—81no—
603C4—82no—
604C4—2442no—
605C4—2444no—
606C4#1683989514yes—
607C4#1684027211yescharacteristic
608C4#1684034111yescharacteristic
609C4#1684041811yescharacteristic
610C4#1684043011yescharacteristic
611C2—44no—
612C2—82no—
613C2—4882no—
614C2#2144916011yescharacteristic
615C2C2×#1683989511yescharacteristic
616C2#2145513111yescharacteristic
617C2#2145763211yescharacteristic
618C2#2145812411yescharacteristic
619C2#2145816211yescharacteristic
620C2#2145840811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2537C2
socleC23×C61C22×D4
hypercenter#2537D61
hypocenterC61#24116
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16840416C2semidirect
2. C2×#16840042C2semidirect
3. C2×C4×#11758199C2semidirect
4. #21447001C2semidirect
5. #21446815C2semidirect
6. #21446779C2semidirect
7. #21446614C2semidirect
8. #21446137C2semidirect
9. #21446006C2semidirect
10. #21445816C2semidirect
11. C61×#2537C2semidirect
60. #14519777D4semidirect
66. C4×D244D4semidirect
121. C2×D4×C61D8semidirect
122. C2×D4×C61#41semidirect
189. C2×#92418#41semidirect
304. #2537D61semidirect
355. C61#24116semidirect
445. C2×D4#11758199semidirect
446. C2×D4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168400422
C2×C4×#117581992
#214470012
#214468152
#214467792
#214466142
#214461372
#214460062
#214458162
C61×#25372
#2411661