Finite Group Databasebeta

← #29714364

Subgroups — #29714364

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16840047C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4#21447008C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446771C211yesmaximal, characteristic, complemented
7#21446746C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal, complemented
9#21446079C212yesmaximal, complemented
10#21445799C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758201C2212yes—
14#16840312—24no—
15#16840312—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16840047—24no—
24#16839962—24no—
25C2×#14519708C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C22⋊Q8×C61C2211yescharacteristic
37C2×C4×Q8×C61C2211yescharacteristic
38C22×C42×C61C2211yescharacteristic
39C61×#391C2211yescharacteristic
40C61×#389C2211yescharacteristic
41C61×#387—21no—
42C61×#385—21no—
43C61×#380C2211yescharacteristic
44C4×C4⋊C4×C61—21no—
45C4×C4⋊C4×C61C2211yescharacteristic
46C4×C22⋊C4×C61—21no—
47C2×#11758222—28no—
48C2×#11758222—44no—
49C2×#11758222—44no—
50C22×Dic122—21no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C22×C4×D61—22no—
56C2×#11758201—24no—
57C2×#11758201—24no—
58C2×#11758201—42no—
59C22×#92418C2312yes—
60C22×#92418C2312yes—
61#14519777—22no—
62#14519777D414yes—
63#14519774—22no—
64#14519774D414yes—
65#14519765—28no—
66#14519751—28no—
67#14519751—24no—
68#14519751—24no—
69C4×D244—28no—
70C4×D244—24no—
71C4×D244—24no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×#11758198—22no—
75C2×#11758198—22no—
76#14519708—24no—
77C4⋊#92418—22no—
78C4⋊#92418—22no—
79#14519701—24no—
80#14519697—24no—
81#14519697—24no—
82#14519697—42no—
83C22⋊#92418—42no—
84C22×Q8×C61C2311yescharacteristic
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61—21no—
103C2×C42×C61—21no—
104C2×C42×C61D411yescharacteristic
105C2×C42×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#95×C61—21no—
109#95×C61—21no—
110#95×C61—21no—
111#95×C61—21no—
112#95×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61#4112yescomplemented
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—41no—
135C22×C4×C61—41no—
136C22×C4×C61SD1614yescomplemented
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—28no—
144C2×C22⋊D61—44no—
145C2×C22⋊D61—44no—
146C22×Dic61—22no—
147#11758222—432no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—22no—
155C2×D244—42no—
156C2×D244—42no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C2×D414yes—
160C2×C4×D61—28no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×Dic122—24no—
164C2×Dic122—42no—
165C2×Dic122—42no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61#4112yescomplemented
175C22⋊C4×C61—24no—
176C22⋊C4×C61—42no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—42no—
182C42×C61—42no—
183C42×C61C2×D412yes—
184C42×C61C2×D412yes—
185#11758201—48no—
186#11758198—28no—
187#11758198—28no—
188C4⋊Dic61—24no—
189C4⋊Dic61—22no—
190C4⋊Dic61C2×D414yes—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61C2×SD1612yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—24no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—28no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—44no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61#12012yes—
245C2×C4×C61#12012yes—
246C2×C4×C61C2×SD1612yes—
247C2×C4×C61C2×SD1612yes—
248C2×C4×C61AGL(1,ℤ/8)12yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C22⋊D61—416no—
256C2×Dic61—24no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259D244—24no—
260D244—24no—
261D244—48no—
262D244—42no—
263D244—42no—
264D244—48no—
265C4×D61—416no—
266Dic122—48no—
267#92418—42no—
268#92418—42no—
269#24154—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C2×#12011yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61#53711yescharacteristic
280C22×C61C22×SD1611yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61#57511yescharacteristic
283C22×C61#57511yescharacteristic
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61#45014yes—
300C4×C61#54611yescharacteristic
301C4×C61#54711yescharacteristic
302C4×C61#54811yescharacteristic
303C4×C61#55011yescharacteristic
304Dic61—44no—
305C2×#532—611no—
306C2×#475—611no—
307C2×#460—611no—
308C4×C22⋊Q8D6111yescharacteristic, complemented
309#2034—612no—
310#2000—612no—
311#1988—611no—
312#1982—611no—
313#1877—612no—
314#1864—612no—
315#1797—611no—
316C2×C61—42no—
317C2×C61#240711yescharacteristic
318C2×C61C2×#45011yescharacteristic
319C2×C61#300511yescharacteristic
320C2×C61#310311yescharacteristic
321C2×C61#311511yescharacteristic
322C2×C61#324611yescharacteristic
323C2×C61#325111yescharacteristic
324D61—84no—
325C22×#41—1221no—
326C22×SD16—612no—
327#532—1224no—
328#532—1224no—
329C2×C4⋊Q8—611no—
330C2×#127—611no—
331C2×#124—1221no—
332C2×C22⋊Q8C2×D6111yescharacteristic
333C2×#121—612no—
334C2×#121—1221no—
335C2×C4×Q8C2×D6111yescharacteristic
336C2×C4×D4—612no—
337C2×C4×D4—1221no—
338C22×C42C2×D6111yescharacteristic
339#475—1224no—
340#460—1224no—
341C2×#106—612no—
342C2×C4⋊C8—611no—
343C2×C4⋊C8—611no—
344C2×#103—612no—
345C2×#102—612no—
346C2×#102—612no—
347C2×#102—612no—
348C2×C22⋊C8—612no—
349#391C2×D6111yescharacteristic
350#389C2×D6111yescharacteristic
351#387—21no—
352#385—21no—
353#380C2×D6111yescharacteristic
354C4×C4⋊C4—21no—
355C4×C4⋊C4C2×D6111yescharacteristic
356C4×C22⋊C4—21no—
357#340—612no—
358#340—612no—
359#336—612no—
360C61#2415411yescharacteristic, complemented, Sylow
361C2×#41—1228no—
362C2×#41—2444no—
363C2×#41—2444no—
364C22×Q8—1221no—
365C22×Q8C22×D6111yescharacteristic
366C22×D4—612no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—1221no—
370C23×C4—1222no—
371C23×C4C22⋊D6111yescharacteristic
372C23×C4C22⋊D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C2×SD16—1224no—
375C2×SD16—1224no—
376C2×SD16—2442no—
377C22×C8—612no—
378C22×C8—612no—
379C4⋊Q8—614no—
380C4⋊Q8—1222no—
381#127—614no—
382#127—1222no—
383#124—1228no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—1224no—
387#121—1224no—
388#121—1228no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—1224no—
392C4×D4—1224no—
393C4×D4—1228no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1222no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22⋊D6111yescharacteristic
411C2×C42C22⋊D6111yescharacteristic
412C2×C42C22×D6111yescharacteristic
413C2×C42C22×D6111yescharacteristic
414#106—1224no—
415C4⋊C8—1222no—
416C4⋊C8—1222no—
417#103—1224no—
418#102—1224no—
419#102—1224no—
420#102—2442no—
421C22⋊C8—2442no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C24—1222no—
428C24C2×C22⋊D6111yescharacteristic
429#41—24432no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1224no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8#1175822212yescomplemented
439C2×D4—614no—
440C2×D4—1228no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2442no—
453C2×D4—2442no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—1228no—
470C22×C4—1222no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4#1175820114yescomplemented
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480SD16—2448no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—614no—
496C4⋊C4—1222no—
497C4⋊C4—1224no—
498C4⋊C4#1175822212yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1228no—
502C22⋊C4—1228no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42D4×D6112yes—
510C42D4×D6112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—1224no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23C2×#1175820112yes—
525C23C22×C22⋊D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2448no—
531D4—1224no—
532D4—1224no—
533D4—2448no—
534D4—24416no—
535D4—2442no—
536D4—2448no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1224no—
567C2×C4—24416no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4#1451974512yes—
571C2×C4#1451974512yes—
572C2×C4C2×#1175820112yes—
573C2×C4C2×#1175820112yes—
574C2×C4#1451980212yes—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4#1451982311yescharacteristic
580C2×C4#1451982411yescharacteristic
581C8—2442no—
582C8—2442no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—4882no—
591C22—4882no—
592C22—4882no—
593C22—4882no—
594C22—2444no—
595C22—2444no—
596C22—2444no—
597C22C2×#1451974511yescharacteristic
598C22#1684027311yescharacteristic
599C22#1684034011yescharacteristic
600C22C22×#1175820111yescharacteristic
601C22C2×#1451980211yescharacteristic
602C22#1684056011yescharacteristic
603C22#1684056011yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—2444no—
613C4#1683991214yes—
614C4#1684040811yescharacteristic
615C4#1684041711yescharacteristic
616C4#1684042111yescharacteristic
617C4#1684043311yescharacteristic
618C2—42no—
619C2—4884no—
620C2#2144916411yescharacteristic
621C2C2×#1683991211yescharacteristic
622C2#2145505911yescharacteristic
623C2#2145585011yescharacteristic
624C2#2145597011yescharacteristic
625C2#2145713111yescharacteristic
626C2#2145718111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenterC4×C22⋊Q8D61
hypocenterC61#24154
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840047C2semidirect
3. C2×#16839962C2semidirect
4. #21447008C2semidirect
5. #21446820C2semidirect
6. #21446771C2semidirect
7. #21446746C2semidirect
8. #21446156C2semidirect
9. #21446079C2semidirect
10. #21445799C2semidirect
11. C4×C22⋊Q8×C61C2semidirect
120. C2×Q8×C61#41semidirect
136. C22×C4×C61SD16semidirect
174. C4⋊C4×C61#41semidirect
308. C4×C22⋊Q8D61semidirect
360. C61#24154semidirect
438. C2×Q8#11758222semidirect
473. C22×C4#11758201semidirect
498. C4⋊C4#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168400472
C2×#168399622
#214470082
#214468202
#214467712
#214467462
#214461562
#214460792
#214457992
C4×C22⋊Q8×C612
#2415461