Finite Group Databasebeta

← #29714796

Subgroups — #29714796

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840434C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21447053C212yesmaximal, complemented
5#21446906C212yesmaximal, complemented
6#21446756C211yesmaximal, characteristic, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal, complemented
9#21446079C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2582C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758201C2211yescharacteristic
14C22×#11758201C2212yes—
15#16840434—24no—
16#16840434—24no—
17C2×#14519774C2212yes—
18C2×#14519774C2211yescharacteristic
19C2×#14519752—21no—
20C2×#14519752—22no—
21C2×C4×D244C2212yes—
22C2×C4×D244C2211yescharacteristic
23#16839962—24no—
24C4×#11758201—24no—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#398—22no—
40C61×#391—21no—
41C61×#391C2212yes—
42C61×#389—21no—
43C61×#389C2212yes—
44C61×#384C2211yescharacteristic
45C2×#11758226—28no—
46C2×#11758226—44no—
47C2×#11758226—44no—
48C22×D244—22no—
49C22×D244—21no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244C2312yes—
53C22×D244C2311yescharacteristic
54C22×C4×D61—22no—
55C22×C4×D61—21no—
56C2×#11758201—22no—
57C2×#11758201—22no—
58C2×#11758201—24no—
59C2×#11758201—24no—
60C2×#11758201—22no—
61C2×#11758201—42no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2312yes—
64C22×#92418C2311yescharacteristic
65#14519774—24no—
66#14519774—24no—
67#14519774—22no—
68#14519774D414yes—
69#14519752—28no—
70#14519752—216no—
71C4×D244—24no—
72C4×D244—22no—
73C4×D244—24no—
74C4×D244D414yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758198—22no—
77C2×#11758198—21no—
78#14519708—22no—
79C4⋊#92418—22no—
80#14519701—22no—
81#14519697—24no—
82#14519697—22no—
83#14519697—24no—
84#14519697—42no—
85#14519697—42no—
86C4×#92418—22no—
87C22×Q8×C61D411yescharacteristic
88C22×Q8×C61C2311yescharacteristic
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D412yes—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104#95×C61—22no—
105#95×C61—22no—
106#95×C61—22no—
107C23×D61—22no—
108C23×D61—21no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61SD1614yescomplemented
117C2×Q8×C61#4112yescomplemented
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127#11758226—432no—
128C2×D244—22no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244—21no—
135C2×D244—22no—
136C2×D244—28no—
137C2×D244—24no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244—28no—
141C2×D244—24no—
142C2×D244—44no—
143C2×D244—44no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244—81no—
147C2×D244—81no—
148C2×D244—82no—
149C2×D244—82no—
150C2×D244C2×D414yes—
151C2×D244C2×D412yes—
152C2×C4×D61—28no—
153C2×C4×D61—24no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C4⋊C4×C61—24no—
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—42no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61—82no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—42no—
171C42×C61C2×D412yes—
172C42×C61C2×D412yes—
173#11758201—28no—
174#11758201—48no—
175#11758198—28no—
176#11758198—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61C2×D412yes—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—21no—
185C2×#92418—22no—
186C2×#92418—21no—
187C2×#92418#4112yescomplemented
188C23×C61C22×D411yescharacteristic
189C22×D61—24no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—41no—
195C22×D61—42no—
196C22×D61—41no—
197C22×D61—42no—
198C22×D61—41no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—41no—
203Q8×C61—28no—
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—41no—
207Q8×C61—41no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12012yes—
226C2×C4×C61C2×SD1614yes—
227C2×C4×C61C2×SD1612yes—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238D244—22no—
239D244—22no—
240D244—24no—
241D244—24no—
242D244—416no—
243D244—42no—
244D244—42no—
245D244—41no—
246D244—41no—
247D244—48no—
248D244—84no—
249D244—88no—
250C4×D61—416no—
251C4×D61—48no—
252#92418—22no—
253#92418—41no—
254#92418—42no—
255#24188—611nomaximal, Sylow
256C22×C61C2×#12011yescharacteristic
257C22×C61#54811yescharacteristic
258C22×C61#55011yescharacteristic
259C22×C61C22×SD1611yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—42no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—42no—
267C2×D61—44no—
268C2×D61—42no—
269C2×D61—81no—
270C2×D61—82no—
271C2×D61—82no—
272C2×D61—81no—
273C2×D61—82no—
274C2×D61—81no—
275C2×D61—82no—
276C2×D61—81no—
277C4×C61—24no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61—81no—
283C4×C61#45014yes—
284C4×C61#52511yescharacteristic
285C4×C61#53711yescharacteristic
286C4×C61#54611yescharacteristic
287C4×C61#54711yescharacteristic
288Dic61—42no—
289C2×#550—611no—
290C2×#460—611no—
291C2×C4×SD16—611no—
292#2582D6111yescharacteristic, complemented
293#2041—612no—
294#2014—612no—
295#1984—611no—
296#1957—611no—
297#1877—612no—
298#1864—612no—
299#1802—611no—
300C2×C61#240711yescharacteristic
301C2×C61C2×#45011yescharacteristic
302C2×C61#301211yescharacteristic
303C2×C61#336211yescharacteristic
304C2×C61#336811yescharacteristic
305C2×C61#339211yescharacteristic
306C2×C61#339911yescharacteristic
307D61—82no—
308D61—84no—
309C22×#41—1221no—
310C22×SD16—611no—
311C22×SD16—612no—
312#550—1224no—
313#550—1224no—
314C2×#127—611no—
315C2×#127—612no—
316C2×#121—1221no—
317C2×#121—1222no—
318C2×C4×Q8C2×D6111yescharacteristic
319C2×C4×Q8C2×D6111yescharacteristic
320C2×C4×D4—611no—
321C2×C4×D4—612no—
322#460—1224no—
323C4×SD16—1224no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—611no—
330C2×#102—612no—
331C2×#102—612no—
332C2×C4×C8—611no—
333#398—22no—
334#391—21no—
335#391C2×D6112yes—
336#389—21no—
337#389C2×D6112yes—
338#384C2×D6111yescharacteristic
339#340—612no—
340#336—612no—
341#336—612no—
342C61#2418811yescharacteristic, complemented, Sylow
343C2×#41—1228no—
344C2×#41—2444no—
345C2×#41—2444no—
346C22×Q8C22⋊D6111yescharacteristic
347C22×Q8C22×D6111yescharacteristic
348C22×D4—611no—
349C22×D4—612no—
350C22×D4—1222no—
351C22×D4—1221no—
352C22×D4—1222no—
353C22×D4—1221no—
354C23×C4—1222no—
355C23×C4—1221no—
356C2×SD16—1222no—
357C2×SD16—1222no—
358C2×SD16—1222no—
359C2×SD16—1224no—
360C2×SD16—1224no—
361C2×SD16—2442no—
362C22×C8—611no—
363C22×C8—612no—
364C22×C8—611no—
365#127—614no—
366#127—1224no—
367#127—1224no—
368#127—1222no—
369#121—1228no—
370#121—12216no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—614no—
376C4×D4—1224no—
377C4×D4—1224no—
378C4×D4—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4C22⋊D6111yescharacteristic
385C2×C4⋊C4C22⋊D6112yes—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6112yes—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42C22×D6112yes—
392C2×C42C22×D6111yescharacteristic
393#106—1222no—
394C4⋊C8—1222no—
395#103—1222no—
396#102—1224no—
397#102—1222no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C4×C8—1222no—
402#95—22no—
403#95—22no—
404#95—22no—
405C24—1222no—
406C24—1221no—
407#41—24432no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8#1175820114yescomplemented
416C2×Q8#1175822612yescomplemented
417C2×D4—612no—
418C2×D4—614no—
419C2×D4—1228no—
420C2×D4—1228no—
421C2×D4—1222no—
422C2×D4—1221no—
423C2×D4—1222no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1222no—
427C2×D4—1221no—
428C2×D4—1222no—
429C2×D4—1221no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—4881no—
434C2×D4—4881no—
435C2×D4—4882no—
436C2×D4—4882no—
437C2×D4—2444no—
438C2×D4—2444no—
439C2×D4—2442no—
440C2×D4—2442no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—1228no—
445C22×C4—1221no—
446C22×C4—1224no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—1228no—
457SD16—2448no—
458C2×C8—612no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—1221no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—612no—
478C4⋊C4—1221no—
479C22⋊C4—1228no—
480C22⋊C4—1224no—
481C42—22no—
482C42—22no—
483C42—42no—
484C42D4×D6112yes—
485C42D4×D6112yes—
486C23—1224no—
487C23—1222no—
488C23—2442no—
489C23—2441no—
490C23—2442no—
491C23—2441no—
492C23—2442no—
493C23—2441no—
494C23—2442no—
495C23—2441no—
496C23—2441no—
497C23—2442no—
498C23—2442no—
499C23—2441no—
500C23C22×C22⋊D6111yescharacteristic
501Q8—28no—
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—1222no—
507D4—1222no—
508D4—1224no—
509D4—1224no—
510D4—4884no—
511D4—4888no—
512D4—24416no—
513D4—2442no—
514D4—2442no—
515D4—2441no—
516D4—2441no—
517D4—2448no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—1222no—
535C2×C4—24416no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2448no—
539C2×C4#1451974512yes—
540C2×C4#1451974512yes—
541C2×C4C2×#1175820114yes—
542C2×C4C2×#1175820112yes—
543C2×C4#1451980212yes—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4C2×#1175822611yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982511yescharacteristic
550C8—1222no—
551C8—2441no—
552C8—2442no—
553C22—4882no—
554C22—4881no—
555C22—4882no—
556C22—4881no—
557C22—4882no—
558C22—4881no—
559C22—4882no—
560C22—4881no—
561C22—2444no—
562C22—2442no—
563C22—2444no—
564C22—2442no—
565C22—2442no—
566C22—2444no—
567C22C2×#1451974511yescharacteristic
568C22#1684042311yescharacteristic
569C22#1684043111yescharacteristic
570C22C22×#1175820111yescharacteristic
571C22C2×#1451980211yescharacteristic
572C22#1684056511yescharacteristic
573C22#1684057011yescharacteristic
574C4—22no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—82no—
580C4—2442no—
581C4#1683991214yes—
582C4#1684027511yescharacteristic
583C4#1684034211yescharacteristic
584C4#1684040311yescharacteristic
585C4#1684041411yescharacteristic
586C2—4884no—
587C2—4882no—
588C2#2144916611yescharacteristic
589C2C2×#1683991211yescharacteristic
590C2#2145512011yescharacteristic
591C2#2145849011yescharacteristic
592C2#2145854811yescharacteristic
593C2#2145879611yescharacteristic
594C2#2145885711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2582C2
socleC23×C61C22×D4
prime radicalC2×#16840434C2
hypercenter#2582D61
hypocenterC61#24188
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. C2×C4×#11758201C2semidirect
4. #21447053C2semidirect
5. #21446906C2semidirect
6. #21446756C2semidirect
7. #21446624C2semidirect
8. #21446156C2semidirect
9. #21446079C2semidirect
10. #21445816C2semidirect
11. C61×#2582C2semidirect
116. C2×Q8×C61SD16semidirect
117. C2×Q8×C61#41semidirect
187. C2×#92418#41semidirect
292. #2582D61semidirect
342. C61#24188semidirect
415. C2×Q8#11758201semidirect
416. C2×Q8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
C2×#168399622
C2×C4×#117582012
#214470532
#214469062
#214467562
#214466242
#214461562
#214460792
#214458162
C61×#25822
#2418861