Finite Group Databasebeta

← #29695479

Subgroups — #29695479

Order 15616.

1542 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×#16839827C211yesmaximal, characteristic, complemented
3C2×#16839757C212yesmaximal, complemented
4#21448770C211yesmaximal, characteristic, complemented
5#21446745C211yesmaximal, characteristic, complemented
6#21446624C211yesmaximal, characteristic, complemented
7#21446246C212yesmaximal, complemented
8#21446140C212yesmaximal, complemented
9#21445758C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758226C2212yes—
13C22×C8×D61C2212yes—
14#16840434—24no—
15#16840434C418yescomplemented
16C2×#14519774—21no—
17C2×#14519774C2211yescharacteristic
18C2×#14519752—22no—
19C2×#14519752C2212yes—
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yes—
23#16839827—24no—
24#16839757—28no—
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839663—21no—
34#16839652—22no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C2×#11758226—24no—
46C2×#11758226—24no—
47C2×#11758226—24no—
48C2×#11758226—42no—
49C2×#11758226—42no—
50C2×#11758226C2×C418yes—
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244D411yescharacteristic
56C22×D244C2312yes—
57C22×C4×D61—22no—
58C22×C4×D61C2312yes—
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C2×C8×D61—24no—
62C2×C8×D61—24no—
63C2×C8×D61—42no—
64#14519774—28no—
65#14519774—22no—
66#14519774C2×C414yes—
67#14519752—24no—
68#14519752—216no—
69#14519752C2×C418yes—
70C4×D244—28no—
71C4×D244C2×C414yes—
72C4×D244D414yes—
73#14519735—24no—
74#14519735—42no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—22no—
84#14519701—22no—
85#14519701D414yescomplemented
86#14519699—24no—
87#14519699—42no—
88#14519697—24no—
89#14519697—42no—
90#14519689—24no—
91C4×#92418—22no—
92#14519684—21no—
93#14519684—22no—
94C22×Q8×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C4×Q8×C61—22no—
97C4×Q8×C61C2×C414yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—24no—
107C23×D61—22no—
108C23×D61C2×D412yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61C22×C412yes—
113C2×Q8×C61C2×D412yes—
114C2×Q8×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61C2×D412yes—
124#11758226—216no—
125#11758226—48no—
126#11758226—48no—
127#11758226—48no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—28no—
135C2×D244—24no—
136C2×D244—22no—
137C2×D244—28no—
138C2×D244—24no—
139C2×D244—44no—
140C2×D244—44no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—81no—
144C2×D244—81no—
145C2×D244—82no—
146C2×D244—82no—
147C2×D244C22⋊C414yes—
148C2×D244C22×C414yes—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—28no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61C22×C414yes—
157C2×C4×D61C2×D414yes—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61C22×C412yes—
168C42×C61—24no—
169C42×C61C22×C412yes—
170C42×C61C2×D412yes—
171#11758198—28no—
172#11758198—44no—
173#11758198—82no—
174#11758198C22⋊C418yes—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22⋊C414yes—
179C4⋊Dic61C2×D412yes—
180C4×Dic61—44no—
181C4×Dic61—42no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418C2×D412yes—
188C8×D61—48no—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—24no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61C2×C22⋊C414yes—
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61C4×D414yes—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61#13512yes—
222C2×C4×C61#13512yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61C2×C22⋊C414yes—
238D244—24no—
239D244—24no—
240D244—28no—
241D244—416no—
242D244—42no—
243D244—42no—
244D244—48no—
245D244—84no—
246D244—88no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—24no—
251C4×D61—416no—
252C8×C61—42no—
253#92418—22no—
254#92418—41no—
255#22418—611nomaximal, Sylow
256C22×C61C22×C22⋊C411yescharacteristic
257C22×C61D4211yescharacteristic
258C22×C61#54611yescharacteristic
259C22×C61C2×#13511yescharacteristic
260C22×C61C2×#13511yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—24no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—44no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—24no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—82no—
281C4×C61#41614yescomplemented
282C4×C61C2×C4×D411yescharacteristic
283C4×C61#51811yescharacteristic
284C4×C61D4211yescharacteristic
285C4×C61#54611yescharacteristic
286Dic61—24no—
287Dic61—42no—
288Dic61—81no—
289C2×#550—611no—
290C2×C8×D4—611no—
291C2×#416—612no—
292#2354—611no—
293#1982—611no—
294#1957—611no—
295#1893—612no—
296#1875—612no—
297#1803D6111yescharacteristic, complemented
298#1788—612no—
299C4×#103—611no—
300C2×C61D4×C22⋊C411yescharacteristic
301C2×C61C2×#41611yescharacteristic
302C2×C61#289711yescharacteristic
303C2×C61#328411yescharacteristic
304C2×C61#328711yescharacteristic
305C2×C61#330411yescharacteristic
306C2×C61#330511yescharacteristic
307D61—44no—
308D61—84no—
309C22×#41—612no—
310C23×C8—612no—
311#550—618no—
312#550—1224no—
313C2×#127—611no—
314C2×#127—1221no—
315C2×#121—612no—
316C2×#121—1222no—
317C2×C4×Q8C2×D6111yescharacteristic
318C2×C4×D4—611no—
319C2×C4×D4—1221no—
320C2×#117—612no—
321C8×D4—1224no—
322C2×C4⋊C8C2×D6111yescharacteristic
323#416—1228no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103C2×D6112yes—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—612no—
330C2×#102—612no—
331C2×C22⋊C8—612no—
332C2×C4×C8—611no—
333#390—1221no—
334#388—1222no—
335#384C2×D6111yescharacteristic
336#380—612no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—1221no—
339#340—611no—
340#340—611no—
341#340C2×D6112yes—
342C61#2241811yescharacteristic, complemented, Sylow
343C2×#41—618no—
344C2×#41—1224no—
345C2×#41—1224no—
346C2×#41—1224no—
347C2×#41—2442no—
348C2×#41—2442no—
349C22×Q8C22×D6111yescharacteristic
350C22×D4—611no—
351C22×D4—612no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4—1221no—
355C22×D4—1221no—
356C23×C4—612no—
357C23×C4—1222no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—1224no—
361C22×C8—1224no—
362C22×C8—2442no—
363C22×C8C22×D6112yes—
364#127—614no—
365#127—1228no—
366#127—1222no—
367#121—618no—
368#121—12216no—
369#121—1224no—
370C4×Q8—22no—
371C4×Q8C4×D6114yes—
372C4×D4—614no—
373C4×D4—614no—
374C4×D4—1228no—
375#117—1224no—
376#117—2442no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4C22×D6112yes—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—612no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—21no—
390C2×C42—1222no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—22no—
394#103—24no—
395#103—614no—
396#103—1222no—
397#102—1224no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C22⋊C8—1224no—
402C4×C8—1222no—
403#95—1222no—
404#95—1221no—
405C24—612no—
406C24—1222no—
407#41—12216no—
408#41—2448no—
409#41—2448no—
410#41—2448no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8C2×C4×D6112yes—
415C2×Q8D4×D6112yes—
416C2×Q8#1175822612yes—
417C2×D4—614no—
418C2×D4—614no—
419C2×D4—1228no—
420C2×D4—1228no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—4881no—
431C2×D4—4881no—
432C2×D4—4882no—
433C2×D4—4882no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4—2442no—
437C2×D4—2442no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—1228no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—612no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—2448no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—614no—
473C4⋊C4—612no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4C2×C4×D6112yes—
478C22⋊C4—618no—
479C22⋊C4—1228no—
480C22⋊C4—4882no—
481C22⋊C4—2444no—
482C42—24no—
483C42—2444no—
484C42—2442no—
485C42C2×C4×D6112yes—
486C42D4×D6112yes—
487C23—614no—
488C23—1222no—
489C23—1222no—
490C23—1222no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—1224no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23—2442no—
501C23C2×D4×D6111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8#1451973914yes—
506D4—1224no—
507D4—1224no—
508D4—1228no—
509D4—4884no—
510D4—4888no—
511D4—24416no—
512D4—2448no—
513D4—2442no—
514D4—2442no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—614no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1222no—
534C2×C4—24416no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4C22⋊C4×D6112yes—
541C2×C4C22⋊C4×D6112yes—
542C2×C4#1451979812yes—
543C2×C4#1451980012yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C2×#1175822611yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982311yescharacteristic
550C8—42no—
551C8—1222no—
552C8—2441no—
553C22—1224no—
554C22—1224no—
555C22—1224no—
556C22—4882no—
557C22—4882no—
558C22—4882no—
559C22—4882no—
560C22—2444no—
561C22—2444no—
562C22—2444no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×C22⋊C4×D6111yescharacteristic
568C22#1684039611yescharacteristic
569C22#1684040311yescharacteristic
570C22C2×#1451979811yescharacteristic
571C22C2×#1451980011yescharacteristic
572C22#1684056611yescharacteristic
573C22#1684057111yescharacteristic
574C4—22no—
575C4—24no—
576C4—42no—
577C4—82no—
578C4—1224no—
579C4—4881no—
580C4—2442no—
581C4#1683975714yescomplemented
582C4C2×#1451973911yescharacteristic
583C4#1684023611yescharacteristic
584C4#1684039611yescharacteristic
585C4#1684040311yescharacteristic
586C2—4884no—
587C2—2444no—
588C2#2144863911yescharacteristic
589C2C2×#1683975711yescharacteristic
590C2#2145434311yescharacteristic
591C2#2145754411yescharacteristic
592C2#2145757411yescharacteristic
593C2#2145775511yescharacteristic
594C2#2145776611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC2×#16840434C2
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22418
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16839827C2semidirect
3. C2×#16839757C2semidirect
4. #21448770C2semidirect
5. #21446745C2semidirect
6. #21446624C2semidirect
7. #21446246C2semidirect
8. #21446140C2semidirect
9. #21445758C2semidirect
10. C4×#14519701C2semidirect
11. C61×#1803C2semidirect
15. #16840434C4semidirect
85. #14519701D4semidirect
281. C4×C61#416semidirect
297. #1803D61semidirect
342. C61#22418semidirect
581. C4#16839757semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
C2×#168398272
C2×#168397572
#214487702
#214467452
#214466242
#214462462
#214461402
#214457582
C4×#145197012
C61×#18032
#2241861