Finite Group Databasebeta

← #29695481

Subgroups — #29695481

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×#16840432C211yesmaximal, characteristic
2C2×C8×D244C211yesmaximal, characteristic, complemented
3C2×#16839755C212yesmaximal, complemented
4#21448770C211yesmaximal, characteristic, complemented
5#21446747C211yesmaximal, characteristic, complemented
6#21446244C212yesmaximal, complemented
7#21446138C212yesmaximal, complemented
8#21445821C211yesmaximal, characteristic, complemented
9#21445756C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1957C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C22×C8×D61C2212yes—
14#16840432—24no—
15#16840432C418yescomplemented
16C2×#14519773—21no—
17C2×#14519749—22no—
18C2×#14519748C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×D4×Dic61—21no—
22C2×#14519735C2212yes—
23C8×D244—24no—
24C2×C4⋊#92418C2211yescharacteristic
25#16839755—28no—
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519689C2212yes—
32#16839663—21no—
33#16839652—22no—
34#16839619C2212yes—
35C4⋊C4×Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2211yescharacteristic
39#16839524C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C2×#102×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#384C2211yescharacteristic
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—24no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C418yes—
51C22×Dic122C2311yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61—22no—
54C22×C22⋊D61—22no—
55C23×Dic61—22no—
56C22×C4×D61C2312yes—
57C22×#92418C2312yes—
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60C2×C8×D61—42no—
61#14519773—28no—
62#14519749—216no—
63#14519748—24no—
64#14519748C2×C418yes—
65C4×Dic122—22no—
66C4×Dic122C2×C414yes—
67C4×D244C2×C414yes—
68C4×D244D414yes—
69D4×Dic61—28no—
70#14519735—24no—
71#14519735—42no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×C22⋊Dic61—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—22no—
84C4⋊#92418—22no—
85#14519702—22no—
86#14519702D414yescomplemented
87#14519700—24no—
88#14519698—24no—
89#14519698—42no—
90#14519689—24no—
91#14519684—21no—
92#14519684—22no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61C2312yes—
96C22×C8×C61C2311yescharacteristic
97C22×C8×C61C2311yescharacteristic
98#127×C61—22no—
99#127×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104#102×C61—24no—
105#102×C61—42no—
106C4×C8×C61—22no—
107C24×C61—22no—
108C23×D61C2×D412yes—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—81no—
119C2×D4×C61—81no—
120C2×D4×C61C22×C414yes—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—28no—
126C2×C22⋊D61—28no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C2×C22⋊D61—82no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—28no—
135C22×Dic61—21no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61C2×D412yes—
139#11758224—216no—
140#11758224—48no—
141#11758224—48no—
142#11758224—48no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244C22⋊C414yes—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61C22×C414yes—
151C2×C4×D61C2×D414yes—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
156C2×Dic122C2×D412yes—
157C2×Dic122#4112yes—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D412yes—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—81no—
168C42×C61C22×C412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—28no—
171#11758198—44no—
172#11758198—82no—
173#11758198C22⋊C418yes—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22⋊C414yes—
179C4⋊Dic61C22×C412yes—
180C4⋊Dic61C2×D412yes—
181#11758196—24no—
182C4×Dic61—24no—
183C4×Dic61—44no—
184C4×Dic61—42no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C8×D61—48no—
189C23×C61—24no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61C2×C22⋊C414yes—
204D4×C61—24no—
205D4×C61—24no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—48no—
209D4×C61—84no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61#13512yes—
218C2×C4×C61#13512yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C22⋊D61—416no—
226C22⋊D61—88no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—416no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61C2×C22⋊C414yes—
243D244—28no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—24no—
248Dic122—22no—
249Dic122—22no—
250Dic122—22no—
251Dic122C4×D414yes—
252C8×C61—22no—
253C8×C61—41no—
254#92418—42no—
255#22418—611nomaximal, Sylow
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61C22×C22⋊C411yescharacteristic
264C22×C61D4211yescharacteristic
265C22×C61#54611yescharacteristic
266C22×C61C2×#13511yescharacteristic
267C22×C61C2×#13511yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—24no—
271C2×D61—24no—
272C2×D61—24no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C4×C61—22no—
278C4×C61—82no—
279C4×C61#41614yescomplemented
280C4×C61C2×C4×D411yescharacteristic
281C4×C61#51811yescharacteristic
282C4×C61D4211yescharacteristic
283C4×C61#54611yescharacteristic
284Dic61—24no—
285Dic61—24no—
286Dic61—42no—
287Dic61—42no—
288Dic61—81no—
289C2×#550—611no—
290C2×C8×D4—611no—
291C2×#416—612no—
292#2354—611no—
293#1982—611no—
294#1957D6111yescharacteristic, complemented
295#1893—612no—
296#1875—612no—
297#1803—611no—
298#1788—612no—
299C4×#103—611no—
300C2×C61—84no—
301C2×C61D4×C22⋊C411yescharacteristic
302C2×C61C2×#41611yescharacteristic
303C2×C61#289711yescharacteristic
304C2×C61#328411yescharacteristic
305C2×C61#328711yescharacteristic
306C2×C61#330411yescharacteristic
307C2×C61#330511yescharacteristic
308D61—44no—
309C22×#41—612no—
310C23×C8—612no—
311#550—618no—
312#550—1224no—
313C2×#127—1221no—
314C2×#127C2×D6111yescharacteristic
315C2×#121—612no—
316C2×#121—1222no—
317C2×C4×Q8—611no—
318C2×C4×D4—611no—
319C2×C4×D4—1221no—
320C2×#117—612no—
321C8×D4—1224no—
322C2×C4⋊C8—611no—
323#416—1228no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—612no—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102C2×D6112yes—
330C2×#102C2×D6112yes—
331C2×C22⋊C8—612no—
332C2×C4×C8C2×D6111yescharacteristic
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#340—612no—
342C61#2241811yescharacteristic, complemented, Sylow
343C2×#41—618no—
344C2×#41—1224no—
345C2×#41—1224no—
346C2×#41—1224no—
347C2×#41—2442no—
348C2×#41—2442no—
349C22×Q8—611no—
350C22×D4—21no—
351C22×D4—21no—
352C22×D4—611no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4C22×D6112yes—
356C23×C4—612no—
357C23×C4—1222no—
358C22×C8—612no—
359C22×C8—1224no—
360C22×C8—1224no—
361C22×C8—2442no—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364#127—22no—
365#127—1228no—
366#127C4×D6114yes—
367#121—618no—
368#121—12216no—
369#121—1224no—
370C4×Q8—614no—
371C4×Q8—1222no—
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—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—1221no—
390C2×C42—1222no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—1222no—
394#103—614no—
395#103—1224no—
396#103—1222no—
397#102—24no—
398#102—42no—
399#102—1224no—
400#102—2442no—
401C22⋊C8—1224no—
402C4×C8—22no—
403#95—1222no—
404#95—1221no—
405C24—22no—
406C24—612no—
407#41—12216no—
408#41—2448no—
409#41—2448no—
410#41—2448no—
411C2×Q8—612no—
412C2×Q8—612no—
413C2×Q8—612no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—614no—
429C2×D4—1228no—
430C2×D4—1228no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—4882no—
434C2×D4—4882no—
435C2×D4—2444no—
436C2×D4—2444no—
437C2×D4C2×C4×D6114yes—
438C22×C4—22no—
439C22×C4—612no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—1228no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—2448no—
465C2×C8D4×D6112yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—614no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1224no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C22⋊C4—618no—
479C22⋊C4—1228no—
480C22⋊C4—4882no—
481C22⋊C4—2444no—
482C42—1224no—
483C42—2444no—
484C42—2442no—
485C42C2×C4×D6112yes—
486C42D4×D6112yes—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—614no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23C2×D4×D6111yescharacteristic
502Q8—614no—
503Q8—1222no—
504Q8—1222no—
505Q8—1222no—
506D4—24no—
507D4—24no—
508D4—48no—
509D4—42no—
510D4—42no—
511D4—84no—
512D4—1228no—
513D4—4888no—
514D4—24416no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—614no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—24416no—
533C2×C4—2441no—
534C2×C4—2441no—
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#1451979612yes—
543C2×C4#1451979812yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C2×C4#1451982311yescharacteristic
550C8—22no—
551C8—41no—
552C8—2442no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—1224no—
561C22—1224no—
562C22—1224no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×C22⋊C4×D6111yescharacteristic
568C22#1684039711yescharacteristic
569C22#1684040511yescharacteristic
570C22C2×#1451979611yescharacteristic
571C22C2×#1451979811yescharacteristic
572C22#1684056311yescharacteristic
573C22#1684056811yescharacteristic
574C4—22no—
575C4—82no—
576C4—1224no—
577C4—1224no—
578C4—4881no—
579C4—2442no—
580C4—2442no—
581C4#1683975514yescomplemented
582C4C4×D4×D6111yescharacteristic
583C4#1684023411yescharacteristic
584C4#1684039611yescharacteristic
585C4#1684040911yescharacteristic
586C2—84no—
587C2—2444no—
588C2#2144864011yescharacteristic
589C2C2×#1683975511yescharacteristic
590C2#2145434011yescharacteristic
591C2#2145754811yescharacteristic
592C2#2145757711yescharacteristic
593C2#2145775711yescharacteristic
594C2#2145776811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1957C2
socleC23×C61C22×D4
prime radicalC2×#16840432C2
cosocleC22×C4×C61C24
hypercenter#1957D61
hypocenterC61#22418
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×C8×D244C2semidirect
3. C2×#16839755C2semidirect
4. #21448770C2semidirect
5. #21446747C2semidirect
6. #21446244C2semidirect
7. #21446138C2semidirect
8. #21445821C2semidirect
9. #21445756C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1957C2semidirect
15. #16840432C4semidirect
86. #14519702D4semidirect
279. C4×C61#416semidirect
294. #1957D61semidirect
342. C61#22418semidirect
581. C4#16839755semidirect

Maximal subgroups

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

subgroupindex
C2×#168404322
C2×C8×D2442
C2×#168397552
#214487702
#214467472
#214462442
#214461382
#214458212
#214457562
C4×#145197022
C61×#19572
#2241861