Finite Group Databasebeta

← #29713336

Subgroups — #29713336

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3C2×#16840076C211yesmaximal, characteristic
4C2×#16839660C211yesmaximal, characteristic, complemented
5#21447158C212yesmaximal, complemented
6#21447061C212yesmaximal, complemented
7#21446134C211yesmaximal, characteristic
8#21446124C211yesmaximal, characteristic, complemented
9#21446083C212yesmaximal, complemented
10#21446076C212yesmaximal, complemented
11C61×#1817C211yesmaximal, characteristic, complemented
12#16840296—24no—
13#16840296—24no—
14C2×#14519771—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519765C2211yescharacteristic
17C2×#14519760—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519734—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26C22×#11758196—21no—
27#16840093—24no—
28#16840076—24no—
29C2×#14519711C2212yes—
30C2×#14519709C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33C2×#14519699C2212yes—
34C2×#14519699C2212yes—
35#16839660—21no—
36#16839660—21no—
37#16839660C2212yes—
38#16839660C2212yes—
39C2×#14519684—21no—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2211yescharacteristic
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#336C2212yes—
46C61×#336C2212yes—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C23×Dic61—21no—
52C22×C4×D61—21no—
53#14519771—28no—
54#14519769—22no—
55#14519769—22no—
56#14519765—22no—
57#14519765—22no—
58#14519760—28no—
59#14519756—22no—
60#14519756D414yes—
61#14519751—22no—
62#14519751D414yes—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×D244—22no—
66C4×D244—22no—
67C4×C22⋊D61—28no—
68#14519734—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C2×#11758196—22no—
84C2×#11758196—44no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91#14519711—24no—
92#14519709—24no—
93#14519702—24no—
94#14519702—24no—
95#14519699—24no—
96#14519699—24no—
97#14519684—22no—
98#14519684—22no—
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D411yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C22×C8×C61C2312yes—
104C2×C42×C61—22no—
105C2×C42×C61—22no—
106C2×C42×C61D411yescharacteristic
107C2×C42×C61D411yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C22⋊C8×C61—22no—
111C22⋊C8×C61—22no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244C2×D412yes—
148C2×C4×D61—24no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—21no—
154C2×Dic122C2×D412yes—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C22⋊Dic61—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172C4⋊Dic61—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—42no—
179C4⋊Dic61C2×D412yes—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61#4114yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186#11758196—44no—
187#11758196—44no—
188C4×Dic61—24no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#12012yes—
194C23×C61#13512yes—
195C23×C61AGL(1,ℤ/8)12yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—44no—
220C2×C4×C61—44no—
221C2×C4×C61#12012yes—
222C2×C4×C61#13712yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4112yes—
226C2×C4×C612−1+412yes—
227C22⋊D61—48no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—48no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—84no—
245D244—24no—
246D244—42no—
247C4×D61—48no—
248Dic122—24no—
249Dic122—42no—
250C8×C61—42no—
251C8×C61—42no—
252#24066—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61#44914yes—
260C22×C61C2×#12011yescharacteristic
261C22×C61#52511yescharacteristic
262C22×C61#53611yescharacteristic
263C22×C61C2×#13511yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61C2×#13711yescharacteristic
266C22×C61#57511yescharacteristic
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61#54712yes—
280C4×C61#54712yes—
281Dic61—44no—
282Dic61—42no—
283Dic61—42no—
284Dic61—42no—
285Dic61—82no—
286C2×#529—611no—
287C2×#483—611no—
288C2×#480—611no—
289C2×#389—611no—
290#2059—612no—
291#2042—612no—
292#1874—611no—
293#1872—611no—
294#1864—612no—
295#1863—612no—
296#1817D6111yescharacteristic, complemented
297C2×C61—24no—
298C2×C61#240611yescharacteristic
299C2×C61C2×#44911yescharacteristic
300C2×C61#300711yescharacteristic
301C2×C61#310811yescharacteristic
302C2×C61#311411yescharacteristic
303C2×C61#323211yescharacteristic
304C2×C61#323811yescharacteristic
305D61—82no—
306#529—1224no—
307#529—1224no—
308C2×#126—1221no—
309C2×#125—611no—
310C2×#124—611no—
311C2×#123—1221no—
312C2×C22⋊Q8—611no—
313C2×#121—611no—
314C2×C4×Q8—611no—
315C2×C4×D4—611no—
316C2×C4×D4—1221no—
317C2×#117—1221no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—1221no—
321C22×C42C2×D6111yescharacteristic
322#483—1224no—
323#480—1224no—
324C2×#107—612no—
325C2×#106—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#102—612no—
329C2×#102—612no—
330C2×C22⋊C8C2×D6111yescharacteristic
331C2×C22⋊C8C2×D6111yescharacteristic
332#389—612no—
333#389—612no—
334#389—1221no—
335#389—1221no—
336C2×#95—1221no—
337#340—612no—
338#340—612no—
339#336C2×D6112yes—
340#336C2×D6112yes—
341C61#2406611yescharacteristic, complemented, Sylow
342C22×Q8—611no—
343C22×D4—611no—
344C22×D4—1221no—
345C23×C4—1221no—
346C23×C4—1221no—
347C23×C4—1221no—
348C23×C4D24411yescharacteristic
349C23×C4D24411yescharacteristic
350C23×C4C22×D6111yescharacteristic
351C22×C8C22×D6112yes—
352C22×C8C22×D6112yes—
353#126—1228no—
354#125—1222no—
355#125—1222no—
356#124—1222no—
357#124—1222no—
358#123—1228no—
359C22⋊Q8—614no—
360C22⋊Q8—1222no—
361#121—614no—
362#121—1222no—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1228no—
368#117—1228no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—2444no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—1221no—
393C2×C42D24411yescharacteristic
394C2×C42D24411yescharacteristic
395C2×C42C22×D6111yescharacteristic
396C2×C42C22×D6111yescharacteristic
397#107—1224no—
398#106—1224no—
399#103—1224no—
400#103—1224no—
401#102—1224no—
402#102—1224no—
403C22⋊C8—22no—
404C22⋊C8—22no—
405#95—1222no—
406#95—1222no—
407C24—1221no—
408C24C2×D24411yescharacteristic
409C2×Q8—612no—
410C2×Q8—1221no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×D4—612no—
414C2×D4—1224no—
415C2×D4—1221no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—2442no—
419C2×D4—2442no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—1224no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4C2×D24411yescharacteristic
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4C2×D24411yescharacteristic
447C22×C4D4×D6112yes—
448C22×C4D4×D6112yes—
449C22×C4C23×D6111yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—612no—
457C4⋊C4—612no—
458C4⋊C4—614no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1222no—
466C4⋊C4—1224no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—2444no—
470C4⋊C4—2444no—
471C4⋊C4—2442no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—22no—
483C42—1224no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—1222no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23#1451974712yes—
496C23#1451979912yes—
497C23#1451980512yes—
498C23C22×D24411yescharacteristic
499Q8—1224no—
500Q8—2442no—
501D4—1224no—
502D4—2448no—
503D4—2442no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—4884no—
526C2×C4—2448no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2448no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2442no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4#1451974712yes—
540C2×C4#1451981012yes—
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4#41×D6112yes—
544C2×C4#1451982412yes—
545C8—42no—
546C8—42no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—2442no—
558C22—2442no—
559C22—2442no—
560C22#1683990914yes—
561C22C2×#1451974711yescharacteristic
562C22#1684027411yescharacteristic
563C22#1684033811yescharacteristic
564C22C2×#1451979911yescharacteristic
565C22C2×#1451980511yescharacteristic
566C22C2×#1451981011yescharacteristic
567C22#1684056211yescharacteristic
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—4882no—
574C4—2444no—
575C4—2442no—
576C4—2442no—
577C4—2442no—
578C4#1684041012yes—
579C4#1684041012yes—
580C2—24no—
581C2—4882no—
582C2#2144915611yescharacteristic
583C2C2×#1683990911yescharacteristic
584C2#2145508111yescharacteristic
585C2#2145590111yescharacteristic
586C2#2145595911yescharacteristic
587C2#2145698911yescharacteristic
588C2#2145704211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1817C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1817D61
hypocenterC61#24066
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
4. C2×#16839660C2semidirect
5. #21447158C2semidirect
6. #21447061C2semidirect
8. #21446124C2semidirect
9. #21446083C2semidirect
10. #21446076C2semidirect
11. C61×#1817C2semidirect
296. #1817D61semidirect
341. C61#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#168402962
C2×#168400932
C2×#168400762
C2×#168396602
#214471582
#214470612
#214461342
#214461242
#214460832
#214460762
C61×#18172
#2406661