Finite Group Databasebeta

← #29713333

Subgroups — #29713333

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×#16840293C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3C2×#16840073C211yesmaximal, characteristic
4C2×#16839661C211yesmaximal, characteristic, complemented
5#21447156C212yesmaximal, complemented
6#21447059C212yesmaximal, complemented
7#21446126C211yesmaximal, characteristic, complemented
8#21446082C212yesmaximal, complemented
9#21446074C212yesmaximal, complemented
10#21445885C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic
12#16840293—24no—
13#16840293—24no—
14C2×#14519770—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519758—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4⋊Dic61—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×C4×Dic61C2211yescharacteristic
26#16840093—24no—
27#16840073—24no—
28C2×#14519710C2212yes—
29C2×#14519709C2212yes—
30C2×#14519702C2212yes—
31C2×#14519700C2212yes—
32C2×#14519698C2212yes—
33C2×C22⋊#92418C2211yescharacteristic
34#16839661—21no—
35#16839661—21no—
36#16839661C2212yes—
37#16839661C2212yes—
38C2×#14519684—21no—
39#16839524C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×#121×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#340C2212yes—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52C22×#92418C2312yes—
53#14519770—28no—
54#14519768—22no—
55#14519768—22no—
56#14519764—22no—
57#14519764—22no—
58#14519758—28no—
59#14519756—22no—
60#14519756D414yes—
61#14519742—22no—
62#14519742—22no—
63D4×Dic61—28no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—44no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—24no—
91#14519709—24no—
92#14519702—24no—
93#14519700—24no—
94#14519698—24no—
95C22⋊#92418—22no—
96#14519684—22no—
97#14519684—22no—
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103#121×C61—22no—
104#121×C61D414yes—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110#102×C61—24no—
111C22⋊C8×C61—22no—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61C2×D412yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—41no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—21no—
154C2×Dic122C2×D412yes—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61#4114yes—
163C22⋊C4×C61—24no—
164C22⋊Dic61—24no—
165C22⋊Dic61—24no—
166C22⋊Dic61—24no—
167C22⋊Dic61—24no—
168C4⋊Dic61—24no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—44no—
173C4⋊Dic61—44no—
174C4⋊Dic61—42no—
175C4⋊Dic61C2×D412yes—
176#11758196—24no—
177#11758196—24no—
178#11758196—24no—
179C4×Dic61—24no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61#12012yes—
201C23×C61#13512yes—
202C23×C61AGL(1,ℤ/8)12yes—
203C23×C61C22×D411yescharacteristic
204D4×C61—24no—
205D4×C61—48no—
206D4×C61—42no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—48no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61#12012yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C612−1+412yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—44no—
237C2×Dic61—48no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—44no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—84no—
247C2×Dic61#13712yes—
248Dic122—24no—
249Dic122—42no—
250C8×C61—42no—
251#92418—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—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61#44914yes—
267C22×C61C2×#12011yescharacteristic
268C22×C61#52511yescharacteristic
269C22×C61#53611yescharacteristic
270C22×C61C2×#13511yescharacteristic
271C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C61C2×#13711yescharacteristic
273C22×C61#57511yescharacteristic
274C4×C61—22no—
275C4×C61—44no—
276C4×C61—42no—
277C4×C61#54712yes—
278C4×C61#54712yes—
279Dic61—22no—
280Dic61—42no—
281Dic61—41no—
282Dic61—41no—
283Dic61—41no—
284Dic61—42no—
285Dic61—82no—
286C2×#529—611no—
287C2×#483—611no—
288C2×#480—611no—
289C2×#389—611no—
290#2059—612no—
291#2042—612no—
292#1874D6111yescharacteristic
293#1872—611no—
294#1864—612no—
295#1863—612no—
296#1817—611no—
297C2×C61—24no—
298C2×C61—82no—
299C2×C61#240611yescharacteristic
300C2×C61C2×#44911yescharacteristic
301C2×C61#300711yescharacteristic
302C2×C61#310811yescharacteristic
303C2×C61#311411yescharacteristic
304C2×C61#323211yescharacteristic
305C2×C61#323811yescharacteristic
306#529—1224no—
307#529—1224no—
308C2×#126—1221no—
309C2×#125—611no—
310C2×#124—611no—
311C2×#123—1221no—
312C2×C22⋊Q8—611no—
313C2×#121C2×D6111yescharacteristic
314C2×C4×Q8—611no—
315C2×C4×D4—611no—
316C2×C4×D4—1221no—
317C2×#117—1221no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—1221no—
320C22×C4⋊C4C2×D6111yescharacteristic
321C22×C42—611no—
322#483—1224no—
323#480—1224no—
324C2×#107—612no—
325C2×#106—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#102—612no—
329C2×#102C2×D6112yes—
330C2×C22⋊C8—611no—
331C2×C22⋊C8C2×D6111yescharacteristic
332#389—612no—
333#389—612no—
334#389—1221no—
335#389—1221no—
336C2×#95—1221no—
337#340—612no—
338#340C2×D6112yes—
339#336—612no—
340#336—612no—
341C61#2406611yescharacteristic, complemented, Sylow
342C22×Q8—611no—
343C22×D4—21no—
344C22×D4C22×D6111yescharacteristic
345C23×C4—21no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—1221no—
349C23×C4—1221no—
350C23×C4C22×D6111yescharacteristic
351C22×C8—612no—
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—22no—
362#121D24414yes—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×D4—1222no—
366C4×D4—1228no—
367C4×D4—1222no—
368#117—1228no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—2444no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1222no—
396C2×C42—1222no—
397#107—1224no—
398#106—1224no—
399#103—1224no—
400#103—1224no—
401#102—24no—
402#102—1224no—
403C22⋊C8—22no—
404C22⋊C8—1222no—
405#95—1222no—
406#95—1222no—
407C24—21no—
408C24D4×D6111yescharacteristic
409C2×Q8—612no—
410C2×Q8—1221no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×D24412yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1222no—
432C22×C4—1224no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—2442no—
441C22×C4—2441no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4C2×D24412yes—
447C22×C4D4×D6112yes—
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—612no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1224no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—2444no—
468C4⋊C4—2444no—
469C4⋊C4—2442no—
470C4⋊C4C2×D24412yes—
471C4⋊C4#1175822414yes—
472C22⋊C4—24no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C42—1224no—
478C42—1221no—
479C42—1221no—
480C42—1221no—
481C42—1221no—
482C42—1222no—
483C42—1222no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1451974712yes—
496C23#1451979812yes—
497C23#1451980112yes—
498C23C2×D4×D6111yescharacteristic
499Q8—1224no—
500Q8—2442no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—612no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4884no—
530C2×C4—2444no—
531C2×C4—2448no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2444no—
540C2×C4#1451974712yes—
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×D24411yescharacteristic
543C2×C4C2×#1175822412yes—
544C2×C4#1451982412yes—
545C8—42no—
546C8—2442no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1683990814yes—
561C22C2×#1451974711yescharacteristic
562C22#1684026811yescharacteristic
563C22#1684033411yescharacteristic
564C22C2×#1451979811yescharacteristic
565C22C2×#1451980111yescharacteristic
566C22#137×D6111yescharacteristic
567C22#1684055811yescharacteristic
568C4—22no—
569C4—44no—
570C4—42no—
571C4—1222no—
572C4—4882no—
573C4—2441no—
574C4—2441no—
575C4—2442no—
576C4—2441no—
577C4—2442no—
578C4#1684041212yes—
579C4#1684041612yes—
580C2—24no—
581C2—82no—
582C2#2144915311yescharacteristic
583C2C2×#1683990811yescharacteristic
584C2#2145508011yescharacteristic
585C2#2145589411yescharacteristic
586C2#2145595211yescharacteristic
587C2#2145698611yescharacteristic
588C2#2145703811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#1874D61
hypocenterC61#24066
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
4. C2×#16839661C2semidirect
5. #21447156C2semidirect
6. #21447059C2semidirect
7. #21446126C2semidirect
8. #21446082C2semidirect
9. #21446074C2semidirect
10. #21445885C2semidirect
341. C61#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#168402932
C2×#168400932
C2×#168400732
C2×#168396612
#214471562
#214470592
#214461262
#214460822
#214460742
#214458852
C61×#18742
#2406661