Finite Group Databasebeta

← #29679486

Subgroups — #29679486

Order 15616.

1514 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758226C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
3C2×#16839757C212yesmaximal
4#21448225C211yesmaximal, characteristic, complemented
5#21445834C211yesmaximal, characteristic, complemented
6#21445830C211yesmaximal, characteristic, complemented
7#21445826C211yesmaximal, characteristic, complemented
8#21445822C212yesmaximal, complemented
9#21445818C212yesmaximal, complemented
10#21445534C212yesmaximal, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×C8×D61C2212yes—
14C4×#11758226—24no—
15C4×#11758226C418yescomplemented
16C2×Q8×Dic61C2211yescharacteristic
17C2×C4×D244—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519739—21no—
20C2×#14519739C2211yescharacteristic
21C2×#14519735—21no—
22C2×#14519735C2212yes—
23C2×#14519735C2211yescharacteristic
24C2×C42×D61—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C4⋊C8×D61—24no—
29#16839757—28no—
30C2×#14519701C2212yes—
31C2×#14519699C2212yes—
32C2×#14519697C2212yes—
33#16839631C2212yes—
34#16839628C2211yescharacteristic
35#16839621—21no—
36#16839621—21no—
37#16839613—21no—
38#16839613—21no—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C61×#382C2211yescharacteristic
46C61×#340C2212yes—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226C2×C414yes—
51C2×#11758226D414yes—
52C2×#11758226Q814yes—
53C22×D244—21no—
54C22×D244C2311yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2312yes—
62C2×C8×D61—24no—
63C2×C8×D61—24no—
64C2×C8×D61—24no—
65Q8×Dic61—22no—
66Q8×Dic61C2×C414yes—
67C4×D244—28no—
68C4×D244—22no—
69C4×D244C2×C414yes—
70#14519739—28no—
71#14519739—22no—
72#14519739C2×C414yes—
73#14519735—22no—
74#14519735—28no—
75#14519735—24no—
76#14519735—24no—
77#14519735C2×C414yes—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—22no—
82C2×#11758196—22no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—22no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—28no—
94C42×D61C2×C414yes—
95C42×D61D414yes—
96C4⋊#92418—22no—
97#14519706—24no—
98#14519701—24no—
99#14519699—24no—
100#14519697—24no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105C4×Q8×C61—22no—
106C4×Q8×C61C2×C414yes—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C4⋊C8×C61—22no—
113#103×C61—24no—
114#95×C61—21no—
115#95×C61—21no—
116C23×D61—21no—
117C23×D61C2×D411yescharacteristic
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61C22×C412yes—
122C2×Q8×C61C2×D412yes—
123C2×Q8×C61C2×Q812yes—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135#11758226—28no—
136#11758226—28no—
137#11758226—28no—
138#11758226C4⋊C4116yescomplemented
139C2×D244—24no—
140C2×D244—24no—
141C2×D244—24no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×D244C2×D412yes—
147C2×D244C2×Q812yes—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61C22⋊C414yes—
158C2×C4×D61C22⋊C414yes—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×Q812yes—
163C2×C4×D61#4112yes—
164C2×C4×D61#4112yes—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C22×C412yes—
173C42×C61—24no—
174C42×C61C22×C412yes—
175C42×C61C2×D412yes—
176#11758198—24no—
177#11758198—24no—
178#11758198—24no—
179#11758198—44no—
180C4⋊Dic61—21no—
181C4⋊Dic61—24no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61C22×C412yes—
185#11758196—44no—
186#11758196—44no—
187C4×Dic61—24no—
188C4×Dic61—24no—
189C4×Dic61—44no—
190C4×Dic61C22⋊C414yes—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22×C412yes—
193C4×Dic61C2×D412yes—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C8×D61—48no—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61C2×C22⋊C412yes—
207C22×D61#12112yes—
208C22×D61C22⋊Q812yes—
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61C2×C4⋊C414yes—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61#12112yes—
241C2×Dic61C22⋊Q812yes—
242C2×Dic61#13514yes—
243D244—216no—
244D244—22no—
245D244—22no—
246D244—22no—
247D244C2×C4⋊C414yes—
248C4×D61—216no—
249C4×D61—28no—
250C4×D61—44no—
251C4×D61—48no—
252C4×D61—48no—
253C4×D61C2×C4⋊C414yes—
254C4×D61#11714yes—
255C4×D61#12114yes—
256C4×D61C22⋊Q814yes—
257C8×C61—42no—
258#92418—42no—
259#20645—611nomaximal, Sylow
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×C22⋊Q811yescharacteristic
263C22×C61C2×#13512yes—
264C22×C61#57512yes—
265C2×D61—24no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—44no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—44no—
274C2×D61#38014yes—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—24no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61C22×C4⋊C411yescharacteristic
281C4×C61C2×#11711yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×C22⋊Q811yescharacteristic
284Dic61—22no—
285Dic61—42no—
286Dic61—82no—
287Dic61#38014yes—
288Dic61#41614yescomplemented
289C2×C4×#41—611no—
290C22×C4⋊C8—611no—
291C2×#416—612no—
292#2285—611no—
293#1805—611no—
294#1805D6111yescharacteristic, complemented
295#1804—611no—
296#1804—611no—
297#1803—612no—
298#1802—612no—
299#1735—612no—
300C2×C61C2×#38011yescharacteristic
301C2×C61C2×#41611yescharacteristic
302C2×C61#289611yescharacteristic
303C2×C61#303512yes—
304C2×C61#307612yes—
305D61—24no—
306D61—44no—
307C22×#41—611no—
308C23×C8—612no—
309C4×#41—618no—
310C4×#41—1224no—
311C2×C4×Q8—611no—
312C2×C4×Q8C2×D6111yescharacteristic
313C2×C4×D4—611no—
314C2×C4×D4—611no—
315C2×C4×D4—1221no—
316C2×C4×D4—1221no—
317C2×#117—611no—
318C2×#117—612no—
319C2×#117—1221no—
320C22×C42—611no—
321C22×C42—1221no—
322C2×C4⋊C8—612no—
323C2×C4⋊C8—611no—
324C2×C4⋊C8—1224no—
325C2×C4⋊C8C2×D6111yescharacteristic
326#416—1228no—
327C2×#103—612no—
328C2×#103C2×D6112yes—
329C2×#102—612no—
330C2×#102—612no—
331#384—612no—
332#382—611no—
333#382C2×D6111yescharacteristic
334#380—1221no—
335#380—1221no—
336C4×C4⋊C4—1221no—
337C4×C4⋊C4—1221no—
338#340—612no—
339#340—612no—
340#340—612no—
341#340C2×D6112yes—
342C61#2064511yescharacteristic, complemented, Sylow
343C2×#41—614no—
344C2×#41—614no—
345C2×#41—614no—
346C2×#41—1222no—
347C2×#41—1222no—
348C2×#41—1222no—
349C22×Q8C22×D6111yescharacteristic
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4—1221no—
358C22×C8—612no—
359C22×C8—1224no—
360C22×C8—1224no—
361C22×C8—1224no—
362C22×C8C22×D6112yes—
363C4×Q8—22no—
364C4×Q8—614no—
365C4×Q8—1222no—
366C4×Q8C4×D6114yes—
367C4×D4—614no—
368C4×D4—614no—
369C4×D4—1228no—
370C4×D4—1222no—
371C4×D4—1228no—
372C4×D4—1222no—
373#117—614no—
374#117—1228no—
375#117—1222no—
376#117—1224no—
377#117—1224no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4C22×D6112yes—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1222no—
390C2×C42—21no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—614no—
395C2×C42—614no—
396C2×C42—1228no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1222no—
400C2×C42C22×D6111yescharacteristic
401C4⋊C8—22no—
402C4⋊C8—1224no—
403C4⋊C8—1222no—
404#103—24no—
405#103—1224no—
406#102—1224no—
407#102—1224no—
408#95—21no—
409#95—21no—
410#95—1221no—
411#95—1221no—
412C24—611no—
413C24—1221no—
414#41—6116no—
415#41—1228no—
416#41—1228no—
417#41—1228no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C4×D6112yes—
422C2×Q8D4×D6112yes—
423C2×Q8Q8×D6112yes—
424C2×D4—612no—
425C2×D4—612no—
426C2×D4—612no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—611no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1221no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—2442no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4C23×D6111yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—2448no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—612no—
473C4⋊C4—1224no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—2444no—
478C4⋊C4—2444no—
479C4⋊C4C2×C4×D6112yes—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—2444no—
484C42—24no—
485C42—612no—
486C42—612no—
487C42—614no—
488C42—614no—
489C42—1224no—
490C42—1224no—
491C42—2444no—
492C42C2×C4×D6112yes—
493C42D4×D6112yes—
494C23—612no—
495C23—612no—
496C23—612no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—2442no—
503C23—2442no—
504C23C2×D4×D6111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8C4⋊C4×D6114yes—
509D4—614no—
510D4—12216no—
511D4—1222no—
512D4—1222no—
513D4—1222no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—614no—
529C2×C4—614no—
530C2×C4—614no—
531C2×C4—614no—
532C2×C4—614no—
533C2×C4—612no—
534C2×C4—1228no—
535C2×C4—12216no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—2444no—
539C2×C4—2448no—
540C2×C4—2448no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4C22⋊C4×D6112yes—
546C2×C4C22⋊C4×D6112yes—
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×Q8×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4#41×D6111yescharacteristic
553C8—42no—
554C8—2442no—
555C22—614no—
556C22—1222no—
557C22—1224no—
558C22—1222no—
559C22—1224no—
560C22—1224no—
561C22—2442no—
562C22—2444no—
563C22—2442no—
564C22—2444no—
565C22C2×C22⋊C4×D6111yescharacteristic
566C22#121×D6111yescharacteristic
567C22C22⋊Q8×D6111yescharacteristic
568C22#135×D6112yes—
569C22#1684056112yes—
570C4—22no—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—82no—
575C4—614no—
576C4—614no—
577C4—1222no—
578C4—4882no—
579C4—2442no—
580C4C2×C4⋊C4×D6111yescharacteristic
581C4#117×D6111yescharacteristic
582C4#121×D6111yescharacteristic
583C4C22⋊Q8×D6111yescharacteristic
584C2—1224no—
585C2—2444no—
586C2#380×D6111yescharacteristic
587C2#416×D6111yescharacteristic
588C2#2145433711yescharacteristic
589C2#2145527312yes—
590C2#2145560612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#20645
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×D61C2semidirect
4. #21448225C2semidirect
5. #21445834C2semidirect
6. #21445830C2semidirect
7. #21445826C2semidirect
8. #21445822C2semidirect
9. #21445818C2semidirect
10. #21445534C2semidirect
11. C61×#1805C2semidirect
15. C4×#11758226C4semidirect
138. #11758226C4⋊C4semidirect
288. Dic61#416semidirect
294. #1805D61semidirect
342. C61#20645semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582262
C2×C4⋊C8×D612
C2×#168397572
#214482252
#214458342
#214458302
#214458262
#214458222
#214458182
#214455342
C61×#18052
#2064561