Finite Group Databasebeta

← #30622468

Subgroups — #30622468

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462348C211yesmaximal, characteristic, complemented
2#21462337C211yesmaximal, characteristic
3#21461969C212yesmaximal
4#21461946C211yesmaximal, characteristic, complemented
5#21461930C211yesmaximal, characteristic, complemented
6#21461899C212yesmaximal, complemented
7#21461815C212yesmaximal
8#21461616C211yesmaximal, characteristic, complemented
9#21461592C212yesmaximal, complemented
10C4×#14519839C211yesmaximal, characteristic
11#461×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×#14519856—21no—
14C2×#14519846C2212yes—
15C2×#14519840C2212yes—
16C2×#14519840C2212yes—
17C2×#14519839C2211yescharacteristic
18C2×#14519839C2211yescharacteristic
19C2×#14519831C2212yes—
20C4×D61⋊C8C2211yescharacteristic
21(C2×#6923)⋊Q8C2212yes—
22#16840714C2212yes—
23#16840713—21no—
24#16840712—22no—
25#16840711—22no—
26#16840708C2211yescharacteristic
27#16840704—21no—
28C4⋊C4×#6923—21no—
29C4⋊C4×#6923C2212yes—
30C4×C4⋊#6923C2211yescharacteristic
31C4×C4⋊#6923C2211yescharacteristic
32#16840636C2212yes—
33#16840635C2211yescharacteristic
34#16840635C2211yescharacteristic
35(C2×#6923)⋊C8C2212yes—
36C2×SD16×D61C2212yes—
37C4⋊Q8×D61C2211yescharacteristic
38C4×D4×D61C2211yescharacteristic
39#16839968C411yescharacteristic, complemented
40#16839967C411yescharacteristic, complemented
41#16839966C412yescomplemented
42#16839965C412yescomplemented
43#16839964C411yescharacteristic, complemented
44C4⋊C8×D61C2211yescharacteristic
45#103×D61C2212yes—
46C61×#461C411yescharacteristic, complemented
47C2×D61⋊C8C2311yescharacteristic
48C2×D61⋊C8C2311yescharacteristic
49Q8×#6923—24no—
50Q8×#6923—42no—
51#14519856—44no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923D411yescharacteristic
56C2×C4⋊#6923D411yescharacteristic
57C2×C4⋊#6923C2312yes—
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2311yescharacteristic
60C2×C22⋊#6923—22no—
61C2×C22⋊#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923—22no—
64C2×C4×#6923—21no—
65C2×C4×#6923C2312yes—
66#14519846—42no—
67#14519840—24no—
68#14519840—42no—
69#14519839—22no—
70#14519839—22no—
71#14519831—42no—
72C4×#92420—21no—
73C4×#92420—21no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—21no—
78(C2×#6923)⋊C4D412yes—
79C2×Q8×D61C2312yes—
80C2×D4×D61C2311yescharacteristic
81C22×C4×D61—21no—
82C2×#11758190C2×C412yes—
83C2×#11758200C2×C412yes—
84C2×#11758201C2×C412yes—
85SD16×D61—42no—
86SD16×D61—42no—
87C2×C8×D61C2312yes—
88#14519777C2×C411yescharacteristic
89#14519776—21no—
90#14519776—21no—
91#14519775C2×C412yes—
92C4×D244C2×C411yescharacteristic
93D4×Dic61C2×C411yescharacteristic
94#14519739C2×C411yescharacteristic
95C4×C22⋊D61—21no—
96#14519737—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61C2311yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102C4⋊#92418C2×C411yescharacteristic
103#14519706C2×C412yes—
104#14519702C2×C412yes—
105#14519701C2×C412yes—
106#14519700C2×C412yes—
107C2×SD16×C61C2×C412yes—
108C4⋊Q8×C61C2×C411yescharacteristic
109C4×D4×C61C2×C411yescharacteristic
110C4⋊C8×C61C2×C411yescharacteristic
111#103×C61C2×C412yes—
112C23×D61—21no—
113C22×#6923—21no—
114C22×#6923—21no—
115C22×#6923—22no—
116C22×#6923C2×D412yes—
117C22×#6923C2×D412yes—
118C2×Q8×C61C22×C412yes—
119C2×D4×C61C22×C411yescharacteristic
120C22×C4×C61—21no—
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123Q8×D61—22no—
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—22no—
127D4×D61—21no—
128D4×D61—21no—
129D4×D61—21no—
130D4×D61—21no—
131C2×D244C22×C411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2411yescharacteristic
141C2×Dic122—22no—
142C2×Dic122C22×C412yes—
143C22⋊#6923—44no—
144C22⋊#6923—82no—
145C2×#92420—21no—
146C2×#92420—21no—
147C4⋊#6923—22no—
148C4⋊#6923—22no—
149C4⋊#6923—21no—
150C4⋊#6923—22no—
151C4⋊#6923—21no—
152C4⋊#6923—22no—
153C4⋊#6923—21no—
154C4⋊#6923—44no—
155C4⋊#6923—42no—
156C4⋊#6923—81no—
157C4⋊#6923#4112yes—
158C4×#6923—22no—
159C4×#6923—22no—
160C4×#6923—44no—
161C4×#6923—44no—
162C4×#6923—42no—
163D61⋊C8—21no—
164D61⋊C8—21no—
165D61⋊C8—21no—
166D61⋊C8#4112yes—
167SD16×C61—42no—
168C2×C8×C61C22×C412yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C411yescharacteristic
172C22⋊C4×C61—21no—
173C42×C61C22×C411yescharacteristic
174C22⋊Dic61—21no—
175#11758201—42no—
176#11758200—42no—
177#11758198—21no—
178#11758198—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C411yescharacteristic
182#11758196—22no—
183#11758196—22no—
184#11758196—21no—
185C4×Dic61C22×C411yescharacteristic
186C4×Dic61#4111yescharacteristic
187C4×Dic61#4111yescharacteristic
188C2×#92418C22×C412yes—
189#11758190—42no—
190C8×D61—22no—
191C8×D61—22no—
192C23×C61—21no—
193C22×D61—21no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61C22×D411yescharacteristic
201C2×#6923—22no—
202C2×#6923—22no—
203C2×#6923—22no—
204C2×#6923—22no—
205C2×#6923—22no—
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—41no—
209C2×#6923—41no—
210C2×#6923—41no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—41no—
214C2×#6923—42no—
215C2×#6923—41no—
216Q8×C61—22no—
217Q8×C61—22no—
218D4×C61—21no—
219D4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—42no—
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C23×C411yescharacteristic
226C22⋊D61—41no—
227C22⋊D61—41no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—42no—
231C2×Dic61C4×D411yescharacteristic
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C2×#4111yescharacteristic
234D244—21no—
235D244—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61AGL(1,ℤ/8)12yes—
245C4×D61#13712yes—
246C4×D61C2×#4111yescharacteristic
247C4×D61C2×#4111yescharacteristic
248C4×D61H(5,𝔽2)11yescharacteristic
249C4×D61H(5,𝔽2)11yescharacteristic
250Dic122—22no—
251Dic122—22no—
252Dic122—42no—
253Dic122—42no—
254#92420—22no—
255#92420—41no—
256C8×C61—22no—
257#92418—22no—
258#23100—611nomaximal, Sylow
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C2×C4×D411yescharacteristic
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61#54611yescharacteristic
271C2×D61#55011yescharacteristic
272C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
273C2×D61C2×#13711yescharacteristic
274C2×D61#57711yescharacteristic
275C2×D61#57811yescharacteristic
276#6923—42no—
277#6923—42no—
278#6923—42no—
279#6923—82no—
280#6923—81no—
281C4×C61—21no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61#44012yes—
285C4×C61C4×#4111yescharacteristic
286C4×C61#51811yescharacteristic
287Dic61—21no—
288Dic61—42no—
289Dic61—41no—
290Dic61#44012yes—
291Dic61#55311yescharacteristic
292Dic61#55511yescharacteristic
293C2×#461—611no—
294#2365—611no—
295#2354—611no—
296#1973—612no—
297#1956—611no—
298#1947—611no—
299#1928—612no—
300#1880—612no—
301#1770—611no—
302#1763—612no—
303C4×#102—611no—
304C2×C61—81no—
305C2×C61#236311yescharacteristic
306C2×C61C2×#44011yescharacteristic
307C2×C61#294711yescharacteristic
308D61—81no—
309D61#332911yescharacteristic
310D61#333011yescharacteristic
311D61#339711yescharacteristic
312D61#339811yescharacteristic
313C22×SD16—612no—
314C2×C4⋊Q8—611no—
315C2×C4×Q8—612no—
316C2×C4×D4—611no—
317C2×#117—1221no—
318#461—612no—
319#461—611no—
320#461—611no—
321#461—611no—
322#461—612no—
323#461#692311yescharacteristic, complemented
324C2×#107—612no—
325C2×C4⋊C8—611no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#102—611no—
330C2×#102—611no—
331C2×#97—612no—
332C2×C4×C8—611no—
333#391—612no—
334#390—612no—
335#389—1221no—
336#388—1222no—
337#387—1222no—
338#384—611no—
339#380—1221no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—612no—
342C4×C4⋊C4—611no—
343C4×C4⋊C4—1221no—
344#340—612no—
345#340—611no—
346#340—611no—
347#336—612no—
348C61#2310011yescharacteristic, complemented, Sylow
349C22×Q8—612no—
350C22×D4—611no—
351C23×C4—1221no—
352C2×SD16—612no—
353C2×SD16—612no—
354C2×SD16—612no—
355C2×SD16—2442no—
356C2×SD16—2442no—
357C2×SD16C2×#692312yes—
358C22×C8—611no—
359C22×C8—612no—
360C22×C8—611no—
361C4⋊Q8—612no—
362C4⋊Q8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8C2×#692311yescharacteristic
366C4×Q8—1224no—
367C4×Q8—2442no—
368C4×D4—611no—
369C4×D4—611no—
370C4×D4—611no—
371C4×D4—1221no—
372C4×D4—1221no—
373C4×D4C2×#692311yescharacteristic
374#117—2444no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—611no—
390C2×C42—612no—
391C2×C42—1222no—
392C2×C42—1222no—
393C2×C42—1221no—
394#107—2442no—
395C4⋊C8—612no—
396C4⋊C8—611no—
397C4⋊C8C2×#692311yescharacteristic
398#103—612no—
399#103—612no—
400#103—612no—
401#103—1224no—
402#103—2442no—
403#103C2×#692312yes—
404#102—1222no—
405#102—1222no—
406#97—2442no—
407C4×C8—1221no—
408C4×C8—1221no—
409#95—612no—
410#95—1222no—
411#95—1221no—
412#95—1222no—
413#95—1222no—
414C24—1221no—
415C2×Q8—612no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8C22×#692312yes—
422C2×D4—611no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4C22×#692311yescharacteristic
429C22×C4—21no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—2442no—
444C22×C4—2442no—
445SD16—42no—
446SD16—2442no—
447SD16—2442no—
448SD16—2442no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8C22×#692312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—612no—
462C4⋊C4—611no—
463C4⋊C4—1221no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1221no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4—2444no—
478C4⋊C4C22×#692311yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C22⋊C4—4882no—
484C22⋊C4—2444no—
485C42—611no—
486C42—611no—
487C42—611no—
488C42—1222no—
489C42—1222no—
490C42—2442no—
491C42—2444no—
492C42—2444no—
493C42C22×#692311yescharacteristic
494C23—21no—
495C23—611no—
496C23—1221no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503Q8—22no—
504Q8—22no—
505Q8—1222no—
506Q8—1222no—
507Q8—2442no—
508Q8—2442no—
509D4—21no—
510D4—21no—
511D4—1221no—
512D4—1221no—
513D4—2441no—
514D4—2441no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—2442no—
539C2×C4—2441no—
540C2×C4—2442no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4D4×#692311yescharacteristic
554C2×C4D4×#692311yescharacteristic
555C2×C4C23×#692311yescharacteristic
556C8—22no—
557C8—1222no—
558C8—1222no—
559C8—2441no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—611no—
564C22—611no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22—611no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22C2×D4×#692311yescharacteristic
577C4—21no—
578C4—41no—
579C4—42no—
580C4—612no—
581C4—611no—
582C4—611no—
583C4—1221no—
584C4—4882no—
585C4—4881no—
586C4—2442no—
587C4—2442no—
588C4—2442no—
589C4—2441no—
590C4—2442no—
591C4#1684080112yes—
592C4#41×#692311yescharacteristic
593C4#1684082911yescharacteristic
594C2—81no—
595C2—611no—
596C2—611no—
597C2—611no—
598C2—611no—
599C2—4881no—
600C2#2146234611yescharacteristic
601C2C2×#1684080111yescharacteristic
602C2#2146257711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#461C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#461#6923
hypocenterC61#23100
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #21462348C2semidirect
4. #21461946C2semidirect
5. #21461930C2semidirect
6. #21461899C2semidirect
8. #21461616C2semidirect
9. #21461592C2semidirect
39. #16839968C4semidirect
40. #16839967C4semidirect
41. #16839966C4semidirect
42. #16839965C4semidirect
43. #16839964C4semidirect
46. C61×#461C4semidirect
323. #461#6923semidirect
348. C61#23100semidirect

Maximal subgroups

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

subgroupindex
#214623482
#214623372
#214619692
#214619462
#214619302
#214618992
#214618152
#214616162
#214615922
C4×#145198392
#461×D612
#2310061