Finite Group Databasebeta

← #30617064

Subgroups — #30617064

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×#6923C211yesmaximal, characteristic, complemented
2C2×#16840728C211yesmaximal, characteristic
3#21462306C212yesmaximal
4#21461967C211yesmaximal, characteristic, complemented
5#21461925C211yesmaximal, characteristic, complemented
6#21461908C212yesmaximal, complemented
7#21461832C212yesmaximal
8#21461697C211yesmaximal, characteristic
9#106×#6923C211yesmaximal, characteristic, complemented
10#21461592C212yesmaximal, complemented
11#470×D61C211yesmaximal, characteristic
12C22×D61⋊C8C2211yescharacteristic
13C2×Q8×#6923C2212yes—
14C2×#14519856C2211yescharacteristic
15Q16×#6923—24no—
16C2×#14519846C2211yescharacteristic
17C2×#14519845C2211yescharacteristic
18C2×#14519840C2212yes—
19C2×#14519840C2212yes—
20#16840728—24no—
21C2×#14519831C2211yescharacteristic
22C2×#14519831C2211yescharacteristic
23C2×C8×#6923C2211yescharacteristic
24C2×C8×#6923C2211yescharacteristic
25(C2×#6923)⋊Q8C2212yes—
26#16840713—22no—
27#16840713C2212yes—
28#16840712—22no—
29#16840712—22no—
30#16840711—22no—
31#16840704C2212yes—
32C4⋊C4×#6923C2212yes—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36C2×Q16×D61C2211yescharacteristic
37C2×M16×D61C2211yescharacteristic
38C22⋊Q8×D61C2212yes—
39#16840021C411yescharacteristic, complemented
40#16840020C411yescharacteristic, complemented
41#16840019C412yescomplemented
42#16840018C412yescomplemented
43#16840017C411yescharacteristic, complemented
44#106×D61C2211yescharacteristic
45#103×D61C2212yes—
46C61×#470C411yescharacteristic, complemented
47C2×D61⋊C8—21no—
48C2×D61⋊C8—22no—
49C2×D61⋊C8—22no—
50C2×D61⋊C8C2311yescharacteristic
51C2×D61⋊C8C2311yescharacteristic
52C22×#92420—21no—
53Q8×#6923—24no—
54Q8×#6923—24no—
55#14519856—22no—
56#14519856—22no—
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923D411yescharacteristic
60C2×C4⋊#6923D411yescharacteristic
61C2×C4⋊#6923C2312yes—
62C2×C4⋊#6923C2312yes—
63C2×C22⋊#6923—22no—
64C2×C22⋊#6923D411yescharacteristic
65C2×C22⋊#6923D411yescharacteristic
66C2×C4×#6923—22no—
67C2×C4×#6923—22no—
68C2×C4×#6923C2311yescharacteristic
69C2×C4×#6923C2311yescharacteristic
70#14519846—22no—
71#14519845—22no—
72#14519840—24no—
73#14519840—24no—
74#14519831—22no—
75#14519831—22no—
76C8×#6923—22no—
77C8×#6923D414yes—
78(C2×#6923)⋊C4—22no—
79(C2×#6923)⋊C4—22no—
80(C2×#6923)⋊C4—22no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83C2×Q8×D61C2312yes—
84C22×C4×D61C2311yescharacteristic
85C2×Dic244C2×C411yescharacteristic
86C2×#11758202C2×C412yes—
87Q16×D61—22no—
88Q16×D61—22no—
89C2×#11758194C2×C411yescharacteristic
90C2×#11758189C2×C411yescharacteristic
91C2×#11758189C2×C411yescharacteristic
92M16×D61—22no—
93M16×D61—22no—
94C2×C8×D61C2311yescharacteristic
95C2×C8×D61C2311yescharacteristic
96#14519757C2×C412yes—
97#14519756C2×C412yes—
98#14519755—22no—
99#14519754—22no—
100#14519753C2×C412yes—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2312yes—
103C22⋊C4×D61—22no—
104#14519709C2×C411yescharacteristic
105#14519708C2×C412yes—
106#14519702C2×C412yes—
107#14519701C2×C412yes—
108#14519700C2×C412yes—
109C2×Q16×C61C2×C411yescharacteristic
110C2×M16×C61C2×C411yescharacteristic
111C22⋊Q8×C61C2×C412yes—
112#106×C61C2×C411yescharacteristic
113#103×C61C2×C412yes—
114C23×D61C2×D411yescharacteristic
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923C2×D411yescharacteristic
118C22×#6923C2×D411yescharacteristic
119C22×#6923C2×D411yescharacteristic
120C22×#6923C2×D411yescharacteristic
121C2×Q8×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×Dic61#4111yescharacteristic
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61C22×C411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61#4111yescharacteristic
137C2×C4×D61C2411yescharacteristic
138C2×Dic122—22no—
139C2×Dic122C22×C412yes—
140C22⋊#6923—22no—
141C22⋊#6923—22no—
142C22⋊#6923—82no—
143C2×#92420—24no—
144C2×#92420—21no—
145C2×#92420—21no—
146C4⋊#6923—22no—
147C4⋊#6923—22no—
148C4⋊#6923—22no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—44no—
153C4⋊#6923—82no—
154C4×#6923—21no—
155C4×#6923—21no—
156C4×#6923—44no—
157C4×#6923—82no—
158C4×#6923C2×D412yes—
159C4×#6923#4112yes—
160D61⋊C8—24no—
161D61⋊C8—21no—
162D61⋊C8—21no—
163D61⋊C8C2×D412yes—
164D61⋊C8#4112yes—
165Q16×C61—22no—
166M16×C61—22no—
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C22×C411yescharacteristic
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—22no—
172C22⋊Dic61—22no—
173#11758202—24no—
174#11758198—22no—
175#11758198—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C22×C412yes—
178#11758196—22no—
179#11758196—22no—
180#11758196—22no—
181#11758194—22no—
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184Dic244—22no—
185#11758189—22no—
186#11758189—22no—
187C8×D61—21no—
188C8×D61—21no—
189C8×D61C2×D412yes—
190C8×D61#4112yes—
191C23×C61C4×D411yescharacteristic
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61C4×D411yescharacteristic
197C22×D61C22×D411yescharacteristic
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—21no—
201C2×#6923—21no—
202C2×#6923—22no—
203C2×#6923—22no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923#12112yes—
211C2×#6923#12112yes—
212Q8×C61—22no—
213Q8×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C23×C411yescharacteristic
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61C4×D411yescharacteristic
225C2×Dic61C2×#4111yescharacteristic
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61C22×D411yescharacteristic
237C4×D61C2×#4111yescharacteristic
238C4×D61C2×#4111yescharacteristic
239C4×D61C2×#4111yescharacteristic
240Dic122—22no—
241Dic122—22no—
242Dic122—42no—
243Dic122—42no—
244#92420—22no—
245#92420—22no—
246C8×C61—21no—
247C8×C61C4×D412yes—
248#92418—21no—
249#92418C4×D412yes—
250#21537—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—21no—
253C22×C61C2×C4×D411yescharacteristic
254C2×D61—22no—
255C2×D61—21no—
256C2×D61—42no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61C2×#12111yescharacteristic
260C2×D61#53211yescharacteristic
261C2×D61#57712yes—
262C2×D61#57812yes—
263#6923—22no—
264#6923—22no—
265#6923—41no—
266#6923—41no—
267#6923—82no—
268#6923—82no—
269C4×C61—21no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61C2×C4×D411yescharacteristic
273C4×C61C4×#4111yescharacteristic
274Dic61—21no—
275Dic61—42no—
276Dic61—42no—
277Dic61#52511yescharacteristic
278Dic61#52911yescharacteristic
279C2×#470—611no—
280C2×C4×Q16—611no—
281C2×#405—611no—
282#2323—612no—
283#1971—611no—
284#1944—611no—
285#1933—612no—
286#1892—612no—
287#1809—611no—
288C4×#106—611no—
289#1763—612no—
290C2×C61—41no—
291C2×C61C4×#12111yescharacteristic
292C2×C61#294712yes—
293D61—41no—
294D61#305911yescharacteristic
295D61#306011yescharacteristic
296D61#317812yes—
297C22×Q16—611no—
298C22×M16—611no—
299C23×C8—611no—
300C2×C22⋊Q8—612no—
301C2×C4×Q8—612no—
302C2×#117—611no—
303#470—611no—
304#470—611no—
305#470—612no—
306#470—611no—
307#470—612no—
308#470#692311yescharacteristic, complemented
309C4×Q16—1224no—
310C2×#107—611no—
311C2×#106—611no—
312C2×#106—611no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#103—612no—
316#405—1224no—
317C2×#97—611no—
318C2×#97—611no—
319C2×C4×C8—611no—
320C2×C4×C8—611no—
321#391—612no—
322#389—612no—
323#389—1222no—
324#388—1222no—
325#388—1222no—
326#387—1222no—
327#380—612no—
328C4×C4⋊C4—612no—
329#340—612no—
330#340—612no—
331#340—612no—
332C61#2153711yescharacteristic, complemented, Sylow
333C22×Q8—612no—
334C23×C4—611no—
335C2×Q16—612no—
336C2×Q16—611no—
337C2×Q16—1222no—
338C2×Q16—1222no—
339C2×Q16C2×#692311yescharacteristic
340C2×M16—611no—
341C2×M16—611no—
342C2×M16—611no—
343C2×M16—1222no—
344C2×M16—1222no—
345C2×M16C2×#692311yescharacteristic
346C22×C8—611no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—1222no—
351C22×C8—1222no—
352C22×C8—1221no—
353C22×C8—1221no—
354C22⋊Q8—612no—
355C22⋊Q8—612no—
356C22⋊Q8—612no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C22⋊Q8C2×#692312yes—
360C4×Q8—1224no—
361C4×Q8—1224no—
362#117—1222no—
363#117—1222no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C22⋊C4—611no—
373C2×C22⋊C4—611no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—1222no—
379C2×C42—1222no—
380#107—1222no—
381#106—611no—
382#106—612no—
383#106—1222no—
384#106C2×#692311yescharacteristic
385#103—612no—
386#103—612no—
387#103—612no—
388#103—1224no—
389#103—1224no—
390#103C2×#692312yes—
391#97—1222no—
392#97—1222no—
393C4×C8—614no—
394C4×C8—1222no—
395#95—1222no—
396#95—1222no—
397#95—1222no—
398#95—1222no—
399#95—1222no—
400C24—611no—
401C2×Q8—612no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8C22×#692312yes—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4C22×#692311yescharacteristic
426Q16—22no—
427Q16—1224no—
428Q16—1222no—
429M16—22no—
430M16—1222no—
431M16—1222no—
432M16—1222no—
433C2×C8—611no—
434C2×C8—612no—
435C2×C8—612no—
436C2×C8—612no—
437C2×C8—612no—
438C2×C8—611no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1224no—
446C2×C8—1224no—
447C2×C8C22×#692311yescharacteristic
448C2×C8C22×#692311yescharacteristic
449C4⋊C4—22no—
450C4⋊C4—612no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—4882no—
462C4⋊C4—2444no—
463C4⋊C4C22×#692312yes—
464C22⋊C4—22no—
465C22⋊C4—1222no—
466C22⋊C4—1222no—
467C22⋊C4—1222no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C22⋊C4—4882no—
471C42—612no—
472C42—612no—
473C42—1221no—
474C42—1221no—
475C42—4882no—
476C42—2444no—
477C23—611no—
478C23—611no—
479C23—1221no—
480C23—1222no—
481C23—1221no—
482C23—1222no—
483C23D4×#692311yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—1222no—
487Q8—1222no—
488Q8—2442no—
489Q8—2442no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—611no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4D4×#692311yescharacteristic
529C2×C4C23×#692311yescharacteristic
530C8—21no—
531C8—612no—
532C8—1222no—
533C8—1222no—
534C8—1221no—
535C8D4×#692312yes—
536C22—21no—
537C22—22no—
538C22—611no—
539C22—611no—
540C22—612no—
541C22—612no—
542C22—1221no—
543C22—1222no—
544C22—2441no—
545C22—2441no—
546C22—2442no—
547C22C2×D4×#692311yescharacteristic
548C4—21no—
549C4—42no—
550C4—42no—
551C4—611no—
552C4—611no—
553C4—1222no—
554C4—1222no—
555C4—1221no—
556C4—4882no—
557C4—4882no—
558C4—2441no—
559C4—2441no—
560C4—2442no—
561C4—2442no—
562C4C2×D4×#692311yescharacteristic
563C4#41×#692311yescharacteristic
564C2—41no—
565C2—611no—
566C2—611no—
567C2—612no—
568C2—2441no—
569C2#121×#692311yescharacteristic
570C2#2146257712yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#470C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#470#6923
hypocenterC61#21537
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×#6923C2semidirect
4. #21461967C2semidirect
5. #21461925C2semidirect
6. #21461908C2semidirect
9. #106×#6923C2semidirect
10. #21461592C2semidirect
39. #16840021C4semidirect
40. #16840020C4semidirect
41. #16840019C4semidirect
42. #16840018C4semidirect
43. #16840017C4semidirect
46. C61×#470C4semidirect
308. #470#6923semidirect
332. C61#21537semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#69232
C2×#168407282
#214623062
#214619672
#214619252
#214619082
#214618322
#214616972
#106×#69232
#214615922
#470×D612
#2153761