Finite Group Databasebeta

← #30622349

Subgroups — #30622349

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462348C211yesmaximal, characteristic, complemented
2#21462337C211yesmaximal, characteristic
3#21462000C211yesmaximal, characteristic, complemented
4#21461930C211yesmaximal, characteristic, complemented
5#21461909C212yesmaximal
6#21461896C212yesmaximal
7#21461880C212yesmaximal, complemented
8#107×#6923C212yesmaximal, complemented
9#21461612C211yesmaximal, characteristic, complemented
10C4×#14519839C211yesmaximal, characteristic
11#474×D61C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13C2×#14519856—21no—
14C2×#14519846C2212yes—
15C2×#14519839C2211yescharacteristic
16C2×#14519839C2211yescharacteristic
17C4×D61⋊C8C2211yescharacteristic
18C2×C8×#6923C2212yes—
19(C2×#6923)⋊Q8—21no—
20#16840714C2212yes—
21#16840713C2212yes—
22#16840712—22no—
23#16840711—22no—
24#16840708C2211yescharacteristic
25#16840704—21no—
26C4⋊C4×#6923C2212yes—
27C4⋊C4×#6923C2212yes—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30#16840636C2212yes—
31#16840635C2211yescharacteristic
32#16840635C2211yescharacteristic
33#16840635C2212yes—
34#16840635C2212yes—
35(C2×#6923)⋊C8C2212yes—
36C4⋊Q8×D61C2211yescharacteristic
37C4×D4×D61C2211yescharacteristic
38#16840043C411yescharacteristic, complemented
39#16840042C411yescharacteristic, complemented
40#16840041C412yescomplemented
41#16840040C412yescomplemented
42#16840039C411yescharacteristic, complemented
43#107×D61C2212yes—
44C4⋊C8×D61C2211yescharacteristic
45#102×D61C2212yes—
46C61×#474C411yescharacteristic, complemented
47C2×D61⋊C8C2311yescharacteristic
48C2×D61⋊C8C2311yescharacteristic
49Q8×#6923—44no—
50#14519856—44no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923D411yescharacteristic
55C2×C4⋊#6923D411yescharacteristic
56C2×C4⋊#6923C2312yes—
57C2×C4⋊#6923C2311yescharacteristic
58C2×C4⋊#6923C2311yescharacteristic
59C2×C22⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C4×#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923—21no—
64C2×C4×#6923C2312yes—
65#14519846—24no—
66#14519839—22no—
67#14519839—22no—
68C4×#92420—21no—
69C4×#92420—21no—
70C8×#6923—24no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—21no—
75(C2×#6923)⋊C4D412yes—
76C2×Q8×D61—21no—
77C2×D4×D61C2311yescharacteristic
78C22×C4×D61—21no—
79C2×C8×D61C2312yes—
80#14519777C2×C411yescharacteristic
81#14519776—21no—
82#14519776C2×C412yes—
83#14519775—21no—
84C4×D244C2×C411yescharacteristic
85D4×Dic61C2×C411yescharacteristic
86#14519739C2×C411yescharacteristic
87C4×C22⋊D61—21no—
88#14519737—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61C2312yes—
91C4⋊C4×D61C2311yescharacteristic
92C22⋊C4×D61—21no—
93C42×D61C2311yescharacteristic
94#14519711C2×C412yes—
95#14519710C2×C412yes—
96#14519710C2×C412yes—
97C4⋊#92418C2×C411yescharacteristic
98#14519706C2×C412yes—
99#14519699C2×C412yes—
100#14519698C2×C412yes—
101#14519697C2×C412yes—
102C4⋊Q8×C61C2×C411yescharacteristic
103C4×D4×C61C2×C411yescharacteristic
104#107×C61C2×C412yes—
105C4⋊C8×C61C2×C411yescharacteristic
106#102×C61C2×C412yes—
107C23×D61—21no—
108C22×#6923—21no—
109C22×#6923—21no—
110C22×#6923—22no—
111C22×#6923C2×D412yes—
112C22×#6923C2×D412yes—
113C2×Q8×C61—21no—
114C2×D4×C61C22×C411yescharacteristic
115C22×C4×C61—21no—
116C2×C22⋊D61—21no—
117C22×Dic61—21no—
118Q8×D61—42no—
119Q8×D61—42no—
120D4×D61—21no—
121D4×D61—21no—
122D4×D61—21no—
123D4×D61—21no—
124C2×D244C22×C411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—22no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61C2411yescharacteristic
134C2×Dic122—21no—
135C2×Dic122—22no—
136C22⋊#6923—44no—
137C22⋊#6923—82no—
138C2×#92420—21no—
139C2×#92420—21no—
140C4⋊#6923—22no—
141C4⋊#6923—21no—
142C4⋊#6923—22no—
143C4⋊#6923—21no—
144C4⋊#6923—22no—
145C4⋊#6923—21no—
146C4⋊#6923—44no—
147C4⋊#6923—42no—
148C4⋊#6923—81no—
149C4⋊#6923D814yes—
150C4⋊#6923#4112yes—
151C4×#6923—22no—
152C4×#6923—22no—
153C4×#6923—44no—
154C4×#6923—44no—
155C4×#6923—42no—
156D61⋊C8—21no—
157D61⋊C8—21no—
158D61⋊C8—21no—
159D61⋊C8#4112yes—
160C2×C8×C61C22×C412yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C22×C412yes—
163C4⋊C4×C61C22×C411yescharacteristic
164C22⋊C4×C61—21no—
165C42×C61C22×C411yescharacteristic
166C22⋊Dic61—21no—
167#11758198—21no—
168#11758198—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61C22×C412yes—
171C4⋊Dic61C22×C411yescharacteristic
172#11758196—22no—
173#11758196—21no—
174#11758196—22no—
175C4×Dic61C22×C411yescharacteristic
176C4×Dic61#4111yescharacteristic
177C4×Dic61#4111yescharacteristic
178C2×#92418C22×C412yes—
179C8×D61—22no—
180C8×D61—22no—
181C23×C61—21no—
182C22×D61—21no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61C22×D411yescharacteristic
190C2×#6923—22no—
191C2×#6923—22no—
192C2×#6923—22no—
193C2×#6923—22no—
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—41no—
197C2×#6923—41no—
198C2×#6923—41no—
199C2×#6923—42no—
200C2×#6923—41no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—41no—
204C2×#6923C2×D814yes—
205Q8×C61—42no—
206D4×C61—21no—
207D4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—42no—
211C2×C4×C61C4×D411yescharacteristic
212C2×C4×C61C4×D411yescharacteristic
213C2×C4×C61C23×C411yescharacteristic
214C22⋊D61—41no—
215C22⋊D61—41no—
216C2×Dic61—22no—
217C2×Dic61—21no—
218C2×Dic61—42no—
219C2×Dic61C4×D411yescharacteristic
220C2×Dic61C4×D411yescharacteristic
221C2×Dic61C2×#4111yescharacteristic
222D244—21no—
223D244—21no—
224C4×D61—21no—
225C4×D61—21no—
226C4×D61—41no—
227C4×D61—42no—
228C4×D61—41no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61C2×D812yes—
233C4×D61#13512yes—
234C4×D61C2×#4111yescharacteristic
235C4×D61C2×#4111yescharacteristic
236C4×D612−1+411yescharacteristic
237C4×D612−1+411yescharacteristic
238Dic122—42no—
239Dic122—44no—
240#92420—22no—
241#92420—41no—
242C8×C61—22no—
243#92418—22no—
244#23060—611nomaximal, Sylow
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61C2×C4×D411yescharacteristic
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—41no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61#54711yescharacteristic
257C2×D61#54811yescharacteristic
258C2×D61C22×D811yescharacteristic
259C2×D61C2×#13511yescharacteristic
260C2×D61#57711yescharacteristic
261C2×D61#57811yescharacteristic
262#6923—24no—
263#6923—42no—
264#6923—42no—
265#6923—81no—
266#6923—82no—
267C4×C61—21no—
268C4×C61—42no—
269C4×C61—41no—
270C4×C61C4×D812yes—
271C4×C61C4×#4111yescharacteristic
272C4×C61#51911yescharacteristic
273Dic61—21no—
274Dic61—42no—
275Dic61—41no—
276Dic61C4×D812yes—
277Dic61#55211yescharacteristic
278Dic61#55611yescharacteristic
279C2×#474—611no—
280#2365—611no—
281#2354—611no—
282#1990—611no—
283#1947—611no—
284#1934—612no—
285#1925—612no—
286#1918—612no—
287C4×#107—612no—
288C8×C4⋊C4—611no—
289C4×#102—611no—
290C2×C61—81no—
291C2×C61#236211yescharacteristic
292C2×C61C2×C4×D811yescharacteristic
293C2×C61#294711yescharacteristic
294D61—81no—
295D61#333411yescharacteristic
296D61#334011yescharacteristic
297D61#335811yescharacteristic
298D61#336411yescharacteristic
299C2×C4⋊Q8—611no—
300C2×C4×Q8—1221no—
301C2×C4×D4—611no—
302C2×#117—1221no—
303#474—611no—
304#474—611no—
305#474—612no—
306#474—611no—
307#474—612no—
308#474#692311yescharacteristic, complemented
309C2×#107—612no—
310C2×#107—612no—
311C2×C4⋊C8—611no—
312C2×#102—612no—
313C2×#102—611no—
314C2×#102—611no—
315C2×C4×C8—612no—
316C2×C4×C8—611no—
317#391—1221no—
318#390—612no—
319#389—612no—
320#388—1222no—
321#387—1222no—
322#384—611no—
323#380—1221no—
324C4×C4⋊C4—611no—
325C4×C4⋊C4—612no—
326C4×C4⋊C4—612no—
327C4×C4⋊C4—611no—
328#340—611no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340—611no—
333#336—612no—
334C61#2306011yescharacteristic, complemented, Sylow
335C22×Q8—1221no—
336C22×D4—611no—
337C23×C4—1221no—
338C22×C8—611no—
339C22×C8—612no—
340C22×C8—611no—
341C4⋊Q8—612no—
342C4⋊Q8—611no—
343C4⋊Q8—1221no—
344C4⋊Q8—1221no—
345C4⋊Q8C2×#692311yescharacteristic
346C4×Q8—2444no—
347C4×D4—611no—
348C4×D4—611no—
349C4×D4—611no—
350C4×D4—1221no—
351C4×D4—1221no—
352C4×D4C2×#692311yescharacteristic
353#117—2444no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1222no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C42—611no—
369C2×C42—612no—
370C2×C42—1222no—
371C2×C42—1222no—
372C2×C42—1221no—
373#107—612no—
374#107—612no—
375#107—612no—
376#107—1224no—
377#107C2×#692312yes—
378C4⋊C8—612no—
379C4⋊C8—611no—
380C4⋊C8C2×#692311yescharacteristic
381#102—612no—
382#102—612no—
383#102—612no—
384#102—1222no—
385#102—1222no—
386#102C2×#692312yes—
387C4×C8—1221no—
388C4×C8—1221no—
389C4×C8—1224no—
390#95—612no—
391#95—1222no—
392#95—1221no—
393#95—1222no—
394#95—1222no—
395C24—1221no—
396C2×Q8—21no—
397C2×Q8—1222no—
398C2×Q8—1221no—
399C2×Q8—2442no—
400C2×Q8—2442no—
401C2×D4—611no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4C22×#692311yescharacteristic
408C22×C4—21no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—611no—
413C22×C4—612no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—2442no—
423C22×C4—2442no—
424C2×C8—612no—
425C2×C8—612no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8C22×#692312yes—
434C4⋊C4—21no—
435C4⋊C4—614no—
436C4⋊C4—612no—
437C4⋊C4—612no—
438C4⋊C4—611no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1221no—
447C4⋊C4—1222no—
448C4⋊C4—1221no—
449C4⋊C4—4881no—
450C4⋊C4—2442no—
451C4⋊C4—2444no—
452C4⋊C4C22×#692312yes—
453C4⋊C4C22×#692311yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—1221no—
456C22⋊C4—1221no—
457C22⋊C4—1221no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C42—611no—
461C42—611no—
462C42—611no—
463C42—1222no—
464C42—1222no—
465C42—2444no—
466C42—2442no—
467C42—2444no—
468C42C22×#692311yescharacteristic
469C23—21no—
470C23—611no—
471C23—1221no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478Q8—42no—
479Q8—2444no—
480Q8—2442no—
481D4—21no—
482D4—21no—
483D4—1221no—
484D4—1221no—
485D4—2441no—
486D4—2441no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—611no—
491C2×C4—611no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—611no—
498C2×C4—614no—
499C2×C4—611no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4D4×#692311yescharacteristic
526C2×C4D4×#692311yescharacteristic
527C2×C4C23×#692311yescharacteristic
528C8—22no—
529C8—1222no—
530C8—1222no—
531C8—2441no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—611no—
536C22—611no—
537C22—611no—
538C22—611no—
539C22—611no—
540C22—611no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—2441no—
546C22—2441no—
547C22—2441no—
548C22C2×D4×#692311yescharacteristic
549C4—21no—
550C4—41no—
551C4—42no—
552C4—612no—
553C4—611no—
554C4—611no—
555C4—1224no—
556C4—1221no—
557C4—4881no—
558C4—4882no—
559C4—2442no—
560C4—2442no—
561C4—2441no—
562C4—2442no—
563C4D8×#692312yes—
564C4#41×#692311yescharacteristic
565C4#1684083011yescharacteristic
566C2—81no—
567C2—611no—
568C2—611no—
569C2—611no—
570C2—611no—
571C2—4881no—
572C2#2146234511yescharacteristic
573C2C2×D8×#692311yescharacteristic
574C2#2146257711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#474C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#474#6923
hypocenterC61#23060
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #21462348C2semidirect
3. #21462000C2semidirect
4. #21461930C2semidirect
7. #21461880C2semidirect
8. #107×#6923C2semidirect
9. #21461612C2semidirect
38. #16840043C4semidirect
39. #16840042C4semidirect
40. #16840041C4semidirect
41. #16840040C4semidirect
42. #16840039C4semidirect
46. C61×#474C4semidirect
308. #474#6923semidirect
334. C61#23060semidirect

Maximal subgroups

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

subgroupindex
#214623482
#214623372
#214620002
#214619302
#214619092
#214618962
#214618802
#107×#69232
#214616122
C4×#145198392
#474×D612
#2306061