Finite Group Databasebeta

← #30622079

Subgroups — #30622079

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462348C211yesmaximal, characteristic, complemented
2D4×C4⋊#6923C211yesmaximal, characteristic
3#21462000C211yesmaximal, characteristic, complemented
4#21461931C211yesmaximal, characteristic, complemented
5#21461924C212yesmaximal, complemented
6#21461883C212yesmaximal, complemented
7#21461784C212yesmaximal
8#21461616C211yesmaximal, characteristic, complemented
9#21461590C212yesmaximal
10C4×#14519839C211yesmaximal, characteristic
11#474×D61C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13C2×D4×#6923C2212yes—
14C22×C4⋊#6923—21no—
15C2×#14519845C2212yes—
16C2×#14519839C2211yescharacteristic
17C2×#14519839C2211yescharacteristic
18C2×#14519831C2212yes—
19C4×D61⋊C8C2211yescharacteristic
20(C2×#6923)⋊Q8—21no—
21#16840713C2212yes—
22#16840713C2212yes—
23#16840711—22no—
24#16840709—22no—
25#16840708C2211yescharacteristic
26#16840704—21no—
27C4⋊C4×#6923C2212yes—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30#16840636C2212yes—
31#16840635C2212yes—
32#16840635C2211yescharacteristic
33#16840635C2211yescharacteristic
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
47C23×#6923—22no—
48C2×D61⋊C8C2311yescharacteristic
49C2×D61⋊C8C2311yescharacteristic
50Q8×#6923—44no—
51D4×#6923—24no—
52D4×#6923—42no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—44no—
58C2×C4⋊#6923D411yescharacteristic
59C2×C4⋊#6923D411yescharacteristic
60C2×C4⋊#6923C2312yes—
61C2×C4⋊#6923C2311yescharacteristic
62C2×C4⋊#6923C2311yescharacteristic
63C2×C22⋊#6923—22no—
64C2×C4×#6923—22no—
65C2×C4×#6923—21no—
66C2×C4×#6923C2312yes—
67#14519845—42no—
68#14519839—22no—
69#14519839Q814yescomplemented
70#14519831—42no—
71C4×#92420—21no—
72C4×#92420—21no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—21no—
77(C2×#6923)⋊C4D412yes—
78C2×Q8×D61—21no—
79C2×D4×D61C2311yescharacteristic
80C22×C4×D61—21no—
81C2×C8×D61C2312yes—
82#14519777C2×C411yescharacteristic, complemented
83#14519776—21no—
84#14519776C2×C412yescomplemented
85#14519775—21no—
86C4×D244C2×C411yescharacteristic
87D4×Dic61C2×C411yescharacteristic
88#14519739C2×C411yescharacteristic
89C4×C22⋊D61—21no—
90#14519737—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61C2311yescharacteristic
93C4⋊C4×D61C2312yes—
94C22⋊C4×D61—21no—
95C42×D61C2311yescharacteristic
96#14519711C2×C412yescomplemented
97#14519710C2×C412yescomplemented
98#14519710C2×C412yescomplemented
99C4⋊#92418C2×C411yescharacteristic, complemented
100#14519706C2×C412yescomplemented
101#14519699C2×C412yes—
102#14519698C2×C412yes—
103#14519697C2×C412yes—
104C4⋊Q8×C61C2×C411yescharacteristic, complemented
105C4×D4×C61C2×C411yescharacteristic
106#107×C61C2×C412yescomplemented
107C4⋊C8×C61C2×C411yescharacteristic, complemented
108#102×C61C2×C412yes—
109C23×D61—21no—
110C22×#6923—21no—
111C22×#6923—21no—
112C22×#6923—22no—
113C22×#6923—44no—
114C22×#6923—44no—
115C22×#6923—82no—
116C22×#6923C2×D412yes—
117C22×#6923C2×D412yes—
118C2×Q8×C61—21no—
119C2×D4×C61C22×C411yescharacteristic
120C22×C4×C61—21no—
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123Q8×D61—42no—
124Q8×D61—42no—
125D4×D61—21no—
126D4×D61—21no—
127D4×D61C2×Q812yes—
128D4×D61#4112yes—
129C2×D244C22×C411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—22no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×D411yescharacteristic
138C2×C4×D61C2411yescharacteristic
139C2×Dic122—21no—
140C2×Dic122—22no—
141C22⋊#6923—44no—
142C2×#92420—21no—
143C2×#92420—21no—
144C4⋊#6923—22no—
145C4⋊#6923—22no—
146C4⋊#6923—21no—
147C4⋊#6923—22no—
148C4⋊#6923—21no—
149C4⋊#6923—22no—
150C4⋊#6923—21no—
151C4⋊#6923—44no—
152C4⋊#6923—44no—
153C4⋊#6923—42no—
154C4⋊#6923—81no—
155C4⋊#6923C2×Q812yes—
156C4×#6923—22no—
157C4×#6923—22no—
158C4×#6923—44no—
159C4×#6923—42no—
160D61⋊C8—21no—
161D61⋊C8—21no—
162D61⋊C8—21no—
163D61⋊C8C2×Q812yes—
164C2×C8×C61C22×C412yes—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C22×C411yescharacteristic
168C22⋊C4×C61—21no—
169C42×C61C22×C411yescharacteristic
170C22⋊Dic61—21no—
171#11758198—21no—
172#11758198—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C22×C412yes—
175C4⋊Dic61C22×C411yescharacteristic
176#11758196—22no—
177#11758196—21no—
178#11758196—22no—
179C4×Dic61C22×C411yescharacteristic
180C4×Dic61#4111yescharacteristic
181C4×Dic61#4111yescharacteristic
182C2×#92418C22×C412yes—
183C8×D61—22no—
184C8×D61—22no—
185C23×C61—21no—
186C22×D61—21no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61C22×D411yescharacteristic
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—41no—
202C2×#6923—41no—
203C2×#6923—42no—
204C2×#6923—41no—
205C2×#6923—41no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—41no—
209C2×#6923—88no—
210Q8×C61—42no—
211D4×C61—21no—
212D4×C61C4×Q812yescomplemented
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—42no—
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61C23×C411yescharacteristic
219C22⋊D61—22no—
220C22⋊D61—22no—
221C2×Dic61—22no—
222C2×Dic61—21no—
223C2×Dic61—42no—
224C2×Dic61C4×D411yescharacteristic
225C2×Dic61C4×D411yescharacteristic
226C2×Dic61C2×#4111yescharacteristic
227D244—21no—
228D244C4×Q812yescomplemented
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—42no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61AGL(1,ℤ/8)12yes—
238C4×D61AGL(1,ℤ/8)12yes—
239C4×D61C22×Q811yescharacteristic
240C4×D61C2×#4111yescharacteristic
241C4×D61H(5,𝔽2)11yescharacteristic
242C4×D612−1+411yescharacteristic
243Dic122—42no—
244Dic122—44no—
245#92420—22no—
246#92420—41no—
247C8×C61—22no—
248#92418—22no—
249#22968—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61C2×C4×D411yescharacteristic
254C2×D61—22no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61#54611yescharacteristic
262C2×D61D4×Q811yescharacteristic
263C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
264C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
265C2×D61#57711yescharacteristic
266C2×D61#57811yescharacteristic
267#6923—42no—
268#6923—42no—
269#6923—42no—
270#6923—81no—
271#6923—82no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61#44212yes—
276C4×C61C2×C4×Q811yescharacteristic
277C4×C61#52011yescharacteristic
278Dic61—21no—
279Dic61—42no—
280Dic61—41no—
281Dic61#44212yes—
282Dic61#55211yescharacteristic
283Dic61#55411yescharacteristic
284C2×#474—611no—
285#2365—611no—
286D4×C4⋊C4—611no—
287#1990—611no—
288#1948—611no—
289#1943—612no—
290#1921—612no—
291#1864—612no—
292#1770—611no—
293#1762—612no—
294C4×#102—611no—
295C2×C61—42no—
296C2×C61#236011yescharacteristic
297C2×C61C2×#44211yescharacteristic
298C2×C61#294711yescharacteristic
299D61—42no—
300D61#332411yescharacteristic
301D61#333011yescharacteristic
302D61#338411yescharacteristic
303D61#338611yescharacteristic
304C2×C4⋊Q8—611no—
305C2×C4×Q8—1221no—
306C2×C4×D4—611no—
307C2×C4×D4—612no—
308C22×C4⋊C4—1221no—
309#474—611no—
310#474—611no—
311#474—612no—
312#474—611no—
313#474—612no—
314#474#692311yescharacteristic, complemented
315C2×#107—612no—
316C2×#106—612no—
317C2×C4⋊C8—611no—
318C2×#102—611no—
319C2×#102—612no—
320C2×#102—611no—
321C2×#97—612no—
322C2×C4×C8—611no—
323#391—1221no—
324#389—612no—
325#389—612no—
326#387—1222no—
327#385—1222no—
328#384—611no—
329#380—1221no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4—612no—
332C4×C4⋊C4—611no—
333#340—612no—
334#340—611no—
335#340—612no—
336#340—611no—
337#340—612no—
338#336—612no—
339C61#2296811yescharacteristic, complemented, Sylow
340C22×Q8—1221no—
341C22×D4—611no—
342C23×C4—1221no—
343C23×C4—1222no—
344C22×C8—611no—
345C22×C8—612no—
346C22×C8—611no—
347C4⋊Q8—612no—
348C4⋊Q8—611no—
349C4⋊Q8—1221no—
350C4⋊Q8—1221no—
351C4⋊Q8C2×#692311yescharacteristic, complemented
352C4×Q8—2444no—
353C4×D4—611no—
354C4×D4—611no—
355C4×D4—611no—
356C4×D4—1221no—
357C4×D4—1221no—
358C4×D4—1224no—
359C4×D4—2442no—
360C4×D4C2×#692311yescharacteristic
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—2444no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1222no—
376C2×C42—611no—
377C2×C42—612no—
378C2×C42—1222no—
379C2×C42—1221no—
380#107—612no—
381#107—612no—
382#107—612no—
383#107C2×#692312yescomplemented
384#106—2442no—
385C4⋊C8—612no—
386C4⋊C8—611no—
387C4⋊C8C2×#692311yescharacteristic, complemented
388#102—614no—
389#102—612no—
390#102—612no—
391#102—612no—
392#102—1222no—
393#102C2×#692312yes—
394#97—2442no—
395C4×C8—1221no—
396C4×C8—1221no—
397#95—612no—
398#95—1221no—
399#95—1222no—
400#95—1222no—
401#95—1222no—
402C24—1221no—
403C2×Q8—21no—
404C2×Q8—1222no—
405C2×Q8—1221no—
406C2×Q8—2442no—
407C2×Q8—2442no—
408C2×D4—612no—
409C2×D4—612no—
410C2×D4—611no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4C22×#692311yescharacteristic
415C22×C4—21no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—4882no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4—2444no—
433C22×C4—2444no—
434C2×C8—612no—
435C2×C8—612no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8C22×#692312yes—
444C4⋊C4—21no—
445C4⋊C4—612no—
446C4⋊C4—612no—
447C4⋊C4—611no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1221no—
459C4⋊C4—4881no—
460C4⋊C4—2442no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C4⋊C4C22×#692312yes—
464C4⋊C4C22×#692311yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C22⋊C4—2444no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—1222no—
474C42—1222no—
475C42—2442no—
476C42—2444no—
477C42C22×#692311yescharacteristic
478C23—21no—
479C23—611no—
480C23—1222no—
481C23—1222no—
482C23—1221no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487Q8—42no—
488Q8—2444no—
489Q8—2442no—
490D4—21no—
491D4—612no—
492D4—1222no—
493D4—1222no—
494D4—1221no—
495D4Q8×#692312yescomplemented
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—4888no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4D4×#692311yescharacteristic
536C2×C4D4×#692311yescharacteristic
537C2×C4C23×#692311yescharacteristic
538C8—22no—
539C8—1222no—
540C8—1222no—
541C8—2441no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—611no—
546C22—611no—
547C22—611no—
548C22—611no—
549C22—611no—
550C22—611no—
551C22—1222no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2441no—
557C22—2441no—
558C22C2×D4×#692311yescharacteristic
559C4—21no—
560C4—41no—
561C4—42no—
562C4—612no—
563C4—611no—
564C4—611no—
565C4—1221no—
566C4—4882no—
567C4—4881no—
568C4—2442no—
569C4—2442no—
570C4—2442no—
571C4—2441no—
572C4—2442no—
573C4#1684080312yes—
574C4C2×Q8×#692311yescharacteristic
575C4#1684083111yescharacteristic
576C2—42no—
577C2—611no—
578C2—611no—
579C2—611no—
580C2—611no—
581C2—2442no—
582C2#2146234311yescharacteristic
583C2C2×#1684080311yescharacteristic
584C2#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#22968
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. #21461931C2semidirect
5. #21461924C2semidirect
6. #21461883C2semidirect
8. #21461616C2semidirect
38. #16840043C4semidirect
39. #16840042C4semidirect
40. #16840041C4semidirect
41. #16840040C4semidirect
42. #16840039C4semidirect
46. C61×#474C4semidirect
69. #14519839Q8semidirect
82. #14519777C2×C4semidirect
84. #14519776C2×C4semidirect
96. #14519711C2×C4semidirect
97. #14519710C2×C4semidirect
98. #14519710C2×C4semidirect
99. C4⋊#92418C2×C4semidirect
100. #14519706C2×C4semidirect
104. C4⋊Q8×C61C2×C4semidirect
106. #107×C61C2×C4semidirect
107. C4⋊C8×C61C2×C4semidirect
212. D4×C61C4×Q8semidirect
228. D244C4×Q8semidirect
314. #474#6923semidirect
339. C61#22968semidirect
351. C4⋊Q8C2×#6923semidirect
383. #107C2×#6923semidirect
387. C4⋊C8C2×#6923semidirect
495. D4Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623482
D4×C4⋊#69232
#214620002
#214619312
#214619242
#214618832
#214617842
#214616162
#214615902
C4×#145198392
#474×D612
#2296861