Finite Group Databasebeta

← #30603235

Subgroups — #30603235

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519839C212yesmaximal
2C2×#16840708C211yesmaximal, characteristic, complemented
3C2×#16840635C212yesmaximal, complemented
4#21461480C211yesmaximal, characteristic, complemented
5#390×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2212yes—
7C22×C4⋊#6923—22no—
8C22×C4⋊#6923C2212yes—
9C2×#14519839—24no—
10C2×#14519839—24no—
11C2×#14519839—24no—
12#16840708—24no—
13#16840635—24no—
14#16840628—22no—
15C22×D4×D61C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17C2×C22⋊C4×D61—22no—
18C2×C42×D61C2211yescharacteristic
19#16839665C411yescharacteristic, complemented
20#16839664C411yescharacteristic, complemented
21#16839663C411yescharacteristic, complemented
22#16839662—22no—
23C61×#390C411yescharacteristic, complemented
24C23×#6923—22no—
25C2×D61⋊C8—22no—
26C2×D61⋊C8—22no—
27C2×D61⋊C8—22no—
28C2×D61⋊C8—22no—
29C2×D61⋊C8—22no—
30C2×D61⋊C8D414yes—
31C22×#92420—22no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923—28no—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—44no—
39C2×C4⋊#6923—44no—
40C2×C4⋊#6923D414yes—
41#14519839—216no—
42C24×D61—21no—
43C22×D244C2×C411yescharacteristic
44C22×C22⋊D61—21no—
45C2×D4×D61—22no—
46C2×D4×D61—22no—
47C2×D4×D61—22no—
48C2×D4×D61—22no—
49C2×D4×D61—21no—
50C2×D4×D61—21no—
51C2×D4×D61D412yes—
52C2×D4×D61Q812yes—
53C22×C4×D61—21no—
54C22×C4×D61D412yes—
55C22×C4×D61C2311yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C4⋊C4×D61—22no—
59C4⋊C4×D61—22no—
60C2×C22⋊Dic61—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C22⋊C4×D61—44no—
64C22⋊C4×D61—44no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61D412yes—
67C42×D61—22no—
68C42×D61—22no—
69C22×D4×C61C2×C411yescharacteristic
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61—22no—
72C2×C42×C61C2×C411yescharacteristic
73C24×C61—21no—
74C23×D61—21no—
75C23×D61—22no—
76C23×D61—22no—
77C23×D61—41no—
78C23×D61—41no—
79C23×D61—41no—
80C23×D61—41no—
81C23×D61—42no—
82C23×D61—42no—
83C23×D61—42no—
84C23×D61—42no—
85C23×D61C2×D411yescharacteristic
86C22×#6923—24no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—44no—
90C22×#6923—44no—
91C2×D4×C61—21no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61C4⋊C412yescomplemented
95C22×C4×C61—21no—
96C22×C4×C61C22⋊C412yes—
97C22×C4×C61C22×C411yescharacteristic
98C2×C22⋊D61—22no—
99C2×C22⋊D61—22no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C22×Dic61—21no—
103C22×Dic61C22⋊C412yes—
104C22×Dic61C2×D411yescharacteristic
105D4×D61—24no—
106D4×D61—24no—
107D4×D61—24no—
108D4×D61—24no—
109D4×D61—42no—
110D4×D61—42no—
111D4×D61—42no—
112D4×D61—42no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—21no—
116C2×D244C4⋊C412yescomplemented
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—41no—
122C2×C4×D61—41no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61D814yes—
126C2×C4×D61SD1614yes—
127C2×C4×D61C2×D411yescharacteristic
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×Q811yescharacteristic
131C2×C4×D61#4111yescharacteristic
132C2×C4×D61#4112yes—
133C2×C4×D61#4112yes—
134C2×C4×D61#4111yescharacteristic
135C2×#92420—24no—
136C2×#92420—42no—
137C2×#92420—42no—
138C4⋊#6923—24no—
139C4⋊#6923—24no—
140C4⋊#6923—416no—
141C4⋊#6923—42no—
142C4⋊#6923—42no—
143D61⋊C8—24no—
144D61⋊C8—24no—
145D61⋊C8—42no—
146D61⋊C8—42no—
147C4⋊C4×C61—22no—
148C22⋊C4×C61—44no—
149C42×C61—22no—
150C22⋊Dic61—44no—
151#11758198—44no—
152#11758198—44no—
153C4⋊Dic61—22no—
154#11758196—42no—
155C4×Dic61—22no—
156C4×Dic61—24no—
157C23×C61—22no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61C2×C22⋊C411yescharacteristic
163C22×D61—22no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—44no—
173C22×D61—44no—
174C22×D61—44no—
175C22×D61—44no—
176C22×D61—82no—
177C22×D61—82no—
178C22×D61—82no—
179C22×D61—82no—
180C22×D61—82no—
181C22×D61—82no—
182C22×D61—81no—
183C22×D61—81no—
184C22×D61—81no—
185C22×D61—81no—
186C22×D61#12011yescharacteristic
187C22×D61#12111yescharacteristic
188C22×D61C22⋊Q811yescharacteristic
189C22×D61#12311yescharacteristic
190C22×D61#12411yescharacteristic
191C22×D61#12711yescharacteristic
192C22×D61C2×D812yes—
193C22×D61C2×SD1612yes—
194C22×D61AGL(1,ℤ/8)12yes—
195C22×D61#13712yes—
196C2×#6923—44no—
197C2×#6923—48no—
198C2×#6923—84no—
199C2×#6923—82no—
200C2×#6923—82no—
201D4×C61—24no—
202D4×C61—42no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61#10214yescomplemented
208C2×C4×C61C2×C4⋊C411yescharacteristic
209C2×C4×C61#11711yescharacteristic
210C2×C4×C61C4×D412yes—
211C2×C4×C61C4×D412yes—
212C22⋊D61—44no—
213C22⋊D61—44no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61#10214yescomplemented
219C2×Dic61#12111yescharacteristic
220C2×Dic61#12112yes—
221C2×Dic61#12112yes—
222C2×Dic61C22⋊Q811yescharacteristic
223D244—24no—
224D244—42no—
225C4×D61—24no—
226C4×D61—24no—
227C4×D61—24no—
228C4×D61—24no—
229C4×D61—82no—
230C4×D61—81no—
231C4×D61—81no—
232C4×D61#12012yes—
233C4×D61#12112yes—
234C4×D61C22⋊Q812yes—
235C4×D61C22⋊Q812yes—
236C4×D61#12312yes—
237C4×D61#12312yes—
238C4×D61#12412yes—
239C4×D61#12612yes—
240#92420—44no—
241#11049—611nomaximal, Sylow
242C22×C61—44no—
243C22×C61—42no—
244C22×C61—82no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61#38011yescharacteristic
248C22×C61#38611yescharacteristic
249C22×C61#39011yescharacteristic
250C22×C61C2×#10212yes—
251C22×C61#41712yes—
252C2×D61—44no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—84no—
258C2×D61—82no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—84no—
263C2×D61#39211yescharacteristic
264C2×D61#39311yescharacteristic
265C2×D61#39711yescharacteristic
266C2×D61#39911yescharacteristic
267C2×D61#44712yes—
268C2×D61#44812yes—
269C2×D61#45912yes—
270C2×D61#46112yes—
271C2×D61#47412yes—
272C2×D61#47612yes—
273C2×D61#48012yes—
274C2×D61#48112yes—
275C2×D61#48612yes—
276C2×D61#48912yes—
277C2×D61#49211yescharacteristic
278C2×D61#49311yescharacteristic
279C2×D61#49611yescharacteristic
280C2×D61#49711yescharacteristic
281#6923—84no—
282C4×C61—24no—
283C4×C61—81no—
284C4×C61#38512yes—
285C4×C61#38812yes—
286Dic61—24no—
287Dic61—81no—
288Dic61#39412yes—
289Dic61#39612yes—
290C22×#102—612no—
291C2×#390—611no—
292C2×#384—611no—
293C2×#340—612no—
294C2×#336—611no—
295C2×C61—82no—
296C2×C61#143911yescharacteristic
297C2×C61#180212yes—
298C2×C61#187412yes—
299C2×C61#195711yescharacteristic
300C2×C61#195911yescharacteristic
301D61—82no—
302D61#201111yescharacteristic
303D61#201311yescharacteristic
304D61#202312yes—
305D61#207112yes—
306D61#209311yescharacteristic
307D61#209411yescharacteristic
308C23×D4—611no—
309C23×C8—612no—
310C22×C4⋊C4—612no—
311C22×C4⋊C4—611no—
312C22×C4⋊C4—1222no—
313C22×C22⋊C4—1222no—
314C22×C42—611no—
315C2×#102—1224no—
316C2×#102—1224no—
317C2×#102—1224no—
318#390—611no—
319#390—611no—
320#390—611no—
321#390—1222no—
322#390#692311yescharacteristic, complemented
323#384—1224no—
324#340—1224no—
325#336—1222no—
326C61#1104911yescharacteristic, complemented, Sylow
327C25—1221no—
328C22×D4—612no—
329C22×D4—612no—
330C22×D4—611no—
331C22×D4—1222no—
332C22×D4—1222no—
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—1221no—
338C22×D4C2×#692311yescharacteristic
339C23×C4—611no—
340C23×C4—612no—
341C23×C4—1221no—
342C23×C4—1222no—
343C22×C8—614no—
344C22×C8—1222no—
345C22×C8—1222no—
346C22×C8—1222no—
347C22×C8—1222no—
348C22×C8—1222no—
349C22×C8—1222no—
350C2×C4⋊C4—614no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1228no—
361C2×C4⋊C4—2444no—
362C2×C4⋊C4—2444no—
363C2×C4⋊C4C2×#692311yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—2444no—
369C2×C22⋊C4—2444no—
370C2×C42—612no—
371C2×C42—611no—
372C2×C42—1222no—
373C2×C42—1222no—
374C2×C42C2×#692311yescharacteristic
375#102—12216no—
376C24—21no—
377C24—611no—
378C24—1222no—
379C24—1222no—
380C24—1221no—
381C24—2442no—
382C24—2442no—
383C24—2441no—
384C24—2441no—
385C24—2441no—
386C24—2441no—
387C24—2442no—
388C24—2442no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—612no—
393C2×D4—1224no—
394C2×D4—1224no—
395C2×D4—1224no—
396C2×D4—1224no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2442no—
406C2×D4—2442no—
407C2×D4—2442no—
408C2×D4C4⋊#692312yescomplemented
409C22×C4—21no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4C22⋊#692312yes—
437C22×C4C22×#692311yescharacteristic
438C2×C8—1224no—
439C2×C8—1224no—
440C2×C8—1224no—
441C2×C8—2442no—
442C2×C8—2442no—
443C2×C8—2442no—
444C2×C8—2442no—
445C4⋊C4—22no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1222no—
449C4⋊C4—2442no—
450C4⋊C4—2442no—
451C4⋊C4—2442no—
452C4⋊C4—24416no—
453C22⋊C4—44no—
454C22⋊C4—2444no—
455C22⋊C4—2444no—
456C22⋊C4—2444no—
457C42—22no—
458C42—1224no—
459C42—1222no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—611no—
466C23—611no—
467C23—611no—
468C23—611no—
469C23—611no—
470C23—611no—
471C23—612no—
472C23—612no—
473C23—612no—
474C23—612no—
475C23—1222no—
476C23—4882no—
477C23—4882no—
478C23—4882no—
479C23—4882no—
480C23—4881no—
481C23—4882no—
482C23—4881no—
483C23—4881no—
484C23—4882no—
485C23—4881no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23—2444no—
491C23—2444no—
492C23—2444no—
493C23—2444no—
494C23—2441no—
495C23—2441no—
496C23—2442no—
497C23—2442no—
498C23C2×C22⋊#692311yescharacteristic
499D4—24no—
500D4—42no—
501D4—1224no—
502D4—2444no—
503D4—2444no—
504D4—2442no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—614no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—4881no—
529C2×C4—4882no—
530C2×C4—4881no—
531C2×C4—4884no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—2444no—
535C2×C4—2448no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4#1451983914yescomplemented
539C2×C4C2×C4⋊#692311yescharacteristic
540C2×C4#1451985611yescharacteristic
541C2×C4D4×#692312yes—
542C2×C4D4×#692312yes—
543C8—2444no—
544C22—42no—
545C22—44no—
546C22—82no—
547C22—81no—
548C22—81no—
549C22—611no—
550C22—611no—
551C22—611no—
552C22—611no—
553C22—612no—
554C22—612no—
555C22—612no—
556C22—612no—
557C22—612no—
558C22—612no—
559C22—612no—
560C22—612no—
561C22—611no—
562C22—612no—
563C22—611no—
564C22—611no—
565C22—612no—
566C22—611no—
567C22—4882no—
568C22—4884no—
569C22—4882no—
570C22—4882no—
571C22—4882no—
572C22—4884no—
573C22—4882no—
574C22—4881no—
575C22—4881no—
576C22—2442no—
577C22—2444no—
578C22#1684070411yescharacteristic
579C22#1684071011yescharacteristic
580C22#1684071411yescharacteristic
581C22C2×#1451983912yes—
582C22#1684075312yes—
583C4—24no—
584C4—81no—
585C4—612no—
586C4—612no—
587C4—1224no—
588C4—4884no—
589C4—4881no—
590C4#1684070912yes—
591C4#1684071212yes—
592C2—82no—
593C2—611no—
594C2—612no—
595C2—611no—
596C2—611no—
597C2—611no—
598C2—612no—
599C2—4882no—
600C2#2146100211yescharacteristic
601C2#2146168712yes—
602C2#2146180612yes—
603C2#2146194711yescharacteristic
604C2#2146194911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#11049
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16840708C2semidirect
3. C2×#16840635C2semidirect
4. #21461480C2semidirect
19. #16839665C4semidirect
20. #16839664C4semidirect
21. #16839663C4semidirect
23. C61×#390C4semidirect
94. C2×D4×C61C4⋊C4semidirect
116. C2×D244C4⋊C4semidirect
207. C2×C4×C61#102semidirect
218. C2×Dic61#102semidirect
322. #390#6923semidirect
326. C61#11049semidirect
408. C2×D4C4⋊#6923semidirect
538. C2×C4#14519839semidirect

Maximal subgroups

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

subgroupindex
C22×#145198392
C2×#168407082
C2×#168406352
#214614802
#390×D612
#1104961