Finite Group Databasebeta

← #30615917

Subgroups — #30615917

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840772C211yesmaximal, characteristic, complemented
2C2×#16840754C211yesmaximal, characteristic
3#21462292C211yesmaximal, characteristic
4#21462257C211yesmaximal, characteristic, complemented
5#21461897C214yesmaximal, complemented
6#21461896C214yesmaximal, complemented
7#21461506C211yesmaximal, characteristic, complemented
8#21461484C211yesmaximal, characteristic, complemented
9C2×#102×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14#16840772—24no—
15C2×#14519846C2212yescomplemented
16C2×#14519845C2212yescomplemented
17#16840754—24no—
18C2×#14519839C2214yes—
19#16840714—22no—
20#16840714C2214yes—
21#16840703—22no—
22C4⋊C4×#6923—22no—
23C4⋊C4×#6923C2214yescomplemented
24C4×C22⋊#6923—22no—
25#16840635C2214yescomplemented
26(C2×#6923)⋊C8C2212yescomplemented
27(C2×#6923)⋊C8C2212yescomplemented
28C22×D4×D61C2211yescharacteristic
29C22×C8×D61C2211yescharacteristic
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×#14519699C411yescharacteristic, complemented
32C2×#14519698C411yescharacteristic, complemented
33C2×#14519697C411yescharacteristic, complemented
34#102×D61C412yescomplemented
35#102×D61C412yescomplemented
36#102×D61C2212yes—
37#102×D61C2212yes—
38C2×#102×C61C411yescharacteristic, complemented
39C2×#11758213—21no—
40D61⋊M16—22no—
41D61⋊M16—22no—
42C2×#11758217—21no—
43C2×D61⋊C8C2312yes—
44#14519856—22no—
45#14519856—22no—
46#14519856—22no—
47#14519856—22no—
48#14519856—22no—
49#14519856D414yes—
50C2×C4⋊#6923D412yes—
51C2×C4⋊#6923C2312yes—
52C2×C4⋊#6923C2312yes—
53C2×C22⋊#6923—21no—
54C2×C22⋊#6923—21no—
55C2×C22⋊#6923—24no—
56C2×C22⋊#6923—24no—
57C2×C22⋊#6923D412yes—
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923—24no—
61C2×C4×#6923C2312yes—
62#14519846—24no—
63#14519845—24no—
64#14519839—28no—
65(C2×#6923)⋊C4—24no—
66C24×D61—21no—
67C22×D244C2×C411yescharacteristic
68C22×C22⋊D61—21no—
69C2×D4×D61—22no—
70C2×D4×D61—21no—
71C2×D4×D61—22no—
72C2×D4×D61—22no—
73C2×D4×D61—22no—
74C2×D4×D61—21no—
75C2×D4×D61C2×C412yes—
76C2×D4×D61C2312yes—
77C22×C4×D61—21no—
78C22×C4×D61C2311yescharacteristic
79C22×#92418C2×C411yescharacteristic
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61C2×C411yescharacteristic
83C2×C8×D61C2×C411yescharacteristic
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758196—21no—
88C4⋊C4×D61—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61C2×C412yes—
91C4⋊C4×D61C2312yes—
92#14519699C2×C412yes—
93#14519699C2×C412yes—
94#14519698C2×C412yes—
95#14519698C2×C412yes—
96#14519697C2×C412yes—
97#14519697C2×C412yes—
98C22×D4×C61C2×C411yescharacteristic
99C22×C8×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2×C411yescharacteristic
101#102×C61C2×C412yes—
102#102×C61C2×C412yes—
103C24×C61—21no—
104C23×D61—21no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61—82no—
110C23×D61—82no—
111C23×D61C2×D411yescharacteristic
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923C2×D414yes—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—21no—
126C22×Dic61#4111yescharacteristic
127D4×D61—24no—
128D4×D61—24no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—22no—
133D4×D61—42no—
134D4×D61—42no—
135D4×D61—42no—
136D4×D61—42no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244C22×C412yes—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—82no—
144C2×C4×D61C22×C411yescharacteristic
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61#4112yes—
148C2×C4×D61#4112yes—
149C2×C4×D61#4112yes—
150C2×C4×D61#4111yescharacteristic
151C2×C4×D61C2411yescharacteristic
152C22⋊#6923—24no—
153C22⋊#6923—24no—
154C22⋊#6923—42no—
155C22⋊#6923—84no—
156C22⋊#6923—84no—
157#11758217—42no—
158C2×#92420—22no—
159C4⋊#6923—24no—
160C4⋊#6923—22no—
161C4⋊#6923—22no—
162C4⋊#6923—22no—
163C4⋊#6923C2×D414yes—
164C4×#6923—24no—
165C4×#6923—22no—
166C4×#6923—22no—
167C4×#6923—42no—
168C4×#6923—84no—
169#11758213—42no—
170D61⋊C8—22no—
171D61⋊C8—22no—
172C2×C8×C61—22no—
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C412yes—
177C4⋊Dic61—21no—
178C4⋊Dic61C22×C412yes—
179#11758196—22no—
180#11758196—22no—
181C2×#92418—22no—
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184C8×D61—22no—
185C8×D61—22no—
186C8×D61—21no—
187C8×D61—21no—
188C8×D61—21no—
189C8×D61—21no—
190C23×C61—22no—
191C23×C61—42no—
192C23×C61—81no—
193C23×C61—81no—
194C23×C61C4×D411yescharacteristic
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—82no—
205C22×D61—82no—
206C22×D61—88no—
207C22×D61—88no—
208C22×D61—81no—
209C22×D61—81no—
210C22×D61C4×D411yescharacteristic
211C22×D61#12112yes—
212C22×D61#12312yes—
213C22×D61C22×D411yescharacteristic
214C2×#6923—24no—
215C2×#6923—24no—
216C2×#6923—24no—
217C2×#6923—24no—
218C2×#6923—42no—
219C2×#6923—42no—
220C2×#6923—42no—
221C2×#6923—42no—
222C2×#6923—42no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—42no—
226C2×C4×C61—22no—
227C2×C4×C61—81no—
228C2×C4×C61#11712yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—42no—
233C22⋊D61—42no—
234C22⋊D61—44no—
235C2×Dic61—22no—
236C2×Dic61—81no—
237C2×Dic61C4×D411yescharacteristic
238C2×Dic61#12412yes—
239C2×Dic61#12612yes—
240C2×Dic61C2×#4111yescharacteristic
241D244—22no—
242D244—22no—
243D244—42no—
244C4×D61—44no—
245C4×D61—82no—
246C4×D61#11712yes—
247C4×D61C4×D412yes—
248C4×D61#12112yes—
249C4×D61#12312yes—
250C4×D61#12412yes—
251C4×D61#12612yes—
252C4×D61C22×D411yescharacteristic
253C4×D61C2×#4111yescharacteristic
254C4×D61C2×#4111yescharacteristic
255C4×D61C2×#4111yescharacteristic
256#92420—42no—
257C8×C61—21no—
258C8×C61—21no—
259#92418—21no—
260#92418—21no—
261#21065—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—84no—
268C22×C61#38812yes—
269C22×C61C2×C4×D411yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—24no—
273C2×D61—24no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—88no—
278C2×D61—84no—
279C2×D61—84no—
280C2×D61—84no—
281C2×D61#38812yes—
282C2×D61C2×#12111yescharacteristic
283C2×D61C2×#12311yescharacteristic
284C2×D61#57612yes—
285C2×D61#57712yes—
286#6923—44no—
287#6923—44no—
288#6923—82no—
289C4×C61—42no—
290C4×C61#38812yes—
291C4×C61C2×#11711yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293Dic61—42no—
294Dic61#38812yes—
295Dic61C2×#12411yescharacteristic
296Dic61C2×#12611yescharacteristic
297C2×#427—611no—
298C2×#418—611no—
299C22×#102—611no—
300#2309—611no—
301#2274—611no—
302#1926—614no—
303#1925—614no—
304#1736—611no—
305#1729—611no—
306C2×C61—22no—
307C2×C61—82no—
308C2×C61C2×#38811yescharacteristic
309C2×C61#294811yescharacteristic
310C2×C61#294911yescharacteristic
311D61—22no—
312D61—82no—
313D61#304512yes—
314D61#309612yes—
315C23×D4—611no—
316C22×M16—611no—
317C23×C8—611no—
318C2×#117—611no—
319C2×#117—611no—
320C2×#117—611no—
321C22×C4⋊C4—611no—
322#427—1224no—
323C2×#107—612no—
324C2×#106—612no—
325#418—1224no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102—612no—
332C2×#102—612no—
333C2×#102—614no—
334C2×#102#692311yescharacteristic, complemented
335#390—614no—
336#390—1222no—
337#379—1222no—
338C4×C4⋊C4—614no—
339C4×C4⋊C4—1222no—
340C4×C22⋊C4—1222no—
341#340—614no—
342#336—612no—
343#336—612no—
344C61#2106511yescharacteristic, complemented, Sylow
345C25—1221no—
346C22×D4—612no—
347C22×D4—612no—
348C22×D4—611no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4—1221no—
356C22×D4C2×#692311yescharacteristic
357C23×C4—611no—
358C23×C4—1221no—
359C2×M16—1221no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1221no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—612no—
369C22×C8—1222no—
370C22×C8—1222no—
371C22×C8C2×#692311yescharacteristic
372#117—614no—
373#117—1222no—
374#117—1222no—
375#117—1222no—
376#117—1222no—
377#117—1222no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C2×#692311yescharacteristic
388C2×C22⋊C4—612no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1221no—
393C2×C42—612no—
394C2×C42—1221no—
395C2×C42—1224no—
396C2×C42—1221no—
397#107—1224no—
398#106—1224no—
399#102—612no—
400#102—612no—
401#102—612no—
402#102—612no—
403#102—612no—
404#102—612no—
405#102—1228no—
406#102C2×#692312yes—
407#102C2×#692312yes—
408#95—1224no—
409C24—21no—
410C24—611no—
411C24—1222no—
412C24—1222no—
413C24—1221no—
414C24—4882no—
415C24—4882no—
416C24—2442no—
417C24—2442no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—612no—
422C2×D4—1222no—
423C2×D4—1224no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1224no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1221no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4C22×#692312yes—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—614no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—4882no—
457C22×C4C22×#692311yescharacteristic
458M16—2442no—
459M16—2442no—
460C2×C8—22no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8C22×#692311yescharacteristic
474C2×C8C22×#692311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—614no—
477C4⋊C4—612no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1224no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4C22×#692312yes—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—4884no—
489C22⋊C4—4884no—
490C22⋊C4—2442no—
491C42—1222no—
492C42—1222no—
493C42—1224no—
494C42—4884no—
495C42—2442no—
496C23—22no—
497C23—42no—
498C23—81no—
499C23—81no—
500C23—611no—
501C23—612no—
502C23—612no—
503C23—611no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—4881no—
508C23—4881no—
509C23—4882no—
510C23—4882no—
511C23—4888no—
512C23—4888no—
513C23—2444no—
514C23—2444no—
515C23—2442no—
516C23—2442no—
517C23—2444no—
518C23—2442no—
519C23D4×#692311yescharacteristic
520D4—22no—
521D4—22no—
522D4—42no—
523D4—1222no—
524D4—1222no—
525D4—2442no—
526D4—2442no—
527D4—2444no—
528D4—2442no—
529C2×C4—22no—
530C2×C4—81no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—612no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—4882no—
551C2×C4—4881no—
552C2×C4—2444no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4#1451985612yes—
559C2×C4D4×#692311yescharacteristic
560C2×C4D4×#692312yes—
561C2×C4C23×#692311yescharacteristic
562C8—21no—
563C8—21no—
564C8—1221no—
565C8—1221no—
566C8—2442no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—84no—
573C22—611no—
574C22—611no—
575C22—612no—
576C22—612no—
577C22—612no—
578C22—1224no—
579C22—1224no—
580C22—1221no—
581C22—1221no—
582C22—4884no—
583C22—4884no—
584C22—4884no—
585C22—4888no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22#1684071212yes—
590C22C2×D4×#692311yescharacteristic
591C4—42no—
592C4—611no—
593C4—611no—
594C4—612no—
595C4—4882no—
596C4—2444no—
597C4—2444no—
598C4—2442no—
599C4#1684071212yes—
600C4C2×#1451985611yescharacteristic
601C4C2×D4×#692311yescharacteristic
602C2—22no—
603C2—82no—
604C2—612no—
605C2—612no—
606C2—1222no—
607C2—4882no—
608C2C2×#1684071211yescharacteristic
609C2#2146257811yescharacteristic
610C2#2146257911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#102×C61C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenterC2×#102#6923
hypocenterC61#21065
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16840772C2semidirect
4. #21462257C2semidirect
5. #21461897C2semidirect
6. #21461896C2semidirect
7. #21461506C2semidirect
8. #21461484C2semidirect
15. C2×#14519846C22semidirect
16. C2×#14519845C22semidirect
23. C4⋊C4×#6923C22semidirect
25. #16840635C22semidirect
26. (C2×#6923)⋊C8C22semidirect
27. (C2×#6923)⋊C8C22semidirect
31. C2×#14519699C4semidirect
32. C2×#14519698C4semidirect
33. C2×#14519697C4semidirect
34. #102×D61C4semidirect
35. #102×D61C4semidirect
38. C2×#102×C61C4semidirect
334. C2×#102#6923semidirect
344. C61#21065semidirect

Maximal subgroups

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

subgroupindex
C2×#168407722
C2×#168407542
#214622922
#214622572
#214618972
#214618962
#214615062
#214614842
C2×#102×D612
#2106561