Finite Group Databasebeta

← #30602594

Subgroups — #30602594

Order 15616.

1426 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519841C212yesmaximal
2C2×#16840632C212yesmaximal, complemented
3#21461483C211yesmaximal, characteristic, complemented
4C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×#11758224C211yesmaximal, characteristic
6C22×#11758213C2212yes—
7C22×C4×#6923C2212yes—
8C2×#14519841—24no—
9C2×#14519841—24no—
10C2×#14519841C418yescomplemented
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4—21no—
13#16840632—24no—
14#16840628—21no—
15#16840628—21no—
16C22×#11758224C2211yescharacteristic
17C4×#11758224—24no—
18C4×#11758224—24no—
19C2×C4×Dic122C411yescharacteristic, complemented
20C2×#14519742C411yescharacteristic, complemented
21C2×D4×Dic61C411yescharacteristic, complemented
22C2×C4×C22⋊D61—21no—
23C2×#14519737—21no—
24C2×#14519735C2211yescharacteristic
25C2×#14519733—21no—
26C22×C4×Dic61—21no—
27C2×C42×D61C2211yescharacteristic
28C2×C4×D4×C61C411yescharacteristic, complemented
29C23×#6923—22no—
30C2×#11758213—22no—
31C2×#11758213—22no—
32C2×#11758213—22no—
33C2×#11758213—22no—
34C2×#11758213—22no—
35C2×#11758213C2×C414yescomplemented
36C22×#92420—22no—
37C2×C4×#6923—22no—
38C2×C4×#6923—22no—
39C2×C4×#6923—22no—
40C2×C4×#6923—22no—
41C2×C4×#6923—22no—
42C2×C4×#6923C2×C414yescomplemented
43#14519841—216no—
44(C2×#6923)⋊C4—24no—
45(C2×#6923)⋊C4—24no—
46C2×#11758224—22no—
47C2×#11758224—21no—
48C2×#11758224—21no—
49C2×#11758224—22no—
50C2×#11758224C2×C414yes—
51C2×#11758224D414yes—
52C2×#11758224D412yes—
53C2×#11758224Q812yes—
54C22×Dic122C2×C411yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C23×Dic61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C4×Dic122—22no—
62C4×Dic122—22no—
63#14519742—22no—
64#14519742—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C4×C22⋊D61—28no—
68#14519737—28no—
69#14519735—22no—
70#14519735D414yes—
71#14519733—28no—
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—28no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C42×D61—22no—
85C42×D61—22no—
86C22×D4×C61C2×C411yescharacteristic
87C23×C4×C61—21no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2×C411yescharacteristic
93C24×C61—21no—
94C23×D61C2×D411yescharacteristic
95C22×#6923—24no—
96C22×#6923—24no—
97C22×#6923—24no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—42no—
101C22×#6923—42no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61C4212yes—
106C2×D4×C61C22⋊C412yes—
107C2×D4×C61C4⋊C412yes—
108C22×C4×C61—24no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C2×C22⋊D61—24no—
116C2×C22⋊D61—24no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C22×Dic61—21no—
120C22×Dic61—24no—
121C22×Dic61—24no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—24no—
125C22×Dic61—21no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C22⋊C411yescharacteristic
129C22×Dic61C22⋊C411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131#11758224—24no—
132#11758224—24no—
133#11758224—24no—
134#11758224—24no—
135#11758224—24no—
136#11758224—24no—
137#11758224C22⋊C418yes—
138#11758224C4⋊C418yes—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61C22×C412yes—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×Q811yescharacteristic
150C2×C4×D61#4111yescharacteristic
151C2×C4×D61#4111yescharacteristic
152C2×C4×D61#4112yes—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C4212yes—
157C2×Dic122C22⋊C412yes—
158C2×Dic122C4⋊C412yes—
159C2×#92420—24no—
160C2×#92420—42no—
161C2×#92420—42no—
162C4×#6923—24no—
163C4×#6923—24no—
164C4×#6923—42no—
165C4×#6923—42no—
166#11758213—24no—
167#11758213—24no—
168#11758213—42no—
169#11758213—42no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22⋊C412yes—
172C22⋊C4×C61—24no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—24no—
176#11758198—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22⋊C412yes—
179#11758196—24no—
180#11758196—24no—
181C4×Dic61—24no—
182C4×Dic61—24no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—24no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C2×C22⋊C411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61C4×D412yes—
201C22×D61#12012yes—
202C22×D61#12111yescharacteristic
203C22×D61C22⋊Q811yescharacteristic
204C2×#6923—28no—
205C2×#6923—44no—
206C2×#6923—44no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—44no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61#9514yescomplemented
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—48no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#10412yescomplemented
225C2×C4×C61C2×C4211yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C4⋊C411yescharacteristic
229C2×C4×C61#11711yescharacteristic
230C2×C4×C61#11711yescharacteristic
231C22⋊D61—28no—
232C22⋊D61—28no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—28no—
239C2×Dic61—28no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—48no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61#10412yescomplemented
248C2×Dic61C4×D412yes—
249C2×Dic61#12012yes—
250C2×Dic61#12111yescharacteristic
251C2×Dic61C22⋊Q811yescharacteristic
252C4×D61—44no—
253C4×D61—44no—
254C4×D61C2×C22⋊C412yes—
255C4×D61C2×C4⋊C412yes—
256C4×D61C4×D412yes—
257C4×D61C4×D412yes—
258C4×D61#12112yes—
259C4×D61C22⋊Q812yes—
260C4×D61#12312yes—
261C4×D61#12312yes—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265Dic122#9514yescomplemented
266#92420—44no—
267#10790—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61C4×C22⋊C411yescharacteristic
276C22×C61#37911yescharacteristic
277C22×C61#38011yescharacteristic
278C22×C61C2×#10411yescharacteristic
279C22×C61C2×#10411yescharacteristic
280C22×C61#42111yescharacteristic
281C22×C61#42111yescharacteristic
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61#38512yes—
291C2×D61#38612yes—
292#6923—48no—
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61C2×#9511yescharacteristic
297C4×C61C4×C22⋊C411yescharacteristic
298C4×C61#38011yescharacteristic
299C4×C61#38111yescharacteristic
300Dic61—24no—
301Dic61—22no—
302Dic61—41no—
303Dic61—42no—
304Dic61#38512yes—
305Dic61#38612yes—
306C2×C4×#41—611no—
307C22×#104—612no—
308C2×#339—612no—
309#1728—611no—
310C4×#95—611no—
311C2×C61—44no—
312C2×C61C22⋊#9511yescharacteristic
313C2×C61C4×#10411yescharacteristic
314C2×C61#175911yescharacteristic
315C2×C61#178911yescharacteristic
316C2×C61#178911yescharacteristic
317C2×C61#180611yescharacteristic
318C2×C61#180611yescharacteristic
319D61—22no—
320D61—22no—
321C22×#41—611no—
322C22×M16—612no—
323C4×#41—1224no—
324C4×#41—1224no—
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×D4—611no—
328C2×C4×D4—1221no—
329C2×C4×D4—1221no—
330C2×C4×D4#692311yescharacteristic, complemented
331C2×#117—611no—
332C2×#117—1221no—
333C22×C42—611no—
334C22×C42—612no—
335C22×C42—1221no—
336C2×#104—618no—
337C2×#104—1224no—
338C2×#104—1224no—
339C2×#95—1221no—
340C2×#95—1221no—
341#339—1224no—
342#336—1221no—
343#336—1221no—
344C61#1079011yescharacteristic, complemented, Sylow
345C2×#41—614no—
346C2×#41—612no—
347C2×#41—612no—
348C2×#41—614no—
349C2×#41—1222no—
350C2×#41—1221no—
351C2×#41—1222no—
352C2×#41—1221no—
353C22×Q8—611no—
354C22×D4—1221no—
355C22×D4C2×#692311yescharacteristic
356C23×C4—21no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4—1222no—
363C2×M16—614no—
364C2×M16—1222no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1222no—
369C22×C8—1222no—
370C4×Q8—1222no—
371C4×Q8—1222no—
372C4×Q8—1222no—
373C4×Q8—1222no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—1228no—
377C4×D4—1222no—
378C4×D4—1228no—
379C4×D4—1222no—
380#117—614no—
381#117—1228no—
382#117—1222no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C2×#692311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—614no—
394C2×C42—1222no—
395C2×C42—1222no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1228no—
399C2×C42—1221no—
400C2×C42—1222no—
401C2×C42—1222no—
402C2×C42—1222no—
403C2×C42—1222no—
404C2×C42—1222no—
405C2×C42C2×#692311yescharacteristic
406#104—12216no—
407#95—1224no—
408#95—1224no—
409C24—21no—
410C24—611no—
411#41—618no—
412#41—618no—
413#41—1224no—
414#41—1224no—
415#41—1224no—
416#41—1224no—
417#41—1224no—
418#41—1224no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—612no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4C4×#692312yes—
433C2×D4C4⋊#692312yes—
434C2×D4C22⋊#692312yes—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—611no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1224no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—1224no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4C22⋊#692311yescharacteristic
473C22×C4C22⋊#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475M16—1224no—
476M16—1224no—
477M16—2442no—
478M16—2442no—
479C2×C8—1224no—
480C2×C8—2442no—
481C2×C8—2442no—
482C4⋊C4—21no—
483C4⋊C4—612no—
484C4⋊C4—1224no—
485C4⋊C4—1224no—
486C4⋊C4—1221no—
487C4⋊C4C22⋊#692312yes—
488C22⋊C4—24no—
489C22⋊C4—1224no—
490C22⋊C4—1224no—
491C42—21no—
492C42—21no—
493C42—1222no—
494C42—1222no—
495C42—1224no—
496C42—1224no—
497C42—1224no—
498C42—1224no—
499C42—1224no—
500C42—1221no—
501C42—1221no—
502C42—2442no—
503C42—2442no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—611no—
512C23—611no—
513C23—612no—
514C23—612no—
515C23—1221no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23C2×C22⋊#692311yescharacteristic
520Q8—614no—
521Q8—1222no—
522Q8—1222no—
523Q8—1222no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4—1228no—
528D4—1228no—
529D4(C2×#6923)⋊C414yescomplemented
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1228no—
553C2×C4—1228no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1228no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1221no—
564C2×C4—2444no—
565C2×C4—2444no—
566C2×C4—2448no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4#1451984112yescomplemented
576C2×C4C2×C4×#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C2×C22⋊#692311yescharacteristic
579C2×C4C2×C4⋊#692311yescharacteristic
580C2×C4#1451985611yescharacteristic
581C2×C4#1451985611yescharacteristic
582C8—2444no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—612no—
591C22—612no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1221no—
597C22—1221no—
598C22—1221no—
599C22—1221no—
600C22C4×C22⋊#692311yescharacteristic
601C22#1684070311yescharacteristic
602C22#1684070411yescharacteristic
603C22C2×#1451984111yescharacteristic
604C22D61⋊#10411yescharacteristic
605C22#1684075911yescharacteristic
606C22#1684076011yescharacteristic
607C4—22no—
608C4—42no—
609C4—41no—
610C4—612no—
611C4—612no—
612C4—1224no—
613C4—1222no—
614C4—2448no—
615C4—2441no—
616C4—2442no—
617C4C2×(C2×#6923)⋊C411yescharacteristic
618C4C4×C22⋊#692311yescharacteristic
619C4#1684070411yescharacteristic
620C4#1684070511yescharacteristic
621C2—44no—
622C2—1222no—
623C2—1222no—
624C2#2146099311yescharacteristic
625C2C4×#1451984111yescharacteristic
626C2#2146158211yescharacteristic
627C2#2146166011yescharacteristic
628C2#2146166111yescharacteristic
629C2#2146169111yescharacteristic
630C2#2146169311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758224C22
cosocleC22×C4×D61C23
hypercenterC2×C4×D4#6923
hypocenterC61#10790
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16840632C2semidirect
3. #21461483C2semidirect
4. C4×(C2×#6923)⋊C4C2semidirect
10. C2×#14519841C4semidirect
19. C2×C4×Dic122C4semidirect
20. C2×#14519742C4semidirect
21. C2×D4×Dic61C4semidirect
28. C2×C4×D4×C61C4semidirect
35. C2×#11758213C2×C4semidirect
42. C2×C4×#6923C2×C4semidirect
214. D4×C61#95semidirect
224. C2×C4×C61#104semidirect
247. C2×Dic61#104semidirect
265. Dic122#95semidirect
330. C2×C4×D4#6923semidirect
344. C61#10790semidirect
529. D4(C2×#6923)⋊C4semidirect
575. C2×C4#14519841semidirect

Maximal subgroups

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

subgroupindex
C22×#145198412
C2×#168406322
#214614832
C4×(C2×#6923)⋊C42
C2×C4×#117582242
#1079061