Finite Group Databasebeta

← #30602961

Subgroups — #30602961

Order 15616.

1266 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊#6923C211yesmaximal, characteristic
2C2×#16840640C212yesmaximal
3C2×#16840640C212yesmaximal
4#21460993C211yesmaximal, characteristic
5C2×#16839633C211yesmaximal, characteristic
6C22×C22⋊#6923C2212yes—
7C22×C22⋊#6923C2212yes—
8C22×C4×#6923—21no—
9C4×C22⋊#6923—24no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4—21no—
13#16840640—24no—
14#16840640C418yescomplemented
15C23×C4×D61C2211yescharacteristic
16C22×#11758196C411yescharacteristic, complemented
17C22×#11758198C2211yescharacteristic
18C22×#11758198C2211yescharacteristic
19#16839633—21no—
20#16839633—21no—
21#16839633—21no—
22#16839633—21no—
23C2×#14519684C411yescharacteristic, complemented
24C2×#14519684C411yescharacteristic, complemented
25C22×C4⋊C4×C61C411yescharacteristic, complemented
26C23×#6923—22no—
27C23×#6923—22no—
28C2×C22⋊#6923—22no—
29C2×C22⋊#6923—22no—
30C2×C22⋊#6923—22no—
31C2×C22⋊#6923—22no—
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923—22no—
34C2×C22⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923C2×C414yes—
37C2×C22⋊#6923C2×C414yes—
38C2×C22⋊#6923D414yes—
39C2×C22⋊#6923Q814yes—
40C2×C4×#6923—24no—
41C2×C4×#6923—42no—
42C2×C4×#6923—42no—
43(C2×#6923)⋊C4—24no—
44(C2×#6923)⋊C4—42no—
45(C2×#6923)⋊C4—42no—
46C24×D61C2311yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C23×Dic61C2×C411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—42no—
53C22×C4×D61—42no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—42no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198C2×C414yes—
69#14519684—21no—
70#14519684—21no—
71#14519684—21no—
72#14519684—21no—
73C23×C4×C61C2×C411yescharacteristic
74C23×C4×C61C2×C411yescharacteristic
75C23×C4×C61C2×C411yescharacteristic
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—42no—
79C24×C61C22×C411yescharacteristic
80C23×D61—41no—
81C23×D61—41no—
82C23×D61—41no—
83C23×D61—41no—
84C23×D61C22×C412yes—
85C23×D61C2×D411yescharacteristic
86C23×D61C2×D411yescharacteristic
87C23×D61C2×D412yes—
88C23×D61C2×Q812yes—
89C23×D61#4112yes—
90C23×D61#4112yes—
91C23×D61#4112yes—
92C22×#6923—24no—
93C22×#6923—24no—
94C22×#6923—24no—
95C22×#6923—24no—
96C22×#6923—42no—
97C22×#6923—42no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—42no—
101C22×#6923—42no—
102C22×#6923—42no—
103C22×#6923—42no—
104C22×#6923—42no—
105C22×#6923—42no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61C4212yescomplemented
118C22×C4×C61C22⋊C411yescharacteristic
119C22×C4×C61C22⋊C411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61C4212yescomplemented
132C22×Dic61C22⋊C411yescharacteristic
133C22×Dic61C22⋊C411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—21no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—48no—
145C2×C4×D61—48no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C22⋊#6923—24no—
151C22⋊#6923—24no—
152C22⋊#6923—24no—
153C22⋊#6923—24no—
154C22⋊#6923—24no—
155C22⋊#6923—42no—
156C22⋊#6923—42no—
157C22⋊#6923C4⋊C418yescomplemented
158C4×#6923—48no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161#11758198—24no—
162#11758198—24no—
163#11758198—41no—
164#11758198—42no—
165#11758198—41no—
166#11758196—42no—
167#11758196—42no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61C2×C4211yescharacteristic
173C23×C61C2×C22⋊C411yescharacteristic
174C23×C61#11711yescharacteristic
175C23×C61C4×D412yes—
176C23×C61C4×Q812yes—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61C2×C4⋊C412yes—
198C22×D61#11712yes—
199C22×D61C4×D412yes—
200C22×D61C4×D412yes—
201C22×D61C4×D412yes—
202C22×D61C4×Q812yes—
203C22×D61#12011yescharacteristic
204C22×D61#12112yes—
205C22×D61C22⋊Q812yes—
206C22×D61C22⋊Q812yes—
207C22×D61#12311yescharacteristic
208C22×D61#12312yes—
209C22×D61#12411yescharacteristic
210C22×D61#12512yes—
211C22×D61#12612yes—
212C22×D61C4⋊Q811yescharacteristic
213C2×#6923—28no—
214C2×#6923—44no—
215C2×#6923—44no—
216C2×#6923—44no—
217C2×#6923—44no—
218C2×#6923—44no—
219C2×#6923—44no—
220C2×#6923—44no—
221C2×#6923—44no—
222C2×#6923—44no—
223C2×#6923—82no—
224C2×#6923—82no—
225C2×#6923—82no—
226C2×#6923—82no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61#9912yes—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—44no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—82no—
258C2×Dic61#9912yes—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61—42no—
264C4×D61—42no—
265#10912—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61C4×C22⋊C411yescharacteristic
275C22×C61C4×C4⋊C412yes—
276C22×C61#38611yescharacteristic
277C22×C61#38712yes—
278C22×C61#39111yescharacteristic
279C22×C61C2×#9911yescharacteristic
280C22×C61C2×#9911yescharacteristic
281C22×C61#41011yescharacteristic
282C22×C61#41011yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—22no—
286C2×D61—21no—
287C2×D61—22no—
288C2×D61—21no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61#38512yes—
308C2×D61#38712yes—
309C2×D61#38812yes—
310C2×D61#38912yes—
311#6923—48no—
312#6923—84no—
313C4×C61—22no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—81no—
317Dic61—22no—
318Dic61—41no—
319Dic61—42no—
320Dic61—81no—
321C2×#385—611no—
322C2×C4×C22⋊C4—611no—
323C2×#342—612no—
324C2×#342—612no—
325C22⋊#95—611no—
326C2×C61—22no—
327C2×C61—22no—
328C2×C61#143611yescharacteristic
329C2×C61C4×#9911yescharacteristic
330C2×C61C4×#9911yescharacteristic
331C2×C61#189811yescharacteristic
332C2×C61#189911yescharacteristic
333C2×C61#199611yescharacteristic
334C2×C61#199611yescharacteristic
335D61—22no—
336D61—22no—
337D61—22no—
338D61—22no—
339C24×C4—611no—
340C22×C4⋊C4—611no—
341C22×C4⋊C4#692311yescharacteristic, complemented
342C22×C22⋊C4—612no—
343C22×C22⋊C4—611no—
344C22×C22⋊C4—612no—
345C22×C22⋊C4—611no—
346C22×C42—1221no—
347#385—1221no—
348#385—1221no—
349#385—1221no—
350#385—1221no—
351C4×C22⋊C4—1224no—
352C2×#95—611no—
353C2×#95—611no—
354C2×#95—1221no—
355C2×#95—1221no—
356C2×#95—1221no—
357#342—618no—
358#342—1224no—
359C61#1091211yescharacteristic, complemented, Sylow
360C25—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—1222no—
369C23×C4—1222no—
370C23×C4—1222no—
371C23×C4—1222no—
372C23×C4—2442no—
373C23×C4—2442no—
374C23×C4C2×#692311yescharacteristic
375C23×C4C2×#692311yescharacteristic
376C23×C4C2×#692311yescharacteristic
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—2442no—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—614no—
386C2×C22⋊C4—614no—
387C2×C22⋊C4—614no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C42—1224no—
404C2×C42—2442no—
405C2×C42—2442no—
406#95—1221no—
407#95—1221no—
408#95—1224no—
409#95—1221no—
410#95—1221no—
411#95—2442no—
412#95—2442no—
413C24—611no—
414C24—611no—
415C24—612no—
416C24—612no—
417C24—612no—
418C24—612no—
419C24—612no—
420C24—612no—
421C24—2441no—
422C24—2441no—
423C24—2441no—
424C24—2441no—
425C24C22×#692311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1221no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1221no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4—2441no—
464C22×C4—2448no—
465C22×C4—2448no—
466C22×C4—2441no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2441no—
480C22×C4—2441no—
481C22×C4C4×#692312yescomplemented
482C22×C4C22⋊#692311yescharacteristic
483C22×C4C22⋊#692311yescharacteristic
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C22⋊C4—618no—
489C22⋊C4—1224no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—2441no—
497C22⋊C4—2442no—
498C22⋊C4—2441no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—2448no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—611no—
507C23—612no—
508C23—611no—
509C23—612no—
510C23—611no—
511C23—611no—
512C23—612no—
513C23—612no—
514C23—612no—
515C23—612no—
516C23—612no—
517C23—612no—
518C23—612no—
519C23—612no—
520C23—612no—
521C23—612no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2442no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23—2442no—
540C23—2442no—
541C23—2442no—
542C23C2×C4×#692311yescharacteristic
543C23C2×C22⋊#692311yescharacteristic
544C23#1451985611yescharacteristic
545C23D4×#692312yes—
546C23Q8×#692312yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—82no—
562C2×C4—612no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1228no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—4882no—
573C2×C4—4882no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—4882no—
577C2×C4—2444no—
578C2×C4—2444no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2442no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4—2444no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4#1451983512yes—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—612no—
608C22—612no—
609C22—612no—
610C22—612no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—1221no—
616C22—1221no—
617C22—1222no—
618C22—1222no—
619C22—1221no—
620C22—1221no—
621C22—1221no—
622C22—1222no—
623C22—1221no—
624C22—1221no—
625C22—1222no—
626C22—1221no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22C4×C22⋊#692311yescharacteristic
636C22C4⋊C4×#692312yes—
637C22#1684071011yescharacteristic
638C22#1684071112yes—
639C22(C2×#6923)⋊Q811yescharacteristic
640C22C2×#1451983511yescharacteristic
641C22D61⋊#9911yescharacteristic
642C22#1684074111yescharacteristic
643C22#1684074311yescharacteristic
644C4—22no—
645C4—42no—
646C4—41no—
647C4—81no—
648C4—1222no—
649C4—4884no—
650C4—4881no—
651C4—2448no—
652C4—2442no—
653C4—2441no—
654C2—22no—
655C2—22no—
656C2—1222no—
657C2—1222no—
658C2—1222no—
659C2—1222no—
660C2#2146099811yescharacteristic
661C2C4×#1451983511yescharacteristic
662C2#2146156411yescharacteristic
663C2#2146184011yescharacteristic
664C2#2146184311yescharacteristic
665C2#2146200811yescharacteristic
666C2#2146201011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radical, cosocleC24×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#10912
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #16840640C4semidirect
16. C22×#11758196C4semidirect
23. C2×#14519684C4semidirect
24. C2×#14519684C4semidirect
25. C22×C4⋊C4×C61C4semidirect
117. C22×C4×C61C42semidirect
131. C22×Dic61C42semidirect
157. C22⋊#6923C4⋊C4semidirect
341. C22×C4⋊C4#6923semidirect
359. C61#10912semidirect
481. C22×C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#69232
C2×#168406402
#214609932
C2×#168396332
#1091261