Finite Group Databasebeta

← #14685166

Subgroups — #14685166

Order 2816.

3654 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66977C211yesmaximal, characteristic, complemented
2C2×#10622560C212yesmaximal, complemented
3C2×#10622860C212yesmaximal
4#13066861C212yesmaximal, complemented
5#13066859C214yesmaximal, complemented
6#13066839C212yesmaximal, complemented
7#13066839C211yesmaximal, characteristic, complemented
8C23×C11×C22⋊C4C211yesmaximal, characteristic
9C26×C11C2211yescharacteristic
10C24×C4×C11C2211yescharacteristic
11C24×C4×C11C2211yescharacteristic
12C23×Dic22C2211yescharacteristic, complemented
13C22×C11×C22⋊C4—24no—
14C22×C11×C22⋊C4—28no—
15C22×C11×C22⋊C4C2212yes—
16C22×C11×C22⋊C4C2212yes—
17C22×C22⋊Dic11C2212yes—
18C22×C22⋊Dic11C2214yes—
19C22×C22⋊Dic11C2214yes—
20C22×C22⋊Dic11C2212yes—
21C22×C4⋊Dic11C2211yescharacteristic, complemented
22C2×#66977—24no—
23C2×#66977—28no—
24C22×Dic11⋊C4C2212yescomplemented
25#10622860—28no—
26C2×C11⋊#95C2214yescomplemented
27C2×C11⋊#95C2212yescomplemented
28C2×C11⋊#95C2212yescomplemented
29C2×C11⋊#95C2214yescomplemented
30#10622560—28no—
31C25×C11—24no—
32C25×C11—22no—
33C25×C11—24no—
34C25×C11—22no—
35C25×C11—24no—
36C25×C11—24no—
37C25×C11—42no—
38C25×C11—42no—
39C25×C11Q814yescomplemented
40C25×C11C2312yes—
41C25×C11C2311yescharacteristic
42C23×C4×C11—24no—
43C23×C4×C11—22no—
44C23×C4×C11—24no—
45C23×C4×C11—24no—
46C23×C4×C11—24no—
47C23×C4×C11—42no—
48C23×C4×C11—42no—
49C23×C4×C11D414yes—
50C23×C4×C11C2311yescharacteristic
51C23×C4×C11C2312yes—
52C23×C4×C11C2311yescharacteristic
53C23×Dic11C2314yes—
54C23×Dic11C2312yes—
55C23×Dic11C2312yes—
56C22×Dic22—22no—
57C22×Dic22—24no—
58C22×Dic22—24no—
59C22×Dic22—42no—
60C2×C11×C22⋊C4—232no—
61C2×C11×C22⋊C4—24no—
62C2×C11×C22⋊C4—24no—
63C2×C11×C22⋊C4—24no—
64C2×C11×C22⋊C4—24no—
65C2×C11×C22⋊C4—24no—
66C2×C11×C22⋊C4D418yes—
67C2×C22⋊Dic11—24no—
68C2×C22⋊Dic11—24no—
69C2×C22⋊Dic11—28no—
70C2×C22⋊Dic11—24no—
71C2×C22⋊Dic11—28no—
72C2×C22⋊Dic11—24no—
73C2×C22⋊Dic11—24no—
74C2×C22⋊Dic11—28no—
75C2×C22⋊Dic11—28no—
76C2×C22⋊Dic11—44no—
77C2×C22⋊Dic11—44no—
78C2×C22⋊Dic11D418yes—
79C2×C4⋊Dic11—22no—
80C2×C4⋊Dic11—24no—
81#66977—232no—
82C2×Dic11⋊C4—24no—
83C2×Dic11⋊C4—24no—
84C2×Dic11⋊C4—24no—
85C11⋊#95—28no—
86C11⋊#95—24no—
87C11⋊#95—44no—
88C11⋊#95—42no—
89#14717—111nomaximal, Sylow
90C24×C11—22no—
91C24×C11—22no—
92C24×C11—22no—
93C24×C11—22no—
94C24×C11—28no—
95C24×C11—28no—
96C24×C11—216no—
97C24×C11—28no—
98C24×C11—216no—
99C24×C11—24no—
100C24×C11—22no—
101C24×C11—22no—
102C24×C11—22no—
103C24×C11—22no—
104C24×C11—24no—
105C24×C11—24no—
106C24×C11—24no—
107C24×C11—216no—
108C24×C11—48no—
109C24×C11—416no—
110C24×C11—416no—
111C24×C11—416no—
112C24×C11—416no—
113C24×C11—44no—
114C24×C11—44no—
115C24×C11—48no—
116C24×C11—44no—
117C24×C11—48no—
118C24×C11—44no—
119C24×C11C2×D414yes—
120C24×C11C2×D414yes—
121C24×C11C2×D412yes—
122C24×C11C2×Q814yes—
123C24×C11C2×Q812yes—
124C24×C11#4114yes—
125C24×C11#4112yes—
126C24×C11#4114yes—
127C24×C11C2411yescharacteristic
128C22×C4×C11—22no—
129C22×C4×C11—22no—
130C22×C4×C11—28no—
131C22×C4×C11—28no—
132C22×C4×C11—22no—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—22no—
136C22×C4×C11—24no—
137C22×C4×C11—22no—
138C22×C4×C11—24no—
139C22×C4×C11—28no—
140C22×C4×C11—44no—
141C22×C4×C11—416no—
142C22×C4×C11—416no—
143C22×C4×C11—44no—
144C22×C4×C11—44no—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2×D412yes—
147C22×C4×C11C2×D414yes—
148C22×C4×C11C2×D412yes—
149C22×Dic11—24no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—22no—
153C22×Dic11—22no—
154C22×Dic11—24no—
155C22×Dic11—24no—
156C22×Dic11—24no—
157C22×Dic11—24no—
158C22×Dic11—24no—
159C22×Dic11—24no—
160C22×Dic11—24no—
161C22×Dic11—24no—
162C22×Dic11—24no—
163C22×Dic11—24no—
164C22×Dic11C2×D414yes—
165C22×Dic11C2×D414yes—
166C2×Dic22—28no—
167C2×Dic22—416no—
168C2×Dic22—44no—
169C2×Dic22—44no—
170C11×C22⋊C4—216no—
171C11×C22⋊C4—216no—
172C22⋊Dic11—216no—
173C22⋊Dic11—416no—
174C22⋊Dic11—416no—
175C22⋊Dic11—48no—
176C4⋊Dic11—28no—
177Dic11⋊C4—216no—
178C22×C22⋊Q8—111no—
179C23×C22⋊C4D1111yescharacteristic
180C2×#393—112no—
181C2×#379—112no—
182#1439—114no—
183#1439—112no—
184C22⋊#95—112no—
185C22⋊#95—111no—
186C23×C11—22no—
187C23×C11—24no—
188C23×C11—22no—
189C23×C11—24no—
190C23×C11—22no—
191C23×C11—22no—
192C23×C11—24no—
193C23×C11—24no—
194C23×C11—28no—
195C23×C11—28no—
196C23×C11—28no—
197C23×C11—28no—
198C23×C11—232no—
199C23×C11—24no—
200C23×C11—28no—
201C23×C11—24no—
202C23×C11—24no—
203C23×C11—24no—
204C23×C11—24no—
205C23×C11—24no—
206C23×C11—28no—
207C23×C11—28no—
208C23×C11—28no—
209C23×C11—28no—
210C23×C11—28no—
211C23×C11—464no—
212C23×C11—416no—
213C23×C11—416no—
214C23×C11—416no—
215C23×C11—416no—
216C23×C11—416no—
217C23×C11—464no—
218C23×C11—42no—
219C23×C11—44no—
220C23×C11—42no—
221C23×C11—42no—
222C23×C11—42no—
223C23×C11—44no—
224C23×C11—44no—
225C23×C11—44no—
226C23×C11—44no—
227C23×C11—44no—
228C23×C11—416no—
229C23×C11#12014yes—
230C23×C11C22⋊Q818yes—
231C23×C11C22⋊Q814yes—
232C23×C11C22⋊Q818yescomplemented
233C23×C11#12418yes—
234C23×C11#12414yes—
235C23×C11C22×D412yes—
236C23×C11C22×D411yescharacteristic
237C23×C11C22×D412yes—
238C23×C11C22×Q811yescharacteristic
239C23×C11C2×#4111yescharacteristic
240C23×C11C2×#4112yes—
241C23×C11C2×#4112yes—
242C23×C11H(5,𝔽2)12yes—
243C23×C11H(5,𝔽2)12yes—
244C2×C4×C11—22no—
245C2×C4×C11—22no—
246C2×C4×C11—28no—
247C2×C4×C11—22no—
248C2×C4×C11—22no—
249C2×C4×C11—24no—
250C2×C4×C11—22no—
251C2×C4×C11—28no—
252C2×C4×C11—416no—
253C2×C4×C11—416no—
254C2×C4×C11—42no—
255C2×C4×C11—42no—
256C2×C4×C11—44no—
257C2×C4×C11—44no—
258C2×C4×C11#12014yes—
259C2×Dic11—24no—
260C2×Dic11—24no—
261C2×Dic11—28no—
262C2×Dic11—28no—
263C2×Dic11—42no—
264C2×Dic11—42no—
265C2×Dic11—42no—
266C2×Dic11—42no—
267C2×Dic11—44no—
268C2×Dic11—44no—
269C2×Dic11—44no—
270C2×Dic11—44no—
271C2×Dic11—44no—
272C2×Dic11—44no—
273C2×Dic11—44no—
274C2×Dic11—44no—
275C2×Dic11—44no—
276Dic22—416no—
277C26C2×D1111yescharacteristic
278C23×Q8—111no—
279C24×C4C2×D1111yescharacteristic
280C24×C4C2×D1111yescharacteristic
281C2×C22⋊Q8—228no—
282C2×C22⋊Q8—224no—
283C22×C4⋊C4—112no—
284C22×C4⋊C4—111no—
285C22×C22⋊C4—24no—
286C22×C22⋊C4—28no—
287C22×C22⋊C4—112no—
288C22×C22⋊C4—112no—
289C22×C22⋊C4—114no—
290C22×C22⋊C4—114no—
291C22×C22⋊C4C2×D1112yes—
292C22×C22⋊C4C2×D1112yes—
293#393—228no—
294#379—228no—
295C2×#95—114no—
296C2×#95—112no—
297C2×#95—114no—
298C2×#95—112no—
299C22×C11—24no—
300C22×C11—24no—
301C22×C11—24no—
302C22×C11—24no—
303C22×C11—24no—
304C22×C11—24no—
305C22×C11—216no—
306C22×C11—216no—
307C22×C11—28no—
308C22×C11—216no—
309C22×C11—464no—
310C22×C11—44no—
311C22×C11—44no—
312C22×C11—44no—
313C22×C11—44no—
314C22×C11—44no—
315C22×C11—44no—
316C22×C11—48no—
317C22×C11—48no—
318C22×C11—48no—
319C22×C11#39318yescomplemented
320C22×C11C2×#12011yescharacteristic
321C22×C11C2×#12011yescharacteristic
322C22×C11C2×C22⋊Q812yes—
323C22×C11C2×C22⋊Q811yescharacteristic
324C22×C11C2×C22⋊Q812yes—
325C22×C11#52511yescharacteristic
326C22×C11#52512yes—
327C22×C11C2×#12412yes—
328C22×C11C2×#12411yescharacteristic
329C22×C11#53411yescharacteristic
330C22×C11#53511yescharacteristic
331C22×C11#53812yes—
332C22×C11#54312yes—
333C22×C11D4214yes—
334C22×C11#54614yes—
335C22×C11#54614yes—
336C22×C11#55114yes—
337C4×C11—24no—
338C4×C11—42no—
339C4×C11—44no—
340Dic11—44no—
341Dic11—44no—
342Dic11—84no—
343C25—24no—
344C25—22no—
345C25—24no—
346C25—22no—
347C25—24no—
348C25—24no—
349C25—42no—
350C25—42no—
351C25Dic2214yescomplemented
352C25C22×D1112yes—
353C25C22×D1111yescharacteristic
354C22×Q8—224no—
355C22×Q8—222no—
356C22×Q8—224no—
357C22×Q8—442no—
358C23×C4—24no—
359C23×C4—22no—
360C23×C4—24no—
361C23×C4—24no—
362C23×C4—24no—
363C23×C4—42no—
364C23×C4—42no—
365C23×C4—112no—
366C23×C4—112no—
367C23×C4—114no—
368C23×C4C22⋊D1114yes—
369C23×C4C22×D1111yescharacteristic
370C23×C4C22×D1112yes—
371C23×C4C22×D1111yescharacteristic
372C22⋊Q8—2232no—
373C2×C4⋊C4—224no—
374C2×C4⋊C4—224no—
375C2×C4⋊C4—224no—
376C2×C4⋊C4—224no—
377C2×C4⋊C4—222no—
378C2×C22⋊C4—232no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—118no—
385C2×C22⋊C4—224no—
386C2×C22⋊C4—224no—
387C2×C22⋊C4—228no—
388C2×C22⋊C4—228no—
389C2×C22⋊C4—228no—
390C2×C22⋊C4—224no—
391C2×C22⋊C4—224no—
392C2×C22⋊C4—224no—
393C2×C22⋊C4—228no—
394C2×C22⋊C4—444no—
395C2×C22⋊C4—444no—
396C2×C22⋊C4C22⋊D1118yes—
397#95—228no—
398#95—224no—
399#95—444no—
400#95—442no—
401C2×C11—28no—
402C2×C11—48no—
403C2×C11C2×#39311yescharacteristic
404C2×C11#240311yescharacteristic
405C2×C11#240411yescharacteristic
406C2×C11#243511yescharacteristic
407C2×C11#243714yes—
408C2×C11#247214yes—
409C2×C11#256111yescharacteristic
410C2×C11#262611yescharacteristic
411C2×C11#284811yescharacteristic
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—28no—
417C24—216no—
418C24—28no—
419C24—28no—
420C24—216no—
421C24—22no—
422C24—22no—
423C24—24no—
424C24—24no—
425C24—24no—
426C24—24no—
427C24—22no—
428C24—22no—
429C24—216no—
430C24—48no—
431C24—416no—
432C24—416no—
433C24—416no—
434C24—416no—
435C24—48no—
436C24—44no—
437C24—44no—
438C24—48no—
439C24—44no—
440C24—44no—
441C24C2×Dic2214yes—
442C24C2×Dic2212yes—
443C24#433914yes—
444C24#433912yes—
445C24D4×D1114yes—
446C24#434114yes—
447C24C2×C22⋊D1114yes—
448C24C2×C22⋊D1112yes—
449C24C23×D1111yescharacteristic
450C2×Q8—228no—
451C2×Q8—4416no—
452C2×Q8—444no—
453C2×Q8—444no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—44no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—114no—
472C22×C4—114no—
473C22×C4—224no—
474C22×C4—222no—
475C22×C4—222no—
476C22×C4—224no—
477C22×C4—224no—
478C22×C4—224no—
479C22×C4—222no—
480C22×C4—222no—
481C22×C4—224no—
482C22×C4—224no—
483C22×C4—224no—
484C22×C4—224no—
485C22×C4—224no—
486C22×C4—224no—
487C22×C4—224no—
488C22×C4C2×C22⋊D1114yes—
489C22×C4C2×C22⋊D1112yes—
490C22×C4C2×C22⋊D1114yes—
491C22×C4C2×C22⋊D1112yes—
492C4⋊C4—2216no—
493C4⋊C4—228no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—2216no—
497C22⋊C4—4416no—
498C22⋊C4—4416no—
499C22⋊C4—448no—
500C11#1471711yescharacteristic, complemented, Sylow
501C23—24no—
502C23—28no—
503C23—24no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—22no—
513C23—232no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—464no—
527C23—416no—
528C23—416no—
529C23—464no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—416no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23#6693118yes—
545C23#6693818yes—
546C23#6697714yes—
547C23#6697718yescomplemented
548C23#6698914yes—
549C23#6699014yes—
550C23C22×Dic2211yescharacteristic
551C23C2×#433912yes—
552C23C2×#433911yescharacteristic
553C23C2×D4×D1112yes—
554C23C2×#434112yes—
555C23#6703712yes—
556C23#6703712yes—
557C23C22×C22⋊D1111yescharacteristic
558C23C22×C22⋊D1112yes—
559Q8—4416no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—228no—
575C2×C4—224no—
576C2×C4—224no—
577C2×C4—228no—
578C2×C4—444no—
579C2×C4—442no—
580C2×C4—444no—
581C2×C4—442no—
582C2×C4—442no—
583C2×C4—442no—
584C2×C4—444no—
585C2×C4—444no—
586C2×C4—444no—
587C2×C4—444no—
588C2×C4—444no—
589C2×C4—444no—
590C2×C4—444no—
591C2×C4#6700614yes—
592C22—24no—
593C22—24no—
594C22—216no—
595C22—24no—
596C22—24no—
597C22—216no—
598C22—24no—
599C22—24no—
600C22—216no—
601C22—28no—
602C22—464no—
603C22—44no—
604C22—44no—
605C22—48no—
606C22—48no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22#1062256018yescomplemented
613C22C2×#6693112yes—
614C22#1062301412yes—
615C22#1062302114yes—
616C22C2×#6693812yes—
617C22#1062302512yes—
618C22#1062302614yes—
619C22C2×#6697712yes—
620C22C2×#6697711yescharacteristic
621C22#1062332211yescharacteristic
622C22#1062332212yes—
623C22C2×#6698911yescharacteristic
624C22C2×#6699011yescharacteristic
625C22D4×C22⋊D1114yes—
626C22#1062333511yescharacteristic
627C22#1062333611yescharacteristic
628C22#1062333714yes—
629C22C2×#6700611yescharacteristic
630C4—24no—
631C4—42no—
632C4—44no—
633C4—444no—
634C4—444no—
635C4—884no—
636C2—28no—
637C2—48no—
638C2C2×#1062256011yescharacteristic
639C2#1307192611yescharacteristic
640C2#1307193511yescharacteristic
641C2#1307226311yescharacteristic
642C2#1307227714yes—
643C2#1307273614yes—
644C2#1307377411yescharacteristic
645C2#1307442111yescharacteristic
646C2#1307690311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D11
commutator, socle, cosocleC24×C11C24
FittingC23×C11×C22⋊C4C2
prime radicalC26×C11C22
hypercenterC23×C22⋊C4D11
hypocenterC11#14717
LCSG ▷ C24×C11 ▷ C11
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#66977C2semidirect
2. C2×#10622560C2semidirect
4. #13066861C2semidirect
5. #13066859C2semidirect
6. #13066839C2semidirect
7. #13066839C2semidirect
12. C23×Dic22C22semidirect
21. C22×C4⋊Dic11C22semidirect
24. C22×Dic11⋊C4C22semidirect
26. C2×C11⋊#95C22semidirect
27. C2×C11⋊#95C22semidirect
28. C2×C11⋊#95C22semidirect
29. C2×C11⋊#95C22semidirect
39. C25×C11Q8semidirect
232. C23×C11C22⋊Q8semidirect
319. C22×C11#393semidirect
351. C25Dic22semidirect
500. C11#14717semidirect
547. C23#66977semidirect
612. C22#10622560semidirect

Maximal subgroups

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

subgroupindex
C22×#669772
C2×#106225602
C2×#106228602
#130668612
#130668592
#130668392
C23×C11×C22⋊C42
#1471711