Finite Group Databasebeta

← #14725309

Subgroups — #14725309

Order 2816.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13077631C211yesmaximal, characteristic, complemented
2C4×#66961C211yesmaximal, characteristic
3M16×D44C211yesmaximal, characteristic, complemented
4#13067520C211yesmaximal, characteristic, complemented
5#13067508C211yesmaximal, characteristic, complemented
6#13067482C211yesmaximal, characteristic, complemented
7#13067475C211yesmaximal, characteristic, complemented
8#13067454C211yesmaximal, characteristic, complemented
9#13067447C211yesmaximal, characteristic, complemented
10#13067441C211yesmaximal, characteristic, complemented
11C8×D88C211yesmaximal, characteristic, complemented
12#13067383C211yesmaximal, characteristic, complemented
13#13067060C211yesmaximal, characteristic, complemented
14#13067056C211yesmaximal, characteristic, complemented
15C11×#1455C211yesmaximal, characteristic, complemented
16C2×#66961C411yescharacteristic
17C2×M16×D11—21no—
18C4×#4339C2211yescharacteristic
19C2×C4×D44C2211yescharacteristic
20C4×C11×M16C2211yescharacteristic
21C4×C11×M16C2211yescharacteristic
22C4×C11×M16C2211yescharacteristic
23#10622746C411yescharacteristic
24#10622744C411yescharacteristic
25#10622740—21no—
26#10622740—21no—
27#10622739—21no—
28#10622734C411yescharacteristic
29#10622539C2211yescharacteristic
30#10622537C2211yescharacteristic
31#10622324C411yescharacteristic
32#10622324C411yescharacteristic
33#10622323C411yescharacteristic
34#10622323C411yescharacteristic
35#10622319—21no—
36#10622319—21no—
37#10622318—21no—
38D11×#97—21no—
39C4×D88C2211yescharacteristic
40C4×D88C2211yescharacteristic
41C4×#4314C2211yescharacteristic
42C4×#4314C2211yescharacteristic
43#10622301C2211yescharacteristic
44#10622301C2211yescharacteristic
45C8×D44C2211yescharacteristic
46C8×D44C2211yescharacteristic
47C8×D44C2211yescharacteristic, complemented
48C8×D44C2211yescharacteristic, complemented
49#10622292C2211yescharacteristic, complemented
50#10622292C2211yescharacteristic, complemented
51C11×#322C2211yescharacteristic, complemented
52C11×#322C2211yescharacteristic, complemented
53C11×#322C2211yescharacteristic, complemented
54C82×C11C2211yescharacteristic, complemented
55#10622097C2211yescharacteristic
56#10622097C2211yescharacteristic
57#10622097C2211yescharacteristic
58#10622097C2211yescharacteristic
59#10622097C2211yescharacteristic, complemented
60#10622097C2211yescharacteristic, complemented
61#10622096C2211yescharacteristic, complemented
62#10622096C2211yescharacteristic, complemented
63#10622095C2211yescharacteristic, complemented
64#10622095C2211yescharacteristic, complemented
65#10622094C2211yescharacteristic, complemented
66#10622094C2211yescharacteristic, complemented
67C2×#4339C2×C411yescharacteristic
68C22×D44C2×C411yescharacteristic
69C22×C4×D11—21no—
70C2×C11×M16—21no—
71C2×C11×M16C2×C411yescharacteristic
72C2×C11×M16C2×C411yescharacteristic
73C2×C11×M16D411yescharacteristic
74C2×C11×M16D411yescharacteristic
75C2×C42×C11C2311yescharacteristic
76C4×C22⋊D11—21no—
77C4×C22⋊D11—21no—
78C2×(C2×D11)⋊C4—21no—
79#66979C2×C411yescharacteristic
80C2×C4⋊Dic11C2×C411yescharacteristic
81C2×#4318—21no—
82#66961—24no—
83M16×D11—44no—
84C2×D88C2×C411yescharacteristic
85C2×D88C2×C411yescharacteristic
86C2×#4314C2×C411yescharacteristic
87C2×#4314C2×C411yescharacteristic
88C2×#4313—21no—
89C2×#4313—21no—
90C2×C8×D11—21no—
91C2×C8×D11—21no—
92C2×C8×D11—21no—
93C2×C8×D11—21no—
94C4×D44—21no—
95C4×D44—21no—
96C4×D44C2×C411yescharacteristic
97C4×D44C2×C411yescharacteristic
98C4×D44C2×C411yescharacteristic
99C4×D44C2311yescharacteristic
100C4×D44C2311yescharacteristic
101C4×D44C2311yescharacteristic
102#66925—21no—
103C42×D11—21no—
104C4×Dic22C2×C411yescharacteristic
105C4×Dic22C2311yescharacteristic
106C11×#97—21no—
107C11×#97C2×C411yescharacteristic
108C11×#97C2×C411yescharacteristic
109C11×#97D411yescharacteristic
110C11×#97D411yescharacteristic
111C4×C8×C11C2311yescharacteristic
112C4×C8×C11C2311yescharacteristic
113C4×C8×C11C2311yescharacteristic
114C4×C8×C11C2311yescharacteristic
115C4×C8×C11C2311yescharacteristic
116C4×C8×C11C2311yescharacteristic
117C22⋊(C11⋊C8)—21no—
118#66885C2×C411yescharacteristic
119#66885C2×C411yescharacteristic
120#66885C2×C411yescharacteristic
121#66885C2×C411yescharacteristic
122#66885C2×C411yescharacteristic
123#66885C2×C411yescharacteristic
124(C2×D11)⋊C8—21no—
125(C2×D11)⋊C8—21no—
126(C2×D11)⋊C8—21no—
127(C2×D11)⋊C8—21no—
128(C2×D11)⋊C8—21no—
129(C2×D11)⋊C8—21no—
130#66882C2×C411yescharacteristic
131#66882C2×C411yescharacteristic
132#66881C2×C411yescharacteristic
133#66881C2×C411yescharacteristic
134#66880C2×C411yescharacteristic
135#66880C2×C411yescharacteristic
136#66879—21no—
137#66879—21no—
138Dic11⋊C8—21no—
139Dic11⋊C8—21no—
140C4⋊(C11⋊C8)—21no—
141C4⋊(C11⋊C8)C2311yescharacteristic
142C4⋊(C11⋊C8)C2311yescharacteristic
143C4⋊(C11⋊C8)C2311yescharacteristic
144C4⋊(C11⋊C8)C2311yescharacteristic
145#66867—21no—
146#18046—111nomaximal, Sylow
147C23×D11—21no—
148C22×C4×C11C22×C411yescharacteristic
149C22×C4×C11C2×D411yescharacteristic
150C22×C4×C11#4111yescharacteristic
151C2×C22⋊D11—21no—
152C22×Dic11—21no—
153#4339—22no—
154#4339—41no—
155C2×D44—22no—
156C2×D44—21no—
157C2×D44—41no—
158C2×D44C22×C411yescharacteristic
159C2×D44C22×C411yescharacteristic
160C2×D44C22×C411yescharacteristic
161C2×C4×D11—21no—
162C2×C4×D11—21no—
163C2×C4×D11—21no—
164C2×C4×D11—21no—
165C2×C4×D11—21no—
166C2×C4×D11—21no—
167C2×C4×D11—21no—
168C2×C4×D11—42no—
169C2×C4×D11—42no—
170C2×Dic22C22×C411yescharacteristic
171C11×M16—22no—
172C11×M16—22no—
173C11×M16—41no—
174C11×M16—41no—
175C11×M16—42no—
176C2×C8×C11—21no—
177C2×C8×C11—21no—
178C2×C8×C11C22×C411yescharacteristic
179C2×C8×C11C22×C411yescharacteristic
180C2×C8×C11C22×C411yescharacteristic
181C2×C8×C11C22×C411yescharacteristic
182C2×C8×C11C2×D411yescharacteristic
183C2×C8×C11C2×D411yescharacteristic
184C2×C8×C11C2×D411yescharacteristic
185C2×C8×C11C2×D411yescharacteristic
186C42×C11C22×C411yescharacteristic
187C42×C11C2×D411yescharacteristic
188C42×C11#4111yescharacteristic
189C42×C11C2411yescharacteristic
190C22⋊Dic11—21no—
191(C2×D11)⋊C4—21no—
192(C2×D11)⋊C4—21no—
193(C2×D11)⋊C4—21no—
194(C2×D11)⋊C4—21no—
195(C2×D11)⋊C4—21no—
196(C2×D11)⋊C4—21no—
197C4⋊Dic11—21no—
198C4⋊Dic11C22×C411yescharacteristic
199C4⋊Dic11C22×C411yescharacteristic
200C4⋊Dic11C22×C411yescharacteristic
201C4⋊Dic11C22×C411yescharacteristic
202Dic11⋊C4—21no—
203Dic11⋊C4—21no—
204C4×Dic11—21no—
205C4×Dic11—21no—
206C4×Dic11—21no—
207#4318—42no—
208C2×C11⋊C8—21no—
209C2×C11⋊C8—21no—
210C2×C11⋊C8—21no—
211C2×C11⋊C8—21no—
212D88—22no—
213D88M1614yescomplemented
214#4314—22no—
215#4314—22no—
216#4313—42no—
217#4313—42no—
218C8×D11—42no—
219C8×D11—42no—
220C8×D11—42no—
221C8×D11—42no—
222#2970—111no—
223C4×AGL(1,ℤ/8)—111no—
224D4×M16—111no—
225#1592—111no—
226#1590—111no—
227#1586—111no—
228#1585—111no—
229#1582—111no—
230#1581—111no—
231#1580—111no—
232C8×D8—111no—
233#1569—111no—
234#1490—111no—
235#1489—111no—
236#1455D1111yescharacteristic, complemented
237C23×C11C4×D411yescharacteristic
238C22×D11—21no—
239C22×D11—21no—
240C22×D11—21no—
241C22×D11—42no—
242C22×D11—42no—
243C22×D11—81no—
244C2×C4×C11—21no—
245C2×C4×C11—21no—
246C2×C4×C11—21no—
247C2×C4×C11—21no—
248C2×C4×C11—41no—
249C2×C4×C11C4×D411yescharacteristic
250C2×C4×C11C4×D411yescharacteristic
251C2×C4×C11C4×D411yescharacteristic
252C2×C4×C11C23×C411yescharacteristic
253C2×C4×C11C22×D411yescharacteristic
254C2×C4×C11C2×#4111yescharacteristic
255C22⋊D11—42no—
256C2×Dic11—21no—
257C2×Dic11—21no—
258C2×Dic11—21no—
259C2×Dic11—21no—
260C2×Dic11—21no—
261C2×Dic11—42no—
262D44—21no—
263D44—21no—
264D44—21no—
265D44—21no—
266D44—42no—
267D44C2×M1612yes—
268D44C2×M1612yescomplemented
269C4×D11—42no—
270C4×D11—42no—
271C4×D11—42no—
272C4×D11—41no—
273C4×D11—41no—
274C4×D11—41no—
275C4×D11—41no—
276C4×D11—41no—
277C4×D11—41no—
278C4×D11—81no—
279Dic22—21no—
280Dic22—21no—
281C8×C11—22no—
282C8×C11—21no—
283C8×C11—21no—
284C8×C11—21no—
285C8×C11—21no—
286C8×C11—21no—
287C8×C11—21no—
288C8×C11—41no—
289C8×C11C2×M1612yescomplemented
290C8×C11AGL(1,ℤ/8)12yescomplemented
291C11⋊C8—41no—
292C11⋊C8—41no—
293C11⋊C8—41no—
294C11⋊C8—41no—
295C2×AGL(1,ℤ/8)—111no—
296C22×M16—221no—
297C4×#41—111no—
298C2×C4×D4—111no—
299#446—111no—
300#446—111no—
301#442—111no—
302#442—111no—
303#440—111no—
304#440—111no—
305C4×SD16—111no—
306C4×SD16—111no—
307C4×D8—111no—
308C4×D8—111no—
309#436—111no—
310#436—111no—
311#435—221no—
312#435—221no—
313#435—221no—
314#435—221no—
315C8×D4—111no—
316C8×D4—111no—
317C8×D4—111no—
318C8×D4—111no—
319#432—111no—
320#432—221no—
321#428—111no—
322#423—111no—
323#418—111no—
324#417—111no—
325#407—221no—
326C4×M16C2×D1111yescharacteristic
327C4×M16C2×D1111yescharacteristic
328C4×M16C2×D1111yescharacteristic
329C2×#97—221no—
330#335—111no—
331#335—111no—
332#334—111no—
333#334—111no—
334Q8⋊C8—111no—
335Q8⋊C8—111no—
336D4⋊C8—111no—
337D4⋊C8—111no—
338D4⋊C8—111no—
339D4⋊C8—111no—
340D4⋊C8—111no—
341D4⋊C8—111no—
342#322C2×D1111yescharacteristic, complemented
343#322C2×D1111yescharacteristic, complemented
344#322C2×D1111yescharacteristic, complemented
345C82C2×D1111yescharacteristic, complemented
346C22×C11—21no—
347C22×C11—21no—
348C22×C11—21no—
349C22×C11C2×C4×D411yescharacteristic
350C2×D11—22no—
351C2×D11—22no—
352C2×D11—41no—
353C2×D11—41no—
354C2×D11—41no—
355C2×D11—41no—
356C2×D11—41no—
357C2×D11—41no—
358C2×D11—41no—
359C2×D11—84no—
360C4×C11—21no—
361C4×C11—21no—
362C4×C11—21no—
363C4×C11—21no—
364C4×C11C22×M1611yescharacteristic
365C4×C11#56811yescharacteristic
366C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
367C4×C11#57511yescharacteristic
368Dic11—41no—
369Dic11—41no—
370Dic11—41no—
371Dic11—41no—
372Dic11—41no—
373C2×#41—111no—
374C22×D4—111no—
375C23×C4—221no—
376AGL(1,ℤ/8)—224no—
377C2×SD16—111no—
378C2×SD16—111no—
379C2×D8—111no—
380C2×D8—111no—
381C2×M16—21no—
382C2×M16—221no—
383C2×M16—221no—
384C2×M16—221no—
385C2×M16—444no—
386C2×M16C4×D1111yescharacteristic
387C2×M16C4×D1111yescharacteristic
388C2×M16D4411yescharacteristic
389C2×M16D4411yescharacteristic
390C22×C8—221no—
391C22×C8—221no—
392C22×C8—221no—
393C22×C8—221no—
394C4×Q8—111no—
395C4×Q8—111no—
396C4×D4—111no—
397C4×D4—111no—
398C4×D4—111no—
399C4×D4—111no—
400C4×D4—111no—
401C4×D4—111no—
402C4×D4—221no—
403C4×D4—221no—
404C4×D4—221no—
405C4×D4—221no—
406#117—111no—
407#117—221no—
408C2×C4⋊C4—111no—
409C2×C22⋊C4—221no—
410C2×C42—221no—
411C2×C42C22×D1111yescharacteristic
412#107—111no—
413#107—111no—
414#106—111no—
415#106—111no—
416C4⋊C8—111no—
417C4⋊C8—111no—
418C4⋊C8—111no—
419C4⋊C8—111no—
420C4⋊C8—221no—
421C4⋊C8—221no—
422C4⋊C8—221no—
423#103—111no—
424#103—111no—
425#102—111no—
426#102—111no—
427#102—111no—
428#102—111no—
429#102—111no—
430#102—111no—
431C22⋊C8—221no—
432C22⋊C8—221no—
433C22⋊C8—221no—
434C22⋊C8—221no—
435C22⋊C8—221no—
436C22⋊C8—221no—
437C22⋊C8—221no—
438#97—21no—
439#97—221no—
440#97—221no—
441#97—221no—
442#97C4×D1111yescharacteristic
443#97C4×D1111yescharacteristic
444#97D4411yescharacteristic
445#97D4411yescharacteristic
446C4×C8C22×D1111yescharacteristic
447C4×C8C22×D1111yescharacteristic
448C4×C8C22×D1111yescharacteristic
449C4×C8C22×D1111yescharacteristic
450C4×C8C22×D1111yescharacteristic
451C4×C8C22×D1111yescharacteristic
452C2×C11—41no—
453C2×C11D4×M1611yescharacteristic
454C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
455C2×C11#295611yescharacteristic
456D11—42no—
457D11—42no—
458D11—81no—
459C24—221no—
460#41—222no—
461#41—441no—
462C2×Q8—111no—
463C2×D4—111no—
464C2×D4—111no—
465C2×D4—111no—
466C2×D4—222no—
467C2×D4—221no—
468C2×D4—221no—
469C2×D4—441no—
470C22×C4—221no—
471C22×C4—221no—
472C22×C4—221no—
473C22×C4—221no—
474C22×C4—221no—
475C22×C4—221no—
476C22×C4—221no—
477C22×C4—221no—
478C22×C4—442no—
479C22×C4—442no—
480C22×C4C2×C4×D1111yescharacteristic
481C22×C4C2×D4411yescharacteristic
482C22×C4#433911yescharacteristic
483SD16—222no—
484SD16—222no—
485D8—114no—
486D8—222no—
487M16—22no—
488M16—22no—
489M16—41no—
490M16—41no—
491M16—42no—
492M16—442no—
493M16—442no—
494M16—442no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—221no—
498C2×C8—221no—
499C2×C8—221no—
500C2×C8—221no—
501C2×C8—442no—
502C2×C8—442no—
503C2×C8—442no—
504C2×C8—442no—
505C2×C8C2×C4×D1111yescharacteristic
506C2×C8C2×C4×D1111yescharacteristic
507C2×C8C2×C4×D1111yescharacteristic
508C2×C8C2×C4×D1111yescharacteristic
509C2×C8C2×D4411yescharacteristic
510C2×C8C2×D4411yescharacteristic
511C2×C8C2×D4411yescharacteristic
512C2×C8C2×D4411yescharacteristic
513C4⋊C4—111no—
514C4⋊C4—111no—
515C4⋊C4—111no—
516C4⋊C4—111no—
517C4⋊C4—221no—
518C4⋊C4—221no—
519C4⋊C4—221no—
520C22⋊C4—221no—
521C22⋊C4—221no—
522C22⋊C4—221no—
523C22⋊C4—221no—
524C22⋊C4—221no—
525C22⋊C4—221no—
526C22⋊C4—221no—
527C42—221no—
528C42—221no—
529C42—221no—
530C42C2×C4×D1111yescharacteristic
531C42C2×D4411yescharacteristic
532C42#433911yescharacteristic
533C42C23×D1111yescharacteristic
534C11#1804611yescharacteristic, complemented, Sylow
535C23—221no—
536C23—221no—
537C23—221no—
538C23—442no—
539C23—442no—
540C23—881no—
541C23C4×D4411yescharacteristic
542Q8—221no—
543Q8—221no—
544D4—112no—
545D4—112no—
546D4—221no—
547D4—221no—
548D4—221no—
549D4—221no—
550D4—442no—
551D4—442no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—221no—
558C2×C4—221no—
559C2×C4—221no—
560C2×C4—221no—
561C2×C4—221no—
562C2×C4—442no—
563C2×C4—442no—
564C2×C4—442no—
565C2×C4—442no—
566C2×C4—441no—
567C2×C4—441no—
568C2×C4—441no—
569C2×C4—441no—
570C2×C4—441no—
571C2×C4—441no—
572C2×C4—881no—
573C2×C4C4×D4411yescharacteristic
574C2×C4C4×D4411yescharacteristic
575C2×C4C4×D4411yescharacteristic
576C2×C4C22×C4×D1111yescharacteristic
577C2×C4C22×D4411yescharacteristic
578C2×C4C2×#433911yescharacteristic
579C8—22no—
580C8—21no—
581C8—21no—
582C8—21no—
583C8—21no—
584C8—21no—
585C8—21no—
586C8—41no—
587C8—441no—
588C8—441no—
589C8—441no—
590C8—441no—
591C8M16×D1112yescomplemented
592C8#6696112yescomplemented
593C22—21no—
594C22—21no—
595C22—21no—
596C22—222no—
597C22—222no—
598C22—441no—
599C22—441no—
600C22—441no—
601C22—441no—
602C22—441no—
603C22—441no—
604C22—441no—
605C22—884no—
606C22C2×C4×D4411yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—441no—
612C4—441no—
613C4—441no—
614C4—441no—
615C4—441no—
616C4C2×M16×D1111yescharacteristic
617C4#1062327611yescharacteristic
618C4C2×#6696111yescharacteristic
619C4#1062327911yescharacteristic
620C2—41no—
621C2—442no—
622C2—442no—
623C2—881no—
624C2M16×D4411yescharacteristic
625C2C4×#6696111yescharacteristic
626C2#1307752711yescharacteristic
subgroupG/H
centerC2×C4C22×D44
commutatorC2×C4×C11C23×C4
FrattiniC42C23×D11
FittingC11×#1455C2
socleC22×C11C2×C4×D4
prime radicalC2×#66961C4
cosocleC42×C11C24
hypercenter#1455D11
hypocenterC11#18046
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

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

NH ≅ G/Ntype
1. #13077631C2semidirect
3. M16×D44C2semidirect
4. #13067520C2semidirect
5. #13067508C2semidirect
6. #13067482C2semidirect
7. #13067475C2semidirect
8. #13067454C2semidirect
9. #13067447C2semidirect
10. #13067441C2semidirect
11. C8×D88C2semidirect
12. #13067383C2semidirect
13. #13067060C2semidirect
14. #13067056C2semidirect
15. C11×#1455C2semidirect
47. C8×D44C22semidirect
48. C8×D44C22semidirect
49. #10622292C22semidirect
50. #10622292C22semidirect
51. C11×#322C22semidirect
52. C11×#322C22semidirect
53. C11×#322C22semidirect
54. C82×C11C22semidirect
59. #10622097C22semidirect
60. #10622097C22semidirect
61. #10622096C22semidirect
62. #10622096C22semidirect
63. #10622095C22semidirect
64. #10622095C22semidirect
65. #10622094C22semidirect
66. #10622094C22semidirect
213. D88M16semidirect
236. #1455D11semidirect
268. D44C2×M16semidirect
289. C8×C11C2×M16semidirect
290. C8×C11AGL(1,ℤ/8)semidirect
342. #322C2×D11semidirect
343. #322C2×D11semidirect
344. #322C2×D11semidirect
345. C82C2×D11semidirect
534. C11#18046semidirect
591. C8M16×D11semidirect
592. C8#66961semidirect

Maximal subgroups

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

subgroupindex
#130776312
C4×#669612
M16×D442
#130675202
#130675082
#130674822
#130674752
#130674542
#130674472
#130674412
C8×D882
#130673832
#130670602
#130670562
C11×#14552
#1804611