Finite Group Databasebeta

← #15638833

Subgroups — #15638833

Order 2816.

2086 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic22C211yesmaximal, characteristic, complemented
2#13080904C214yesmaximal, complemented
3#13080893C214yesmaximal, complemented
4#13080282C214yesmaximal, complemented
5#13080258C214yesmaximal, complemented
6#13079561C211yesmaximal, characteristic, complemented
7#13079538C211yesmaximal, characteristic, complemented
8#13079442C214yesmaximal, complemented
9#13078651C211yesmaximal, characteristic, complemented
10#13078643C211yesmaximal, characteristic, complemented
11#13077849C214yesmaximal, complemented
12#13077657C211yesmaximal, characteristic, complemented
13C11×#3506C211yesmaximal, characteristic
14C23×Dic22C2212yes—
15C11×#559—22no—
16C11×#556C2212yes—
17C11×#555C2212yes—
18C11×#553—22no—
19C11×#550C2212yes—
20C11×#547C2212yes—
21C11×#538—22no—
22C11×#535—21no—
23C11×#533C2211yescharacteristic
24C11×#532C2211yescharacteristic
25C11×#525C2214yes—
26C11×#520—21no—
27C2×C4×D4×C11C2211yescharacteristic
28C2×#67004C2218yescomplemented
29C2×D4×Dic11—21no—
30C2×#66977C2212yes—
31D4×Dic22—22no—
32D4×Dic22—22no—
33D4×Dic22C2214yes—
34D4×Dic22C2214yes—
35C2×#66941—21no—
36C2×#66931—22no—
37C2×#66922C2211yescharacteristic
38C2×C4×Dic22C2211yescharacteristic
39#10622852C2218yescomplemented
40#10622820C2214yes—
41#10622707C2212yescomplemented
42#10622706C2212yescomplemented
43#10622706C2212yescomplemented
44#10622705C2212yescomplemented
45#10622704C2212yescomplemented
46#10622703C2212yescomplemented
47#10622689C2212yescomplemented
48#10622688C2212yescomplemented
49#10622665C2218yescomplemented
50#10622663C2214yes—
51#10622663C2212yes—
52#10622663C2218yescomplemented
53#10622657C2212yes—
54#10622655C2218yescomplemented
55#10622653C2212yes—
56#10622653C2214yes—
57#10622653C2218yescomplemented
58#10622647C2212yes—
59#10622644C2214yescomplemented
60C4×#4326C2214yescomplemented
61#10622632C2214yescomplemented
62C4×#4324C2214yescomplemented
63#10622625C2214yes—
64#10622625C2214yes—
65#10622617C2218yescomplemented
66#10622609C2218yescomplemented
67#10622538C2211yescharacteristic
68#10622538C2211yescharacteristic
69C2×C11×#41—21no—
70C2×C11×#41C2314yes—
71C22×D4×C11D411yescharacteristic
72C23×C4×C11C2312yes—
73C23×Dic11—22no—
74C22×Dic22—22no—
75C22×Dic22—24no—
76C22×Dic22—48no—
77C22×Dic22C2311yescharacteristic
78C22×Dic22C2318yes—
79C22×Dic22C2312yes—
80C11×C4⋊Q8C2311yescharacteristic
81C11×C4⋊Q8C2311yescharacteristic
82C11×C4⋊Q8C2311yescharacteristic
83C11×#127—22no—
84C11×#127C2311yescharacteristic
85C11×#126—24no—
86C11×#125—21no—
87C11×#124—22no—
88C11×#124—24no—
89C11×#124—21no—
90C11×#123—24no—
91C11×#123—24no—
92C11×C22⋊Q8—21no—
93C11×C22⋊Q8C2312yes—
94C11×C22⋊Q8C2314yes—
95C11×#121—22no—
96C11×#121—22no—
97C11×#121—21no—
98C11×#121—24no—
99C11×#121C2312yes—
100C11×#121C2314yes—
101C11×#120—24no—
102C4×Q8×C11—22no—
103C4×Q8×C11—21no—
104C4×Q8×C11C2312yes—
105C4×D4×C11—24no—
106C4×D4×C11—24no—
107C4×D4×C11—22no—
108C4×D4×C11—21no—
109C4×D4×C11D412yes—
110C4×D4×C11D412yes—
111C4×D4×C11C2312yes—
112C4×D4×C11C2312yes—
113C4×D4×C11C2312yes—
114C11×#117—21no—
115C11×#117—22no—
116C11×#117C2314yes—
117C2×C11×C4⋊C4—22no—
118C2×C11×C4⋊C4D411yescharacteristic
119C2×C11×C22⋊C4D412yes—
120C2×C42×C11C2311yescharacteristic
121C2×C22⋊Dic11—22no—
122#67004—416no—
123C2×#4326C2314yes—
124C2×#4326C2318yes—
125D4×Dic11—24no—
126D4×Dic11—24no—
127C2×#4324C2314yes—
128C2×#4324C2318yes—
129C2×C4⋊Dic11—21no—
130C2×C4⋊Dic11—21no—
131C2×C4⋊Dic11C2311yescharacteristic
132#66977—24no—
133#66977C2318yes—
134C2×Dic11⋊C4—21no—
135C2×Dic11⋊C4—22no—
136C2×C4×Dic11—21no—
137C2×#4318C2318yes—
138#66941—24no—
139#66941—24no—
140#66931—28no—
141#66931—28no—
142#66922—21no—
143#66922—21no—
144#66922C2312yes—
145#66922C2312yes—
146C4×Dic22—21no—
147C4×Dic22—21no—
148C4×Dic22C2312yes—
149C4×Dic22C2312yes—
150#66899C2314yes—
151#66896C2314yes—
152C22⋊(C11⋊C8)C2312yes—
153C22⋊(C11⋊C8)C2312yes—
154C22⋊(C11⋊C8)C2318yes—
155#66874C2318yes—
156#66874C2318yes—
157#66874C2314yes—
158#66874C2314yes—
159#66872C2314yes—
160#66871C2314yes—
161C4⋊(C11⋊C8)C2312yes—
162C4⋊(C11⋊C8)C2312yes—
163C4×C11⋊C8C2312yes—
164C4×C11⋊C8C2312yes—
165#61584—111nomaximal, Sylow
166C24×C11C2×D412yes—
167C11×#41—48no—
168C11×#41—44no—
169C2×Q8×C11—21no—
170C2×Q8×C11—21no—
171C2×Q8×C11C2412yes—
172C2×D4×C11—22no—
173C2×D4×C11—24no—
174C2×D4×C11—24no—
175C2×D4×C11—24no—
176C2×D4×C11—21no—
177C2×D4×C11—21no—
178C2×D4×C11C2×D412yes—
179C2×D4×C11C2×D412yes—
180C2×D4×C11C2412yes—
181C22×C4×C11—22no—
182C22×C4×C11—24no—
183C22×C4×C11—44no—
184C22×C4×C11C2×D411yescharacteristic
185C22×C4×C11C2×D412yes—
186C22×C4×C11C2×D414yes—
187C22×C4×C11C2×D411yescharacteristic
188C22×C4×C11C2411yescharacteristic
189C22×C4×C11C2414yes—
190C22×Dic11—21no—
191C22×Dic11—28no—
192C22×Dic11—21no—
193C22×Dic11—22no—
194C22×Dic11—48no—
195C2×Dic22—24no—
196C2×Dic22—216no—
197C2×Dic22—21no—
198C2×Dic22—22no—
199C2×Dic22—28no—
200C2×Dic22—48no—
201C2×Dic22—44no—
202C2×Dic22—416no—
203C2×Dic22—48no—
204C2×Dic22C2414yes—
205C2×Dic22C2412yes—
206C11×C4⋊C4—21no—
207C11×C4⋊C4—24no—
208C11×C4⋊C4—21no—
209C11×C4⋊C4—22no—
210C11×C4⋊C4—21no—
211C11×C4⋊C4—21no—
212C11×C4⋊C4—22no—
213C11×C4⋊C4C2×D412yes—
214C11×C4⋊C4C2×D412yes—
215C11×C4⋊C4C2412yes—
216C11×C4⋊C4C2412yes—
217C11×C22⋊C4—22no—
218C11×C22⋊C4—24no—
219C11×C22⋊C4—24no—
220C11×C22⋊C4—24no—
221C11×C22⋊C4—22no—
222C11×C22⋊C4—22no—
223C11×C22⋊C4C2×D414yes—
224C11×C22⋊C4C2×D414yes—
225C42×C11—24no—
226C42×C11—22no—
227C42×C11C2×D411yescharacteristic
228C42×C11C2×D411yescharacteristic
229C42×C11C2411yescharacteristic
230C42×C11C2411yescharacteristic
231C22⋊Dic11—28no—
232#4326—28no—
233#4326—48no—
234#4324—28no—
235#4324—48no—
236C4⋊Dic11—21no—
237C4⋊Dic11—22no—
238C4⋊Dic11—22no—
239C4⋊Dic11—22no—
240C4⋊Dic11—22no—
241C4⋊Dic11C2412yes—
242Dic11⋊C4—22no—
243Dic11⋊C4—22no—
244Dic11⋊C4—28no—
245C4×Dic11—22no—
246C4×Dic11—22no—
247#4318—48no—
248C2×C11⋊C8C2414yes—
249C2×C11⋊C8C2414yes—
250#3506D1111yescharacteristic
251C2×D4×Q8—111no—
252#3329—114no—
253#3328—114no—
254#3278—114no—
255#3276—114no—
256#3209—111no—
257#3206—111no—
258#3196—114no—
259#3104—111no—
260#3103—111no—
261#3004—114no—
262#2975—111no—
263C23×C11—24no—
264C23×C11—22no—
265C23×C11—42no—
266C23×C11—42no—
267C23×C11C22×D414yes—
268C23×C11C22×D411yescharacteristic
269Q8×C11—24no—
270Q8×C11—42no—
271Q8×C11—81no—
272D4×C11—24no—
273D4×C11—24no—
274D4×C11—24no—
275D4×C11—44no—
276D4×C11—42no—
277D4×C11—81no—
278D4×C11—84no—
279D4×C11—84no—
280C2×C4×C11—28no—
281C2×C4×C11—22no—
282C2×C4×C11—22no—
283C2×C4×C11—22no—
284C2×C4×C11—22no—
285C2×C4×C11—44no—
286C2×C4×C11—42no—
287C2×C4×C11—41no—
288C2×C4×C11—44no—
289C2×C4×C11—84no—
290C2×C4×C11#13714yescomplemented
291C2×C4×C11C22×D411yescharacteristic
292C2×C4×C11C22×D411yescharacteristic
293C2×C4×C11C22×D414yes—
294C2×C4×C11C22×D411yescharacteristic
295C2×C4×C11C22×D411yescharacteristic
296C2×C4×C11C22×D411yescharacteristic
297C2×C4×C11C2511yescharacteristic
298C2×Dic11—24no—
299C2×Dic11—22no—
300C2×Dic11—22no—
301C2×Dic11—24no—
302C2×Dic11—416no—
303C2×Dic11—44no—
304C2×Dic11—42no—
305Dic22—24no—
306Dic22—22no—
307Dic22—28no—
308Dic22—416no—
309Dic22—44no—
310Dic22H(5,𝔽2)14yes—
311C11⋊C8—24no—
312C11⋊C8—24no—
313C23×Q8—112no—
314C2×#137—118no—
315#559—22no—
316#556C2×D1112yes—
317#555C2×D1112yes—
318#553—22no—
319#550C2×D1112yes—
320D4×Q8—114no—
321D4×Q8—114no—
322D4×Q8—222no—
323D4×Q8—222no—
324#547C2×D1112yes—
325#538—22no—
326#535—21no—
327#533C2×D1111yescharacteristic
328#532C2×D1111yescharacteristic
329C2×C4⋊Q8—111no—
330C2×C4⋊Q8—221no—
331#525C2×D1114yes—
332C2×C22⋊Q8—112no—
333C2×C22⋊Q8—222no—
334#520—21no—
335C2×C4×Q8—111no—
336C2×C4×D4—221no—
337C2×C4×D4C2×D1111yescharacteristic
338#494—112no—
339#492—112no—
340#487—112no—
341#487—112no—
342#485—118no—
343#484—112no—
344#481—112no—
345#477—112no—
346#475—112no—
347#470—118no—
348#468—118no—
349#462—112no—
350#462—114no—
351#461—114no—
352#460—112no—
353#452—118no—
354#451—118no—
355#451—112no—
356#451—114no—
357#448—118no—
358#448—112no—
359#448—114no—
360C4×Q16—114no—
361C4×SD16—114no—
362#436—114no—
363#436—114no—
364#431—111no—
365#431—111no—
366#419—118no—
367#407—114no—
368C22×C11—21no—
369C22×C11—21no—
370C22×C11—24no—
371C22×C11—24no—
372C22×C11—24no—
373C22×C11—48no—
374C22×C11—42no—
375C22×C11—42no—
376C22×C11—42no—
377C22×C11C2×#13712yes—
378C22×C11C23×D411yescharacteristic
379C4×C11—21no—
380C4×C11—21no—
381C4×C11—21no—
382C4×C11—24no—
383C4×C11—42no—
384C4×C11—42no—
385C4×C11—42no—
386C4×C11C2×#13712yes—
387C4×C11C2×#13712yes—
388C4×C11C2×H(5,𝔽2)12yes—
389Dic11—24no—
390Dic11—42no—
391Dic11—44no—
392C2×#41—21no—
393C2×#41C22×D1114yes—
394C22×Q8—111no—
395C22×Q8—118no—
396C22×Q8—112no—
397C22×Q8—222no—
398C22×Q8—224no—
399C22×Q8—448no—
400C22×D4C22⋊D1111yescharacteristic
401C23×C4—222no—
402C23×C4C22×D1112yes—
403#137—4416no—
404C2×Q16—114no—
405C2×Q16—118no—
406C2×SD16—114no—
407C2×SD16—118no—
408C2×M16—118no—
409C4⋊Q8—112no—
410C4⋊Q8—112no—
411C4⋊Q8—224no—
412C4⋊Q8—224no—
413C4⋊Q8—221no—
414C4⋊Q8—221no—
415C4⋊Q8C22×D1111yescharacteristic
416C4⋊Q8C22×D1111yescharacteristic
417C4⋊Q8C22×D1111yescharacteristic
418#127—22no—
419#127C22×D1111yescharacteristic
420#126—24no—
421#125—21no—
422#124—22no—
423#124—24no—
424#124—21no—
425#123—24no—
426#123—24no—
427C22⋊Q8—21no—
428C22⋊Q8—118no—
429C22⋊Q8—224no—
430C22⋊Q8—228no—
431C22⋊Q8—228no—
432C22⋊Q8C22×D1112yes—
433C22⋊Q8C22×D1114yes—
434#121—22no—
435#121—22no—
436#121—21no—
437#121—24no—
438#121C22×D1112yes—
439#121C22×D1114yes—
440#120—24no—
441C4×Q8—22no—
442C4×Q8—21no—
443C4×Q8—112no—
444C4×Q8—112no—
445C4×Q8—221no—
446C4×Q8—221no—
447C4×Q8C22×D1112yes—
448C4×D4—24no—
449C4×D4—24no—
450C4×D4—22no—
451C4×D4—21no—
452C4×D4—224no—
453C4×D4—224no—
454C4×D4C22⋊D1112yes—
455C4×D4C22⋊D1112yes—
456C4×D4C22×D1112yes—
457C4×D4C22×D1112yes—
458C4×D4C22×D1112yes—
459#117—21no—
460#117—22no—
461#117C22×D1114yes—
462C2×C4⋊C4—22no—
463C2×C4⋊C4—111no—
464C2×C4⋊C4—221no—
465C2×C4⋊C4—221no—
466C2×C4⋊C4—222no—
467C2×C4⋊C4—221no—
468C2×C4⋊C4C22⋊D1111yescharacteristic
469C2×C22⋊C4—222no—
470C2×C22⋊C4C22⋊D1112yes—
471C2×C42—221no—
472C2×C42C22×D1111yescharacteristic
473#107—114no—
474#106—114no—
475C4⋊C8—112no—
476C4⋊C8—112no—
477#103—114no—
478#103—114no—
479#103—114no—
480#103—118no—
481#103—118no—
482#102—114no—
483C22⋊C8—112no—
484C22⋊C8—112no—
485C22⋊C8—118no—
486C4×C8—112no—
487C4×C8—112no—
488C2×C11—22no—
489C2×C11—44no—
490C2×C11—82no—
491C2×C11#348411yescharacteristic
492C2×C11C22×#13711yescharacteristic
493C2×C11#358611yescharacteristic
494C24C2×C22⋊D1112yes—
495#41—48no—
496#41—44no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—112no—
500C2×Q8—114no—
501C2×Q8—224no—
502C2×Q8—2216no—
503C2×Q8—221no—
504C2×Q8—228no—
505C2×Q8—222no—
506C2×Q8—448no—
507C2×Q8—444no—
508C2×Q8—448no—
509C2×Q8—4416no—
510C2×Q8C23×D1112yes—
511C2×D4—22no—
512C2×D4—24no—
513C2×D4—24no—
514C2×D4—24no—
515C2×D4—21no—
516C2×D4—21no—
517C2×D4C2×C22⋊D1112yes—
518C2×D4C2×C22⋊D1112yes—
519C2×D4C23×D1112yes—
520C22×C4—22no—
521C22×C4—24no—
522C22×C4—44no—
523C22×C4—221no—
524C22×C4—221no—
525C22×C4—222no—
526C22×C4—228no—
527C22×C4—448no—
528C22×C4C2×C22⋊D1111yescharacteristic
529C22×C4C2×C22⋊D1112yes—
530C22×C4C2×C22⋊D1114yes—
531C22×C4C2×C22⋊D1111yescharacteristic
532C22×C4C23×D1111yescharacteristic
533C22×C4C23×D1114yes—
534Q16—228no—
535Q16—448no—
536SD16—228no—
537SD16—448no—
538M16—448no—
539C2×C8—114no—
540C2×C8—114no—
541C4⋊C4—21no—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—112no—
549C4⋊C4—221no—
550C4⋊C4—222no—
551C4⋊C4—222no—
552C4⋊C4—222no—
553C4⋊C4—222no—
554C4⋊C4—228no—
555C4⋊C4—222no—
556C4⋊C4—222no—
557C4⋊C4C2×C22⋊D1112yes—
558C4⋊C4C2×C22⋊D1112yes—
559C4⋊C4C23×D1112yes—
560C4⋊C4C23×D1112yes—
561C22⋊C4—22no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—228no—
568C22⋊C4C2×C22⋊D1114yes—
569C22⋊C4C2×C22⋊D1114yes—
570C42—24no—
571C42—22no—
572C42—222no—
573C42—222no—
574C42C2×C22⋊D1111yescharacteristic
575C42C2×C22⋊D1111yescharacteristic
576C42C23×D1111yescharacteristic
577C42C23×D1111yescharacteristic
578C11#6158411yescharacteristic, complemented, Sylow
579C23—24no—
580C23—22no—
581C23—42no—
582C23—42no—
583C23C22×C22⋊D1114yes—
584C23C22×C22⋊D1111yescharacteristic
585Q8—24no—
586Q8—42no—
587Q8—81no—
588Q8—114no—
589Q8—224no—
590Q8—222no—
591Q8—228no—
592Q8—4416no—
593Q8—444no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—44no—
598D4—42no—
599D4—81no—
600D4—84no—
601D4—84no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—84no—
612C2×C4—224no—
613C2×C4—222no—
614C2×C4—222no—
615C2×C4—224no—
616C2×C4—4416no—
617C2×C4—444no—
618C2×C4—442no—
619C2×C4#6700414yescomplemented
620C2×C4C22×C22⋊D1114yes—
621C2×C4C22×C22⋊D1111yescharacteristic
622C2×C4C22×C22⋊D1111yescharacteristic
623C2×C4C22×C22⋊D1111yescharacteristic
624C2×C4C22×C22⋊D1111yescharacteristic
625C2×C4C22×C22⋊D1111yescharacteristic
626C2×C4C24×D1111yescharacteristic
627C8—224no—
628C8—224no—
629C22—21no—
630C22—21no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—48no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22C2×#6700412yes—
639C22C23×C22⋊D1111yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—224no—
648C4—442no—
649C4—444no—
650C4C2×#6700412yes—
651C4C2×#6700412yes—
652C4D11×H(5,𝔽2)12yes—
653C2—22no—
654C2—44no—
655C2—82no—
656C2#1308224711yescharacteristic
657C2C22×#6700411yescharacteristic
658C2#1308340211yescharacteristic
subgroupG/H
centerC22C23×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
Fitting, prime radicalC11×#3506C2
socleC22×C11C23×D4
hypercenter#3506D11
hypocenterC11#61584
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic22C2semidirect
2. #13080904C2semidirect
3. #13080893C2semidirect
4. #13080282C2semidirect
5. #13080258C2semidirect
6. #13079561C2semidirect
7. #13079538C2semidirect
8. #13079442C2semidirect
9. #13078651C2semidirect
10. #13078643C2semidirect
11. #13077849C2semidirect
12. #13077657C2semidirect
28. C2×#67004C22semidirect
39. #10622852C22semidirect
41. #10622707C22semidirect
42. #10622706C22semidirect
43. #10622706C22semidirect
44. #10622705C22semidirect
45. #10622704C22semidirect
46. #10622703C22semidirect
47. #10622689C22semidirect
48. #10622688C22semidirect
49. #10622665C22semidirect
52. #10622663C22semidirect
54. #10622655C22semidirect
57. #10622653C22semidirect
59. #10622644C22semidirect
60. C4×#4326C22semidirect
61. #10622632C22semidirect
62. C4×#4324C22semidirect
65. #10622617C22semidirect
66. #10622609C22semidirect
290. C2×C4×C11#137semidirect
578. C11#61584semidirect
619. C2×C4#67004semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic222
#130809042
#130808932
#130802822
#130802582
#130795612
#130795382
#130794422
#130786512
#130786432
#130778492
#130776572
C11×#35062
#6158411