Finite Group Databasebeta

← #14781761

Subgroups — #14781761

Order 2816.

1570 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66961C211yesmaximal, characteristic
2C2×#10622726C211yesmaximal, characteristic, complemented
3C22×#66885C211yesmaximal, characteristic, complemented
4#13071204C211yesmaximal, characteristic, complemented
5#13071196C211yesmaximal, characteristic, complemented
6#13069934C214yesmaximal, complemented
7#13069927C214yesmaximal, complemented
8#13068336C211yesmaximal, characteristic, complemented
9C2×C11×#424C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C23×C8×C11C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C2×#66979C2211yescharacteristic
16C22×C4⋊Dic11C2211yescharacteristic
17C2×#66961—24no—
18C2×#66961—24no—
19C2×#66961—24no—
20C22×D88C2212yes—
21C22×#4314C2212yes—
22C11×#424—24no—
23C2×C11×C4⋊C8C2214yescomplemented
24#10622726—24no—
25C2×#66885—24no—
26C2×#66885C2214yes—
27C2×#66885C2212yescomplemented
28C2×#66880C2212yescomplemented
29#10622601—22no—
30#10622600—22no—
31#10622571—22no—
32#10622568—22no—
33#10622552C2214yes—
34#10622552C2212yescomplemented
35#10622542C2212yescomplemented
36#10622189C2214yescomplemented
37C24×D11—21no—
38C23×C4×C11C2311yescharacteristic
39C22×C22⋊D11—21no—
40C23×Dic11—21no—
41C2×#4339—22no—
42C2×#4339—22no—
43C2×#4339—22no—
44C2×#4339—22no—
45C2×#4339—22no—
46C2×#4339D414yes—
47C22×D44—22no—
48C22×D44—22no—
49C22×D44—22no—
50C22×D44—22no—
51C22×D44—21no—
52C22×D44—22no—
53C22×D44D414yes—
54C22×D44C2312yes—
55C22×D44C2311yescharacteristic
56C22×C4×D11—21no—
57C22×Dic22C2311yescharacteristic
58C2×C11×M16—22no—
59C2×C11×M16—22no—
60C2×C11×M16D414yes—
61C22×C8×C11—22no—
62C22×C8×C11—22no—
63C22×C8×C11—22no—
64C22×C8×C11C2312yes—
65C22×C8×C11C2312yes—
66C11×#117—22no—
67C11×#117—22no—
68C2×C11×C4⋊C4D412yes—
69C2×C11×C22⋊C4D412yes—
70C2×C42×C11C2312yes—
71C2×C22⋊Dic11—21no—
72C2×(C2×D11)⋊C4—24no—
73C2×(C2×D11)⋊C4—24no—
74C2×(C2×D11)⋊C4—24no—
75#66979—22no—
76#66979—22no—
77C2×C4⋊Dic11—22no—
78C2×C4⋊Dic11—22no—
79C2×C4⋊Dic11—21no—
80C2×C4⋊Dic11C2312yes—
81C2×C4⋊Dic11C2312yes—
82C2×C4×Dic11—21no—
83#66961—232no—
84C2×D88—24no—
85C2×D88—24no—
86C2×D88—24no—
87C2×#4314—24no—
88C2×#4314—24no—
89C2×#4314—24no—
90C11×C4⋊C8—28no—
91C11⋊#95—24no—
92#66885—28no—
93#66885—24no—
94#66880—24no—
95#23544—111nomaximal, Sylow
96C24×C11C2×D411yescharacteristic
97C23×D11—21no—
98C23×D11—22no—
99C23×D11—24no—
100C23×D11—42no—
101C23×D11—42no—
102C23×D11—82no—
103C23×D11—82no—
104C22×C4×C11C2×D414yes—
105C22×C4×C11C2×D412yes—
106C22×C4×C11C2×D412yes—
107C22×C4×C11C2×D412yes—
108C22×C4×C11C2×D411yescharacteristic
109C22×C4×C11#4112yes—
110C22×C4×C11#4112yes—
111C22×C4×C11#4112yes—
112C22×C4×C11C2411yescharacteristic
113C2×C22⋊D11—24no—
114C2×C22⋊D11—42no—
115C2×C22⋊D11—42no—
116C22×Dic11—21no—
117C22×Dic11—22no—
118C22×Dic11—22no—
119C22×Dic11—42no—
120C22×Dic11—82no—
121#4339—28no—
122#4339—28no—
123#4339—44no—
124#4339—44no—
125C2×D44—28no—
126C2×D44—24no—
127C2×D44—28no—
128C2×D44—22no—
129C2×D44—22no—
130C2×D44—21no—
131C2×D44—21no—
132C2×D44—22no—
133C2×D44—21no—
134C2×D44—22no—
135C2×D44—22no—
136C2×D44—21no—
137C2×D44—21no—
138C2×D44—42no—
139C2×D44—42no—
140C2×D44—44no—
141C2×D44—44no—
142C2×D44C2×D414yes—
143C2×D44C2×D412yes—
144C2×C4×D11—24no—
145C2×C4×D11—42no—
146C2×C4×D11—42no—
147C2×Dic22—21no—
148C2×Dic22—21no—
149C2×Dic22—21no—
150C2×Dic22—21no—
151C2×Dic22—21no—
152C2×Dic22C2×D412yes—
153C11×M16—28no—
154C2×C8×C11—22no—
155C2×C8×C11—22no—
156C2×C8×C11—22no—
157C2×C8×C11—22no—
158C2×C8×C11—22no—
159C2×C8×C11—44no—
160C2×C8×C11C2×D414yes—
161C11×C4⋊C4—24no—
162C11×C22⋊C4—24no—
163C42×C11—22no—
164C42×C11—22no—
165C22⋊Dic11—42no—
166(C2×D11)⋊C4—84no—
167(C2×D11)⋊C4—84no—
168(C2×D11)⋊C4—84no—
169C4⋊Dic11—22no—
170C4⋊Dic11—22no—
171C4⋊Dic11—22no—
172C4⋊Dic11—22no—
173C4⋊Dic11—42no—
174C4×Dic11—42no—
175D88—216no—
176#4314—216no—
177C22×AGL(1,ℤ/8)—111no—
178C2×#424D1111yescharacteristic, complemented
179C2×#416—111no—
180C22×#102—111no—
181#2311—111no—
182#2309—111no—
183#2055—114no—
184#2054—114no—
185#1736—111no—
186C23×C11—22no—
187C23×C11—22no—
188C23×C11#12112yes—
189C23×C11#12112yes—
190C23×C11#12112yes—
191C23×C11C22×D411yescharacteristic
192C22×D11—22no—
193C22×D11—24no—
194C22×D11—41no—
195C22×D11—41no—
196C22×D11—42no—
197C22×D11—42no—
198C22×D11—42no—
199C22×D11—44no—
200C22×D11—44no—
201C22×D11—42no—
202C22×D11—48no—
203C22×D11—84no—
204C22×D11—84no—
205C22×D11—88no—
206C22×D11—88no—
207C2×C4×C11—24no—
208C2×C4×C11—24no—
209C2×C4×C11—24no—
210C2×C4×C11#12014yes—
211C2×C4×C11#12112yes—
212C2×C4×C11#12112yes—
213C2×C4×C11#12112yes—
214C2×C4×C11#12314yes—
215C2×C4×C11#12314yes—
216C2×C4×C11#12414yes—
217C2×C4×C11C22×D411yescharacteristic
218C2×C4×C11C22×D411yescharacteristic
219C2×C4×C11C22×D411yescharacteristic
220C2×C4×C11C2×#4111yescharacteristic
221C2×C4×C11C2×#4111yescharacteristic
222C2×C4×C11C2×#4111yescharacteristic
223C22⋊D11—48no—
224C2×Dic11—22no—
225C2×Dic11—41no—
226C2×Dic11—41no—
227C2×Dic11—42no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—44no—
231C2×Dic11—42no—
232C2×Dic11—84no—
233D44—24no—
234D44—22no—
235D44—24no—
236D44—22no—
237D44—42no—
238D44—41no—
239D44—42no—
240D44—41no—
241D44—48no—
242C4×D11—48no—
243Dic22—22no—
244Dic22—22no—
245Dic22—41no—
246Dic22—41no—
247C8×C11—24no—
248C8×C11—42no—
249C22×#41—111no—
250C23×D4—111no—
251C2×AGL(1,ℤ/8)—224no—
252C2×AGL(1,ℤ/8)—224no—
253C2×AGL(1,ℤ/8)—224no—
254C22×SD16—112no—
255C22×D8—112no—
256C22×M16C2×D1111yescharacteristic
257C23×C8C2×D1111yescharacteristic
258C2×#117—111no—
259C2×#117C2×D1111yescharacteristic
260C22×C4⋊C4—111no—
261#424—24no—
262C2×C4⋊C8C2×D1114yescomplemented
263#416—224no—
264C2×#103—112no—
265C2×#102—114no—
266C2×#102—112no—
267C2×#102—224no—
268#391—112no—
269#390—114no—
270#390—112no—
271#390—222no—
272#386—222no—
273#380—222no—
274#379—222no—
275#340—114no—
276C22×C11—22no—
277C22×C11—22no—
278C22×C11—24no—
279C22×C11—24no—
280C22×C11#39414yes—
281C22×C11C2×#12111yescharacteristic
282C22×C11C2×#12111yescharacteristic
283C22×C11C2×#12111yescharacteristic
284C22×C11#57612yes—
285C22×C11#57712yes—
286C2×D11—44no—
287C2×D11—42no—
288C2×D11—44no—
289C2×D11—44no—
290C2×D11—44no—
291C2×D11—816no—
292C2×D11—81no—
293C2×D11—81no—
294C2×D11—84no—
295C2×D11—82no—
296C2×D11—84no—
297C4×C11—44no—
298C4×C11#39414yes—
299C4×C11C2×#12011yescharacteristic
300C4×C11C2×#12311yescharacteristic
301C4×C11C2×#12311yescharacteristic
302C4×C11C2×#12411yescharacteristic
303Dic11—42no—
304Dic11—82no—
305Dic11—82no—
306C25—221no—
307C2×#41—114no—
308C2×#41—222no—
309C2×#41—222no—
310C2×#41—222no—
311C2×#41—222no—
312C2×#41—222no—
313C22×Q8—111no—
314C22×D4—114no—
315C22×D4—111no—
316C22×D4—112no—
317C22×D4—221no—
318C22×D4—221no—
319C22×D4—222no—
320C22×D4—222no—
321C22×D4—222no—
322C22×D4—222no—
323C22×D4—222no—
324C23×C4—221no—
325C23×C4—221no—
326C23×C4C22×D1111yescharacteristic
327AGL(1,ℤ/8)—2232no—
328C2×SD16—224no—
329C2×SD16—224no—
330C2×SD16—224no—
331C2×D8—224no—
332C2×D8—224no—
333C2×D8—224no—
334C2×M16—22no—
335C2×M16—22no—
336C2×M16C22⋊D1114yes—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—22no—
340C22×C8C22×D1112yes—
341C22×C8C22×D1112yes—
342#117—22no—
343#117—22no—
344#117—222no—
345#117—222no—
346C2×C4⋊C4—112no—
347C2×C4⋊C4—112no—
348C2×C4⋊C4—222no—
349C2×C4⋊C4—222no—
350C2×C4⋊C4—221no—
351C2×C4⋊C4D4412yes—
352C2×C22⋊C4—221no—
353C2×C22⋊C4—224no—
354C2×C22⋊C4—224no—
355C2×C22⋊C4—224no—
356C2×C22⋊C4D4412yes—
357C2×C42—221no—
358C2×C42C22×D1112yes—
359C4⋊C8—28no—
360#103—224no—
361#102—228no—
362#102—224no—
363#95—224no—
364C2×C11—24no—
365C2×C11C2×#39411yescharacteristic
366C2×C11#304111yescharacteristic
367C2×C11#304211yescharacteristic
368C2×C11#304512yes—
369C2×C11#306212yes—
370D11—82no—
371D11—84no—
372C24—221no—
373C24—222no—
374C24—224no—
375C24—442no—
376C24—442no—
377C24—882no—
378C24—882no—
379C24C2×D4411yescharacteristic
380#41—228no—
381#41—228no—
382#41—444no—
383#41—444no—
384C2×Q8—112no—
385C2×Q8—221no—
386C2×Q8—221no—
387C2×Q8—221no—
388C2×Q8—221no—
389C2×Q8—221no—
390C2×D4—112no—
391C2×D4—114no—
392C2×D4—224no—
393C2×D4—228no—
394C2×D4—224no—
395C2×D4—228no—
396C2×D4—222no—
397C2×D4—221no—
398C2×D4—221no—
399C2×D4—222no—
400C2×D4—222no—
401C2×D4—221no—
402C2×D4—222no—
403C2×D4—221no—
404C2×D4—221no—
405C2×D4—222no—
406C2×D4—442no—
407C2×D4—442no—
408C2×D4—444no—
409C2×D4—442no—
410C2×D4—442no—
411C2×D4—444no—
412C22×C4—221no—
413C22×C4—224no—
414C22×C4—222no—
415C22×C4—222no—
416C22×C4—442no—
417C22×C4—442no—
418C22×C4—442no—
419C22×C4—882no—
420C22×C4C2×D4414yes—
421C22×C4C2×D4411yescharacteristic
422C22×C4#433912yes—
423C22×C4D4×D1112yes—
424C22×C4D4×D1112yes—
425C22×C4#434312yes—
426C22×C4#434312yes—
427C22×C4C2×C22⋊D1112yes—
428C22×C4C23×D1111yescharacteristic
429SD16—2216no—
430D8—2216no—
431M16—28no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—44no—
438C2×C8C2×C22⋊D1114yes—
439C4⋊C4—24no—
440C4⋊C4—222no—
441C4⋊C4—222no—
442C4⋊C4—222no—
443C4⋊C4—222no—
444C4⋊C4—442no—
445C22⋊C4—24no—
446C22⋊C4—442no—
447C22⋊C4—884no—
448C22⋊C4—884no—
449C22⋊C4—884no—
450C42—22no—
451C42—22no—
452C42—442no—
453C11#2354411yescharacteristic, complemented, Sylow
454C23—22no—
455C23—22no—
456C23—222no—
457C23—224no—
458C23—444no—
459C23—442no—
460C23—442no—
461C23—444no—
462C23—441no—
463C23—442no—
464C23—441no—
465C23—442no—
466C23—448no—
467C23—884no—
468C23—884no—
469C23—888no—
470C23—888no—
471C23#6694812yes—
472C23#6694812yes—
473C23#6698312yes—
474C23C22×D4411yescharacteristic
475Q8—222no—
476Q8—222no—
477Q8—441no—
478Q8—441no—
479D4—224no—
480D4—222no—
481D4—222no—
482D4—224no—
483D4—448no—
484D4—442no—
485D4—441no—
486D4—442no—
487D4—441no—
488D4—448no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—222no—
493C2×C4—441no—
494C2×C4—441no—
495C2×C4—448no—
496C2×C4—442no—
497C2×C4—442no—
498C2×C4—444no—
499C2×C4—442no—
500C2×C4—442no—
501C2×C4—884no—
502C2×C4#6694714yes—
503C2×C4#6694714yes—
504C2×C4#6694812yes—
505C2×C4#6694812yes—
506C2×C4#6698312yes—
507C2×C4#6699014yes—
508C2×C4#6700014yes—
509C2×C4C2×#433911yescharacteristic
510C2×C4C2×D4×D1111yescharacteristic
511C2×C4C2×D4×D1111yescharacteristic
512C2×C4C2×#434311yescharacteristic
513C2×C4C2×#434311yescharacteristic
514C2×C4C22×C22⋊D1111yescharacteristic
515C8—24no—
516C8—42no—
517C22—24no—
518C22—22no—
519C22—22no—
520C22—24no—
521C22—444no—
522C22—444no—
523C22—444no—
524C22—442no—
525C22—444no—
526C22—8816no—
527C22—884no—
528C22—882no—
529C22—881no—
530C22—881no—
531C22—884no—
532C22#1062260414yes—
533C22C2×#6694811yescharacteristic
534C22C2×#6694811yescharacteristic
535C22#1062328212yes—
536C22#1062328312yes—
537C22C2×#6698311yescharacteristic
538C4—44no—
539C4—442no—
540C4—882no—
541C4—882no—
542C4#1062260414yes—
543C4C2×#6694711yescharacteristic
544C4C2×#6694711yescharacteristic
545C4C2×#6699011yescharacteristic
546C4C2×#6700011yescharacteristic
547C2—24no—
548C2—884no—
549C2—882no—
550C2C2×#1062260411yescharacteristic
551C2#1307812011yescharacteristic
552C2#1307812911yescharacteristic
553C2#1307815712yes—
554C2#1307829612yes—
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#424C2
socleC23×C11C22×D4
prime radicalC22×#66961C2
hypercenterC2×#424D11
hypocenterC11#23544
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
2. C2×#10622726C2semidirect
3. C22×#66885C2semidirect
4. #13071204C2semidirect
5. #13071196C2semidirect
6. #13069934C2semidirect
7. #13069927C2semidirect
8. #13068336C2semidirect
9. C2×C11×#424C2semidirect
23. C2×C11×C4⋊C8C22semidirect
27. C2×#66885C22semidirect
28. C2×#66880C22semidirect
34. #10622552C22semidirect
35. #10622542C22semidirect
36. #10622189C22semidirect
178. C2×#424D11semidirect
262. C2×C4⋊C8C2×D11semidirect
453. C11#23544semidirect

Maximal subgroups

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

subgroupindex
C22×#669612
C2×#106227262
C22×#668852
#130712042
#130711962
#130699342
#130699272
#130683362
C2×C11×#4242
#2354411