Finite Group Databasebeta

← #14756212

Subgroups — #14756212

Order 2816.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622734C211yesmaximal, characteristic, complemented
2C2×#10622846C211yesmaximal, characteristic
3#13071208C211yesmaximal, characteristic
4#13070973C211yesmaximal, characteristic, complemented
5#13069259C212yesmaximal, complemented
6#13069252C212yesmaximal, complemented
7#13069245C212yesmaximal, complemented
8#13069238C212yesmaximal, complemented
9C2×#10622167C211yesmaximal, characteristic, complemented
10#13068314C211yesmaximal, characteristic, complemented
11C2×C11×#417C211yesmaximal, characteristic
12C22×C11×#41C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15C2×#66979C2211yescharacteristic
16C2×#66979C2211yescharacteristic
17C22×C4⋊Dic11C2211yescharacteristic
18C23×C11⋊C8C2211yescharacteristic
19C11×#417C418yescomplemented
20C2×C11×#103C2212yes—
21C2×C11×#102C2212yes—
22#10622846—24no—
23#10622843C2212yes—
24C2×#66899C2211yescharacteristic
25C2×#66899C2211yescharacteristic
26#10622834C2212yes—
27#10622833—22no—
28C2×#66896C2211yescharacteristic
29C2×#66896C2211yescharacteristic
30#10622825—21no—
31#10622825—21no—
32C4×C22⋊Dic11—22no—
33#10622734—24no—
34C2×#66882C2212yes—
35C2×#66881C2212yes—
36C4⋊C4×Dic11—21no—
37C4⋊C4×Dic11—21no—
38C4⋊C4×Dic11C2212yes—
39C4⋊C4×Dic11C2212yes—
40#10622188C2212yes—
41#10622188C2212yes—
42#10622167C2212yes—
43#10622167C2212yes—
44C2×C11×#41—22no—
45C2×C11×#41—22no—
46C2×C11×#41—22no—
47C2×C11×#41—22no—
48C2×C11×#41—22no—
49C2×C11×#41C2×C414yes—
50C22×Q8×C11C2311yescharacteristic
51C22×D4×C11—21no—
52C22×D4×C11C2311yescharacteristic
53C23×C4×C11—21no—
54C23×C4×C11—21no—
55C23×C4×C11C2311yescharacteristic
56C23×Dic11—21no—
57C2×C11×M16—22no—
58C2×C11×M16—22no—
59C2×C11×M16C2×C414yes—
60C22×C8×C11C2312yes—
61C2×C11×C4⋊C4—21no—
62C2×C11×C4⋊C4—22no—
63C2×C11×C4⋊C4C2×C414yes—
64C2×C11×C4⋊C4C2312yes—
65C2×C22⋊Dic11—21no—
66C2×C22⋊Dic11—22no—
67C2×C22⋊Dic11—22no—
68C2×C22⋊Dic11D412yes—
69#66979—22no—
70#66979—22no—
71#66979—22no—
72#66979D414yes—
73C2×C4⋊Dic11—22no—
74C2×C4⋊Dic11—22no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11D411yescharacteristic
77C2×C4⋊Dic11D411yescharacteristic
78C2×C4⋊Dic11C2312yes—
79C2×C4⋊Dic11C2311yescharacteristic
80C2×C4⋊Dic11C2311yescharacteristic
81C2×C4×Dic11—21no—
82C2×C4×Dic11—22no—
83C2×C4×Dic11—22no—
84C2×C4×Dic11C2312yes—
85C22×C11⋊C8—22no—
86C22×C11⋊C8—22no—
87C22×C11⋊C8—22no—
88C22×C11⋊C8C2311yescharacteristic
89C22×C11⋊C8C2311yescharacteristic
90C11×#103C2×C418yescomplemented
91C11×#102C2×C418yescomplemented
92#66899—22no—
93#66899—22no—
94#66896—22no—
95#66896—22no—
96C11⋊#95—22no—
97C11⋊#95—22no—
98C11⋊#95—22no—
99C11⋊#95—22no—
100#66882—24no—
101#66881—24no—
102#21062—111nomaximal, Sylow
103C24×C11—21no—
104C24×C11C2×D411yescharacteristic
105C11×#41—28no—
106C11×#41—28no—
107C11×#41—44no—
108C11×#41—44no—
109C2×Q8×C11—21no—
110C2×Q8×C11—21no—
111C2×Q8×C11—21no—
112C2×Q8×C11—21no—
113C2×Q8×C11—21no—
114C2×Q8×C11C22×C412yes—
115C2×D4×C11—24no—
116C2×D4×C11—21no—
117C2×D4×C11—21no—
118C2×D4×C11—21no—
119C2×D4×C11—21no—
120C2×D4×C11—21no—
121C2×D4×C11—42no—
122C2×D4×C11—42no—
123C2×D4×C11C22×C412yes—
124C22×C4×C11—21no—
125C22×C4×C11—22no—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—42no—
129C22×C4×C11—42no—
130C22×C4×C11—81no—
131C22×C4×C11—81no—
132C22×C4×C11C22×C412yes—
133C22×C4×C11C2×D411yescharacteristic
134C22×C4×C11C2×D412yes—
135C22×C4×C11#4112yes—
136C22×C4×C11#4112yes—
137C22×C4×C11#4112yes—
138C22×C4×C11#4112yes—
139C22×C4×C11C2411yescharacteristic
140C22×Dic11—21no—
141C22×Dic11—21no—
142C22×Dic11—22no—
143C22×Dic11—42no—
144C22×Dic11—81no—
145C22×Dic11—81no—
146C22×Dic11C2×D412yes—
147C22×Dic11C2×D412yes—
148C11×M16—28no—
149C2×C8×C11—22no—
150C2×C8×C11—22no—
151C2×C8×C11C22×C414yes—
152C11×C4⋊C4—24no—
153C11×C4⋊C4—22no—
154C11×C4⋊C4C22×C414yes—
155C22⋊Dic11—24no—
156C22⋊Dic11—24no—
157C22⋊Dic11—82no—
158C22⋊Dic11—82no—
159C4⋊Dic11—22no—
160C4⋊Dic11—22no—
161C4⋊Dic11—22no—
162C4⋊Dic11—21no—
163C4⋊Dic11—21no—
164C4⋊Dic11—22no—
165C4⋊Dic11—42no—
166C4⋊Dic11C2×D412yes—
167C4⋊Dic11C2×D412yes—
168C4×Dic11—24no—
169C4×Dic11—22no—
170C4×Dic11—22no—
171C4×Dic11—82no—
172C4×Dic11—82no—
173C2×C11⋊C8—21no—
174C2×C11⋊C8—21no—
175C2×C11⋊C8—21no—
176C2×C11⋊C8—21no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8—21no—
179C2×C11⋊C8—44no—
180C2×#428—111no—
181C2×#417D1111yescharacteristic
182C2×#416—111no—
183#2311—111no—
184#2274—111no—
185#1928—112no—
186#1927—112no—
187#1926—112no—
188#1925—112no—
189C2×#340—111no—
190#1729—111no—
191C23×C11—21no—
192C23×C11—21no—
193C23×C11—21no—
194C23×C11—21no—
195C23×C11—22no—
196C23×C11—41no—
197C23×C11—41no—
198C23×C11—41no—
199C23×C11—41no—
200C23×C11—41no—
201C23×C11—41no—
202C23×C11C4×D412yes—
203C23×C11#12112yes—
204C23×C11#12312yes—
205C23×C11C22×D411yescharacteristic
206Q8×C11—22no—
207Q8×C11—22no—
208Q8×C11—41no—
209Q8×C11—41no—
210D4×C11—22no—
211D4×C11—22no—
212D4×C11—48no—
213D4×C11—41no—
214D4×C11—41no—
215C2×C4×C11—24no—
216C2×C4×C11—22no—
217C2×C4×C11—41no—
218C2×C4×C11—41no—
219C2×C4×C11—42no—
220C2×C4×C11—42no—
221C2×C4×C11—48no—
222C2×C4×C11—84no—
223C2×C4×C11#11714yes—
224C2×C4×C11C4×D414yes—
225C2×C4×C11C4×D412yes—
226C2×C4×C11#12112yes—
227C2×C4×C11#12312yes—
228C2×C4×C11#12414yes—
229C2×C4×C11#12614yes—
230C2×C4×C11C23×C411yescharacteristic
231C2×C4×C11C22×D411yescharacteristic
232C2×C4×C11C2×#4111yescharacteristic
233C2×C4×C11C2×#4111yescharacteristic
234C2×C4×C11C2×#4111yescharacteristic
235C2×C4×C11C2×#4111yescharacteristic
236C2×Dic11—22no—
237C2×Dic11—22no—
238C2×Dic11—22no—
239C2×Dic11—22no—
240C2×Dic11—22no—
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—22no—
244C2×Dic11—41no—
245C2×Dic11—41no—
246C2×Dic11—41no—
247C2×Dic11—41no—
248C2×Dic11—42no—
249C2×Dic11—42no—
250C2×Dic11—42no—
251C2×Dic11—84no—
252C8×C11—24no—
253C11⋊C8—41no—
254C11⋊C8—41no—
255C22×#41C2×D1111yescharacteristic
256C22×M16C2×D1111yescharacteristic
257C23×C8—111no—
258C2×#117—111no—
259C2×#117—111no—
260C22×C4⋊C4—111no—
261C22×C4⋊C4C2×D1111yescharacteristic
262#428—224no—
263C2×#107—112no—
264C2×#106—112no—
265#417Dic1118yescomplemented
266#416—224no—
267C2×#103—111no—
268C2×#103—111no—
269C2×#103C2×D1112yes—
270C2×#102—111no—
271C2×#102—111no—
272C2×#102C2×D1112yes—
273#391—112no—
274#390—112no—
275#386—222no—
276#380—221no—
277#380—221no—
278C4×C4⋊C4—112no—
279C4×C4⋊C4—112no—
280C4×C4⋊C4—221no—
281C4×C4⋊C4—221no—
282C4×C22⋊C4—222no—
283#340—112no—
284#340—112no—
285#336—112no—
286#336—112no—
287C22×C11—22no—
288C22×C11—22no—
289C22×C11—22no—
290C22×C11—22no—
291C22×C11—22no—
292C22×C11—22no—
293C22×C11—42no—
294C22×C11—42no—
295C22×C11—42no—
296C22×C11—81no—
297C22×C11—81no—
298C22×C11—81no—
299C22×C11—81no—
300C22×C11#38814yes—
301C22×C11C2×C4×D411yescharacteristic
302C22×C11C2×#12111yescharacteristic
303C22×C11C2×#12311yescharacteristic
304C22×C11#57611yescharacteristic
305C22×C11#57711yescharacteristic
306C22×C11#57711yescharacteristic
307C22×C11#57811yescharacteristic
308C4×C11—44no—
309C4×C11—42no—
310C4×C11#38814yes—
311C4×C11C2×#11711yescharacteristic
312C4×C11C2×C4×D411yescharacteristic
313C4×C11C2×#12411yescharacteristic
314C4×C11C2×#12611yescharacteristic
315Dic11—42no—
316Dic11—42no—
317Dic11—42no—
318Dic11—42no—
319Dic11—82no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41C2×Dic1114yes—
326C22×Q8C22×D1111yescharacteristic
327C22×D4—21no—
328C22×D4C22×D1111yescharacteristic
329C23×C4—21no—
330C23×C4—21no—
331C23×C4—221no—
332C23×C4C22×D1111yescharacteristic
333C2×M16—22no—
334C2×M16—22no—
335C2×M16C2×Dic1114yes—
336C22×C8—111no—
337C22×C8—111no—
338C22×C8—222no—
339C22×C8—222no—
340C22×C8—222no—
341C22×C8C22×D1112yes—
342#117—114no—
343#117—222no—
344#117—222no—
345#117—222no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—111no—
349C2×C4⋊C4—111no—
350C2×C4⋊C4—111no—
351C2×C4⋊C4—111no—
352C2×C4⋊C4—112no—
353C2×C4⋊C4—222no—
354C2×C4⋊C4—222no—
355C2×C4⋊C4—221no—
356C2×C4⋊C4C2×Dic1114yes—
357C2×C4⋊C4C22×D1112yes—
358C2×C22⋊C4—112no—
359C2×C22⋊C4—221no—
360C2×C22⋊C4—222no—
361C2×C22⋊C4—222no—
362C2×C42—112no—
363C2×C42—221no—
364C2×C42—222no—
365C2×C42—222no—
366#107—224no—
367#106—224no—
368#103—222no—
369#103—222no—
370#103C2×Dic1118yescomplemented
371#102—222no—
372#102—222no—
373#102C2×Dic1118yescomplemented
374#95—222no—
375#95—222no—
376#95—222no—
377#95—222no—
378C2×C11—24no—
379C2×C11—82no—
380C2×C11C2×#38811yescharacteristic
381C2×C11#294611yescharacteristic
382C2×C11#294711yescharacteristic
383C2×C11#304511yescharacteristic
384C2×C11#304611yescharacteristic
385C2×C11#309811yescharacteristic
386C2×C11#309911yescharacteristic
387C24—21no—
388C24D4×D1111yescharacteristic
389#41—28no—
390#41—28no—
391#41—44no—
392#41—44no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C22×Dic1112yes—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C22×Dic1112yes—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—112no—
417C22×C4—112no—
418C22×C4—221no—
419C22×C4—221no—
420C22×C4—222no—
421C22×C4—442no—
422C22×C4—881no—
423C22×C4—881no—
424C22×C4#433912yes—
425C22×C4#433912yes—
426C22×C4D4×D1111yescharacteristic
427C22×C4D4×D1112yes—
428C22×C4#434112yes—
429C22×C4#434112yes—
430C22×C4C22×Dic1112yes—
431C22×C4C23×D1111yescharacteristic
432M16—28no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—221no—
436C2×C8—221no—
437C2×C8—221no—
438C2×C8—221no—
439C2×C8—221no—
440C2×C8—221no—
441C2×C8—444no—
442C2×C8C22×Dic1114yes—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—112no—
446C4⋊C4—112no—
447C4⋊C4—222no—
448C4⋊C4—222no—
449C4⋊C4—221no—
450C4⋊C4—221no—
451C4⋊C4—222no—
452C4⋊C4—222no—
453C4⋊C4—442no—
454C4⋊C4C22×Dic1114yes—
455C22⋊C4—224no—
456C22⋊C4—224no—
457C22⋊C4—882no—
458C22⋊C4—882no—
459C42—224no—
460C42—222no—
461C42—222no—
462C42—882no—
463C42—882no—
464C11#2106211yescharacteristic, complemented, Sylow
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#6693612yes—
477C23#6693712yes—
478C23D4×Dic1112yes—
479C23C2×D4×D1111yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—41no—
483Q8—41no—
484D4—22no—
485D4—22no—
486D4—48no—
487D4—41no—
488D4—41no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—84no—
497C2×C4—222no—
498C2×C4—222no—
499C2×C4—222no—
500C2×C4—222no—
501C2×C4—222no—
502C2×C4—222no—
503C2×C4—222no—
504C2×C4—222no—
505C2×C4—441no—
506C2×C4—441no—
507C2×C4—441no—
508C2×C4—441no—
509C2×C4—442no—
510C2×C4—442no—
511C2×C4—442no—
512C2×C4—884no—
513C2×C4#6693214yes—
514C2×C4#6693612yes—
515C2×C4#6693712yes—
516C2×C4#6693814yes—
517C2×C4#6697914yes—
518C2×C4D4×Dic1114yes—
519C2×C4D4×Dic1112yes—
520C2×C4C2×#433911yescharacteristic
521C2×C4C2×#433911yescharacteristic
522C2×C4C2×D4×D1111yescharacteristic
523C2×C4C2×#434111yescharacteristic
524C2×C4C2×#434111yescharacteristic
525C2×C4C23×Dic1111yescharacteristic
526C8—24no—
527C8—441no—
528C8—441no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#1062256414yes—
543C22C2×#6693611yescharacteristic
544C22C2×#6693711yescharacteristic
545C22#1062330511yescharacteristic
546C22#1062330611yescharacteristic
547C22#1062330911yescharacteristic
548C22#1062331011yescharacteristic
549C22C2×D4×Dic1111yescharacteristic
550C4—44no—
551C4—42no—
552C4—442no—
553C4—442no—
554C4—442no—
555C4—442no—
556C4—882no—
557C4#1062256414yes—
558C4C2×#6693211yescharacteristic
559C4C2×#6693811yescharacteristic
560C4C2×#6697911yescharacteristic
561C4C2×D4×Dic1111yescharacteristic
562C2—24no—
563C2—82no—
564C2C2×#1062256411yescharacteristic
565C2#1307744911yescharacteristic
566C2#1307746011yescharacteristic
567C2#1307815411yescharacteristic
568C2#1307816511yescharacteristic
569C2#1307859311yescharacteristic
570C2#1307860411yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#417C2
socleC23×C11C22×D4
prime radicalC22×C11×#41C22
cosocleC22×C4×C11C24
hypercenterC2×#417D11
hypocenterC11#21062
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#10622734C2semidirect
4. #13070973C2semidirect
5. #13069259C2semidirect
6. #13069252C2semidirect
7. #13069245C2semidirect
8. #13069238C2semidirect
9. C2×#10622167C2semidirect
10. #13068314C2semidirect
19. C11×#417C4semidirect
90. C11×#103C2×C4semidirect
91. C11×#102C2×C4semidirect
265. #417Dic11semidirect
370. #103C2×Dic11semidirect
373. #102C2×Dic11semidirect
464. C11#21062semidirect

Maximal subgroups

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

subgroupindex
C2×#106227342
C2×#106228462
#130712082
#130709732
#130692592
#130692522
#130692452
#130692382
C2×#106221672
#130683142
C2×C11×#4172
#2106211