Finite Group Databasebeta

← #14781158

Subgroups — #14781158

Order 2816.

1306 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13082021C211yesmaximal, characteristic, complemented
2C2×#10622744C211yesmaximal, characteristic
3C2×#10622614C211yesmaximal, characteristic
4#13071196C211yesmaximal, characteristic
5#13069771C218yesmaximal, complemented
6#13068354C211yesmaximal, characteristic, complemented
7#13068326C211yesmaximal, characteristic, complemented
8C11×#1733C211yesmaximal, characteristic, complemented
9C23×D44C2211yescharacteristic
10C22×C11×M16C2211yescharacteristic
11C2×C11×#117C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C2×#66979C2211yescharacteristic
14C2×#66979C2211yescharacteristic
15C22×#4318C2211yescharacteristic
16#10623050—22no—
17#10623050C2214yescomplemented
18#10623048—22no—
19#10623048—22no—
20#10623031—22no—
21#10623014—22no—
22#10623001—22no—
23#10622744—22no—
24#10622744C2214yes—
25C2×#66885—22no—
26#10622614—22no—
27#10622614C2214yes—
28#10622601—22no—
29C2×#66873—22no—
30#10622568—22no—
31#10622552C2214yes—
32C11×#339C2214yescomplemented
33#10622192C2214yescomplemented
34#10622161C2214yescomplemented
35C24×D11—21no—
36C23×C4×C11C2311yescharacteristic
37C22×D44—21no—
38C22×D44—21no—
39C22×D44—21no—
40C22×D44—21no—
41C22×D44—22no—
42C22×D44—22no—
43C22×D44—21no—
44C22×D44—22no—
45C22×D44Q814yescomplemented
46C22×D44C2312yes—
47C2×C11×M16—21no—
48C2×C11×M16—22no—
49C2×C11×M16—22no—
50C2×C11×M16C2312yes—
51C22×C8×C11—21no—
52C11×#117—21no—
53C11×#117—21no—
54C11×#117—22no—
55C11×#117—22no—
56C11×#117D412yes—
57C11×#117C2312yes—
58C2×C11×C4⋊C4—21no—
59C2×C11×C4⋊C4D412yes—
60C2×C11×C22⋊C4—21no—
61C2×C11×C22⋊C4—21no—
62C2×C42×C11—21no—
63C2×C42×C11C2312yes—
64C2×C22⋊Dic11—21no—
65C2×C22⋊Dic11D412yes—
66C2×(C2×D11)⋊C4—28no—
67#66979—21no—
68#66979—21no—
69#66979—21no—
70#66979—21no—
71#66979—21no—
72#66979—21no—
73#66979D412yes—
74#66979C2312yes—
75C2×C4⋊Dic11D412yes—
76C2×C4⋊Dic11C2312yes—
77#66977—28no—
78C2×C4×Dic11—21no—
79C2×C4×Dic11D412yes—
80C2×#4318—21no—
81C2×#4318—22no—
82C2×#4318—22no—
83C2×#4318C2312yes—
84C22×C11⋊C8—21no—
85#66943—24no—
86#66943—24no—
87#66941—24no—
88#66941—24no—
89#66931—28no—
90#66931—28no—
91#66923—24no—
92#66922—24no—
93C4×Dic22—28no—
94#66885—28no—
95#66873—28no—
96#23463—111nomaximal, Sylow
97C24×C11C2×D411yescharacteristic
98C23×D11—22no—
99C23×D11—24no—
100C23×D11—41no—
101C23×D11—41no—
102C23×D11—42no—
103C23×D11—84no—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11C2×D411yescharacteristic
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11C2×D411yescharacteristic
109C22×C4×C11C2×D411yescharacteristic
110C22×C4×C11C2×D411yescharacteristic
111C22×C4×C11C2×Q812yes—
112C22×C4×C11#4112yes—
113C22×C4×C11#4112yes—
114C22×C4×C11#4112yes—
115C22×C4×C11C2411yescharacteristic
116C22×Dic11—22no—
117C22×Dic11C2×D414yes—
118C2×D44—24no—
119C2×D44—24no—
120C2×D44—24no—
121C2×D44—24no—
122C2×D44—24no—
123C2×D44—22no—
124C2×D44—22no—
125C2×D44—22no—
126C2×D44—22no—
127C2×D44—22no—
128C2×D44—41no—
129C2×D44—41no—
130C2×D44—42no—
131C2×D44—42no—
132C2×D44—42no—
133C2×D44—42no—
134C2×D44—42no—
135C2×D44C2×Q814yes—
136C2×Dic22—28no—
137C11×M16—22no—
138C11×M16—22no—
139C11×M16—42no—
140C2×C8×C11—22no—
141C2×C8×C11—42no—
142C11×C4⋊C4—22no—
143C11×C4⋊C4—42no—
144C11×C4⋊C4C2×D414yes—
145C11×C22⋊C4—22no—
146C11×C22⋊C4—22no—
147C11×C22⋊C4—42no—
148C42×C11—22no—
149C42×C11—22no—
150C42×C11—22no—
151C42×C11—22no—
152C22⋊Dic11—22no—
153C22⋊Dic11—22no—
154C22⋊Dic11—22no—
155C22⋊Dic11—22no—
156(C2×D11)⋊C4—88no—
157C4⋊Dic11—28no—
158C4⋊Dic11—22no—
159C4⋊Dic11—22no—
160C4⋊Dic11—22no—
161C4⋊Dic11C2×D414yes—
162Dic11⋊C4—28no—
163Dic11⋊C4—28no—
164C4×Dic11—22no—
165C4×Dic11—22no—
166C4×Dic11—22no—
167C4×Dic11—22no—
168#4318—22no—
169#4318—22no—
170#4318—42no—
171C2×C11⋊C8—22no—
172C2×C11⋊C8—42no—
173#3458—111no—
174C2×#418—111no—
175C2×#418—111no—
176#2309—111no—
177#2027—118no—
178#1742—111no—
179#1733—111no—
180#1733D1111yescharacteristic, complemented
181C23×C11—42no—
182C23×C11#12011yescharacteristic
183C23×C11#12011yescharacteristic
184C23×C11C22⋊Q812yes—
185C23×C11#12412yes—
186C23×C11C22×D411yescharacteristic
187C22×D11—24no—
188C22×D11—48no—
189C22×D11—42no—
190C22×D11—42no—
191C22×D11—44no—
192C22×D11—44no—
193C22×D11—88no—
194C22×D11—84no—
195C22×D11—84no—
196C22×D11—88no—
197C2×C4×C11—24no—
198C2×C4×C11—44no—
199C2×C4×C11—42no—
200C2×C4×C11—82no—
201C2×C4×C11#12011yescharacteristic
202C2×C4×C11#12011yescharacteristic
203C2×C4×C11C22⋊Q812yes—
204C2×C4×C11C22⋊Q812yes—
205C2×C4×C11C22⋊Q812yes—
206C2×C4×C11C22⋊Q812yes—
207C2×C4×C11C22⋊Q812yes—
208C2×C4×C11#12412yes—
209C2×C4×C11#12412yes—
210C2×C4×C11#12412yes—
211C2×C4×C11#12412yes—
212C2×C4×C11#12412yes—
213C2×C4×C11C22×D411yescharacteristic
214C2×C4×C11C22×D411yescharacteristic
215C2×C4×C11C22×Q811yescharacteristic
216C2×C4×C11C2×#4111yescharacteristic
217C2×C4×C11C2×#4111yescharacteristic
218C2×C4×C11C2×#4111yescharacteristic
219C2×Dic11—24no—
220C2×Dic11—24no—
221C2×Dic11—44no—
222C2×Dic11—82no—
223D44—24no—
224D44—24no—
225D44—42no—
226D44—42no—
227D44—44no—
228D44—44no—
229Dic22—88no—
230C8×C11—42no—
231C11⋊C8—42no—
232C23×D4—111no—
233C22×M16—111no—
234C22×M16C2×D1111yescharacteristic
235#544—114no—
236#544—222no—
237#544—222no—
238#543—222no—
239#533—222no—
240#533—222no—
241#533—222no—
242C2×#117—111no—
243C2×#117—111no—
244C2×#117C2×D1111yescharacteristic
245C2×#117C2×D1111yescharacteristic
246#418—114no—
247#418—114no—
248#418—222no—
249#418—222no—
250C2×#102—222no—
251C2×#102—222no—
252#390—114no—
253#390—222no—
254#379—222no—
255#343—114no—
256#339—114no—
257#339C2×D1114yescomplemented
258C22×C11—22no—
259C22×C11—22no—
260C22×C11—22no—
261C22×C11—42no—
262C22×C11—42no—
263C22×C11#39312yes—
264C22×C11#39312yes—
265C22×C11C2×#12011yescharacteristic
266C22×C11C2×C22⋊Q811yescharacteristic
267C22×C11C2×#12411yescharacteristic
268C2×D11—48no—
269C2×D11—44no—
270C2×D11—84no—
271C2×D11—88no—
272C2×D11—88no—
273C2×D11—84no—
274C4×C11—44no—
275C4×C11—82no—
276C4×C11#39312yes—
277C4×C11#39312yes—
278C4×C11C2×C22⋊Q811yescharacteristic
279C4×C11C2×C22⋊Q811yescharacteristic
280C4×C11C2×#12411yescharacteristic
281C4×C11C2×#12411yescharacteristic
282Dic11—44no—
283Dic11—44no—
284C25—221no—
285C22×D4—114no—
286C22×D4—112no—
287C22×D4—221no—
288C22×D4—221no—
289C22×D4—222no—
290C22×D4—222no—
291C22×D4—221no—
292C22×D4—221no—
293C22×D4—221no—
294C22×D4—222no—
295C23×C4C22×D1111yescharacteristic
296C2×M16—21no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—112no—
300C2×M16—221no—
301C2×M16—222no—
302C2×M16—222no—
303C2×M16C22×D1112yes—
304C22×C8—21no—
305C22×C8—221no—
306C4⋊Q8—224no—
307C4⋊Q8—224no—
308C4⋊Q8—224no—
309#125—224no—
310#125—224no—
311#125—224no—
312C22⋊Q8—228no—
313C22⋊Q8—228no—
314C22⋊Q8—228no—
315C4×Q8—228no—
316#117—21no—
317#117—21no—
318#117—22no—
319#117—22no—
320#117—112no—
321#117—112no—
322#117—221no—
323#117—221no—
324#117—221no—
325#117—221no—
326#117—221no—
327#117—221no—
328#117D4412yes—
329#117C22×D1112yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—112no—
332C2×C4⋊C4—112no—
333C2×C4⋊C4D4412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—112no—
337C2×C22⋊C4—221no—
338C2×C22⋊C4—228no—
339C2×C42—21no—
340C2×C42—112no—
341C2×C42—221no—
342C2×C42C22×D1112yes—
343#102—228no—
344#102—228no—
345C2×C11—21no—
346C2×C11—21no—
347C2×C11—22no—
348C2×C11—22no—
349C2×C11C2×#39311yescharacteristic
350C2×C11#302112yes—
351D11—84no—
352C24—222no—
353C24—224no—
354C24—441no—
355C24—441no—
356C24—442no—
357C24—884no—
358C24D4×D1111yescharacteristic
359C2×Q8—228no—
360C2×D4—114no—
361C2×D4—224no—
362C2×D4—224no—
363C2×D4—224no—
364C2×D4—224no—
365C2×D4—224no—
366C2×D4—222no—
367C2×D4—222no—
368C2×D4—222no—
369C2×D4—222no—
370C2×D4—222no—
371C2×D4—442no—
372C2×D4—442no—
373C2×D4—441no—
374C2×D4—441no—
375C2×D4—442no—
376C2×D4—442no—
377C2×D4—442no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—114no—
381C22×C4—222no—
382C22×C4C2×D4411yescharacteristic
383C22×C4C2×D4411yescharacteristic
384C22×C4#433912yes—
385C22×C4#433912yes—
386C22×C4D4×D1111yescharacteristic
387C22×C4D4×D1111yescharacteristic
388C22×C4D4×D1111yescharacteristic
389C22×C4#434112yes—
390C22×C4Q8×D1112yes—
391C22×C4C23×D1111yescharacteristic
392M16—22no—
393M16—22no—
394M16—42no—
395M16—222no—
396M16—222no—
397M16—442no—
398C2×C8—22no—
399C2×C8—42no—
400C2×C8—222no—
401C2×C8—442no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—114no—
405C4⋊C4—228no—
406C4⋊C4—228no—
407C4⋊C4—228no—
408C4⋊C4—222no—
409C4⋊C4—222no—
410C4⋊C4—222no—
411C4⋊C4C2×D4414yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4—222no—
416C22⋊C4—222no—
417C22⋊C4—222no—
418C22⋊C4—222no—
419C22⋊C4—888no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—222no—
425C42—222no—
426C42—222no—
427C42—222no—
428C11#2346311yescharacteristic, complemented, Sylow
429C23—42no—
430C23—224no—
431C23—448no—
432C23—442no—
433C23—442no—
434C23—444no—
435C23—444no—
436C23—884no—
437C23—884no—
438C23—888no—
439C23—888no—
440C23#6693511yescharacteristic
441C23#6693511yescharacteristic
442C23#6693812yes—
443C23#6694912yes—
444C23C2×D4×D1111yescharacteristic
445Q8—888no—
446D4—224no—
447D4—224no—
448D4—442no—
449D4—442no—
450D4—444no—
451D4—444no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—224no—
457C2×C4—224no—
458C2×C4—444no—
459C2×C4—882no—
460C2×C4#6692812yes—
461C2×C4#6692812yes—
462C2×C4#6693511yescharacteristic
463C2×C4#6693511yescharacteristic
464C2×C4#6693812yes—
465C2×C4#6693812yes—
466C2×C4#6693812yes—
467C2×C4#6694912yes—
468C2×C4#6694912yes—
469C2×C4#6694912yes—
470C2×C4#6695012yes—
471C2×C4#6695012yes—
472C2×C4C22×D4411yescharacteristic
473C2×C4C2×#433911yescharacteristic
474C2×C4C2×#433911yescharacteristic
475C2×C4C2×D4×D1111yescharacteristic
476C2×C4C2×#434111yescharacteristic
477C2×C4C2×Q8×D1111yescharacteristic
478C8—42no—
479C8—442no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—448no—
486C22—444no—
487C22—884no—
488C22—888no—
489C22—888no—
490C22—884no—
491C22#1062228612yes—
492C22#1062228612yes—
493C22C2×#6693511yescharacteristic
494C22C2×#6693811yescharacteristic
495C22C2×#6694911yescharacteristic
496C4—44no—
497C4—82no—
498C4—444no—
499C4—444no—
500C4#1062228612yes—
501C4#1062228612yes—
502C4C2×#6692811yescharacteristic
503C4C2×#6693811yescharacteristic
504C4C2×#6694911yescharacteristic
505C4C2×#6695011yescharacteristic
506C2—21no—
507C2—21no—
508C2—22no—
509C2—22no—
510C2—884no—
511C2C2×#1062228611yescharacteristic
512C2#1307798912yes—
subgroupG/H
centerC22C2×#66935
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1733C2
socleC22×C11C2×#120
prime radicalC23×D44C22
hypercenter#1733D11
hypocenterC11#23463
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. #13082021C2semidirect
5. #13069771C2semidirect
6. #13068354C2semidirect
7. #13068326C2semidirect
8. C11×#1733C2semidirect
17. #10623050C22semidirect
32. C11×#339C22semidirect
33. #10622192C22semidirect
34. #10622161C22semidirect
45. C22×D44Q8semidirect
180. #1733D11semidirect
257. #339C2×D11semidirect
428. C11#23463semidirect

Maximal subgroups

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

subgroupindex
#130820212
C2×#106227442
C2×#106226142
#130711962
#130697712
#130683542
#130683262
C11×#17332
#2346311