Finite Group Databasebeta

← #14780874

Subgroups — #14780874

Order 2816.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67002C211yesmaximal, characteristic
2C23×#4314C211yesmaximal, characteristic, complemented
3C2×#10623356C211yesmaximal, characteristic, complemented
4C2×#10622617C211yesmaximal, characteristic, complemented
5#13071196C211yesmaximal, characteristic, complemented
6#13069700C214yesmaximal, complemented
7#13069695C214yesmaximal, complemented
8#13068312C211yesmaximal, characteristic, complemented
9C2×C11×#416C211yesmaximal, characteristic, complemented
10C22×C11×#41C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C23×Dic22C2211yescharacteristic
13C23×C8×C11C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15#10623356—24no—
16#10623356—24no—
17C2×#67002—24no—
18C2×#67002—24no—
19C2×#67002—24no—
20C2×#66997C2212yescomplemented
21C22×#4325C2212yes—
22C2×#66989C2212yescomplemented
23C2×#66988—22no—
24C22×#4323C2212yes—
25C2×#66979C2211yescharacteristic
26C2×#66977—22no—
27C22×#4318C2211yescharacteristic
28C22×#4314—24no—
29C22×#4314—24no—
30C22×#4314—24no—
31C22×#4314C2214yes—
32C11×#416—24no—
33C2×C11×#103C2212yescomplemented
34C2×C11×#102C2212yescomplemented
35#10622617—24no—
36#10622601C2214yes—
37C2×#66874C2214yescomplemented
38#10622568—22no—
39#10622552—22no—
40#10622188C2214yescomplemented
41C24×D11—21no—
42C2×C11×#41—22no—
43C2×C11×#41—22no—
44C2×C11×#41—22no—
45C2×C11×#41—22no—
46C2×C11×#41—22no—
47C2×C11×#41D414yes—
48C22×Q8×C11C2311yescharacteristic
49C22×D4×C11—21no—
50C22×D4×C11C2311yescharacteristic
51C23×C4×C11—21no—
52C23×C4×C11C2311yescharacteristic
53C23×Dic11—21no—
54C22×D44—22no—
55C22×D44—22no—
56C22×D44—22no—
57C22×D44—22no—
58C22×D44—21no—
59C22×D44—22no—
60C22×D44D414yes—
61C22×D44C2312yes—
62C22×Dic22—22no—
63C22×Dic22—22no—
64C22×Dic22—22no—
65C22×Dic22—22no—
66C22×Dic22—21no—
67C22×Dic22—22no—
68C22×Dic22D414yes—
69C22×Dic22C2312yes—
70C22×C8×C11—22no—
71C22×C8×C11—22no—
72C22×C8×C11D414yes—
73C22×C8×C11C2312yes—
74C11×#117—22no—
75C11×#117—22no—
76C2×C11×C4⋊C4C2312yes—
77C2×C11×C22⋊C4—21no—
78C2×C42×C11—21no—
79C2×C22⋊Dic11—24no—
80C2×C22⋊Dic11D412yes—
81#67002—232no—
82#66997—24no—
83#66997—24no—
84C2×#4325—24no—
85C2×#4325—24no—
86C2×#4325—24no—
87#66989—24no—
88#66989—24no—
89#66988—216no—
90C2×#4323—24no—
91C2×#4323—24no—
92C2×#4323—24no—
93C2×(C2×D11)⋊C4—28no—
94#66979—22no—
95#66979—22no—
96C2×C4⋊Dic11D412yes—
97#66977—216no—
98C2×Dic11⋊C4—24no—
99C2×C4×Dic11C2312yes—
100C2×#4318—22no—
101C2×#4318—22no—
102C2×#4318D414yes—
103C22×C11⋊C8C2312yes—
104C2×#4314—232no—
105C2×#4314—28no—
106C2×#4314—28no—
107C2×#4314—28no—
108C11×#103—24no—
109C11×#102—24no—
110#66874—28no—
111#23424—111nomaximal, Sylow
112C24×C11—21no—
113C24×C11C2×D411yescharacteristic
114C23×D11—22no—
115C23×D11—24no—
116C23×D11—42no—
117C23×D11—42no—
118C23×D11—84no—
119C11×#41—28no—
120C11×#41—28no—
121C11×#41—44no—
122C11×#41—44no—
123C2×Q8×C11—21no—
124C2×Q8×C11—21no—
125C2×Q8×C11—21no—
126C2×Q8×C11—21no—
127C2×Q8×C11—21no—
128C2×Q8×C11C2×D412yes—
129C2×D4×C11—24no—
130C2×D4×C11—21no—
131C2×D4×C11—21no—
132C2×D4×C11—21no—
133C2×D4×C11—21no—
134C2×D4×C11—21no—
135C2×D4×C11—42no—
136C2×D4×C11—42no—
137C2×D4×C11C2×D412yes—
138C22×C4×C11—21no—
139C22×C4×C11—24no—
140C22×C4×C11—22no—
141C22×C4×C11—42no—
142C22×C4×C11—42no—
143C22×C4×C11C2×D412yes—
144C22×C4×C11C2×D412yes—
145C22×C4×C11C2×D412yes—
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2×D412yes—
148C22×C4×C11C2×D412yes—
149C22×C4×C11#4112yes—
150C22×C4×C11C2411yescharacteristic
151C22×Dic11—24no—
152C22×Dic11—22no—
153C22×Dic11—42no—
154C22×Dic11—42no—
155C22×Dic11C2×D414yes—
156C2×D44—28no—
157C2×D44—28no—
158C2×D44—24no—
159C2×D44—22no—
160C2×D44—22no—
161C2×D44—22no—
162C2×D44—22no—
163C2×D44—22no—
164C2×D44—42no—
165C2×D44—44no—
166C2×D44—44no—
167C2×D44—42no—
168C2×D44C2×D414yes—
169C2×Dic22—28no—
170C2×Dic22—24no—
171C2×Dic22—28no—
172C2×Dic22—22no—
173C2×Dic22—22no—
174C2×Dic22—22no—
175C2×Dic22—22no—
176C2×Dic22—22no—
177C2×Dic22—44no—
178C2×Dic22—42no—
179C2×Dic22—44no—
180C2×Dic22—42no—
181C2×Dic22C2×D414yes—
182C2×C8×C11—28no—
183C2×C8×C11—22no—
184C2×C8×C11—22no—
185C2×C8×C11C2×D414yes—
186C11×C4⋊C4—22no—
187C11×C4⋊C4—22no—
188C11×C22⋊C4—42no—
189C42×C11—42no—
190C22⋊Dic11—216no—
191C22⋊Dic11—24no—
192#4325—216no—
193#4323—216no—
194(C2×D11)⋊C4—88no—
195C4⋊Dic11—24no—
196Dic11⋊C4—216no—
197C4×Dic11—22no—
198C4×Dic11—22no—
199#4318—28no—
200C2×C11⋊C8—22no—
201C2×C11⋊C8—22no—
202C2×C11⋊C8C2×D414yes—
203#4314—232no—
204C22×AGL(1,ℤ/8)—111no—
205C23×SD16—111no—
206C2×#536—111no—
207C2×#419—111no—
208C2×#416D1111yescharacteristic, complemented
209#2309—111no—
210#2013—114no—
211#2012—114no—
212#1729—111no—
213C23×C11—24no—
214C23×C11—22no—
215C23×C11—41no—
216C23×C11—41no—
217C23×C11—44no—
218C23×C11#12012yes—
219C23×C11#12112yes—
220C23×C11#12712yes—
221C23×C11C22×D411yescharacteristic
222C22×D11—24no—
223C22×D11—42no—
224C22×D11—42no—
225C22×D11—48no—
226C22×D11—48no—
227C22×D11—88no—
228C22×D11—88no—
229C22×D11—88no—
230Q8×C11—22no—
231Q8×C11—22no—
232Q8×C11—41no—
233Q8×C11—41no—
234D4×C11—22no—
235D4×C11—22no—
236D4×C11—48no—
237D4×C11—41no—
238D4×C11—41no—
239C2×C4×C11—22no—
240C2×C4×C11—41no—
241C2×C4×C11—41no—
242C2×C4×C11—48no—
243C2×C4×C11—42no—
244C2×C4×C11—44no—
245C2×C4×C11#12014yes—
246C2×C4×C11#12012yes—
247C2×C4×C11#12014yes—
248C2×C4×C11#12114yes—
249C2×C4×C11#12112yes—
250C2×C4×C11#12114yes—
251C2×C4×C11#12712yes—
252C2×C4×C11C22×D411yescharacteristic
253C2×C4×C11C22×D411yescharacteristic
254C2×C4×C11C22×D411yescharacteristic
255C2×C4×C11C22×D411yescharacteristic
256C2×C4×C11C22×D411yescharacteristic
257C2×C4×C11C2×#4111yescharacteristic
258C2×Dic11—24no—
259C2×Dic11—24no—
260C2×Dic11—28no—
261C2×Dic11—42no—
262C2×Dic11—42no—
263C2×Dic11—48no—
264D44—24no—
265D44—24no—
266D44—42no—
267D44—42no—
268D44—48no—
269Dic22—24no—
270Dic22—24no—
271Dic22—42no—
272Dic22—42no—
273Dic22—48no—
274C8×C11—24no—
275C11⋊C8—24no—
276C22×#41C2×D1111yescharacteristic
277C23×Q8—111no—
278C23×D4—111no—
279C2×AGL(1,ℤ/8)—224no—
280C2×AGL(1,ℤ/8)—224no—
281C2×AGL(1,ℤ/8)—224no—
282C22×SD16—112no—
283C22×SD16—114no—
284C22×SD16—224no—
285C22×SD16—224no—
286C22×SD16—224no—
287C22×D8—112no—
288C22×M16—111no—
289C23×C8C2×D1111yescharacteristic
290#536—224no—
291#536—224no—
292C2×C4⋊Q8—112no—
293C2×#124—112no—
294C2×#123—222no—
295C2×C22⋊Q8—222no—
296C2×#117—111no—
297C2×#117C2×D1111yescharacteristic
298#419—224no—
299#416—24no—
300C2×#103—114no—
301C2×#103C2×D1112yescomplemented
302C2×#102C2×D1112yescomplemented
303#390—114no—
304#390—222no—
305#379—222no—
306#336—114no—
307C22×C11—24no—
308C22×C11—24no—
309C22×C11—24no—
310C22×C11—42no—
311C22×C11—44no—
312C22×C11—82no—
313C22×C11—82no—
314C22×C11#39214yes—
315C22×C11C2×#12011yescharacteristic
316C22×C11C2×#12111yescharacteristic
317C22×C11C2×#12711yescharacteristic
318C22×C11#57714yes—
319C2×D11—44no—
320C2×D11—48no—
321C2×D11—816no—
322C2×D11—84no—
323C2×D11—84no—
324C4×C11—42no—
325C4×C11—82no—
326C4×C11#39214yes—
327C4×C11C2×#12011yescharacteristic
328C4×C11C2×#12011yescharacteristic
329C4×C11C2×#12111yescharacteristic
330C4×C11C2×#12111yescharacteristic
331Dic11—44no—
332Dic11—44no—
333C25—221no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41D4414yes—
340C22×Q8—114no—
341C22×Q8—112no—
342C22×Q8—222no—
343C22×Q8—221no—
344C22×Q8—222no—
345C22×Q8—222no—
346C22×Q8—222no—
347C22×Q8—222no—
348C22×Q8C22×D1111yescharacteristic
349C22×D4—21no—
350C22×D4—114no—
351C22×D4—112no—
352C22×D4—222no—
353C22×D4—222no—
354C22×D4—222no—
355C22×D4—222no—
356C22×D4—221no—
357C22×D4—222no—
358C22×D4C22×D1111yescharacteristic
359C23×C4—21no—
360C23×C4—221no—
361C23×C4C22×D1111yescharacteristic
362AGL(1,ℤ/8)—2232no—
363C2×SD16—228no—
364C2×SD16—228no—
365C2×SD16—2232no—
366C2×SD16—224no—
367C2×SD16—224no—
368C2×SD16—224no—
369C2×SD16—228no—
370C2×D8—224no—
371C2×D8—224no—
372C2×D8—224no—
373C2×M16—114no—
374C2×M16—222no—
375C2×M16—222no—
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—112no—
379C22×C8C22⋊D1114yes—
380C22×C8C22×D1112yes—
381C4⋊Q8—224no—
382C4⋊Q8—224no—
383#124—224no—
384#124—224no—
385#123—2216no—
386C22⋊Q8—2216no—
387#117—22no—
388#117—22no—
389#117—222no—
390#117—222no—
391C2×C4⋊C4—112no—
392C2×C4⋊C4—224no—
393C2×C4⋊C4C22×D1112yes—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—112no—
396C2×C22⋊C4—224no—
397C2×C22⋊C4—228no—
398C2×C42—21no—
399C2×C42—112no—
400#103—24no—
401#103—228no—
402#102—24no—
403C2×C11—24no—
404C2×C11—82no—
405C2×C11C2×#39211yescharacteristic
406C2×C11#301012yes—
407C2×C11#304212yes—
408C2×C11#315412yes—
409D11—84no—
410C24—21no—
411C24—222no—
412C24—224no—
413C24—442no—
414C24—442no—
415C24—884no—
416C24D4×D1111yescharacteristic
417#41—28no—
418#41—28no—
419#41—44no—
420#41—44no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—114no—
427C2×Q8—224no—
428C2×Q8—222no—
429C2×Q8—228no—
430C2×Q8—228no—
431C2×Q8—222no—
432C2×Q8—222no—
433C2×Q8—222no—
434C2×Q8—222no—
435C2×Q8—444no—
436C2×Q8—442no—
437C2×Q8—444no—
438C2×Q8—442no—
439C2×Q8C2×D4412yes—
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—114no—
449C2×D4—228no—
450C2×D4—224no—
451C2×D4—228no—
452C2×D4—222no—
453C2×D4—222no—
454C2×D4—222no—
455C2×D4—222no—
456C2×D4—222no—
457C2×D4—442no—
458C2×D4—444no—
459C2×D4—444no—
460C2×D4—442no—
461C2×D4C2×D4412yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—114no—
468C22×C4—224no—
469C22×C4—222no—
470C22×C4—442no—
471C22×C4—442no—
472C22×C4C2×D4412yes—
473C22×C4#433912yes—
474C22×C4D4×D1111yescharacteristic
475C22×C4D4×D1112yes—
476C22×C4D4×D1112yes—
477C22×C4D4×D1112yes—
478C22×C4C2×C22⋊D1112yes—
479C22×C4C23×D1111yescharacteristic
480SD16—2232no—
481SD16—2216no—
482D8—2216no—
483M16—228no—
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—114no—
488C2×C8—222no—
489C2×C8—222no—
490C2×C8C2×C22⋊D1114yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—2216no—
494C4⋊C4—224no—
495C22⋊C4—42no—
496C22⋊C4—2216no—
497C22⋊C4—224no—
498C22⋊C4—888no—
499C42—42no—
500C42—222no—
501C42—222no—
502C11#2342411yescharacteristic, complemented, Sylow
503C23—24no—
504C23—22no—
505C23—41no—
506C23—44no—
507C23—41no—
508C23—224no—
509C23—442no—
510C23—448no—
511C23—442no—
512C23—448no—
513C23—888no—
514C23—888no—
515C23—888no—
516C23#6693512yes—
517C23#6693712yes—
518C23#6699312yes—
519C23C2×D4×D1111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—224no—
525Q8—224no—
526Q8—442no—
527Q8—442no—
528Q8—448no—
529D4—22no—
530D4—22no—
531D4—48no—
532D4—41no—
533D4—41no—
534D4—224no—
535D4—224no—
536D4—442no—
537D4—442no—
538D4—448no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—224no—
546C2×C4—224no—
547C2×C4—228no—
548C2×C4—442no—
549C2×C4—442no—
550C2×C4—448no—
551C2×C4#6693514yes—
552C2×C4#6693512yes—
553C2×C4#6693714yes—
554C2×C4#6693712yes—
555C2×C4#6698314yes—
556C2×C4#6699014yes—
557C2×C4#6699312yes—
558C2×C4C22×D4411yescharacteristic
559C2×C4C2×#433911yescharacteristic
560C2×C4C2×D4×D1111yescharacteristic
561C2×C4C2×D4×D1111yescharacteristic
562C2×C4C2×D4×D1111yescharacteristic
563C2×C4C22×C22⋊D1111yescharacteristic
564C8—24no—
565C8—224no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—44no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—448no—
574C22—444no—
575C22—8816no—
576C22—884no—
577C22—884no—
578C22#1062257314yes—
579C22C2×#6693511yescharacteristic
580C22C2×#6693711yescharacteristic
581C22#1062330114yes—
582C22C2×#6699311yescharacteristic
583C4—42no—
584C4—82no—
585C4—444no—
586C4—444no—
587C4#1062257314yes—
588C4C2×#6693511yescharacteristic
589C4C2×#6693711yescharacteristic
590C4C2×#6698311yescharacteristic
591C4C2×#6699011yescharacteristic
592C2—24no—
593C2—82no—
594C2—884no—
595C2C2×#1062257311yescharacteristic
596C2#1307790512yes—
597C2#1307813112yes—
598C2#1307909312yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#416C2
socleC23×C11C22×D4
prime radicalC22×#67002C2
hypercenterC2×#416D11
hypocenterC11#23424
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C23×#4314C2semidirect
3. C2×#10623356C2semidirect
4. C2×#10622617C2semidirect
5. #13071196C2semidirect
6. #13069700C2semidirect
7. #13069695C2semidirect
8. #13068312C2semidirect
9. C2×C11×#416C2semidirect
20. C2×#66997C22semidirect
22. C2×#66989C22semidirect
33. C2×C11×#103C22semidirect
34. C2×C11×#102C22semidirect
37. C2×#66874C22semidirect
40. #10622188C22semidirect
208. C2×#416D11semidirect
301. C2×#103C2×D11semidirect
302. C2×#102C2×D11semidirect
502. C11#23424semidirect

Maximal subgroups

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

subgroupindex
C22×#670022
C23×#43142
C2×#106233562
C2×#106226172
#130711962
#130697002
#130696952
#130683122
C2×C11×#4162
#2342411