Finite Group Databasebeta

← #14780933

Subgroups — #14780933

Order 2816.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67004C211yesmaximal, characteristic, complemented
2C22×#66989C211yesmaximal, characteristic, complemented
3C2×#10622733C211yesmaximal, characteristic, complemented
4C2×#10622828C211yesmaximal, characteristic, complemented
5#13071107C211yesmaximal, characteristic, complemented
6#13069708C214yesmaximal, complemented
7#13069691C214yesmaximal, complemented
8#13068309C211yesmaximal, characteristic, complemented
9C22×C11×AGL(1,ℤ/8)C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C23×D4×C11C2211yescharacteristic
12C23×Dic22C2211yescharacteristic
13C2×C11×AGL(1,ℤ/8)—24no—
14C2×C11×AGL(1,ℤ/8)—24no—
15C2×C11×AGL(1,ℤ/8)—24no—
16C22×C11×SD16C2212yes—
17C22×C11×D8C2212yes—
18C22×C11×M16C2211yescharacteristic
19C22×C22⋊Dic11—22no—
20C2×#67004—24no—
21C2×#67004—24no—
22C2×#67004—24no—
23C22×#4326C2212yescomplemented
24C2×#66989—24no—
25C2×#66989—24no—
26C2×#66989—22no—
27C2×#66989C2214yes—
28C22×#4324C2212yescomplemented
29C2×#66979C2211yescharacteristic
30C22×C4×Dic11C2211yescharacteristic
31C22×#4318C2211yescharacteristic
32#10622843C2212yescomplemented
33#10622834C2212yescomplemented
34#10622833—22no—
35#10622828—24no—
36C2×#66896C2214yes—
37C4×C22⋊Dic11—22no—
38#10622733—24no—
39C2×#66880C2214yescomplemented
40#10622188C2214yescomplemented
41C25×C11—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×C11—22no—
51C22×D4×C11—22no—
52C22×D4×C11—22no—
53C22×D4×C11—22no—
54C22×D4×C11—21no—
55C22×D4×C11—22no—
56C22×D4×C11D414yes—
57C22×D4×C11C2311yescharacteristic
58C22×D4×C11C2312yes—
59C23×C4×C11—21no—
60C23×C4×C11C2311yescharacteristic
61C23×Dic11—21no—
62C23×Dic11D412yes—
63C22×Dic22—22no—
64C22×Dic22—22no—
65C22×Dic22—22no—
66C22×Dic22—22no—
67C22×Dic22—21no—
68C22×Dic22—22no—
69C22×Dic22D414yes—
70C22×Dic22C2312yes—
71C11×AGL(1,ℤ/8)—232no—
72C2×C11×SD16—24no—
73C2×C11×SD16—24no—
74C2×C11×SD16—24no—
75C2×C11×D8—24no—
76C2×C11×D8—24no—
77C2×C11×D8—24no—
78C2×C11×M16—22no—
79C2×C11×M16—22no—
80C2×C11×M16D414yes—
81C22×C8×C11C2312yes—
82C2×C22⋊Dic11—216no—
83C2×C22⋊Dic11—28no—
84C2×C22⋊Dic11—21no—
85C2×C22⋊Dic11—24no—
86#67004—232no—
87C2×#4326—24no—
88C2×#4326—24no—
89C2×#4326—24no—
90#66989—216no—
91#66989—28no—
92#66989—28no—
93C2×#4324—24no—
94C2×#4324—24no—
95C2×#4324—24no—
96#66979—22no—
97#66979—22no—
98C2×C4⋊Dic11C2312yes—
99C2×C4×Dic11—22no—
100C2×C4×Dic11—22no—
101C2×C4×Dic11—21no—
102C2×C4×Dic11D412yes—
103C2×C4×Dic11C2312yes—
104C2×#4318—22no—
105C2×#4318—22no—
106C2×#4318D414yes—
107C22×C11⋊C8C2312yes—
108#66896—28no—
109C11⋊#95—24no—
110#66880—28no—
111#23431—111nomaximal, Sylow
112C24×C11—24no—
113C24×C11—21no—
114C24×C11—22no—
115C24×C11—48no—
116C24×C11—42no—
117C24×C11—42no—
118C24×C11C2×D411yescharacteristic
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—28no—
131C2×D4×C11—24no—
132C2×D4×C11—28no—
133C2×D4×C11—22no—
134C2×D4×C11—21no—
135C2×D4×C11—21no—
136C2×D4×C11—21no—
137C2×D4×C11—21no—
138C2×D4×C11—22no—
139C2×D4×C11—22no—
140C2×D4×C11—22no—
141C2×D4×C11—22no—
142C2×D4×C11—21no—
143C2×D4×C11—42no—
144C2×D4×C11—42no—
145C2×D4×C11—44no—
146C2×D4×C11—44no—
147C2×D4×C11—42no—
148C2×D4×C11—42no—
149C2×D4×C11C2×D412yes—
150C2×D4×C11C2×D414yes—
151C22×C4×C11—21no—
152C22×C4×C11—24no—
153C22×C4×C11—42no—
154C22×C4×C11—42no—
155C22×C4×C11C2×D412yes—
156C22×C4×C11C2×D412yes—
157C22×C4×C11C2×D412yes—
158C22×C4×C11C2×D411yescharacteristic
159C22×C4×C11C2×D412yes—
160C22×C4×C11C2×D412yes—
161C22×C4×C11#4112yes—
162C22×C4×C11C2411yescharacteristic
163C22×Dic11—24no—
164C22×Dic11—24no—
165C22×Dic11—22no—
166C22×Dic11—22no—
167C22×Dic11—44no—
168C22×Dic11—42no—
169C22×Dic11—42no—
170C22×Dic11C2×D414yes—
171C2×Dic22—28no—
172C2×Dic22—24no—
173C2×Dic22—28no—
174C2×Dic22—22no—
175C2×Dic22—22no—
176C2×Dic22—22no—
177C2×Dic22—22no—
178C2×Dic22—22no—
179C2×Dic22—44no—
180C2×Dic22—44no—
181C2×Dic22—42no—
182C2×Dic22—42no—
183C2×Dic22C2×D414yes—
184C11×SD16—216no—
185C11×D8—216no—
186C11×M16—28no—
187C2×C8×C11—22no—
188C2×C8×C11—22no—
189C2×C8×C11C2×D414yes—
190C22⋊Dic11—232no—
191C22⋊Dic11—48no—
192C22⋊Dic11—42no—
193#4326—216no—
194#4324—216no—
195C4⋊Dic11—22no—
196C4⋊Dic11—22no—
197C4×Dic11—24no—
198C4×Dic11—22no—
199C4×Dic11—22no—
200C4×Dic11—42no—
201#4318—28no—
202C2×C11⋊C8—22no—
203C2×C11⋊C8—22no—
204C2×C11⋊C8C2×D414yes—
205C22×#137—111no—
206C22×AGL(1,ℤ/8)D1111yescharacteristic
207C22×#124—111no—
208C2×#419—111no—
209C2×#418—111no—
210#2297—111no—
211#2015—114no—
212#2012—114no—
213#1728—111no—
214C23×C11—24no—
215C23×C11—24no—
216C23×C11—22no—
217C23×C11—416no—
218C23×C11—41no—
219C23×C11—42no—
220C23×C11—41no—
221C23×C11—42no—
222C23×C11—44no—
223C23×C11—48no—
224C23×C11—48no—
225C23×C11—88no—
226C23×C11—88no—
227C23×C11#12012yes—
228C23×C11#12112yes—
229C23×C11#12712yes—
230C23×C11C22×D411yescharacteristic
231Q8×C11—22no—
232Q8×C11—22no—
233Q8×C11—41no—
234Q8×C11—41no—
235D4×C11—22no—
236D4×C11—22no—
237D4×C11—24no—
238D4×C11—24no—
239D4×C11—48no—
240D4×C11—42no—
241D4×C11—42no—
242D4×C11—41no—
243D4×C11—41no—
244D4×C11—48no—
245C2×C4×C11—22no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—48no—
249C2×C4×C11#12014yes—
250C2×C4×C11#12012yes—
251C2×C4×C11#12014yes—
252C2×C4×C11#12114yes—
253C2×C4×C11#12114yes—
254C2×C4×C11#12112yes—
255C2×C4×C11#12712yes—
256C2×C4×C11C22×D411yescharacteristic
257C2×C4×C11C22×D411yescharacteristic
258C2×C4×C11C22×D411yescharacteristic
259C2×C4×C11C22×D411yescharacteristic
260C2×C4×C11C22×D411yescharacteristic
261C2×C4×C11C2×#4111yescharacteristic
262C2×Dic11—24no—
263C2×Dic11—24no—
264C2×Dic11—24no—
265C2×Dic11—24no—
266C2×Dic11—48no—
267C2×Dic11—42no—
268C2×Dic11—42no—
269C2×Dic11—48no—
270C2×Dic11—42no—
271C2×Dic11—44no—
272Dic22—24no—
273Dic22—24no—
274Dic22—42no—
275Dic22—42no—
276Dic22—48no—
277C8×C11—24no—
278C11⋊C8—24no—
279C22×#41C2×D1111yescharacteristic
280C23×Q8—111no—
281C23×D4C2×D1111yescharacteristic
282C2×#137—224no—
283C2×#137—224no—
284C2×#137—224no—
285C2×AGL(1,ℤ/8)—24no—
286C2×AGL(1,ℤ/8)—24no—
287C2×AGL(1,ℤ/8)—24no—
288C22×Q16—112no—
289C22×SD16—112no—
290C22×SD16C2×D1112yes—
291C22×D8C2×D1112yes—
292C22×M16—111no—
293C22×M16C2×D1111yescharacteristic
294C2×#124—114no—
295C2×#124—222no—
296C2×#124—224no—
297C2×#124—224no—
298C2×#117—111no—
299C22×C22⋊C4—222no—
300C22×C42—111no—
301#419—224no—
302#418—224no—
303C2×#103—114no—
304C2×#102—114no—
305#391—112no—
306#390—112no—
307#386—222no—
308C4×C22⋊C4—222no—
309#336—114no—
310C22×C11—24no—
311C22×C11—24no—
312C22×C11—24no—
313C22×C11—42no—
314C22×C11—44no—
315C22×C11—48no—
316C22×C11—44no—
317C22×C11—816no—
318C22×C11—82no—
319C22×C11—84no—
320C22×C11—82no—
321C22×C11—84no—
322C22×C11#39214yes—
323C22×C11C2×#12011yescharacteristic
324C22×C11C2×#12111yescharacteristic
325C22×C11C2×#12711yescharacteristic
326C22×C11#57514yes—
327C4×C11—42no—
328C4×C11#39214yes—
329C4×C11C2×#12011yescharacteristic
330C4×C11C2×#12011yescharacteristic
331C4×C11C2×#12111yescharacteristic
332C4×C11C2×#12111yescharacteristic
333Dic11—44no—
334Dic11—44no—
335Dic11—82no—
336C25—21no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41C22⋊D1114yes—
343C22×Q8—114no—
344C22×Q8—112no—
345C22×Q8—221no—
346C22×Q8—222no—
347C22×Q8—222no—
348C22×Q8—222no—
349C22×Q8—222no—
350C22×Q8—222no—
351C22×Q8C22×D1111yescharacteristic
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—21no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4C22⋊D1114yes—
360C22×D4C22×D1112yes—
361C22×D4C22×D1111yescharacteristic
362C23×C4—21no—
363C23×C4—112no—
364C23×C4—221no—
365C23×C4C22×D1111yescharacteristic
366#137—2232no—
367AGL(1,ℤ/8)—232no—
368C2×Q16—224no—
369C2×Q16—224no—
370C2×Q16—224no—
371C2×SD16—24no—
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16—224no—
375C2×SD16—224no—
376C2×SD16—224no—
377C2×D8—24no—
378C2×D8—24no—
379C2×D8—24no—
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—114no—
383C2×M16—222no—
384C2×M16—222no—
385C2×M16C22⋊D1114yes—
386C22×C8—112no—
387C22×C8C22×D1112yes—
388#124—228no—
389#124—2216no—
390#124—228no—
391#117—222no—
392#117—222no—
393C2×C4⋊C4—112no—
394C2×C22⋊C4—228no—
395C2×C22⋊C4—2216no—
396C2×C22⋊C4—224no—
397C2×C22⋊C4—221no—
398C2×C42—112no—
399C2×C42—112no—
400C2×C42—222no—
401C2×C42—222no—
402C2×C42—221no—
403#103—228no—
404#102—228no—
405#95—224no—
406C2×C11—24no—
407C2×C11—82no—
408C2×C11—84no—
409C2×C11C2×#39211yescharacteristic
410C2×C11#300512yes—
411C2×C11#304012yes—
412C2×C11#315312yes—
413C24—24no—
414C24—21no—
415C24—22no—
416C24—48no—
417C24—42no—
418C24—42no—
419C24D4×D1111yescharacteristic
420#41—28no—
421#41—28no—
422#41—44no—
423#41—44no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—114no—
430C2×Q8—224no—
431C2×Q8—222no—
432C2×Q8—228no—
433C2×Q8—228no—
434C2×Q8—222no—
435C2×Q8—222no—
436C2×Q8—222no—
437C2×Q8—222no—
438C2×Q8—442no—
439C2×Q8—444no—
440C2×Q8—442no—
441C2×Q8—444no—
442C2×Q8C2×C22⋊D1112yes—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4C2×C22⋊D1114yes—
464C2×D4C2×C22⋊D1112yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—114no—
470C22×C4—224no—
471C22×C4—224no—
472C22×C4—222no—
473C22×C4—222no—
474C22×C4—444no—
475C22×C4—442no—
476C22×C4—442no—
477C22×C4D4×D1112yes—
478C22×C4D4×D1111yescharacteristic
479C22×C4D4×D1112yes—
480C22×C4#434112yes—
481C22×C4C2×C22⋊D1112yes—
482C22×C4C2×C22⋊D1112yes—
483C22×C4C2×C22⋊D1112yes—
484C22×C4C23×D1111yescharacteristic
485Q16—2216no—
486SD16—216no—
487SD16—2216no—
488D8—216no—
489M16—28no—
490M16—228no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—114no—
494C2×C8—222no—
495C2×C8—222no—
496C2×C8C2×C22⋊D1114yes—
497C4⋊C4—222no—
498C4⋊C4—222no—
499C22⋊C4—2232no—
500C22⋊C4—448no—
501C22⋊C4—442no—
502C42—224no—
503C42—222no—
504C42—222no—
505C42—442no—
506C11#2343111yescharacteristic, complemented, Sylow
507C23—24no—
508C23—24no—
509C23—22no—
510C23—416no—
511C23—44no—
512C23—48no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—48no—
518C23—88no—
519C23—88no—
520C23#6699012yes—
521C23#6699212yes—
522C23#6699312yes—
523C23C2×D4×D1111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528Q8—224no—
529Q8—224no—
530Q8—442no—
531Q8—448no—
532Q8—442no—
533D4—22no—
534D4—24no—
535D4—22no—
536D4—24no—
537D4—48no—
538D4—42no—
539D4—42no—
540D4—41no—
541D4—41no—
542D4—48no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—224no—
548C2×C4—224no—
549C2×C4—224no—
550C2×C4—224no—
551C2×C4—448no—
552C2×C4—442no—
553C2×C4—442no—
554C2×C4—448no—
555C2×C4—444no—
556C2×C4—442no—
557C2×C4#6699014yes—
558C2×C4#6699012yes—
559C2×C4#6699114yes—
560C2×C4#6699214yes—
561C2×C4#6699212yes—
562C2×C4#6699312yes—
563C2×C4#6700614yes—
564C2×C4C2×D4×D1111yescharacteristic
565C2×C4C2×D4×D1111yescharacteristic
566C2×C4C2×#434111yescharacteristic
567C2×C4C22×C22⋊D1111yescharacteristic
568C2×C4C22×C22⋊D1111yescharacteristic
569C2×C4C22×C22⋊D1111yescharacteristic
570C8—24no—
571C8—224no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—44no—
576C22—48no—
577C22—42no—
578C22—44no—
579C22—816no—
580C22—84no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22#1062283614yes—
585C22#1062330414yes—
586C22C2×#6699011yescharacteristic
587C22C2×#6699211yescharacteristic
588C22C2×#6699311yescharacteristic
589C4—42no—
590C4—444no—
591C4—444no—
592C4—882no—
593C4#1062283614yes—
594C4C2×#6699011yescharacteristic
595C4C2×#6699111yescharacteristic
596C4C2×#6699211yescharacteristic
597C4C2×#6700611yescharacteristic
598C2—24no—
599C2—84no—
600C2—82no—
601C2C2×#1062283611yescharacteristic
602C2#1307785512yes—
603C2#1307810812yes—
604C2#1307907812yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC22×C11×AGL(1,ℤ/8)C2
socleC23×C11C22×D4
hypercenterC22×AGL(1,ℤ/8)D11
hypocenterC11#23431
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#67004C2semidirect
2. C22×#66989C2semidirect
3. C2×#10622733C2semidirect
4. C2×#10622828C2semidirect
5. #13071107C2semidirect
6. #13069708C2semidirect
7. #13069691C2semidirect
8. #13068309C2semidirect
23. C22×#4326C22semidirect
28. C22×#4324C22semidirect
32. #10622843C22semidirect
33. #10622834C22semidirect
39. C2×#66880C22semidirect
40. #10622188C22semidirect
506. C11#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#670042
C22×#669892
C2×#106227332
C2×#106228282
#130711072
#130697082
#130696912
#130683092
C22×C11×AGL(1,ℤ/8)2
#2343111