Finite Group Databasebeta

← #14754781

Subgroups — #14754781

Order 2816.

1458 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
1C2×#10623358C211yesmaximal, characteristic
2C2×#10622734C211yesmaximal, characteristic, complemented
3C2×#10622846C211yesmaximal, characteristic
4#13071234C211yesmaximal, characteristic, complemented
5#13068886C212yesmaximal, complemented
6#13068879C212yesmaximal, complemented
7#13068872C212yesmaximal, complemented
8#13068865C212yesmaximal, complemented
9C2×#10622167C211yesmaximal, characteristic, complemented
10#13068314C211yesmaximal, characteristic, complemented
11C2×C11×#417C211yesmaximal, characteristic
12C22×C11×#41C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15#10623358—24no—
16#10623358—24no—
17C2×Q8×Dic11C2212yes—
18C2×D4×Dic11—22no—
19C2×D4×Dic11C2212yes—
20C2×#66979—21no—
21C2×#66979C2211yescharacteristic
22C2×#66979C2211yescharacteristic
23C22×C4⋊Dic11—21no—
24C22×C4⋊Dic11C2211yescharacteristic
25C23×C11⋊C8C2211yescharacteristic
26C11×#417C418yescomplemented
27C2×C11×#103C2212yes—
28C2×C11×#102C2212yes—
29#10622846—24no—
30C2×#66899C2211yescharacteristic
31C2×#66899C2211yescharacteristic
32C2×#66896C2211yescharacteristic
33C2×#66896C2211yescharacteristic
34#10622825—22no—
35#10622734—24no—
36C2×#66882C2212yes—
37C2×#66881C2212yes—
38#10622598—21no—
39#10622598—21no—
40#10622598C2212yes—
41#10622598C2212yes—
42#10622188C2212yes—
43#10622188C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C2×C11×#41—22no—
47C2×C11×#41—22no—
48C2×C11×#41—22no—
49C2×C11×#41C2×C414yes—
50C2×C11×#41D414yes—
51C2×C11×#41Q814yes—
52C22×Q8×C11C2311yescharacteristic
53C22×D4×C11—21no—
54C22×D4×C11C2311yescharacteristic
55C23×C4×C11—21no—
56C23×C4×C11—21no—
57C23×C4×C11C2311yescharacteristic
58C23×Dic11—21no—
59C23×Dic11—22no—
60C2×C11×M16—22no—
61C2×C11×M16—22no—
62C2×C11×M16C2×C414yescomplemented
63C22×C8×C11C2312yes—
64C2×C11×C4⋊C4—21no—
65C2×C11×C4⋊C4—22no—
66C2×C11×C4⋊C4C2×C414yescomplemented
67C2×C11×C4⋊C4C2312yes—
68C2×C22⋊Dic11—22no—
69C2×C22⋊Dic11—21no—
70C2×C22⋊Dic11D412yes—
71Q8×Dic11—24no—
72Q8×Dic11—24no—
73D4×Dic11—216no—
74D4×Dic11—24no—
75D4×Dic11—24no—
76#66979—28no—
77#66979—22no—
78#66979—22no—
79#66979—22no—
80#66979—22no—
81C2×C4⋊Dic11—28no—
82C2×C4⋊Dic11—21no—
83C2×C4⋊Dic11—22no—
84C2×C4⋊Dic11—22no—
85C2×C4⋊Dic11—22no—
86C2×C4⋊Dic11—22no—
87C2×C4⋊Dic11D411yescharacteristic
88C2×C4⋊Dic11D411yescharacteristic
89C2×C4⋊Dic11D414yes—
90C2×C4⋊Dic11C2311yescharacteristic
91C2×C4⋊Dic11C2311yescharacteristic
92C2×C4⋊Dic11C2312yes—
93C2×C4×Dic11—22no—
94C2×C4×Dic11—21no—
95C2×C4×Dic11C2312yes—
96C22×C11⋊C8—22no—
97C22×C11⋊C8—22no—
98C22×C11⋊C8—22no—
99C22×C11⋊C8C2311yescharacteristic
100C22×C11⋊C8C2311yescharacteristic
101C11×#103C2×C418yescomplemented
102C11×#102C2×C418yescomplemented
103#66899—22no—
104#66899—22no—
105#66896—22no—
106#66896—22no—
107C11⋊#95—22no—
108C11⋊#95—22no—
109#66882—24no—
110#66881—24no—
111#20878—111nomaximal, Sylow
112C24×C11—21no—
113C24×C11C2×D411yescharacteristic
114C11×#41—28no—
115C11×#41—28no—
116C11×#41—28no—
117C11×#41C4⋊C4116yescomplemented
118C2×Q8×C11—21no—
119C2×Q8×C11—21no—
120C2×Q8×C11—21no—
121C2×Q8×C11C22×C412yes—
122C2×Q8×C11C2×D412yes—
123C2×Q8×C11C2×Q812yes—
124C2×D4×C11—24no—
125C2×D4×C11—24no—
126C2×D4×C11—24no—
127C2×D4×C11—21no—
128C2×D4×C11—21no—
129C2×D4×C11—21no—
130C2×D4×C11C22×C412yes—
131C2×D4×C11C2×D412yes—
132C2×D4×C11C2×Q812yes—
133C22×C4×C11—21no—
134C22×C4×C11—24no—
135C22×C4×C11—24no—
136C22×C4×C11—24no—
137C22×C4×C11—22no—
138C22×C4×C11—24no—
139C22×C4×C11—81no—
140C22×C4×C11—81no—
141C22×C4×C11C22×C412yes—
142C22×C4×C11C2×D412yes—
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11C2×D412yes—
145C22×C4×C11C2×Q812yes—
146C22×C4×C11#4112yes—
147C22×C4×C11#4112yes—
148C22×C4×C11C2411yescharacteristic
149C22×Dic11—28no—
150C22×Dic11—24no—
151C22×Dic11—22no—
152C22×Dic11—21no—
153C22×Dic11—21no—
154C22×Dic11—81no—
155C22×Dic11—81no—
156C22×Dic11—82no—
157C22×Dic11—82no—
158C22×Dic11C2×D412yes—
159C22×Dic11C2×D412yes—
160C11×M16—28no—
161C2×C8×C11—22no—
162C2×C8×C11—22no—
163C2×C8×C11C22×C414yes—
164C11×C4⋊C4—24no—
165C11×C4⋊C4—22no—
166C11×C4⋊C4C22×C414yes—
167C22⋊Dic11—28no—
168C22⋊Dic11—24no—
169C22⋊Dic11—42no—
170C4⋊Dic11—28no—
171C4⋊Dic11—22no—
172C4⋊Dic11—24no—
173C4⋊Dic11—22no—
174C4⋊Dic11—22no—
175C4⋊Dic11—21no—
176C4⋊Dic11—21no—
177C4⋊Dic11—21no—
178C4⋊Dic11—21no—
179C4⋊Dic11—82no—
180C4⋊Dic11—82no—
181C4⋊Dic11C2×D414yes—
182C4×Dic11—28no—
183C4×Dic11—22no—
184C4×Dic11—22no—
185C4×Dic11—42no—
186C2×C11⋊C8—21no—
187C2×C11⋊C8—21no—
188C2×C11⋊C8—21no—
189C2×C11⋊C8—21no—
190C2×C11⋊C8—21no—
191C2×C11⋊C8—21no—
192C2×C11⋊C8—44no—
193C2×#520—111no—
194C2×#428—111no—
195C2×#417D1111yescharacteristic
196C2×#416—111no—
197#2316—111no—
198#1865—112no—
199#1864—112no—
200#1863—112no—
201#1862—112no—
202C2×#340—111no—
203#1729—111no—
204C23×C11—21no—
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11—22no—
209C23×C11—22no—
210C23×C11—22no—
211C23×C11—41no—
212C23×C11—41no—
213C23×C11—41no—
214C23×C11—41no—
215C23×C11C4×D412yes—
216C23×C11#12012yes—
217C23×C11C22⋊Q812yes—
218C23×C11C22×D411yescharacteristic
219Q8×C11—22no—
220Q8×C11—22no—
221Q8×C11—22no—
222Q8×C11C2×C4⋊C414yes—
223D4×C11—216no—
224D4×C11—22no—
225D4×C11—22no—
226D4×C11—22no—
227D4×C11C2×C4⋊C414yes—
228C2×C4×C11—22no—
229C2×C4×C11—22no—
230C2×C4×C11—22no—
231C2×C4×C11—216no—
232C2×C4×C11—24no—
233C2×C4×C11—42no—
234C2×C4×C11—42no—
235C2×C4×C11—84no—
236C2×C4×C11C2×C4⋊C414yes—
237C2×C4×C11C4×D414yes—
238C2×C4×C11C4×D412yes—
239C2×C4×C11#12012yes—
240C2×C4×C11C22⋊Q814yes—
241C2×C4×C11C22⋊Q812yes—
242C2×C4×C11#12314yes—
243C2×C4×C11C23×C411yescharacteristic
244C2×C4×C11C22×D411yescharacteristic
245C2×C4×C11C22×D411yescharacteristic
246C2×C4×C11C22×Q811yescharacteristic
247C2×C4×C11C2×#4111yescharacteristic
248C2×C4×C11C2×#4111yescharacteristic
249C2×Dic11—22no—
250C2×Dic11—22no—
251C2×Dic11—24no—
252C2×Dic11—22no—
253C2×Dic11—22no—
254C2×Dic11—22no—
255C2×Dic11—22no—
256C2×Dic11—42no—
257C2×Dic11—42no—
258C2×Dic11—41no—
259C2×Dic11—41no—
260C2×Dic11—41no—
261C2×Dic11—41no—
262C2×Dic11—41no—
263C2×Dic11—41no—
264C2×Dic11—84no—
265C2×Dic11—88no—
266C8×C11—24no—
267C11⋊C8—41no—
268C11⋊C8—41no—
269C22×#41C2×D1111yescharacteristic
270C22×M16C2×D1111yescharacteristic
271C23×C8—111no—
272#520—224no—
273#520—224no—
274C2×C4×Q8—112no—
275C2×C4×D4—112no—
276C2×C4×D4—222no—
277C2×#117—111no—
278C2×#117—111no—
279C2×#117—221no—
280C22×C4⋊C4—111no—
281C22×C4⋊C4—221no—
282C22×C4⋊C4C2×D1111yescharacteristic
283#428—224no—
284C2×#107—112no—
285C2×#106—112no—
286#417Dic1118yescomplemented
287#416—224no—
288C2×#103—111no—
289C2×#103—111no—
290C2×#103C2×D1112yes—
291C2×#102—111no—
292C2×#102—111no—
293C2×#102C2×D1112yes—
294#389—112no—
295#389—112no—
296#389—221no—
297#389—221no—
298#380—222no—
299#340—112no—
300#340—112no—
301#336—112no—
302#336—112no—
303C22×C11—22no—
304C22×C11—22no—
305C22×C11—22no—
306C22×C11—22no—
307C22×C11—22no—
308C22×C11—22no—
309C22×C11—24no—
310C22×C11—42no—
311C22×C11—42no—
312C22×C11—42no—
313C22×C11—42no—
314C22×C11—42no—
315C22×C11—42no—
316C22×C11#38514yes—
317C22×C11C2×C4×D411yescharacteristic
318C22×C11C2×#12011yescharacteristic
319C22×C11C2×C22⋊Q811yescharacteristic
320C22×C11#57611yescharacteristic
321C22×C11#57711yescharacteristic
322C22×C11#57711yescharacteristic
323C22×C11#57811yescharacteristic
324C4×C11—24no—
325C4×C11—44no—
326C4×C11#38514yes—
327C4×C11C22×C4⋊C411yescharacteristic
328C4×C11C2×C4×D411yescharacteristic
329C4×C11C2×C22⋊Q811yescharacteristic
330C4×C11C2×#12311yescharacteristic
331Dic11—42no—
332Dic11—44no—
333Dic11—42no—
334Dic11—81no—
335Dic11—81no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41Dic2214yes—
340C2×#41D4414yes—
341C2×#41C2×Dic1114yes—
342C22×Q8C22×D1111yescharacteristic
343C22×D4—21no—
344C22×D4C22×D1111yescharacteristic
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—222no—
348C23×C4—221no—
349C23×C4C22×D1111yescharacteristic
350C2×M16—22no—
351C2×M16—22no—
352C2×M16C2×Dic1114yescomplemented
353C22×C8—111no—
354C22×C8—111no—
355C22×C8—222no—
356C22×C8—222no—
357C22×C8—222no—
358C22×C8C22×D1112yes—
359C4×Q8—224no—
360C4×Q8—224no—
361C4×D4—224no—
362C4×D4—2216no—
363C4×D4—224no—
364#117—228no—
365#117—222no—
366#117—222no—
367#117—222no—
368#117—222no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—114no—
372C2×C4⋊C4—112no—
373C2×C4⋊C4—111no—
374C2×C4⋊C4—111no—
375C2×C4⋊C4—111no—
376C2×C4⋊C4—111no—
377C2×C4⋊C4—222no—
378C2×C4⋊C4—228no—
379C2×C4⋊C4—222no—
380C2×C4⋊C4—221no—
381C2×C4⋊C4—222no—
382C2×C4⋊C4—222no—
383C2×C4⋊C4C2×Dic1114yescomplemented
384C2×C4⋊C4C22×D1112yes—
385C2×C22⋊C4—112no—
386C2×C22⋊C4—222no—
387C2×C22⋊C4—221no—
388C2×C42—112no—
389C2×C42—222no—
390C2×C42—221no—
391#107—224no—
392#106—224no—
393#103—222no—
394#103—222no—
395#103C2×Dic1118yescomplemented
396#102—222no—
397#102—222no—
398#102C2×Dic1118yescomplemented
399#95—222no—
400#95—222no—
401C2×C11—24no—
402C2×C11—44no—
403C2×C11C2×#38511yescharacteristic
404C2×C11#294611yescharacteristic
405C2×C11#294711yescharacteristic
406C2×C11#301211yescharacteristic
407C2×C11#301311yescharacteristic
408C2×C11#308311yescharacteristic
409C2×C11#308411yescharacteristic
410C24—21no—
411C24D4×D1111yescharacteristic
412#41—28no—
413#41—28no—
414#41—28no—
415#41C4⋊Dic11116yescomplemented
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8C2×Dic2212yes—
420C2×Q8C2×D4412yes—
421C2×Q8C22×Dic1112yes—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×Dic2212yes—
429C2×D4C2×D4412yes—
430C2×D4C22×Dic1112yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—112no—
440C22×C4—112no—
441C22×C4—222no—
442C22×C4—228no—
443C22×C4—224no—
444C22×C4—221no—
445C22×C4—221no—
446C22×C4—882no—
447C22×C4—882no—
448C22×C4—881no—
449C22×C4—881no—
450C22×C4C2×Dic2212yes—
451C22×C4C2×D4412yes—
452C22×C4D4×D1111yescharacteristic
453C22×C4D4×D1112yes—
454C22×C4#434112yes—
455C22×C4#434112yes—
456C22×C4C22×Dic1112yes—
457C22×C4C23×D1111yescharacteristic
458M16—28no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—221no—
462C2×C8—221no—
463C2×C8—221no—
464C2×C8—221no—
465C2×C8—221no—
466C2×C8—221no—
467C2×C8—444no—
468C2×C8C22×Dic1114yes—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—114no—
472C4⋊C4—228no—
473C4⋊C4—224no—
474C4⋊C4—222no—
475C4⋊C4—222no—
476C4⋊C4—222no—
477C4⋊C4—221no—
478C4⋊C4—221no—
479C4⋊C4—221no—
480C4⋊C4—221no—
481C4⋊C4—882no—
482C4⋊C4—882no—
483C4⋊C4C22×Dic1114yes—
484C22⋊C4—228no—
485C22⋊C4—224no—
486C22⋊C4—442no—
487C42—228no—
488C42—222no—
489C42—222no—
490C42—442no—
491C11#2087811yescharacteristic, complemented, Sylow
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#6693112yes—
504C23#6693512yes—
505C23D4×Dic1112yes—
506C23C2×D4×D1111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×C4⋊Dic1114yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×C4⋊Dic1114yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—222no—
525C2×C4—224no—
526C2×C4—222no—
527C2×C4—222no—
528C2×C4—222no—
529C2×C4—222no—
530C2×C4—222no—
531C2×C4—442no—
532C2×C4—442no—
533C2×C4—441no—
534C2×C4—441no—
535C2×C4—441no—
536C2×C4—441no—
537C2×C4—441no—
538C2×C4—441no—
539C2×C4—884no—
540C2×C4—888no—
541C2×C4#6693114yes—
542C2×C4#6693112yes—
543C2×C4#6693512yes—
544C2×C4#6693914yes—
545C2×C4C2×C4⋊Dic1114yes—
546C2×C4D4×Dic1114yes—
547C2×C4D4×Dic1112yes—
548C2×C4C22×Dic2211yescharacteristic
549C2×C4C22×D4411yescharacteristic
550C2×C4C2×D4×D1111yescharacteristic
551C2×C4C2×#434111yescharacteristic
552C2×C4C2×#434111yescharacteristic
553C2×C4C23×Dic1111yescharacteristic
554C8—24no—
555C8—441no—
556C8—441no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1062256314yes—
571C22C2×#6693111yescharacteristic
572C22C2×#6693511yescharacteristic
573C22#1062330511yescharacteristic
574C22#1062330611yescharacteristic
575C22#1062330911yescharacteristic
576C22#1062331011yescharacteristic
577C22C2×D4×Dic1111yescharacteristic
578C4—24no—
579C4—44no—
580C4—442no—
581C4—444no—
582C4—442no—
583C4—881no—
584C4—881no—
585C4#1062256314yes—
586C4C2×#6693111yescharacteristic
587C4C2×#6693911yescharacteristic
588C4C22×C4⋊Dic1111yescharacteristic
589C4C2×D4×Dic1111yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1062256311yescharacteristic
593C2#1307744911yescharacteristic
594C2#1307746011yescharacteristic
595C2#1307792211yescharacteristic
596C2#1307793311yescharacteristic
597C2#1307847111yescharacteristic
598C2#1307848211yescharacteristic
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#20878
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
2. C2×#10622734C2semidirect
4. #13071234C2semidirect
5. #13068886C2semidirect
6. #13068879C2semidirect
7. #13068872C2semidirect
8. #13068865C2semidirect
9. C2×#10622167C2semidirect
10. #13068314C2semidirect
26. C11×#417C4semidirect
62. C2×C11×M16C2×C4semidirect
66. C2×C11×C4⋊C4C2×C4semidirect
101. C11×#103C2×C4semidirect
102. C11×#102C2×C4semidirect
117. C11×#41C4⋊C4semidirect
286. #417Dic11semidirect
352. C2×M16C2×Dic11semidirect
383. C2×C4⋊C4C2×Dic11semidirect
395. #103C2×Dic11semidirect
398. #102C2×Dic11semidirect
415. #41C4⋊Dic11semidirect
491. C11#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#106233582
C2×#106227342
C2×#106228462
#130712342
#130688862
#130688792
#130688722
#130688652
C2×#106221672
#130683142
C2×C11×#4172
#2087811