Finite Group Databasebeta

← #14816693

Subgroups — #14816693

Order 2816.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622622C211yesmaximal, characteristic, complemented
2C2×#10622474C211yesmaximal, characteristic, complemented
3#13072331C211yesmaximal, characteristic, complemented
4#13071628C211yesmaximal, characteristic
5#13070147C211yesmaximal, characteristic, complemented
6#13070117C211yesmaximal, characteristic, complemented
7#13070002C211yesmaximal, characteristic, complemented
8#13069955C211yesmaximal, characteristic, complemented
9#13069376C211yesmaximal, characteristic, complemented
10#13069185C211yesmaximal, characteristic, complemented
11#13069051C211yesmaximal, characteristic
12#13068935C211yesmaximal, characteristic
13#13068810C211yesmaximal, characteristic, complemented
14#13068800C211yesmaximal, characteristic, complemented
15C11×#1864C211yesmaximal, characteristic, complemented
16C2×C4×D4×C11C2211yescharacteristic
17C2×#66991C2211yescharacteristic
18C2×C4×C22⋊D11—21no—
19C22×#4313C2211yescharacteristic
20C2×#66945C2211yescharacteristic
21C2×C4⋊C4×D11C2211yescharacteristic
22C2×#66923C2211yescharacteristic
23C2×C11×#106C2211yescharacteristic
24C2×C11×#102C2211yescharacteristic
25C2×C11×#102C2211yescharacteristic
26C11×#389C2211yescharacteristic
27#10622834C2211yescharacteristic
28C2×#66896C2211yescharacteristic
29C2×#66896C2211yescharacteristic
30#10622823—21no—
31C2×(C2×D11)⋊C8C2211yescharacteristic
32C2×#66882C2211yescharacteristic
33C2×#66881C2211yescharacteristic
34C2×Dic11⋊C8C2211yescharacteristic
35#10622622—24no—
36#10622605—21no—
37#10622600C2211yescharacteristic
38#10622589C2211yescharacteristic
39C4⋊C4×Dic11C2211yescharacteristic
40#10622586C2211yescharacteristic
41#10622585—21no—
42C2×#66872C2211yescharacteristic
43C2×#66872C2211yescharacteristic
44C2×#66871C2211yescharacteristic
45#10622569—21no—
46#10622558—21no—
47C4×(C2×D11)⋊C4C2211yescharacteristic
48#10622541C2211yescharacteristic
49C2×C4⋊(C11⋊C8)C2211yescharacteristic
50#10622474—24no—
51#10622289—21no—
52#10622265—21no—
53C11×#340C2211yescharacteristic
54C11×#336C2211yescharacteristic
55#10622189C2211yescharacteristic
56#10622189C2211yescharacteristic
57#10622188C2211yescharacteristic
58#10622167C2211yescharacteristic
59#10622167C2211yescharacteristic
60#10622167C2211yescharacteristic
61C22×D4×C11C2311yescharacteristic
62C23×C4×C11—21no—
63C22×C22⋊D11—21no—
64C22×C4×D11—21no—
65C22×C4×D11C2311yescharacteristic
66C22×C8×C11C2311yescharacteristic
67C22×C8×C11C2311yescharacteristic
68C4×D4×C11—22no—
69C4×D4×C11—41no—
70C2×C11×C4⋊C4—21no—
71C2×C11×C4⋊C4C2311yescharacteristic
72C2×C11×C4⋊C4C2311yescharacteristic
73C2×C11×C4⋊C4C2311yescharacteristic
74C2×C11×C22⋊C4—21no—
75C2×C42×C11C2311yescharacteristic
76C2×C22⋊Dic11—21no—
77C2×C22⋊Dic11—21no—
78C2×C22⋊Dic11—21no—
79#66991—22no—
80#66991Q814yes—
81C4×C22⋊D11—44no—
82C2×(C2×D11)⋊C4—21no—
83C2×(C2×D11)⋊C4—21no—
84C2×(C2×D11)⋊C4D411yescharacteristic
85C2×(C2×D11)⋊C4D411yescharacteristic
86C2×C4⋊Dic11C2311yescharacteristic
87C2×C4⋊Dic11C2311yescharacteristic
88C2×C4⋊Dic11C2311yescharacteristic
89C2×C4⋊Dic11C2311yescharacteristic
90C2×Dic11⋊C4—21no—
91C2×Dic11⋊C4—21no—
92C2×Dic11⋊C4—21no—
93C2×Dic11⋊C4—21no—
94C2×Dic11⋊C4—21no—
95C2×Dic11⋊C4—21no—
96C2×Dic11⋊C4—21no—
97C2×C4×Dic11—21no—
98C2×C4×Dic11—21no—
99C2×C4×Dic11C2311yescharacteristic
100C22×C11⋊C8C2311yescharacteristic
101C22×C11⋊C8C2311yescharacteristic
102C2×#4313—22no—
103C2×#4313—22no—
104C2×#4313—22no—
105#66945—22no—
106#66945—41no—
107C4⋊C4×D11—22no—
108C4⋊C4×D11—41no—
109#66923—22no—
110#66923—41no—
111C11×#106—22no—
112C11×#102—22no—
113C11×#102—41no—
114C11×#95—21no—
115#66896—22no—
116#66896—22no—
117C11⋊#95—21no—
118C11⋊#95—21no—
119C11⋊#95—21no—
120C11⋊#95—21no—
121C11⋊#95D411yescharacteristic
122C11⋊#95D411yescharacteristic
123(C2×D11)⋊C8—22no—
124#66882—22no—
125#66881—41no—
126Dic11⋊C8—22no—
127#66872—22no—
128#66872—22no—
129#66871D414yescomplemented
130C4⋊(C11⋊C8)—22no—
131#26366—111nomaximal, Sylow
132C24×C11—21no—
133C23×D11C2×D411yescharacteristic
134C2×D4×C11—21no—
135C2×D4×C11—21no—
136C2×D4×C11—21no—
137C2×D4×C11—21no—
138C2×D4×C11—21no—
139C2×D4×C11C2×Q812yes—
140C22×C4×C11—21no—
141C22×C4×C11—21no—
142C22×C4×C11—21no—
143C22×C4×C11—42no—
144C22×C4×C11—42no—
145C22×C4×C11—81no—
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2×D411yescharacteristic
148C22×C4×C11C2411yescharacteristic
149C2×C22⋊D11—24no—
150C2×C22⋊D11—42no—
151C2×C22⋊D11—42no—
152C22×Dic11—21no—
153C22×Dic11—21no—
154C22×Dic11—21no—
155C22×Dic11—21no—
156C22×Dic11C2×D411yescharacteristic
157C22×Dic11C2×D411yescharacteristic
158C22×Dic11C2×D411yescharacteristic
159C2×C4×D11—21no—
160C2×C4×D11—21no—
161C2×C4×D11—21no—
162C2×C4×D11—21no—
163C2×C4×D11—42no—
164C2×C4×D11—42no—
165C2×C4×D11—81no—
166C2×C4×D11C2×Q812yes—
167C2×C4×D11#4112yes—
168C2×C8×C11—21no—
169C2×C8×C11—21no—
170C2×C8×C11—21no—
171C2×C8×C11—21no—
172C2×C8×C11—21no—
173C2×C8×C11#4112yes—
174C11×C4⋊C4—21no—
175C11×C4⋊C4—21no—
176C11×C4⋊C4—21no—
177C11×C4⋊C4—21no—
178C11×C4⋊C4—21no—
179C11×C4⋊C4—81no—
180C11×C4⋊C4C2×D412yes—
181C11×C22⋊C4—42no—
182C42×C11—21no—
183C42×C11—21no—
184C22⋊Dic11—24no—
185C22⋊Dic11—42no—
186C22⋊Dic11—81no—
187(C2×D11)⋊C4—22no—
188(C2×D11)⋊C4—41no—
189(C2×D11)⋊C4—42no—
190(C2×D11)⋊C4—81no—
191C4⋊Dic11—21no—
192C4⋊Dic11—21no—
193C4⋊Dic11—21no—
194C4⋊Dic11—21no—
195C4⋊Dic11—21no—
196C4⋊Dic11—21no—
197C4⋊Dic11C2×D412yes—
198C4⋊Dic11C2×Q812yes—
199Dic11⋊C4—42no—
200Dic11⋊C4—42no—
201Dic11⋊C4—42no—
202Dic11⋊C4—42no—
203Dic11⋊C4—42no—
204Dic11⋊C4—42no—
205Dic11⋊C4—81no—
206C4×Dic11—21no—
207C4×Dic11—21no—
208C4×Dic11—42no—
209C4×Dic11—42no—
210C2×C11⋊C8—21no—
211C2×C11⋊C8—21no—
212C2×C11⋊C8—21no—
213C2×C11⋊C8—21no—
214C2×C11⋊C8—21no—
215C2×C11⋊C8C2×D412yes—
216#4313—28no—
217C2×#478—111no—
218C2×#428—111no—
219#2441—111no—
220#2361—111no—
221#2094—111no—
222#2088—111no—
223#2067—111no—
224#2059—111no—
225#1948—111no—
226#1918—111no—
227#1895—111no—
228#1875—111no—
229#1864D1111yescharacteristic, complemented
230#1853—111no—
231#1851—111no—
232C23×C11—22no—
233C23×C11—41no—
234C23×C11—41no—
235C23×C11—41no—
236C23×C11—41no—
237C23×C11—41no—
238C23×C11—41no—
239C23×C11C22×D411yescharacteristic
240C22×D11—21no—
241C22×D11—21no—
242C22×D11—21no—
243C22×D11—21no—
244C22×D11—21no—
245C22×D11—21no—
246C22×D11C22⋊Q812yes—
247D4×C11—22no—
248D4×C11—22no—
249D4×C11—41no—
250D4×C11—41no—
251C2×C4×C11—21no—
252C2×C4×C11—21no—
253C2×C4×C11—22no—
254C2×C4×C11—21no—
255C2×C4×C11—21no—
256C2×C4×C11—21no—
257C2×C4×C11—21no—
258C2×C4×C11—41no—
259C2×C4×C11—41no—
260C2×C4×C11—41no—
261C2×C4×C11—41no—
262C2×C4×C11—41no—
263C2×C4×C11—41no—
264C2×C4×C11—41no—
265C2×C4×C11—41no—
266C2×C4×C11—84no—
267C2×C4×C11C22×D411yescharacteristic
268C2×C4×C11C22×Q811yescharacteristic
269C2×C4×C11C2×#4111yescharacteristic
270C2×C4×C11C2×#4111yescharacteristic
271C2×C4×C11H(5,𝔽2)11yescharacteristic
272C2×C4×C112−1+411yescharacteristic
273C22⋊D11—48no—
274C2×Dic11—21no—
275C2×Dic11—21no—
276C2×Dic11—21no—
277C2×Dic11—21no—
278C2×Dic11—22no—
279C2×Dic11—22no—
280C2×Dic11—21no—
281C2×Dic11—21no—
282C2×Dic11—21no—
283C2×Dic11—41no—
284C2×Dic11—41no—
285C2×Dic11—41no—
286C2×Dic11—41no—
287C2×Dic11—41no—
288C2×Dic11—41no—
289C2×Dic11—41no—
290C2×Dic11—41no—
291C2×Dic11—41no—
292C2×Dic11—41no—
293C2×Dic11C22⋊Q812yes—
294C2×Dic11AGL(1,ℤ/8)12yes—
295C4×D11—22no—
296C4×D11—22no—
297C4×D11—22no—
298C4×D11—22no—
299C4×D11—84no—
300C8×C11—22no—
301C8×C11—41no—
302C11⋊C8—22no—
303C11⋊C8—41no—
304C22×M16—111no—
305C2×#125—111no—
306C2×#121—111no—
307C2×C4×D4—221no—
308C2×C4×D4C2×D1111yescharacteristic
309C2×#117—111no—
310C22×C4⋊C4—111no—
311#478—224no—
312#428—224no—
313C2×#107—111no—
314C2×#107—111no—
315C2×#106—111no—
316C2×#106—111no—
317C2×#106—111no—
318C2×#106C2×D1111yescharacteristic
319C2×C4⋊C8—111no—
320C2×C4⋊C8—111no—
321C2×#102—111no—
322C2×#102—111no—
323C2×#102C2×D1111yescharacteristic
324C2×#102C2×D1111yescharacteristic
325C2×C22⋊C8—111no—
326#399—221no—
327#396—221no—
328#390—111no—
329#389—111no—
330#389—111no—
331#389—221no—
332#389C2×D1111yescharacteristic
333#388—221no—
334#387—221no—
335#386—221no—
336#385—221no—
337#384—111no—
338#380—111no—
339C4×C4⋊C4—111no—
340C4×C22⋊C4—111no—
341#340—111no—
342#340—111no—
343#340—111no—
344#340—111no—
345#340—111no—
346#340C2×D1111yescharacteristic
347#336—111no—
348#336C2×D1111yescharacteristic
349C22×C11—42no—
350C22×C11—42no—
351C22×C11—42no—
352C22×C11—81no—
353C22×C11—81no—
354C22×C11—81no—
355C22×C11—81no—
356C22×C11C2×C22⋊Q811yescharacteristic
357C22×C11D4211yescharacteristic
358C22×C11#54711yescharacteristic
359C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C11#57511yescharacteristic
361C22×C11#57711yescharacteristic
362C22×C11#57711yescharacteristic
363C2×D11—22no—
364C2×D11—22no—
365C2×D11—22no—
366C2×D11—41no—
367C2×D11—41no—
368C2×D11—41no—
369C2×D11—41no—
370C4×C11—41no—
371C4×C11—41no—
372C4×C11—42no—
373C4×C11—81no—
374C4×C11—81no—
375C4×C11#52511yescharacteristic
376C4×C11#53911yescharacteristic
377C4×C11D4×Q811yescharacteristic
378C4×C11#55411yescharacteristic
379Dic11—22no—
380Dic11—22no—
381Dic11—42no—
382Dic11—41no—
383Dic11—42no—
384Dic11—81no—
385Dic11—81no—
386C22×D4—221no—
387C22×D4C22×D1111yescharacteristic
388C23×C4—21no—
389C23×C4—111no—
390C23×C4—221no—
391C2×M16—222no—
392C2×M16—222no—
393C2×M16—222no—
394C22×C8—111no—
395C22×C8—111no—
396C22×C8C22×D1111yescharacteristic
397C22×C8C22×D1111yescharacteristic
398#125—222no—
399#125—441no—
400#121—114no—
401#121—222no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—444no—
405#117—222no—
406#117—441no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—111no—
409C2×C4⋊C4—111no—
410C2×C4⋊C4—111no—
411C2×C4⋊C4—111no—
412C2×C4⋊C4—222no—
413C2×C4⋊C4—221no—
414C2×C4⋊C4—221no—
415C2×C4⋊C4—221no—
416C2×C4⋊C4—221no—
417C2×C4⋊C4—221no—
418C2×C4⋊C4—221no—
419C2×C4⋊C4—221no—
420C2×C4⋊C4—441no—
421C2×C4⋊C4C22×D1111yescharacteristic
422C2×C4⋊C4C22×D1111yescharacteristic
423C2×C4⋊C4C22×D1111yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—111no—
426C2×C22⋊C4—111no—
427C2×C22⋊C4—221no—
428C2×C22⋊C4—221no—
429C2×C22⋊C4—221no—
430C2×C22⋊C4—221no—
431C2×C22⋊C4—221no—
432C2×C42—111no—
433C2×C42—221no—
434C2×C42—221no—
435C2×C42C22×D1111yescharacteristic
436#107—114no—
437#107—222no—
438#106—22no—
439#106—222no—
440#106—222no—
441#106—441no—
442C4⋊C8—222no—
443C4⋊C8—222no—
444#102—22no—
445#102—41no—
446#102—222no—
447#102—222no—
448C22⋊C8—222no—
449#95—21no—
450#95—111no—
451#95—111no—
452#95—221no—
453#95—221no—
454#95—221no—
455#95—221no—
456C2×C11—82no—
457C2×C11#244911yescharacteristic
458C2×C11#307711yescharacteristic
459C2×C11#308411yescharacteristic
460C2×C11#327111yescharacteristic
461C2×C11#327211yescharacteristic
462C2×C11#334511yescharacteristic
463C2×C11#335111yescharacteristic
464D11—42no—
465C24—21no—
466C24—111no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—224no—
473C2×D4—442no—
474C2×D4—442no—
475C2×D4Q8×D1112yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—111no—
483C22×C4—111no—
484C22×C4—112no—
485C22×C4—112no—
486C22×C4—111no—
487C22×C4—221no—
488C22×C4—221no—
489C22×C4—221no—
490C22×C4—221no—
491C22×C4—221no—
492C22×C4—221no—
493C22×C4—221no—
494C22×C4—221no—
495C22×C4—442no—
496C22×C4—442no—
497C22×C4—881no—
498C22×C4D4×D1111yescharacteristic
499C22×C4D4×D1111yescharacteristic
500C22×C4C23×D1111yescharacteristic
501M16—228no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—112no—
508C2×C8—221no—
509C2×C8—221no—
510C2×C8—221no—
511C2×C8—221no—
512C2×C8—221no—
513C2×C8#433912yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—81no—
520C4⋊C4—112no—
521C4⋊C4—112no—
522C4⋊C4—221no—
523C4⋊C4—221no—
524C4⋊C4—221no—
525C4⋊C4—221no—
526C4⋊C4—221no—
527C4⋊C4—221no—
528C4⋊C4—442no—
529C4⋊C4—442no—
530C4⋊C4—442no—
531C4⋊C4—442no—
532C4⋊C4—442no—
533C4⋊C4—442no—
534C4⋊C4—881no—
535C4⋊C4D4×D1112yes—
536C22⋊C4—42no—
537C22⋊C4—222no—
538C22⋊C4—224no—
539C22⋊C4—441no—
540C22⋊C4—442no—
541C22⋊C4—442no—
542C22⋊C4—881no—
543C22⋊C4—881no—
544C42—21no—
545C42—21no—
546C42—221no—
547C42—221no—
548C42—442no—
549C42—442no—
550C11#2636611yescharacteristic, complemented, Sylow
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—112no—
559C23—221no—
560C23—221no—
561C23—221no—
562C23—221no—
563C23—221no—
564C23—221no—
565C23C2×D4×D1111yescharacteristic
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—448no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—112no—
588C2×C4—112no—
589C2×C4—221no—
590C2×C4—221no—
591C2×C4—221no—
592C2×C4—221no—
593C2×C4—222no—
594C2×C4—222no—
595C2×C4—222no—
596C2×C4—222no—
597C2×C4—221no—
598C2×C4—221no—
599C2×C4—221no—
600C2×C4—222no—
601C2×C4—222no—
602C2×C4—441no—
603C2×C4—441no—
604C2×C4—441no—
605C2×C4—441no—
606C2×C4—441no—
607C2×C4—441no—
608C2×C4—441no—
609C2×C4—441no—
610C2×C4—441no—
611C2×C4—441no—
612C2×C4—884no—
613C2×C4C2×#433911yescharacteristic
614C2×C4C2×D4×D1111yescharacteristic
615C2×C4#6703711yescharacteristic
616C2×C4C2×Q8×D1111yescharacteristic
617C2×C4#41×D1111yescharacteristic
618C2×C4#6704311yescharacteristic
619C8—22no—
620C8—41no—
621C8—222no—
622C8—441no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—222no—
631C22—222no—
632C22—222no—
633C22—441no—
634C22—441no—
635C22—441no—
636C22—441no—
637C22#1062308211yescharacteristic
638C22#1062312111yescharacteristic
639C22D11×C22⋊Q811yescharacteristic
640C22#1062329311yescharacteristic
641C22#1062329911yescharacteristic
642C22D11×AGL(1,ℤ/8)11yescharacteristic
643C22#1062330611yescharacteristic
644C4—41no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—222no—
650C4—222no—
651C4—441no—
652C4—442no—
653C4—442no—
654C4—881no—
655C4—881no—
656C4#1062304211yescharacteristic
657C4#1062307811yescharacteristic
658C4#1062316311yescharacteristic
659C4#1062319111yescharacteristic
660C2—82no—
661C2—442no—
662C2#1307242411yescharacteristic
663C2#1307841111yescharacteristic
664C2#1307847311yescharacteristic
665C2#1308019411yescharacteristic
666C2#1308021411yescharacteristic
667C2#1308109111yescharacteristic
668C2#1308118111yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1864C2
socleC23×C11C22×D4
prime radicalC2×#66991C22
hypercenter#1864D11
hypocenterC11#26366
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
1. C2×#10622622C2semidirect
2. C2×#10622474C2semidirect
3. #13072331C2semidirect
5. #13070147C2semidirect
6. #13070117C2semidirect
7. #13070002C2semidirect
8. #13069955C2semidirect
9. #13069376C2semidirect
10. #13069185C2semidirect
13. #13068810C2semidirect
14. #13068800C2semidirect
15. C11×#1864C2semidirect
129. #66871D4semidirect
229. #1864D11semidirect
550. C11#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#106226222
C2×#106224742
#130723312
#130716282
#130701472
#130701172
#130700022
#130699552
#130693762
#130691852
#130690512
#130689352
#130688102
#130688002
C11×#18642
#2636611