Finite Group Databasebeta

← #14814002

Subgroups — #14814002

Order 2816.

904 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622688C211yesmaximal, characteristic
2C2×#10622631C211yesmaximal, characteristic, complemented
3#13071536C211yesmaximal, characteristic, complemented
4#13070137C211yesmaximal, characteristic, complemented
5#13070107C211yesmaximal, characteristic, complemented
6#13070029C211yesmaximal, characteristic, complemented
7#13069847C211yesmaximal, characteristic
8#13069316C211yesmaximal, characteristic, complemented
9#13069182C211yesmaximal, characteristic, complemented
10#13068974C211yesmaximal, characteristic, complemented
11#13068974C211yesmaximal, characteristic, complemented
12#13068974C211yesmaximal, characteristic, complemented
13#13068883C211yesmaximal, characteristic, complemented
14#13068877C211yesmaximal, characteristic, complemented
15C11×#2445C211yesmaximal, characteristic
16C2×C11×#127C2211yescharacteristic
17C2×C11×#123—21no—
18C2×C4×D4×C11C2211yescharacteristic
19C22×#4324C2211yescharacteristic
20C22×#4324C2211yescharacteristic
21C2×C4×Dic22C2211yescharacteristic
22C2×C4×Dic22C2211yescharacteristic
23C2×C4×Dic22C2211yescharacteristic
24C11×#394—21no—
25C11×#394—21no—
26C11×#390C2211yescharacteristic
27C11×#390C2211yescharacteristic
28C11×#390C2211yescharacteristic
29C11×#389C2211yescharacteristic
30C11×#386—21no—
31C4×C11×C4⋊C4C2211yescharacteristic
32C2×#66896C2211yescharacteristic
33C2×#66896C2211yescharacteristic
34C2×#66896C2211yescharacteristic
35C2×#66896C2211yescharacteristic
36C2×#66896C2211yescharacteristic
37C2×#66896C2211yescharacteristic
38#10622688—24no—
39#10622631—24no—
40#10622585—21no—
41#10622585—21no—
42C2×#66874C2211yescharacteristic
43C2×#66874C2211yescharacteristic
44C2×#66872C2211yescharacteristic
45C2×#66871C2211yescharacteristic
46#10622549C2211yescharacteristic
47#10622541C2211yescharacteristic
48#10622541C2211yescharacteristic
49C2×C4⋊(C11⋊C8)C2211yescharacteristic
50C2×#66867C2211yescharacteristic
51#10622263—21no—
52#10622262—21no—
53#10622167C2211yescharacteristic
54#10622167C2211yescharacteristic
55#10622167C2211yescharacteristic
56#10622167C2211yescharacteristic
57#10622162C2211yescharacteristic
58#10622162C2211yescharacteristic
59#10622162C2211yescharacteristic
60#10622162C2211yescharacteristic
61C22×D4×C11—21no—
62C22×D4×C11—21no—
63C22×D4×C11C2311yescharacteristic
64C22×D4×C11C2311yescharacteristic
65C23×C4×C11—21no—
66C22×Dic22C2311yescharacteristic
67C11×#127—22no—
68C11×#127—41no—
69C11×#123—44no—
70C4×D4×C11—22no—
71C4×D4×C11—41no—
72C2×C11×C4⋊C4—21no—
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C4⋊C4D411yescharacteristic
75C2×C11×C4⋊C4C2311yescharacteristic
76C2×C11×C4⋊C4C2311yescharacteristic
77C2×C11×C22⋊C4—21no—
78C2×C11×C22⋊C4—21no—
79C2×C11×C22⋊C4—21no—
80C2×C11×C22⋊C4—21no—
81C2×C11×C22⋊C4—21no—
82C2×C11×C22⋊C4—21no—
83C2×C11×C22⋊C4—21no—
84C2×C42×C11C2311yescharacteristic
85C2×C42×C11C2311yescharacteristic
86C2×C42×C11C2311yescharacteristic
87C2×#4324—22no—
88C2×#4324—22no—
89C2×#4324—22no—
90C2×#4324—22no—
91C2×#4324—22no—
92C2×#4324D414yes—
93C2×C4⋊Dic11C2311yescharacteristic
94C2×C4⋊Dic11C2311yescharacteristic
95C2×C4⋊Dic11C2311yescharacteristic
96C2×Dic11⋊C4—21no—
97C2×Dic11⋊C4—21no—
98C2×Dic11⋊C4—21no—
99C2×Dic11⋊C4—21no—
100C2×Dic11⋊C4—21no—
101C2×C4×Dic11—21no—
102C2×C4×Dic11—21no—
103C2×C4×Dic11—21no—
104C22×C11⋊C8C2311yescharacteristic
105C22×C11⋊C8C2311yescharacteristic
106C22×C11⋊C8C2311yescharacteristic
107C22×C11⋊C8C2311yescharacteristic
108C4×Dic22—22no—
109C4×Dic22—22no—
110C4×Dic22—22no—
111C4×Dic22—22no—
112C4×Dic22—41no—
113C4×Dic22—41no—
114C11×#95—21no—
115C11×#95D411yescharacteristic
116C11×#95D411yescharacteristic
117#66896—22no—
118#66896—22no—
119#66896—22no—
120#66896—22no—
121#66896—22no—
122#66896—41no—
123C11⋊#95—21no—
124C11⋊#95—21no—
125C11⋊#95—21no—
126C11⋊#95—21no—
127#66874—22no—
128#66874—22no—
129#66872—41no—
130#66871—22no—
131C4⋊(C11⋊C8)—22no—
132#66867—22no—
133#26175—111nomaximal, Sylow
134C24×C11—21no—
135C24×C11—21no—
136C2×D4×C11—21no—
137C2×D4×C11—21no—
138C2×D4×C11—21no—
139C2×D4×C11—21no—
140C2×D4×C11—21no—
141C2×D4×C11—21no—
142C2×D4×C11—21no—
143C2×D4×C11—21no—
144C2×D4×C11—21no—
145C2×D4×C11—21no—
146C2×D4×C11—21no—
147C2×D4×C11—42no—
148C2×D4×C11—42no—
149C2×D4×C11—42no—
150C2×D4×C11—42no—
151C2×D4×C11—81no—
152C2×D4×C11—81no—
153C2×D4×C11C2×D412yes—
154C22×C4×C11—21no—
155C22×C4×C11—21no—
156C22×C4×C11—42no—
157C22×C4×C11—42no—
158C22×C4×C11—81no—
159C22×C4×C11C2×D411yescharacteristic
160C22×C4×C11C2×D411yescharacteristic
161C22×C4×C11C2×D411yescharacteristic
162C22×C4×C11C2×D411yescharacteristic
163C22×C4×C11C2×D411yescharacteristic
164C22×C4×C11C2×D411yescharacteristic
165C22×C4×C11C2411yescharacteristic
166C22×Dic11—21no—
167C22×Dic11—21no—
168C22×Dic11—21no—
169C22×Dic11—21no—
170C2×Dic22—21no—
171C2×Dic22—21no—
172C2×Dic22—21no—
173C2×Dic22C2×D412yes—
174C2×Dic22#4112yes—
175C2×Dic22#4112yes—
176C11×C4⋊C4—22no—
177C11×C4⋊C4—21no—
178C11×C4⋊C4—21no—
179C11×C4⋊C4—21no—
180C11×C4⋊C4—21no—
181C11×C4⋊C4—41no—
182C11×C4⋊C4—42no—
183C11×C22⋊C4—42no—
184C11×C22⋊C4—42no—
185C11×C22⋊C4—42no—
186C11×C22⋊C4—42no—
187C11×C22⋊C4—81no—
188C11×C22⋊C4—81no—
189C11×C22⋊C4—81no—
190C42×C11—21no—
191C42×C11—21no—
192C42×C11—21no—
193C42×C11—21no—
194C42×C11—21no—
195C42×C11—21no—
196#4324—28no—
197#4324—44no—
198C4⋊Dic11—21no—
199C4⋊Dic11—21no—
200C4⋊Dic11—21no—
201C4⋊Dic11—21no—
202C4⋊Dic11—21no—
203C4⋊Dic11#4112yes—
204Dic11⋊C4—24no—
205Dic11⋊C4—42no—
206Dic11⋊C4—42no—
207Dic11⋊C4—42no—
208Dic11⋊C4—42no—
209C4×Dic11—24no—
210C4×Dic11—42no—
211C4×Dic11—42no—
212C2×C11⋊C8—21no—
213C2×C11⋊C8—21no—
214C2×C11⋊C8—21no—
215C2×C11⋊C8—21no—
216C2×C11⋊C8—21no—
217C2×C11⋊C8—21no—
218C2×C11⋊C8—21no—
219C2×C11⋊C8—21no—
220C2×C11⋊C8—21no—
221C2×C11⋊C8—21no—
222C2×C11⋊C8—21no—
223C2×C11⋊C8C2×D412yes—
224C2×#460—111no—
225C2×#440—111no—
226#2445D1111yescharacteristic
227#2351—111no—
228#2092—111no—
229#2086—111no—
230#2073—111no—
231#2041—111no—
232#1936—111no—
233#1917—111no—
234#1880—111no—
235#1880—111no—
236#1880—111no—
237#1865—111no—
238#1864—111no—
239C23×C11—22no—
240C23×C11—41no—
241C23×C11—41no—
242C23×C11—41no—
243C23×C11—41no—
244C23×C11—41no—
245C23×C11—41no—
246C23×C11—41no—
247C23×C11—41no—
248C23×C11—41no—
249C23×C11—41no—
250C23×C11—41no—
251C23×C11—41no—
252C23×C11—41no—
253C23×C11C22×D411yescharacteristic
254D4×C11—22no—
255D4×C11—22no—
256D4×C11—41no—
257D4×C11—41no—
258D4×C11—41no—
259D4×C11—41no—
260D4×C11—41no—
261D4×C11—41no—
262D4×C11—84no—
263D4×C11—84no—
264C2×C4×C11—21no—
265C2×C4×C11—21no—
266C2×C4×C11—21no—
267C2×C4×C11—21no—
268C2×C4×C11—21no—
269C2×C4×C11—21no—
270C2×C4×C11—21no—
271C2×C4×C11—21no—
272C2×C4×C11—21no—
273C2×C4×C11—21no—
274C2×C4×C11—21no—
275C2×C4×C11—21no—
276C2×C4×C11—21no—
277C2×C4×C11—21no—
278C2×C4×C11—21no—
279C2×C4×C11—21no—
280C2×C4×C11—21no—
281C2×C4×C11—41no—
282C2×C4×C11—41no—
283C2×C4×C11—41no—
284C2×C4×C11—41no—
285C2×C4×C11—41no—
286C2×C4×C11—41no—
287C2×C4×C11—84no—
288C2×C4×C11AGL(1,ℤ/8)12yes—
289C2×C4×C11C22×D411yescharacteristic
290C2×C4×C11C2×#4111yescharacteristic
291C2×C4×C11C2×#4111yescharacteristic
292C2×C4×C11C2×#4111yescharacteristic
293C2×C4×C11H(5,𝔽2)11yescharacteristic
294C2×C4×C112−1+411yescharacteristic
295C2×Dic11—22no—
296C2×Dic11—22no—
297C2×Dic11—22no—
298C2×Dic11—22no—
299C2×Dic11—41no—
300C2×Dic11—41no—
301C2×Dic11—41no—
302C2×Dic11—41no—
303C2×Dic11—41no—
304C2×Dic11—41no—
305C2×Dic11—41no—
306C2×Dic11—41no—
307Dic22—22no—
308Dic22—22no—
309Dic22—22no—
310Dic22#12314yescomplemented
311C11⋊C8—22no—
312C11⋊C8—41no—
313C11⋊C8—41no—
314C11⋊C8—41no—
315C22×SD16—111no—
316C22×SD16—111no—
317C2×#127C2×D1111yescharacteristic
318C2×#123—21no—
319C2×C4×Q8—111no—
320C2×C4×Q8—111no—
321C2×C4×Q8—111no—
322C2×C4×D4C2×D1111yescharacteristic
323#460—224no—
324#440—224no—
325C2×#107—111no—
326C2×#106—111no—
327C2×C4⋊C8—111no—
328C2×#103—111no—
329C2×#103—111no—
330C2×#102—111no—
331C2×#102—111no—
332C2×#102—111no—
333C2×#102—111no—
334C2×#102—111no—
335C2×#102—111no—
336C2×#97—111no—
337#394—21no—
338#394—21no—
339#390C2×D1111yescharacteristic
340#390C2×D1111yescharacteristic
341#390C2×D1111yescharacteristic
342#389—111no—
343#389—111no—
344#389—221no—
345#389—221no—
346#389C2×D1111yescharacteristic
347#386—21no—
348#382—111no—
349#382—221no—
350C4×C4⋊C4—221no—
351C4×C4⋊C4C2×D1111yescharacteristic
352#340—111no—
353#340—111no—
354#340—111no—
355#340—111no—
356#336—111no—
357#336—111no—
358#336—111no—
359#336—111no—
360C22×C11—42no—
361C22×C11—42no—
362C22×C11—42no—
363C22×C11—81no—
364C22×C11—81no—
365C22×C11—81no—
366C22×C11—81no—
367C22×C11—81no—
368C22×C11—81no—
369C22×C11—81no—
370C22×C11—81no—
371C22×C11—81no—
372C22×C11—81no—
373C22×C11—81no—
374C22×C11#52511yescharacteristic
375C22×C11D4211yescharacteristic
376C22×C11#54811yescharacteristic
377C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C11#57511yescharacteristic
379C22×C11#57711yescharacteristic
380C22×C11#57711yescharacteristic
381C4×C11—22no—
382C4×C11—41no—
383C4×C11—41no—
384C4×C11—41no—
385C4×C11—41no—
386C4×C11—41no—
387C4×C11—81no—
388C4×C11—81no—
389C4×C11C2×#12311yescharacteristic
390C4×C11#53911yescharacteristic
391C4×C11#54711yescharacteristic
392C4×C11#55311yescharacteristic
393Dic11—24no—
394Dic11—42no—
395Dic11—81no—
396Dic11—81no—
397C22×Q8—111no—
398C22×D4—21no—
399C22×D4—21no—
400C22×D4C22×D1111yescharacteristic
401C22×D4C22×D1111yescharacteristic
402C23×C4—21no—
403C2×SD16—114no—
404C2×SD16—222no—
405C2×SD16—222no—
406C2×SD16—222no—
407C2×SD16—222no—
408C2×SD16—222no—
409C22×C8—111no—
410C22×C8—111no—
411C22×C8—111no—
412C22×C8—111no—
413#127—22no—
414#127—41no—
415#123—44no—
416C4×Q8—222no—
417C4×Q8—222no—
418C4×Q8—222no—
419C4×Q8—222no—
420C4×Q8—441no—
421C4×Q8—441no—
422C4×D4—22no—
423C4×D4—41no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—111no—
426C2×C4⋊C4—111no—
427C2×C4⋊C4—111no—
428C2×C4⋊C4—221no—
429C2×C4⋊C4—221no—
430C2×C4⋊C4—221no—
431C2×C4⋊C4—221no—
432C2×C4⋊C4—221no—
433C2×C4⋊C4C22⋊D1111yescharacteristic
434C2×C4⋊C4C22⋊D1111yescharacteristic
435C2×C4⋊C4C22×D1111yescharacteristic
436C2×C4⋊C4C22×D1111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C42—221no—
445C2×C42—221no—
446C2×C42—221no—
447C2×C42C22×D1111yescharacteristic
448C2×C42C22×D1111yescharacteristic
449C2×C42C22×D1111yescharacteristic
450#107—222no—
451#106—441no—
452C4⋊C8—222no—
453#103—222no—
454#103—222no—
455#102—222no—
456#102—222no—
457#102—222no—
458#102—222no—
459#102—222no—
460#102—441no—
461#97—222no—
462#95—21no—
463#95—221no—
464#95—221no—
465#95—221no—
466#95—221no—
467#95C22⋊D1111yescharacteristic
468#95C22⋊D1111yescharacteristic
469C2×C11—82no—
470C2×C11—161no—
471C2×C11#244311yescharacteristic
472C2×C11#311511yescharacteristic
473C2×C11#312611yescharacteristic
474C2×C11#326511yescharacteristic
475C2×C11#327511yescharacteristic
476C2×C11#336811yescharacteristic
477C2×C11#337111yescharacteristic
478C24—21no—
479C24—21no—
480C2×Q8—112no—
481C2×Q8—112no—
482C2×Q8—112no—
483C2×Q8—221no—
484C2×Q8—221no—
485C2×Q8—221no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—81no—
502C2×D4—81no—
503C2×D4D4×D1112yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—81no—
509C22×C4—221no—
510C22×C4—221no—
511C22×C4—221no—
512C22×C4—221no—
513C22×C4C2×C22⋊D1111yescharacteristic
514C22×C4C2×C22⋊D1111yescharacteristic
515C22×C4C2×C22⋊D1111yescharacteristic
516C22×C4C2×C22⋊D1111yescharacteristic
517C22×C4C2×C22⋊D1111yescharacteristic
518C22×C4C2×C22⋊D1111yescharacteristic
519C22×C4C23×D1111yescharacteristic
520SD16—228no—
521SD16—444no—
522C2×C8—112no—
523C2×C8—221no—
524C2×C8—221no—
525C2×C8—221no—
526C2×C8—221no—
527C2×C8—221no—
528C2×C8—221no—
529C2×C8—221no—
530C2×C8—221no—
531C2×C8—221no—
532C2×C8—221no—
533C2×C8—221no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—41no—
540C4⋊C4—42no—
541C4⋊C4—112no—
542C4⋊C4—224no—
543C4⋊C4—221no—
544C4⋊C4—221no—
545C4⋊C4—221no—
546C4⋊C4—221no—
547C4⋊C4—221no—
548C4⋊C4—442no—
549C4⋊C4—442no—
550C4⋊C4—442no—
551C4⋊C4—442no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—81no—
557C22⋊C4—81no—
558C22⋊C4—81no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—224no—
566C42—442no—
567C42—442no—
568C11#2617511yescharacteristic, complemented, Sylow
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23C22×C22⋊D1111yescharacteristic
584Q8—114no—
585Q8—222no—
586Q8—222no—
587Q8—222no—
588D4—22no—
589D4—22no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—222no—
623C2×C4—222no—
624C2×C4—222no—
625C2×C4—222no—
626C2×C4—441no—
627C2×C4—441no—
628C2×C4—441no—
629C2×C4—441no—
630C2×C4—441no—
631C2×C4—441no—
632C2×C4—441no—
633C2×C4—441no—
634C2×C4#6698512yes—
635C2×C4C2×D4×D1111yescharacteristic
636C2×C4#6703711yescharacteristic
637C2×C4#41×D1111yescharacteristic
638C2×C4#41×D1111yescharacteristic
639C2×C4#41×D1111yescharacteristic
640C2×C4#6704311yescharacteristic
641C8—222no—
642C8—441no—
643C8—441no—
644C8—441no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#6698511yescharacteristic
660C22D4×C22⋊D1111yescharacteristic
661C22#1062335311yescharacteristic
662C22#1062335911yescharacteristic
663C22#1062336211yescharacteristic
664C22#1062336711yescharacteristic
665C22#1062336711yescharacteristic
666C4—22no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—81no—
673C4—81no—
674C4—224no—
675C4—442no—
676C4—881no—
677C4—881no—
678C4D11×#12311yescharacteristic
679C4#1062318611yescharacteristic
680C4#1062319111yescharacteristic
681C4#1062319211yescharacteristic
682C2—82no—
683C2—161no—
684C2#1307234811yescharacteristic
685C2#1307876111yescharacteristic
686C2#1307890211yescharacteristic
687C2#1308010711yescharacteristic
688C2#1308024311yescharacteristic
689C2#1308134111yescharacteristic
690C2#1308137811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2445C2
socleC23×C11C22×D4
prime radicalC2×C11×#127C22
hypercenter#2445D11
hypocenterC11#26175
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2445

Complements

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

NH ≅ G/Ntype
2. C2×#10622631C2semidirect
3. #13071536C2semidirect
4. #13070137C2semidirect
5. #13070107C2semidirect
6. #13070029C2semidirect
8. #13069316C2semidirect
9. #13069182C2semidirect
10. #13068974C2semidirect
11. #13068974C2semidirect
12. #13068974C2semidirect
13. #13068883C2semidirect
14. #13068877C2semidirect
310. Dic22#123semidirect
568. C11#26175semidirect

Maximal subgroups

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

subgroupindex
C2×#106226882
C2×#106226312
#130715362
#130701372
#130701072
#130700292
#130698472
#130693162
#130691822
#130689742
#130688832
#130688772
C11×#24452
#2617511