Finite Group Databasebeta

← #14756804

Subgroups — #14756804

Order 2816.

1042 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×#10622627C211yesmaximal, characteristic
2#13071080C211yesmaximal, characteristic, complemented
3#13069243C211yesmaximal, characteristic, complemented
4#13069236C211yesmaximal, characteristic, complemented
5#13069201C211yesmaximal, characteristic, complemented
6#13069194C211yesmaximal, characteristic, complemented
7#13068964C211yesmaximal, characteristic
8#13068964C211yesmaximal, characteristic
9#13068877C211yesmaximal, characteristic, complemented
10#13068863C211yesmaximal, characteristic, complemented
11#13068514C211yesmaximal, characteristic, complemented
12#13068444C211yesmaximal, characteristic, complemented
13#13068441C211yesmaximal, characteristic, complemented
14#13068365C211yesmaximal, characteristic, complemented
15C42×D4×C11C211yesmaximal, characteristic, complemented
16C2×C4×D4×C11C2211yescharacteristic
17C2×C4×D4×C11C2211yescharacteristic
18C2×C4×D4×C11C2211yescharacteristic
19C22×C42×C11—21no—
20C22×#4323C2211yescharacteristic
21C2×C4×D44C2211yescharacteristic
22C2×#66925—21no—
23C4×C11×C4⋊C4C2211yescharacteristic
24C4×C11×C4⋊C4C2211yescharacteristic
25C4×C11×C4⋊C4C2211yescharacteristic
26C4×C11×C22⋊C4—21no—
27C4×C11×C22⋊C4—21no—
28C4×C11×C22⋊C4—21no—
29C43×C11C2211yescharacteristic
30C2×#66896C2211yescharacteristic
31C2×#66896C2211yescharacteristic
32C2×#66896C2211yescharacteristic
33#10622627C418yescomplemented
34C2×#66873C2211yescharacteristic
35C2×#66873C2211yescharacteristic
36C2×#66873C2211yescharacteristic
37C2×#66872C2211yescharacteristic
38C2×#66872C2211yescharacteristic
39C2×#66871C2211yescharacteristic
40#10622553—21no—
41#10622553—21no—
42#10622552C2211yescharacteristic
43#10622552C2211yescharacteristic
44C4×(C2×D11)⋊C4—21no—
45C4×C4⋊Dic11C2211yescharacteristic
46#10622541C2211yescharacteristic
47#10622541C2211yescharacteristic
48C2×#66867C2211yescharacteristic
49C2×#66867C2211yescharacteristic
50#10622167C2211yescharacteristic
51#10622167C2211yescharacteristic
52#10622167C2211yescharacteristic
53#10622167C2211yescharacteristic
54#10622167C2211yescharacteristic
55#10622167C2211yescharacteristic
56#10622162C2211yescharacteristic
57#10622162C2211yescharacteristic
58#10622162C2211yescharacteristic
59#10622162C2211yescharacteristic
60C22×D4×C11C2311yescharacteristic
61C23×C4×C11—21no—
62C23×C4×C11—21no—
63C23×C4×C11—21no—
64C22×D44C2311yescharacteristic
65C22×C4×D11—21no—
66C4×D4×C11—22no—
67C4×D4×C11—22no—
68C4×D4×C11—22no—
69C4×D4×C11—22no—
70C4×D4×C11—22no—
71C4×D4×C11C2×C414yes—
72C2×C11×C4⋊C4C2311yescharacteristic
73C2×C11×C4⋊C4C2311yescharacteristic
74C2×C11×C4⋊C4C2311yescharacteristic
75C2×C11×C22⋊C4—21no—
76C2×C11×C22⋊C4—21no—
77C2×C11×C22⋊C4—21no—
78C2×C42×C11—21no—
79C2×C42×C11—21no—
80C2×C42×C11—21no—
81C2×C42×C11—44no—
82C2×C42×C11D411yescharacteristic
83C2×C42×C11D411yescharacteristic
84C2×C42×C11D411yescharacteristic
85C2×C42×C11D411yescharacteristic
86C2×C42×C11C2311yescharacteristic
87C2×C42×C11C2311yescharacteristic
88C2×C42×C11C2311yescharacteristic
89C2×#4323—22no—
90C2×#4323—22no—
91C2×#4323C2×C414yes—
92C2×(C2×D11)⋊C4—21no—
93C2×(C2×D11)⋊C4—21no—
94C2×(C2×D11)⋊C4—21no—
95C2×(C2×D11)⋊C4—21no—
96C2×(C2×D11)⋊C4—21no—
97C2×C4⋊Dic11C2311yescharacteristic
98C2×C4⋊Dic11C2311yescharacteristic
99C2×C4⋊Dic11C2311yescharacteristic
100C2×Dic11⋊C4—21no—
101C2×Dic11⋊C4—21no—
102C2×C4×Dic11—21no—
103C22×C11⋊C8C2311yescharacteristic
104C22×C11⋊C8C2311yescharacteristic
105C22×C11⋊C8C2311yescharacteristic
106C22×C11⋊C8C2311yescharacteristic
107C4×D44—22no—
108C4×D44C2×C414yescomplemented
109#66925—44no—
110C11×#95—21no—
111C11×#95—21no—
112C11×#95—21no—
113#66896—22no—
114#66896—22no—
115#66896C2×C414yes—
116C11⋊#95—21no—
117C11⋊#95—21no—
118C11⋊#95—21no—
119#66873—22no—
120#66873—22no—
121#66873C2×C414yescomplemented
122#66872—22no—
123#66872C2×C414yescomplemented
124#66871—22no—
125#66867—22no—
126#66867C2×C414yescomplemented
127#21122—111nomaximal, Sylow
128C24×C11—21no—
129C23×D11—21no—
130C2×D4×C11—21no—
131C2×D4×C11—21no—
132C2×D4×C11—21no—
133C2×D4×C11C22×C412yes—
134C2×D4×C11#4112yes—
135C2×D4×C11#4112yes—
136C22×C4×C11—24no—
137C22×C4×C11—24no—
138C22×C4×C11—21no—
139C22×C4×C11—21no—
140C22×C4×C11—21no—
141C22×C4×C11—24no—
142C22×C4×C11—42no—
143C22×C4×C11—42no—
144C22×C4×C11—42no—
145C22×C4×C11—42no—
146C22×C4×C11—42no—
147C22×C4×C11—42no—
148C22×C4×C11C2×D411yescharacteristic
149C22×C4×C11C2×D411yescharacteristic
150C22×C4×C11C2×D411yescharacteristic
151C22×C4×C11C2×D411yescharacteristic
152C22×C4×C11C2×D411yescharacteristic
153C22×C4×C11C2×D411yescharacteristic
154C22×C4×C11C2411yescharacteristic
155C22×Dic11—21no—
156C22×Dic11—21no—
157C22×Dic11—21no—
158C2×D44—21no—
159C2×D44—21no—
160C2×D44—21no—
161C2×D44—21no—
162C2×D44—21no—
163C2×D44C22×C412yes—
164C2×C4×D11—24no—
165C2×C4×D11—42no—
166C2×C4×D11—42no—
167C11×C4⋊C4—21no—
168C11×C4⋊C4—21no—
169C11×C4⋊C4—21no—
170C11×C4⋊C4C22×C412yes—
171C11×C4⋊C4#4112yes—
172C11×C4⋊C4#4112yes—
173C11×C22⋊C4—24no—
174C11×C22⋊C4—24no—
175C11×C22⋊C4—24no—
176C42×C11—22no—
177C42×C11—22no—
178C42×C11—21no—
179C42×C11—21no—
180C42×C11—21no—
181C42×C11—21no—
182C42×C11—41no—
183C42×C11—41no—
184C42×C11—42no—
185C42×C11—42no—
186C42×C11—42no—
187C42×C11C22×C412yes—
188C42×C11#4112yes—
189#4323—28no—
190(C2×D11)⋊C4—24no—
191(C2×D11)⋊C4—42no—
192(C2×D11)⋊C4—42no—
193(C2×D11)⋊C4—42no—
194(C2×D11)⋊C4—42no—
195C4⋊Dic11—21no—
196C4⋊Dic11—21no—
197C4⋊Dic11—21no—
198C4⋊Dic11—21no—
199C4⋊Dic11—21no—
200C4⋊Dic11C22×C412yes—
201Dic11⋊C4—42no—
202Dic11⋊C4—42no—
203C4×Dic11—42no—
204C2×C11⋊C8—21no—
205C2×C11⋊C8—21no—
206C2×C11⋊C8—21no—
207C2×C11⋊C8—21no—
208C2×C11⋊C8—21no—
209C2×C11⋊C8—21no—
210C2×C11⋊C8—21no—
211C2×C11⋊C8—21no—
212C2×C11⋊C8—21no—
213C2×C11⋊C8—21no—
214C2×C11⋊C8C22×C412yes—
215C2×C11⋊C8C22×C412yes—
216C2×#442—111no—
217#2293—111no—
218C42×D4D1111yescharacteristic, complemented
219#1926—111no—
220#1925—111no—
221#1920—111no—
222#1919—111no—
223#1879—111no—
224#1879—111no—
225#1864—111no—
226#1862—111no—
227#1776—111no—
228#1762—111no—
229#1762—111no—
230#1745—111no—
231C23×C11—22no—
232C23×C11—22no—
233C23×C11—22no—
234C23×C11—41no—
235C23×C11—41no—
236C23×C11—41no—
237C23×C11—41no—
238C23×C11C22×D411yescharacteristic
239C22×D11—22no—
240C22×D11—41no—
241C22×D11—41no—
242C22×D11—41no—
243C22×D11—41no—
244C22×D11—41no—
245C22×D11—41no—
246D4×C11—22no—
247D4×C11—22no—
248D4×C11—22no—
249D4×C11#11714yescomplemented
250C2×C4×C11—21no—
251C2×C4×C11—21no—
252C2×C4×C11—21no—
253C2×C4×C11—21no—
254C2×C4×C11—21no—
255C2×C4×C11—21no—
256C2×C4×C11—21no—
257C2×C4×C11—21no—
258C2×C4×C11—22no—
259C2×C4×C11—21no—
260C2×C4×C11—21no—
261C2×C4×C11—22no—
262C2×C4×C11—22no—
263C2×C4×C11—21no—
264C2×C4×C11—21no—
265C2×C4×C11—21no—
266C2×C4×C11—21no—
267C2×C4×C11—48no—
268C2×C4×C11—48no—
269C2×C4×C11—48no—
270C2×C4×C11—41no—
271C2×C4×C11—41no—
272C2×C4×C11—41no—
273C2×C4×C11—41no—
274C2×C4×C11—41no—
275C2×C4×C11—41no—
276C2×C4×C11C4×D412yes—
277C2×C4×C11C4×D412yes—
278C2×C4×C11AGL(1,ℤ/8)12yes—
279C2×C4×C11AGL(1,ℤ/8)12yes—
280C2×C4×C11C23×C411yescharacteristic
281C2×C4×C11C2×#4111yescharacteristic
282C2×C4×C11C2×#4111yescharacteristic
283C2×C4×C11C2×#4111yescharacteristic
284C2×C4×C11C2×#4111yescharacteristic
285C2×C4×C11C2×#4111yescharacteristic
286C2×Dic11—22no—
287C2×Dic11—41no—
288C2×Dic11—41no—
289C2×Dic11—41no—
290C2×Dic11—41no—
291C2×Dic11—41no—
292C2×Dic11—41no—
293C2×Dic11—41no—
294C2×Dic11—41no—
295D44—22no—
296D44—22no—
297D44—41no—
298D44—41no—
299C4×D11—48no—
300C11⋊C8—22no—
301C11⋊C8—22no—
302C11⋊C8—41no—
303C11⋊C8—41no—
304C22×D8—111no—
305C2×C4×D4—111no—
306C2×C4×D4C2×D1111yescharacteristic
307C2×C4×D4C2×D1111yescharacteristic
308C2×C4×D4C2×D1111yescharacteristic
309C2×#117—221no—
310C22×C42—21no—
311#442—118no—
312C2×#107—111no—
313C2×#106—111no—
314C2×#106—111no—
315C2×#102—111no—
316C2×#102—111no—
317C2×#102—111no—
318C2×#102—111no—
319C2×#102—111no—
320C2×#102—111no—
321C2×#97—111no—
322C2×#97—111no—
323#390—111no—
324#390—111no—
325#389—111no—
326#389—111no—
327#388—221no—
328#388—221no—
329C4×C4⋊C4—111no—
330C4×C4⋊C4C2×D1111yescharacteristic
331C4×C4⋊C4C2×D1111yescharacteristic
332C4×C4⋊C4C2×D1111yescharacteristic
333C4×C22⋊C4—21no—
334C4×C22⋊C4—21no—
335C4×C22⋊C4—21no—
336C4×C22⋊C4—221no—
337C43C2×D1111yescharacteristic
338#340—111no—
339#340—111no—
340#340—111no—
341#340—111no—
342#340—111no—
343#340—111no—
344#336—111no—
345#336—111no—
346#336—111no—
347#336—111no—
348C22×C11—24no—
349C22×C11—42no—
350C22×C11—42no—
351C22×C11—42no—
352C22×C11—42no—
353C22×C11—42no—
354C22×C11—42no—
355C22×C11C2×C4×D411yescharacteristic
356C22×C11#52511yescharacteristic
357C22×C11#52511yescharacteristic
358C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C11#57511yescharacteristic
361C22×C11#57511yescharacteristic
362C2×D11—42no—
363C2×D11—42no—
364C2×D11—42no—
365C2×D11—81no—
366C2×D11—81no—
367C2×D11—81no—
368C2×D11—81no—
369C4×C11—22no—
370C4×C11—22no—
371C4×C11—22no—
372C4×C11—22no—
373C4×C11—42no—
374C4×C11—41no—
375C4×C11—42no—
376C4×C11—41no—
377C4×C11—42no—
378C4×C11C2×#11711yescharacteristic
379C4×C11C4×#4111yescharacteristic
380C4×C11#52911yescharacteristic
381C4×C11#52911yescharacteristic
382Dic11—42no—
383Dic11—81no—
384Dic11—81no—
385C22×D4—111no—
386C22×D4C22×D1111yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—221no—
391C2×D8—114no—
392C2×D8—222no—
393C2×D8—222no—
394C22×C8—111no—
395C22×C8—111no—
396C22×C8—111no—
397C22×C8—111no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—114no—
404C4×D4—222no—
405C4×D4C4×D1114yes—
406#117—444no—
407C2×C4⋊C4—111no—
408C2×C4⋊C4—111no—
409C2×C4⋊C4—111no—
410C2×C4⋊C4—221no—
411C2×C4⋊C4—221no—
412C2×C4⋊C4C22×D1111yescharacteristic
413C2×C4⋊C4C22×D1111yescharacteristic
414C2×C4⋊C4C22×D1111yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—221no—
419C2×C22⋊C4—221no—
420C2×C22⋊C4—221no—
421C2×C22⋊C4—221no—
422C2×C22⋊C4—221no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—44no—
427C2×C42—221no—
428C2×C42C22⋊D1111yescharacteristic
429C2×C42C22⋊D1111yescharacteristic
430C2×C42C22⋊D1111yescharacteristic
431C2×C42C22⋊D1111yescharacteristic
432C2×C42C22×D1111yescharacteristic
433C2×C42C22×D1111yescharacteristic
434C2×C42C22×D1111yescharacteristic
435#107—222no—
436#106—114no—
437#106—222no—
438#102—114no—
439#102—114no—
440#102—222no—
441#102—222no—
442#102—222no—
443#102—222no—
444#97—114no—
445#97—222no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—221no—
450#95—221no—
451#95—221no—
452C2×C11—44no—
453C2×C11#229011yescharacteristic
454C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
455C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
456C2×C11#310911yescharacteristic
457C2×C11#311111yescharacteristic
458C2×C11#311411yescharacteristic
459C2×C11#311511yescharacteristic
460D11—82no—
461C24—21no—
462C24—221no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—112no—
467C2×D4—221no—
468C2×D4—221no—
469C2×D4—221no—
470C2×D4—221no—
471C2×D4—221no—
472C2×D4C2×C4×D1112yes—
473C2×D4#433912yes—
474C2×D4#433912yes—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—221no—
488C22×C4—224no—
489C22×C4—221no—
490C22×C4—221no—
491C22×C4—442no—
492C22×C4—442no—
493C22×C4C2×C22⋊D1111yescharacteristic
494C22×C4C2×C22⋊D1111yescharacteristic
495C22×C4C2×C22⋊D1111yescharacteristic
496C22×C4C2×C22⋊D1111yescharacteristic
497C22×C4C2×C22⋊D1111yescharacteristic
498C22×C4C2×C22⋊D1111yescharacteristic
499C22×C4C23×D1111yescharacteristic
500D8—228no—
501C2×C8—112no—
502C2×C8—112no—
503C2×C8—221no—
504C2×C8—221no—
505C2×C8—221no—
506C2×C8—221no—
507C2×C8—221no—
508C2×C8—221no—
509C2×C8—221no—
510C2×C8—221no—
511C2×C8—221no—
512C2×C8—221no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—112no—
517C4⋊C4—221no—
518C4⋊C4—221no—
519C4⋊C4—221no—
520C4⋊C4—221no—
521C4⋊C4—221no—
522C4⋊C4—442no—
523C4⋊C4—442no—
524C4⋊C4C2×C4×D1112yes—
525C4⋊C4#433912yes—
526C4⋊C4#433912yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—224no—
531C22⋊C4—442no—
532C22⋊C4—442no—
533C22⋊C4—442no—
534C22⋊C4—442no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—41no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—442no—
547C42C2×C4×D1112yes—
548C42#433912yes—
549C11#2112211yescharacteristic, complemented, Sylow
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—222no—
558C23—441no—
559C23—441no—
560C23—441no—
561C23—441no—
562C23—441no—
563C23—441no—
564C23C22×C22⋊D1111yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—222no—
569D4—222no—
570D4—441no—
571D4—441no—
572D4#6692514yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—222no—
600C2×C4—448no—
601C2×C4—441no—
602C2×C4—441no—
603C2×C4—441no—
604C2×C4—441no—
605C2×C4—441no—
606C2×C4—441no—
607C2×C4—441no—
608C2×C4—441no—
609C2×C4C4×C22⋊D1112yes—
610C2×C4C4×C22⋊D1112yes—
611C2×C4#6698512yes—
612C2×C4#6700212yes—
613C2×C4C22×C4×D1111yescharacteristic
614C2×C4C2×#433911yescharacteristic
615C2×C4C2×#433911yescharacteristic
616C2×C4C2×#433911yescharacteristic
617C2×C4C2×#433911yescharacteristic
618C2×C4C2×#433911yescharacteristic
619C8—222no—
620C8—222no—
621C8—441no—
622C8—441no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—442no—
631C22—442no—
632C22—442no—
633C22—881no—
634C22—881no—
635C22—881no—
636C22—881no—
637C22C2×C4×C22⋊D1111yescharacteristic
638C22#1062332211yescharacteristic
639C22#1062332211yescharacteristic
640C22C2×#6698511yescharacteristic
641C22C2×#6700211yescharacteristic
642C22#1062335311yescharacteristic
643C22#1062335311yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—442no—
654C4—881no—
655C4—881no—
656C4C2×#6692511yescharacteristic
657C4C4×#433911yescharacteristic
658C4#1062301011yescharacteristic
659C4#1062301011yescharacteristic
660C2—44no—
661C2—882no—
662C2#1307105911yescharacteristic
663C2C4×#6700211yescharacteristic
664C2C4×#6698511yescharacteristic
665C2#1307870711yescharacteristic
666C2#1307872511yescharacteristic
667C2#1307875411yescharacteristic
668C2#1307876711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC42×D4×C11C2
socleC23×C11C22×D4
prime radicalC22×#4323C22
cosocleC22×C4×C11C24
hypercenterC42×D4D11
hypocenterC11#21122
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #13071080C2semidirect
3. #13069243C2semidirect
4. #13069236C2semidirect
5. #13069201C2semidirect
6. #13069194C2semidirect
9. #13068877C2semidirect
10. #13068863C2semidirect
11. #13068514C2semidirect
12. #13068444C2semidirect
13. #13068441C2semidirect
14. #13068365C2semidirect
15. C42×D4×C11C2semidirect
33. #10622627C4semidirect
108. C4×D44C2×C4semidirect
121. #66873C2×C4semidirect
123. #66872C2×C4semidirect
126. #66867C2×C4semidirect
218. C42×D4D11semidirect
249. D4×C11#117semidirect
549. C11#21122semidirect
572. D4#66925semidirect

Maximal subgroups

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

subgroupindex
C2×#106226272
#130710802
#130692432
#130692362
#130692012
#130691942
#130689642
#130688772
#130688632
#130685142
#130684442
#130684412
#130683652
C42×D4×C112
#2112211