Finite Group Databasebeta

← #14790146

Subgroups — #14790146

Order 2816.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D44C211yesmaximal, characteristic, complemented
2C2×#10622620C211yesmaximal, characteristic, complemented
3C2×C4×#4323C211yesmaximal, characteristic
4#13069944C212yesmaximal, complemented
5#13069752C212yesmaximal, complemented
6#13069576C211yesmaximal, characteristic, complemented
7#13069553C211yesmaximal, characteristic, complemented
8#13068967C212yesmaximal
9#13068862C212yesmaximal, complemented
10#13068613C211yesmaximal, characteristic, complemented
11C11×#2538C211yesmaximal, characteristic, complemented
12C22×Q8×D11—21no—
13C2×C11×C22⋊Q8—21no—
14C2×C4×Q8×C11C2211yescharacteristic
15C2×C4×D4×C11C2211yescharacteristic
16C22×#4323C2211yescharacteristic
17Q8×D44—24no—
18Q8×D44—24no—
19C2×#66950—21no—
20C2×#66950—22no—
21C2×C4×D44C2212yes—
22C2×C4×D44C2211yescharacteristic
23C2×#66922C2212yes—
24C2×#66922C2211yescharacteristic
25C11×#396—22no—
26C11×#390C2212yes—
27C11×#389C2212yes—
28C11×#384C2211yescharacteristic
29C11×#380—21no—
30C2×#66896C2212yes—
31C2×#66896C2211yescharacteristic
32C4×#4323—24no—
33#10622620—24no—
34C2×#66873C2211yescharacteristic
35C2×#66873C2212yes—
36C2×#66871C2212yes—
37C2×#66871C2211yescharacteristic
38C2×C4⋊(C11⋊C8)C2211yescharacteristic
39C2×C4×C11⋊C8C2211yescharacteristic
40#10622167C2212yes—
41#10622167C2212yes—
42#10622167C2212yes—
43#10622162C2212yes—
44#10622162C2212yes—
45C22×Q8×C11D411yescharacteristic
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11—21no—
48C2×Q8×D11—28no—
49C2×Q8×D11—44no—
50C2×Q8×D11—44no—
51C22×D44C2311yescharacteristic
52C22×C4×D11—22no—
53C22×C4×D11—21no—
54C22×Dic22—22no—
55C22×Dic22—21no—
56C11×C22⋊Q8—44no—
57C4×Q8×C11—22no—
58C4×Q8×C11—22no—
59C4×D4×C11—22no—
60C4×D4×C11—22no—
61C2×C11×C4⋊C4—21no—
62C2×C11×C4⋊C4—22no—
63C2×C11×C4⋊C4—21no—
64C2×C11×C4⋊C4D411yescharacteristic
65C2×C11×C4⋊C4D412yes—
66C2×C11×C4⋊C4C2311yescharacteristic
67C2×C11×C4⋊C4C2312yes—
68C2×C11×C22⋊C4—21no—
69C2×C11×C22⋊C4—22no—
70C2×C11×C22⋊C4—22no—
71C2×C42×C11C2312yes—
72C2×C42×C11C2311yescharacteristic
73C2×#4323—22no—
74C2×#4323—22no—
75C2×#4323Q814yescomplemented
76C2×(C2×D11)⋊C4—22no—
77C2×(C2×D11)⋊C4—21no—
78C2×C4⋊Dic11—22no—
79C2×C4⋊Dic11—21no—
80C2×C4⋊Dic11C2311yescharacteristic
81C2×C4⋊Dic11C2312yes—
82C22×C11⋊C8C2311yescharacteristic
83C22×C11⋊C8C2312yes—
84C22×C11⋊C8C2311yescharacteristic
85#66950—28no—
86#66950—216no—
87C4×D44—24no—
88C4×D44—22no—
89C4×D44—24no—
90C4×D44D414yes—
91#66922—22no—
92#66922—24no—
93#66922—24no—
94#66922D414yes—
95C11×#95—22no—
96#66896—24no—
97#66896—22no—
98#66873—22no—
99#66873—24no—
100#66871—24no—
101#66871—22no—
102C4⋊(C11⋊C8)—22no—
103C4×C11⋊C8—22no—
104#24326—111nomaximal, Sylow
105C24×C11—21no—
106C23×D11—21no—
107C2×Q8×C11—22no—
108C2×Q8×C11—22no—
109C2×Q8×C11D814yescomplemented
110C2×D4×C11—21no—
111C2×D4×C11—21no—
112C2×D4×C11—21no—
113C2×D4×C11—21no—
114C2×D4×C11—21no—
115C2×D4×C11C2×Q812yescomplemented
116C22×C4×C11—21no—
117C22×C4×C11—22no—
118C22×C4×C11—42no—
119C22×C4×C11—42no—
120C22×C4×C11—42no—
121C22×C4×C11C2×D412yes—
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2×D411yescharacteristic
125C22×C4×C11C2411yescharacteristic
126C22×Dic11—21no—
127C22×Dic11—22no—
128Q8×D11—432no—
129C2×D44—21no—
130C2×D44—21no—
131C2×D44—21no—
132C2×D44C2×D412yes—
133C2×D44C2×Q812yes—
134C2×D44#4112yes—
135C2×C4×D11—28no—
136C2×C4×D11—24no—
137C2×C4×D11—44no—
138C2×C4×D11—44no—
139C2×C4×D11—42no—
140C2×C4×D11—42no—
141C2×Dic22—28no—
142C2×Dic22—24no—
143C2×Dic22—44no—
144C2×Dic22—44no—
145C2×Dic22—42no—
146C2×Dic22—42no—
147C11×C4⋊C4—24no—
148C11×C4⋊C4—22no—
149C11×C4⋊C4—21no—
150C11×C4⋊C4—21no—
151C11×C4⋊C4—22no—
152C11×C4⋊C4—22no—
153C11×C4⋊C4—42no—
154C11×C4⋊C4—44no—
155C11×C4⋊C4—81no—
156C11×C22⋊C4—42no—
157C11×C22⋊C4—44no—
158C11×C22⋊C4—82no—
159C42×C11—22no—
160C42×C11—22no—
161C42×C11C2×D412yes—
162C42×C11C2×D412yes—
163#4323—28no—
164(C2×D11)⋊C4—28no—
165(C2×D11)⋊C4—24no—
166C4⋊Dic11—28no—
167C4⋊Dic11—24no—
168C4⋊Dic11—22no—
169C4⋊Dic11—21no—
170C4⋊Dic11C2×D412yes—
171C4⋊Dic11C2×D414yes—
172C2×C11⋊C8—21no—
173C2×C11⋊C8—21no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—21no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—21no—
180C2×C11⋊C8C2×Q812yescomplemented
181C2×D4×Q8—111no—
182C2×#474—111no—
183C2×C4×D8—111no—
184#2538D1111yescharacteristic, complemented
185#2057—112no—
186#2023—112no—
187#1990—111no—
188#1984—111no—
189#1879—112no—
190#1862—112no—
191#1802—111no—
192C23×C11—22no—
193C23×C11—41no—
194C23×C11—41no—
195C23×C11—41no—
196C23×C11—41no—
197C23×C11—41no—
198C23×C11—41no—
199C23×C11C22×D411yescharacteristic
200C22×D11—22no—
201C22×D11—22no—
202C22×D11—22no—
203C22×D11—41no—
204C22×D11—41no—
205C22×D11—41no—
206C22×D11—41no—
207Q8×C11—28no—
208D4×C11—22no—
209D4×C11—22no—
210D4×C11—41no—
211D4×C11—41no—
212C2×C4×C11—22no—
213C2×C4×C11—22no—
214C2×C4×C11—21no—
215C2×C4×C11—21no—
216C2×C4×C11—22no—
217C2×C4×C11—22no—
218C2×C4×C11—22no—
219C2×C4×C11—41no—
220C2×C4×C11—42no—
221C2×C4×C11—42no—
222C2×C4×C11—41no—
223C2×C4×C11—41no—
224C2×C4×C11—42no—
225C2×C4×C11—84no—
226C2×C4×C11#12012yes—
227C2×C4×C11#12012yes—
228C2×C4×C11C2×D814yes—
229C2×C4×C11C2×D812yes—
230C2×C4×C11AGL(1,ℤ/8)12yes—
231C2×C4×C11C22×D411yescharacteristic
232C2×C4×C11C22×D411yescharacteristic
233C2×C4×C11C22×Q811yescharacteristic
234C2×C4×C11C2×#4111yescharacteristic
235C2×C4×C112−1+411yescharacteristic
236C2×C4×C112−1+411yescharacteristic
237C2×Dic11—22no—
238C2×Dic11—24no—
239C2×Dic11—41no—
240C2×Dic11—42no—
241C2×Dic11—41no—
242C2×Dic11—42no—
243D44—22no—
244D44—22no—
245D44—22no—
246D44C22⋊Q814yescomplemented
247C4×D11—416no—
248C4×D11—48no—
249Dic22—416no—
250Dic22—48no—
251C11⋊C8—22no—
252C11⋊C8—41no—
253C11⋊C8—42no—
254C23×Q8—221no—
255C22×D8—111no—
256D4×Q8—224no—
257D4×Q8—224no—
258C2×C4⋊Q8—111no—
259C2×C4⋊Q8—112no—
260C2×C22⋊Q8—21no—
261C2×C22⋊Q8—221no—
262C2×C22⋊Q8—222no—
263C2×C4×Q8C2×D1111yescharacteristic
264C2×C4×D4—111no—
265C2×C4×D4—112no—
266C2×C4×D4C2×D1111yescharacteristic
267#474—224no—
268C4×D8—224no—
269C2×#107—111no—
270C2×#107—112no—
271C2×C4⋊C8—111no—
272C2×#102—112no—
273C2×#102—111no—
274C2×#102—112no—
275C2×#102—111no—
276C2×C4×C8—111no—
277#396—22no—
278#390C2×D1112yes—
279#389C2×D1112yes—
280#384C2×D1111yescharacteristic
281#380—21no—
282#340—112no—
283#340—112no—
284#340—112no—
285#336—112no—
286#336—112no—
287C22×C11—42no—
288C22×C11—42no—
289C22×C11—42no—
290C22×C11—81no—
291C22×C11—81no—
292C22×C11—81no—
293C22×C11—81no—
294C22×C11C2×#12011yescharacteristic
295C22×C11#54811yescharacteristic
296C22×C11D4×Q811yescharacteristic
297C22×C11C22×D811yescharacteristic
298C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C11#57711yescharacteristic
300C22×C11#57811yescharacteristic
301C2×D11—24no—
302C2×D11—42no—
303C2×D11—42no—
304C2×D11—42no—
305C2×D11—42no—
306C2×D11—42no—
307C2×D11—42no—
308C4×C11—24no—
309C4×C11—22no—
310C4×C11—42no—
311C4×C11—82no—
312C4×C11—81no—
313C4×C11#44714yes—
314C4×C11C2×C22⋊Q811yescharacteristic
315C4×C11#53611yescharacteristic
316C4×C11#54811yescharacteristic
317C4×C11D4×Q811yescharacteristic
318Dic11—42no—
319Dic11—44no—
320C22×Q8—228no—
321C22×Q8—222no—
322C22×Q8—221no—
323C22×Q8—444no—
324C22×Q8—444no—
325C22×Q8C22⋊D1111yescharacteristic
326C22×D4—111no—
327C22×D4C22×D1111yescharacteristic
328C23×C4—21no—
329C23×C4—222no—
330C23×C4—221no—
331C2×D8—114no—
332C2×D8—222no—
333C2×D8—222no—
334C22×C8—111no—
335C22×C8—111no—
336C22×C8—112no—
337C4⋊Q8—114no—
338C4⋊Q8—224no—
339C4⋊Q8—224no—
340C4⋊Q8—222no—
341C22⋊Q8—44no—
342C22⋊Q8—228no—
343C22⋊Q8—2216no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—114no—
349C4×D4—224no—
350C4×D4—224no—
351C4×D4—222no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—111no—
356C2×C4⋊C4—112no—
357C2×C4⋊C4—222no—
358C2×C4⋊C4—221no—
359C2×C4⋊C4C22⋊D1111yescharacteristic
360C2×C4⋊C4C22⋊D1112yes—
361C2×C4⋊C4C22×D1111yescharacteristic
362C2×C4⋊C4C22×D1112yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—222no—
367C2×C22⋊C4—221no—
368C2×C42C22×D1112yes—
369C2×C42C22×D1111yescharacteristic
370#107—222no—
371#107—224no—
372C4⋊C8—222no—
373#102—224no—
374#102—222no—
375#102—222no—
376#102—224no—
377C4×C8—222no—
378#95—22no—
379C2×C11—82no—
380C2×C11#240911yescharacteristic
381C2×C11C2×#44711yescharacteristic
382C2×C11#301311yescharacteristic
383C2×C11#335811yescharacteristic
384C2×C11#336711yescharacteristic
385C2×C11Q8×D811yescharacteristic
386C2×C11#338411yescharacteristic
387D11—44no—
388C24—21no—
389C24—221no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—228no—
393C2×Q8—224no—
394C2×Q8—444no—
395C2×Q8—4432no—
396C2×Q8—444no—
397C2×Q8—442no—
398C2×Q8—442no—
399C2×Q8#432314yescomplemented
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—112no—
406C2×D4—112no—
407C2×D4—112no—
408C2×D4—221no—
409C2×D4—221no—
410C2×D4—221no—
411C2×D4Q8×D1112yescomplemented
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—228no—
418C22×C4—222no—
419C22×C4—221no—
420C22×C4—224no—
421C22×C4—444no—
422C22×C4—444no—
423C22×C4—442no—
424C22×C4—442no—
425C22×C4C2×C22⋊D1112yes—
426C22×C4C2×C22⋊D1112yes—
427C22×C4C2×C22⋊D1111yescharacteristic
428C22×C4C2×C22⋊D1111yescharacteristic
429C22×C4C23×D1111yescharacteristic
430D8—228no—
431C2×C8—112no—
432C2×C8—221no—
433C2×C8—221no—
434C2×C8—221no—
435C2×C8—222no—
436C2×C8—221no—
437C2×C8—222no—
438C2×C8—222no—
439C2×C8—221no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—81no—
449C4⋊C4—112no—
450C4⋊C4—114no—
451C4⋊C4—228no—
452C4⋊C4—224no—
453C4⋊C4—222no—
454C4⋊C4—221no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—82no—
458C22⋊C4—228no—
459C22⋊C4—224no—
460C42—22no—
461C42—22no—
462C42D4×D1112yes—
463C42D4×D1112yes—
464C11#2432611yescharacteristic, complemented, Sylow
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—222no—
473C23—222no—
474C23—222no—
475C23—441no—
476C23—441no—
477C23—441no—
478C23—441no—
479C23C22×C22⋊D1111yescharacteristic
480Q8—28no—
481Q8—4416no—
482Q8—448no—
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—114no—
488D4—222no—
489D4—222no—
490D4—222no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—224no—
506C2×C4—222no—
507C2×C4—4416no—
508C2×C4—442no—
509C2×C4—441no—
510C2×C4—442no—
511C2×C4—441no—
512C2×C4—448no—
513C2×C4C2×#432314yes—
514C2×C4C2×#432312yes—
515C2×C4#6699012yes—
516C2×C4#6699012yes—
517C2×C4#6700212yes—
518C2×C4C2×D4×D1111yescharacteristic
519C2×C4C2×D4×D1111yescharacteristic
520C2×C4C2×Q8×D1111yescharacteristic
521C2×C4#6704011yescharacteristic
522C2×C4#41×D1111yescharacteristic
523C2×C4#6704311yescharacteristic
524C8—222no—
525C8—441no—
526C8—442no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—224no—
535C22—442no—
536C22—442no—
537C22—442no—
538C22—442no—
539C22—442no—
540C22—442no—
541C22C22×#432311yescharacteristic
542C22C2×#6699011yescharacteristic
543C22Q8×C22⋊D1111yescharacteristic
544C22C2×#6700211yescharacteristic
545C22#1062336211yescharacteristic
546C22#1062336711yescharacteristic
547C22#1062336911yescharacteristic
548C4—24no—
549C4—22no—
550C4—42no—
551C4—81no—
552C4—82no—
553C4—444no—
554C4—442no—
555C4#1062264914yes—
556C4D11×C22⋊Q811yescharacteristic
557C4#1062315911yescharacteristic
558C4#1062316311yescharacteristic
559C4#1062316411yescharacteristic
560C2—82no—
561C2—444no—
562C2#1307197711yescharacteristic
563C2C2×#1062264911yescharacteristic
564C2#1307792811yescharacteristic
565C2#1308124711yescharacteristic
566C2#1308133011yescharacteristic
567C2Q8×#432311yescharacteristic
568C2#1308151511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2538C2
socleC23×C11C22×D4
prime radicalC22×#4323C22
hypercenter#2538D11
hypocenterC11#24326
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D44C2semidirect
2. C2×#10622620C2semidirect
4. #13069944C2semidirect
5. #13069752C2semidirect
6. #13069576C2semidirect
7. #13069553C2semidirect
9. #13068862C2semidirect
10. #13068613C2semidirect
11. C11×#2538C2semidirect
75. C2×#4323Q8semidirect
109. C2×Q8×C11D8semidirect
115. C2×D4×C11C2×Q8semidirect
180. C2×C11⋊C8C2×Q8semidirect
184. #2538D11semidirect
246. D44C22⋊Q8semidirect
399. C2×Q8#4323semidirect
411. C2×D4Q8×D11semidirect
464. C11#24326semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D442
C2×#106226202
C2×C4×#43232
#130699442
#130697522
#130695762
#130695532
#130689672
#130688622
#130686132
C11×#25382
#2432611