Finite Group Databasebeta

← #14732260

Subgroups — #14732260

Order 2816.

1020 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13079541C211yesmaximal, characteristic, complemented
2#13078647C211yesmaximal, characteristic, complemented
3#13078080C211yesmaximal, characteristic
4#13067823C214yesmaximal, complemented
5#13067688C214yesmaximal, complemented
6#13067297C211yesmaximal, characteristic, complemented
7#13067045C211yesmaximal, characteristic, complemented
8#13067007C211yesmaximal, characteristic, complemented
9C2×C11×#554C211yesmaximal, characteristic, complemented
10C11×#554—22no—
11C11×#554—22no—
12C11×#554—22no—
13C11×#554C2214yes—
14C2×C11×C4⋊Q8C2211yescharacteristic
15C2×C11×#125—21no—
16C2×C11×C22⋊Q8—21no—
17C2×C11×C22⋊Q8—22no—
18C2×C4×Q8×C11C2211yescharacteristic
19C2×C4×D4×C11—21no—
20C2×C4×D4×C11C2211yescharacteristic
21C22×C11×C4⋊C4—21no—
22C2×#66985—21no—
23#10623008C2211yescharacteristic
24#10622840—21no—
25#10622840—22no—
26#10622698C2212yescomplemented
27#10622696—22no—
28#10622694C2212yescomplemented
29#10622640C2212yescomplemented
30C4×#4325—22no—
31#10622633C2212yescomplemented
32#10622632C2212yes—
33#10622629—22no—
34#10622628C2212yes—
35#10622581—21no—
36#10622579—21no—
37#10622538C2211yescharacteristic
38#10622538C2211yescharacteristic
39#10622537C2211yescharacteristic
40#10622184C2214yescomplemented
41Q8⋊(C11⋊C8)C2214yescomplemented
42D4⋊(C11⋊C8)C2214yes—
43C22×Q8×C11D411yescharacteristic
44C22×Q8×C11D412yes—
45C22×D4×C11D411yescharacteristic
46C23×C4×C11—22no—
47C23×C4×C11—21no—
48C2×#4339—21no—
49C11×C4⋊Q8—21no—
50C11×C4⋊Q8—21no—
51C11×C4⋊Q8D412yes—
52C11×C4⋊Q8C2312yes—
53C11×#125—24no—
54C11×#125—24no—
55C11×C22⋊Q8—24no—
56C11×C22⋊Q8—28no—
57C11×C22⋊Q8—24no—
58C11×C22⋊Q8—28no—
59C4×Q8×C11—21no—
60C4×Q8×C11—21no—
61C4×Q8×C11D412yes—
62C4×Q8×C11C2312yes—
63C4×D4×C11—24no—
64C4×D4×C11—24no—
65C4×D4×C11—21no—
66C4×D4×C11—21no—
67C4×D4×C11D412yes—
68C4×D4×C11C2312yes—
69C2×C11×C4⋊C4—22no—
70C2×C11×C4⋊C4—24no—
71C2×C11×C4⋊C4—22no—
72C2×C11×C4⋊C4—21no—
73C2×C11×C4⋊C4—21no—
74C2×C11×C4⋊C4—21no—
75C2×C11×C4⋊C4—24no—
76C2×C11×C4⋊C4D411yescharacteristic
77C2×C11×C4⋊C4D411yescharacteristic
78C2×C11×C22⋊C4—22no—
79C2×C11×C22⋊C4—21no—
80C2×C42×C11—21no—
81C2×C42×C11C2311yescharacteristic
82C2×#4325—22no—
83C2×#4325—24no—
84C2×#4324—22no—
85#66985—44no—
86C2×#4323—22no—
87#66983—21no—
88#66983—21no—
89C2×#4318—21no—
90#66928—21no—
91#66927C2311yescharacteristic
92C4×D44—21no—
93C4×D44—21no—
94#66922C2311yescharacteristic
95#66899—22no—
96#66899—24no—
97C22⋊(C11⋊C8)—21no—
98C22⋊(C11⋊C8)—21no—
99#66874—22no—
100#66873—22no—
101#66873—22no—
102#66872—22no—
103C4⋊(C11⋊C8)—21no—
104C4⋊(C11⋊C8)—21no—
105C4⋊(C11⋊C8)C2312yes—
106C4⋊(C11⋊C8)C2312yes—
107C4×C11⋊C8—21no—
108C4×C11⋊C8C2312yes—
109#18748—111nomaximal, Sylow
110C24×C11—21no—
111C2×Q8×C11—21no—
112C2×Q8×C11—22no—
113C2×Q8×C11—42no—
114C2×Q8×C11—82no—
115C2×Q8×C11C2×D412yes—
116C2×Q8×C11C2×D414yes—
117C2×D4×C11—21no—
118C2×D4×C11—22no—
119C2×D4×C11SD1614yescomplemented
120C2×D4×C11C2×D412yes—
121C22×C4×C11—21no—
122C22×C4×C11—21no—
123C22×C4×C11—21no—
124C22×C4×C11—22no—
125C22×C4×C11—28no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—44no—
129C22×C4×C11—84no—
130C22×C4×C11C2×D411yescharacteristic
131C22×C4×C11C2×D411yescharacteristic
132C22×C4×C11C2×D411yescharacteristic
133C2×C22⋊D11—41no—
134#4339—42no—
135#4339—42no—
136C2×D44—21no—
137C2×D44—21no—
138C2×D44—21no—
139C2×C4×D11—41no—
140C2×Dic22—21no—
141C11×C4⋊C4—28no—
142C11×C4⋊C4—28no—
143C11×C4⋊C4—21no—
144C11×C4⋊C4—22no—
145C11×C4⋊C4—22no—
146C11×C4⋊C4—22no—
147C11×C4⋊C4—21no—
148C11×C4⋊C4—22no—
149C11×C4⋊C4—22no—
150C11×C4⋊C4—22no—
151C11×C4⋊C4C2×D412yes—
152C11×C4⋊C4C2×D412yes—
153C11×C22⋊C4—28no—
154C11×C22⋊C4—22no—
155C11×C22⋊C4—22no—
156C42×C11—22no—
157C42×C11—22no—
158C42×C11C2×D411yescharacteristic
159C42×C11C2×D411yescharacteristic
160C42×C11C2×D411yescharacteristic
161C42×C11C2411yescharacteristic
162#4325—44no—
163#4325—84no—
164#4324—44no—
165#4323—44no—
166(C2×D11)⋊C4—41no—
167(C2×D11)⋊C4—41no—
168C4⋊Dic11—21no—
169C4⋊Dic11—21no—
170#4318—42no—
171C2×C11⋊C8—22no—
172C2×C11⋊C8—22no—
173C2×C11⋊C8—22no—
174C2×#554D1111yescharacteristic, complemented
175#3206—111no—
176#3103—111no—
177#3036—111no—
178#1643—114no—
179#1622—114no—
180#1548—111no—
181#1487—111no—
182#1478—111no—
183C23×C11—22no—
184C23×C11—24no—
185C23×C11—42no—
186C23×C11—42no—
187C23×C11#12011yescharacteristic
188C22×D11—41no—
189Q8×C11—22no—
190Q8×C11—22no—
191Q8×C11—44no—
192Q8×C11—42no—
193Q8×C11—82no—
194D4×C11—22no—
195D4×C11—24no—
196D4×C11C2×SD1614yes—
197C2×C4×C11—22no—
198C2×C4×C11—22no—
199C2×C4×C11—22no—
200C2×C4×C11—22no—
201C2×C4×C11—24no—
202C2×C4×C11—41no—
203C2×C4×C11—44no—
204C2×C4×C11—42no—
205C2×C4×C11—42no—
206C2×C4×C11—44no—
207C2×C4×C11—42no—
208C2×C4×C11—44no—
209C2×C4×C11—88no—
210C2×C4×C11#12011yescharacteristic
211C2×C4×C11#12011yescharacteristic
212C2×C4×C11#12011yescharacteristic
213C2×C4×C11C2×SD1612yes—
214C2×C4×C11#13712yes—
215C2×C4×C11C22×D411yescharacteristic
216C2×C4×C11C22×D411yescharacteristic
217C2×C4×C11C22×D411yescharacteristic
218C22⋊D11—82no—
219C2×Dic11—41no—
220D44—42no—
221D44—41no—
222D44—41no—
223D44—42no—
224C4×D11—82no—
225Dic22—41no—
226Dic22—41no—
227C11⋊C8—42no—
228C11⋊C8—42no—
229C11⋊C8—42no—
230C2×AGL(1,ℤ/8)—221no—
231#554—22no—
232#554—22no—
233#554—22no—
234#554C2×D1114yes—
235#532—111no—
236C2×C4⋊Q8C2×D1111yescharacteristic
237C2×#125—21no—
238C2×C22⋊Q8—21no—
239C2×C22⋊Q8—22no—
240C2×C4×Q8C2×D1111yescharacteristic
241C2×C4×D4—21no—
242C2×C4×D4C2×D1111yescharacteristic
243C22×C4⋊C4—21no—
244#492—112no—
245#487—112no—
246#481—221no—
247#475—112no—
248#463—222no—
249#461—112no—
250#461—222no—
251#460—112no—
252#459—112no—
253#448—221no—
254#448—222no—
255C4×SD16—222no—
256#431—111no—
257#431—111no—
258#423—111no—
259#417—221no—
260#333—114no—
261Q8⋊C8—114no—
262D4⋊C8—114no—
263C22×C11—21no—
264C22×C11—21no—
265C22×C11—24no—
266C22×C11—48no—
267C22×C11—42no—
268C22×C11—42no—
269C22×C11#44812yes—
270C22×C11C2×#12011yescharacteristic
271C2×D11—81no—
272C2×D11—81no—
273C2×D11—81no—
274C4×C11—21no—
275C4×C11—21no—
276C4×C11—21no—
277C4×C11—42no—
278C4×C11—42no—
279C4×C11—42no—
280C4×C11—44no—
281C4×C11#44812yes—
282C4×C11C22×SD1611yescharacteristic
283C4×C11C2×#13711yescharacteristic
284C4×C11#57512yes—
285Dic11—81no—
286C2×#41—221no—
287C22×Q8C22⋊D1111yescharacteristic
288C22×Q8C22⋊D1112yes—
289C22×D4C22⋊D1111yescharacteristic
290C23×C4—22no—
291C23×C4—21no—
292AGL(1,ℤ/8)—444no—
293C2×SD16—222no—
294C2×SD16—222no—
295C2×SD16—224no—
296C2×D8—222no—
297C2×M16—221no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—111no—
301C4⋊Q8C22⋊D1112yes—
302C4⋊Q8C22×D1112yes—
303#127—111no—
304#125—24no—
305#125—24no—
306#124—221no—
307C22⋊Q8—24no—
308C22⋊Q8—28no—
309C22⋊Q8—24no—
310C22⋊Q8—28no—
311#121—221no—
312#121—221no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22⋊D1112yes—
316C4×Q8C22×D1112yes—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—221no—
322C4×D4—221no—
323C4×D4C22⋊D1112yes—
324C4×D4C22×D1112yes—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4C22⋊D1111yescharacteristic
333C2×C4⋊C4C22⋊D1111yescharacteristic
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C42—21no—
337C2×C42C22×D1111yescharacteristic
338#106—222no—
339C4⋊C8—112no—
340C4⋊C8—112no—
341C4⋊C8—221no—
342C4⋊C8—221no—
343#103—222no—
344#103—224no—
345#103—222no—
346#102—222no—
347#102—222no—
348C22⋊C8—221no—
349C22⋊C8—221no—
350C4×C8—112no—
351C4×C8—221no—
352C2×C11—22no—
353C2×C11—44no—
354C2×C11C2×#44811yescharacteristic
355C2×C11#300511yescharacteristic
356C2×C11#301911yescharacteristic
357D11—161no—
358C24—21no—
359#41—442no—
360#41—442no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—82no—
365C2×Q8—221no—
366C2×Q8C2×C22⋊D1112yes—
367C2×Q8C2×C22⋊D1114yes—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—221no—
371C2×D4—221no—
372C2×D4—221no—
373C2×D4—441no—
374C2×D4#432514yescomplemented
375C2×D4C2×C22⋊D1112yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—84no—
385C22×C4—441no—
386C22×C4C2×C22⋊D1111yescharacteristic
387C22×C4C2×C22⋊D1111yescharacteristic
388C22×C4C2×C22⋊D1111yescharacteristic
389SD16—444no—
390SD16—444no—
391SD16—884no—
392D8—444no—
393M16—442no—
394C2×C8—222no—
395C2×C8—222no—
396C2×C8—222no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—221no—
408C4⋊C4—221no—
409C4⋊C4C2×C22⋊D1112yes—
410C4⋊C4C2×C22⋊D1112yes—
411C22⋊C4—28no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—441no—
415C22⋊C4—441no—
416C42—22no—
417C42—22no—
418C42C2×C22⋊D1111yescharacteristic
419C42C2×C22⋊D1111yescharacteristic
420C42C2×C22⋊D1111yescharacteristic
421C42C23×D1111yescharacteristic
422C11#1874811yescharacteristic, complemented, Sylow
423C23—24no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—441no—
428C23#6700611yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—44no—
432Q8—42no—
433Q8—82no—
434Q8—441no—
435Q8—441no—
436D4—22no—
437D4—24no—
438D4—442no—
439D4—441no—
440D4—441no—
441D4—442no—
442D4—882no—
443D4C2×#432514yes—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—88no—
457C2×C4—441no—
458C2×C4—882no—
459C2×C4C2×#432512yes—
460C2×C4#6699512yes—
461C2×C4#6700611yescharacteristic
462C2×C4#6700611yescharacteristic
463C2×C4#6700611yescharacteristic
464C2×C4C22×C22⋊D1111yescharacteristic
465C2×C4C22×C22⋊D1111yescharacteristic
466C2×C4C22×C22⋊D1111yescharacteristic
467C8—442no—
468C8—442no—
469C8—442no—
470C22—21no—
471C22—21no—
472C22—24no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—881no—
477C22—881no—
478C22—881no—
479C22#1062284012yes—
480C22C2×#6700611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—881no—
489C4#1062284012yes—
490C4C22×#432511yescharacteristic
491C4C2×#6699511yescharacteristic
492C4#1062335312yes—
493C2—22no—
494C2—44no—
495C2—1761no—
496C2C2×#1062284011yescharacteristic
497C2#1307785811yescharacteristic
498C2#1307797311yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingC2×C11×#554C2
socleC22×C11C2×#120
prime radical#13078080C2
hypercenterC2×#554D11
hypocenterC11#18748
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #13079541C2semidirect
2. #13078647C2semidirect
4. #13067823C2semidirect
5. #13067688C2semidirect
6. #13067297C2semidirect
7. #13067045C2semidirect
8. #13067007C2semidirect
9. C2×C11×#554C2semidirect
26. #10622698C22semidirect
28. #10622694C22semidirect
29. #10622640C22semidirect
31. #10622633C22semidirect
40. #10622184C22semidirect
41. Q8⋊(C11⋊C8)C22semidirect
119. C2×D4×C11SD16semidirect
174. C2×#554D11semidirect
374. C2×D4#4325semidirect
422. C11#18748semidirect

Maximal subgroups

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

subgroupindex
#130795412
#130786472
#130780802
#130678232
#130676882
#130672972
#130670452
#130670072
C2×C11×#5542
#1874811