Finite Group Databasebeta

← #14726689

Subgroups — #14726689

Order 2816.

1454 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622653C212yesmaximal, complemented
2#13071981C211yesmaximal, characteristic, complemented
3#13067585C214yesmaximal, complemented
4#13067568C214yesmaximal, complemented
5#13067184C212yesmaximal, complemented
6#13066934C211yesmaximal, characteristic, complemented
7C11×#3496C211yesmaximal, characteristic
8C2×C11×H(5,𝔽2)—21no—
9C23×Dic22C2211yescharacteristic, complemented
10C11×#547—22no—
11C11×#546—24no—
12D42×C11—22no—
13C11×#537C2212yes—
14C11×#534—21no—
15C11×#534C2212yes—
16C11×#532—21no—
17C2×C11×#121—21no—
18C2×C11×#121C2212yes—
19C2×C4×D4×C11C2211yescharacteristic
20C22×#4324—22no—
21C2×#66977—21no—
22#10622833—21no—
23#10622825C2211yescharacteristic, complemented
24C2×C22⋊(C11⋊C8)C2212yescomplemented
25#10622653—24no—
26#10622653C2218yescomplemented
27#10622592C2212yescomplemented
28C2×#66874—22no—
29#10622560C2212yescomplemented
30#10622559—21no—
31#10622542—21no—
32#10622175C2218yescomplemented
33#10622166C2212yescomplemented
34#10622163C2212yescomplemented
35C11×H(5,𝔽2)—44no—
36C11×H(5,𝔽2)—44no—
37C2×C11×#41—24no—
38C2×C11×#41—22no—
39C22×D4×C11—24no—
40C22×D4×C11—22no—
41C22×D4×C11—22no—
42C22×D4×C11—21no—
43C22×D4×C11D411yescharacteristic
44C22×D4×C11D412yes—
45C23×C4×C11D411yescharacteristic
46C23×C4×C11C2311yescharacteristic
47C23×Dic11—21no—
48C22×Dic22—22no—
49C22×Dic22—21no—
50C22×Dic22—22no—
51C22×Dic22—44no—
52C22×Dic22C2312yes—
53C11×C4⋊Q8—21no—
54C11×#127—21no—
55C11×#124—22no—
56C11×#123—24no—
57C11×#123—24no—
58C11×C22⋊Q8—24no—
59C11×#121—22no—
60C11×#121—22no—
61C11×#121—24no—
62C11×#121—22no—
63C11×#121—22no—
64C11×#121—22no—
65C11×#121—22no—
66C11×#121D414yes—
67C11×#121C2314yes—
68C11×#120—24no—
69C11×#120—24no—
70C4×D4×C11—22no—
71C4×D4×C11—22no—
72C4×D4×C11—22no—
73C4×D4×C11—22no—
74C2×C11×C4⋊C4—21no—
75C2×C11×C4⋊C4D412yes—
76C2×C11×C4⋊C4C2311yescharacteristic
77C2×C11×C22⋊C4—22no—
78C2×C11×C22⋊C4—22no—
79C2×C11×C22⋊C4D411yescharacteristic
80C2×C11×C22⋊C4C2311yescharacteristic
81C2×C42×C11D411yescharacteristic
82C2×C22⋊Dic11—22no—
83C2×C22⋊Dic11—21no—
84C2×#4324—28no—
85C2×#4324—48no—
86C2×C4⋊Dic11—21no—
87#66977—44no—
88C2×Dic11⋊C4—22no—
89C22×C11⋊C8—22no—
90C11⋊#95—22no—
91C11⋊#95—22no—
92C11⋊#95C2312yes—
93C22⋊(C11⋊C8)—22no—
94C22⋊(C11⋊C8)C2314yes—
95#66874—28no—
96#18195—111nomaximal, Sylow
97C24×C11—22no—
98C24×C11—22no—
99C24×C11C2×D411yescharacteristic
100C24×C11C2×D411yescharacteristic
101C11×#41—44no—
102C11×#41—416no—
103C11×#41—44no—
104C2×Q8×C11—22no—
105C2×D4×C11—22no—
106C2×D4×C11—22no—
107C2×D4×C11—24no—
108C2×D4×C11—22no—
109C2×D4×C11—24no—
110C2×D4×C11—24no—
111C2×D4×C11—22no—
112C2×D4×C11—22no—
113C2×D4×C11—22no—
114C2×D4×C11—44no—
115C2×D4×C11—416no—
116C2×D4×C11—44no—
117C2×D4×C11—44no—
118C2×D4×C11—44no—
119C2×D4×C11—44no—
120C2×D4×C11—82no—
121C2×D4×C11SD1614yescomplemented
122C2×D4×C11C2×D414yes—
123C22×C4×C11—22no—
124C22×C4×C11—24no—
125C22×C4×C11—22no—
126C22×C4×C11—21no—
127C22×C4×C11—21no—
128C22×C4×C11—42no—
129C22×C4×C11—42no—
130C22×C4×C11C2×D411yescharacteristic
131C22×C4×C11C2×D412yes—
132C22×C4×C11C2×D411yescharacteristic
133C22×C4×C11C2×D411yescharacteristic
134C22×C4×C11C2×D411yescharacteristic
135C22×C4×C11C2×D412yes—
136C22×C4×C11C2411yescharacteristic
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—44no—
140C22×Dic11—42no—
141C2×Dic22—28no—
142C2×Dic22—24no—
143C2×Dic22—24no—
144C2×Dic22—44no—
145C2×Dic22—42no—
146C2×Dic22—44no—
147C2×Dic22—48no—
148C2×Dic22—41no—
149C2×Dic22—41no—
150C11×C4⋊C4—22no—
151C11×C4⋊C4—22no—
152C11×C4⋊C4—22no—
153C11×C4⋊C4—22no—
154C11×C4⋊C4C2×D414yes—
155C11×C22⋊C4—24no—
156C11×C22⋊C4—24no—
157C11×C22⋊C4—22no—
158C11×C22⋊C4—24no—
159C11×C22⋊C4—24no—
160C11×C22⋊C4—22no—
161C42×C11—21no—
162C42×C11—21no—
163C22⋊Dic11—44no—
164C22⋊Dic11—42no—
165#4324—416no—
166C4⋊Dic11—42no—
167Dic11⋊C4—44no—
168C2×C11⋊C8—24no—
169C2×C11⋊C8—44no—
170#3496D1111yescharacteristic
171C2×#448—112no—
172#2409—111no—
173#1604—114no—
174#1601—114no—
175#1519—112no—
176#1459—111no—
177C23×C11—21no—
178C23×C11—21no—
179C23×C11—24no—
180C23×C11—41no—
181C23×C11—41no—
182C23×C11—41no—
183C23×C11—41no—
184C23×C11—44no—
185C23×C11—44no—
186C23×C11—82no—
187C23×C11—82no—
188C23×C11—82no—
189C23×C11—82no—
190C23×C11#12012yes—
191C23×C11C2×SD1612yes—
192C23×C11C2×SD1612yescomplemented
193C23×C11AGL(1,ℤ/8)12yes—
194C23×C11C22×D411yescharacteristic
195Q8×C11—44no—
196D4×C11—28no—
197D4×C11—44no—
198D4×C11—44no—
199D4×C11—48no—
200D4×C11—44no—
201D4×C11—44no—
202D4×C11—48no—
203D4×C11—44no—
204D4×C11—82no—
205D4×C11—82no—
206C2×C4×C11—21no—
207C2×C4×C11—21no—
208C2×C4×C11—24no—
209C2×C4×C11—22no—
210C2×C4×C11—21no—
211C2×C4×C11—21no—
212C2×C4×C11—24no—
213C2×C4×C11—42no—
214C2×C4×C11—41no—
215C2×C4×C11—48no—
216C2×C4×C11—44no—
217C2×C4×C11—44no—
218C2×C4×C11—42no—
219C2×C4×C11—41no—
220C2×C4×C11—41no—
221C2×C4×C11—81no—
222C2×C4×C11#12011yescharacteristic
223C2×C4×C11#12011yescharacteristic
224C2×C4×C11C2×SD1612yescomplemented
225C2×C4×C11C22×D412yes—
226C2×Dic11—24no—
227C2×Dic11—48no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11—42no—
233Dic22—28no—
234Dic22—44no—
235Dic22—44no—
236Dic22—44no—
237C11⋊C8—44no—
238C2×H(5,𝔽2)—21no—
239C23×Q8—111no—
240C22×SD16—222no—
241#547—22no—
242#546—24no—
243D42—22no—
244#537C2×D1112yes—
245#534—21no—
246#534C2×D1112yes—
247#532—21no—
248C2×C22⋊Q8—221no—
249C2×#121—21no—
250C2×#121C2×D1112yes—
251C2×C4×D4C2×D1111yescharacteristic
252#448—118no—
253#448—224no—
254C2×#103—222no—
255C2×C22⋊C8—112no—
256#395—112no—
257#393—112no—
258#391—221no—
259#386—221no—
260#385—221no—
261#380—111no—
262#327—118no—
263#324—112no—
264#323—112no—
265C22×C11—21no—
266C22×C11—22no—
267C22×C11—21no—
268C22×C11—22no—
269C22×C11—22no—
270C22×C11—42no—
271C22×C11—42no—
272C22×C11—42no—
273C22×C11—41no—
274C22×C11—41no—
275C22×C11—44no—
276C22×C11—44no—
277C22×C11—44no—
278C22×C11—81no—
279C22×C11—164no—
280C22×C11—164no—
281C22×C11#45014yes—
282C22×C11C2×#12011yescharacteristic
283C22×C11C22×SD1611yescharacteristic
284C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C11—22no—
286C4×C11—21no—
287C4×C11—22no—
288C4×C11—44no—
289C4×C11—41no—
290C4×C11—41no—
291C4×C11B(4,𝔽2)12yes—
292Dic11—44no—
293Dic11—82no—
294H(5,𝔽2)—44no—
295H(5,𝔽2)—44no—
296C2×#41—24no—
297C2×#41—22no—
298C22×Q8—112no—
299C22×Q8—221no—
300C22×Q8—222no—
301C22×Q8—222no—
302C22×Q8—444no—
303C22×D4—24no—
304C22×D4—22no—
305C22×D4—22no—
306C22×D4—21no—
307C22×D4C22⋊D1111yescharacteristic
308C22×D4C22⋊D1112yes—
309C23×C4—221no—
310C23×C4C22⋊D1111yescharacteristic
311C23×C4C22×D1111yescharacteristic
312C2×SD16—228no—
313C2×SD16—448no—
314C22×C8—222no—
315C4⋊Q8—21no—
316#127—21no—
317#124—22no—
318#123—24no—
319#123—24no—
320C22⋊Q8—24no—
321C22⋊Q8—444no—
322#121—22no—
323#121—22no—
324#121—24no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121C22⋊D1114yes—
330#121C22×D1114yes—
331#120—24no—
332#120—24no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—221no—
339C2×C4⋊C4—222no—
340C2×C4⋊C4C22⋊D1112yes—
341C2×C4⋊C4C22×D1111yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—222no—
345C2×C22⋊C4—221no—
346C2×C22⋊C4C22⋊D1111yescharacteristic
347C2×C22⋊C4C22×D1111yescharacteristic
348C2×C42C22⋊D1111yescharacteristic
349#103—228no—
350C22⋊C8—114no—
351C22⋊C8—222no—
352#95—112no—
353#95—222no—
354#95—222no—
355C2×C11—21no—
356C2×C11—21no—
357C2×C11—24no—
358C2×C11—42no—
359C2×C11—84no—
360C2×C11C2×#45011yescharacteristic
361C2×C11#302211yescharacteristic
362C2×C11C2×B(4,𝔽2)11yescharacteristic
363C24—22no—
364C24—22no—
365C24C2×C22⋊D1111yescharacteristic
366C24C2×C22⋊D1111yescharacteristic
367#41—44no—
368#41—416no—
369#41—44no—
370C2×Q8—22no—
371C2×Q8—224no—
372C2×Q8—228no—
373C2×Q8—224no—
374C2×Q8—442no—
375C2×Q8—444no—
376C2×Q8—444no—
377C2×Q8—448no—
378C2×Q8—441no—
379C2×Q8—441no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—44no—
390C2×D4—416no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—82no—
396C2×D4#432414yescomplemented
397C2×D4C2×C22⋊D1114yes—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—222no—
406C22×C4—222no—
407C22×C4—444no—
408C22×C4—442no—
409C22×C4C2×C22⋊D1111yescharacteristic
410C22×C4C2×C22⋊D1111yescharacteristic
411C22×C4C2×C22⋊D1112yes—
412C22×C4C2×C22⋊D1111yescharacteristic
413C22×C4C2×C22⋊D1111yescharacteristic
414C22×C4C2×C22⋊D1112yes—
415C22×C4C23×D1111yescharacteristic
416SD16—4416no—
417C2×C8—224no—
418C2×C8—444no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—442no—
424C4⋊C4—444no—
425C4⋊C4C2×C22⋊D1114yes—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—444no—
433C22⋊C4—442no—
434C42—21no—
435C42—21no—
436C11#1819511yescharacteristic, complemented, Sylow
437C23—21no—
438C23—21no—
439C23—24no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23#6698512yes—
451C23C2×#432412yes—
452C23C2×#432412yescomplemented
453C23#6700612yes—
454C23C22×C22⋊D1111yescharacteristic
455Q8—44no—
456Q8—228no—
457Q8—444no—
458Q8—444no—
459Q8—444no—
460D4—28no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—44no—
465D4—44no—
466D4—48no—
467D4—44no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—224no—
487C2×C4—442no—
488C2×C4—448no—
489C2×C4—442no—
490C2×C4—442no—
491C2×C4—442no—
492C2×C4—442no—
493C2×C4C2×#432412yescomplemented
494C2×C4#6700611yescharacteristic
495C2×C4#6700611yescharacteristic
496C2×C4C22×C22⋊D1112yes—
497C8—444no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—164no—
513C22—164no—
514C22#1062283114yes—
515C22C2×#6698511yescharacteristic
516C22C22×#432411yescharacteristic
517C22C2×#6700611yescharacteristic
518C4—22no—
519C4—22no—
520C4—21no—
521C4—44no—
522C4—41no—
523C4—41no—
524C4—444no—
525C4—882no—
526C4#1062285712yes—
527C2—21no—
528C2—21no—
529C2—24no—
530C2—42no—
531C2—84no—
532C2C2×#1062283111yescharacteristic
533C2#1307799511yescharacteristic
534C2C2×#1062285711yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC11×#3496C2
socleC22×C11C2×#120
hypercenter#3496D11
hypocenterC11#18195
LCSG ▷ C22×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#10622653C2semidirect
2. #13071981C2semidirect
3. #13067585C2semidirect
4. #13067568C2semidirect
5. #13067184C2semidirect
6. #13066934C2semidirect
9. C23×Dic22C22semidirect
23. #10622825C22semidirect
24. C2×C22⋊(C11⋊C8)C22semidirect
26. #10622653C22semidirect
27. #10622592C22semidirect
29. #10622560C22semidirect
32. #10622175C22semidirect
33. #10622166C22semidirect
34. #10622163C22semidirect
121. C2×D4×C11SD16semidirect
192. C23×C11C2×SD16semidirect
224. C2×C4×C11C2×SD16semidirect
396. C2×D4#4324semidirect
436. C11#18195semidirect
452. C23C2×#4324semidirect
493. C2×C4C2×#4324semidirect

Maximal subgroups

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

subgroupindex
C2×#106226532
#130719812
#130675852
#130675682
#130671842
#130669342
C11×#34962
#1819511