Finite Group Databasebeta

← #14788565

Subgroups — #14788565

Order 2816.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623073C211yesmaximal, characteristic
2C2×#10622644C211yesmaximal, characteristic, complemented
3#13073986C211yesmaximal, characteristic, complemented
4#13069855C212yesmaximal
5#13069708C212yesmaximal
6#13069552C211yesmaximal, characteristic, complemented
7#13069418C211yesmaximal, characteristic, complemented
8#13068954C212yesmaximal, complemented
9#13068881C212yesmaximal, complemented
10#13068616C211yesmaximal, characteristic
11C2×C4×C11×SD16C211yesmaximal, characteristic
12C22×C11×SD16C2211yescharacteristic
13C2×C4×Q8×C11C2211yescharacteristic
14C2×C4×D4×C11C2211yescharacteristic
15C22×#4326C2212yes—
16C2×#66989C2212yes—
17C2×#66979—21no—
18C2×#66977—21no—
19#10623073—24no—
20#10623073—24no—
21C2×#66940C2212yes—
22C2×#66932—22no—
23C2×#66922C2211yescharacteristic
24C2×C4×Dic22C2211yescharacteristic
25C4×C11×SD16—24no—
26C2×C11×#106C2211yescharacteristic
27C2×C11×#103C2211yescharacteristic
28C2×C11×#102C2211yescharacteristic
29C2×C4×C8×C11C2211yescharacteristic
30#10622843C2212yes—
31C2×#66896C2212yes—
32C2×#66880C2212yes—
33C2×#66880C2212yes—
34#10622644—24no—
35#10622597—22no—
36#10622586C2212yes—
37C2×#66874C2212yes—
38#10622548C2211yescharacteristic
39#10622542—21no—
40#10622541—21no—
41C2×C4⋊(C11⋊C8)C2211yescharacteristic
42#10622188C2212yes—
43#10622188C2212yes—
44#10622167C2212yes—
45C22×Q8×C11C2311yescharacteristic
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11—21no—
48C22×Dic22D411yescharacteristic
49C22×Dic22C2312yes—
50C2×C11×SD16—22no—
51C2×C11×SD16—22no—
52C2×C11×SD16—22no—
53C22×C8×C11C2311yescharacteristic
54C22×C8×C11C2311yescharacteristic
55C4×Q8×C11—22no—
56C4×Q8×C11—22no—
57C4×D4×C11—22no—
58C4×D4×C11D414yes—
59C2×C11×C4⋊C4—21no—
60C2×C11×C4⋊C4C2311yescharacteristic
61C2×C11×C4⋊C4C2311yescharacteristic
62C2×C11×C22⋊C4—21no—
63C2×C42×C11—21no—
64C2×C42×C11C2311yescharacteristic
65C2×C22⋊Dic11—22no—
66C2×C22⋊Dic11—22no—
67C2×#4326—24no—
68C2×#4326—24no—
69C2×#4326—42no—
70#66989—24no—
71#66989—24no—
72#66979—28no—
73C2×C4⋊Dic11—21no—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11—21no—
77C2×C4⋊Dic11D411yescharacteristic
78C2×C4⋊Dic11C2312yes—
79#66977—28no—
80C2×Dic11⋊C4—22no—
81C2×Dic11⋊C4—22no—
82C2×Dic11⋊C4D412yes—
83C2×C4×Dic11—22no—
84C2×C4×Dic11C2312yes—
85C22×C11⋊C8C2312yes—
86#66940—24no—
87#66940—24no—
88#66932—216no—
89#66922—22no—
90#66922D414yes—
91C4×Dic22—22no—
92C4×Dic22—22no—
93C11×#106—22no—
94C11×#103—22no—
95C11×#102—22no—
96C4×C8×C11—22no—
97#66896—42no—
98C11⋊#95—22no—
99C11⋊#95—22no—
100C11⋊#95—22no—
101#66880—24no—
102#66880—42no—
103#66874—24no—
104C4⋊(C11⋊C8)—22no—
105#24189—111nomaximal, Sylow
106C24×C11—21no—
107C2×Q8×C11—21no—
108C2×Q8×C11—21no—
109C2×Q8×C11—21no—
110C2×Q8×C11—21no—
111C2×Q8×C11—21no—
112C2×Q8×C11#4112yes—
113C2×D4×C11—21no—
114C2×D4×C11—21no—
115C2×D4×C11—21no—
116C2×D4×C11—21no—
117C2×D4×C11—21no—
118C2×D4×C11C2×D412yes—
119C22×C4×C11—21no—
120C22×C4×C11—21no—
121C22×C4×C11—24no—
122C22×C4×C11—21no—
123C22×C4×C11—42no—
124C22×C4×C11—42no—
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2×D411yescharacteristic
127C22×C4×C11C2411yescharacteristic
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11C2×D412yes—
131C22×Dic11C2×D412yes—
132C2×Dic22—22no—
133C2×Dic22—22no—
134C2×Dic22—22no—
135C2×Dic22—22no—
136C2×Dic22—22no—
137C2×Dic22—22no—
138C2×Dic22—22no—
139C2×Dic22—22no—
140C2×Dic22C2×D414yes—
141C11×SD16—28no—
142C2×C8×C11—21no—
143C2×C8×C11—21no—
144C2×C8×C11—21no—
145C2×C8×C11—21no—
146C2×C8×C11—21no—
147C2×C8×C11#4112yes—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—21no—
150C11×C4⋊C4—21no—
151C11×C4⋊C4—42no—
152C11×C4⋊C4C2×D412yes—
153C11×C22⋊C4—24no—
154C42×C11—42no—
155C42×C11C2×D412yes—
156C42×C11C2×D412yes—
157C22⋊Dic11—28no—
158C22⋊Dic11—28no—
159#4326—48no—
160C4⋊Dic11—22no—
161C4⋊Dic11—24no—
162C4⋊Dic11—24no—
163C4⋊Dic11—22no—
164C4⋊Dic11—22no—
165C4⋊Dic11—42no—
166C4⋊Dic11—81no—
167Dic11⋊C4—28no—
168Dic11⋊C4—24no—
169Dic11⋊C4—82no—
170C4×Dic11—28no—
171C4×Dic11—22no—
172C4×Dic11—22no—
173C2×C11⋊C8—22no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—22no—
176C2×#556—111no—
177C2×#462—111no—
178C2×C4×SD16D1111yescharacteristic
179#2582—111no—
180#2042—112no—
181#2015—112no—
182#1984—111no—
183#1956—111no—
184#1878—112no—
185#1865—112no—
186#1802—111no—
187C23×C11—22no—
188C23×C11—41no—
189C23×C11—41no—
190C23×C11—41no—
191C23×C11—41no—
192C23×C11—41no—
193C23×C11—41no—
194C23×C11C22×D411yescharacteristic
195Q8×C11—22no—
196Q8×C11—22no—
197Q8×C11—41no—
198Q8×C11—41no—
199D4×C11—22no—
200D4×C11—22no—
201D4×C11—41no—
202D4×C11—41no—
203C2×C4×C11—21no—
204C2×C4×C11—21no—
205C2×C4×C11—22no—
206C2×C4×C11—22no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—41no—
210C2×C4×C11—41no—
211C2×C4×C11—48no—
212C2×C4×C11—41no—
213C2×C4×C11—41no—
214C2×C4×C11—41no—
215C2×C4×C11#12012yes—
216C2×C4×C11#12012yes—
217C2×C4×C11#13512yes—
218C2×C4×C11#13712yes—
219C2×C4×C11C22×D411yescharacteristic
220C2×C4×C11C22×D411yescharacteristic
221C2×C4×C11C2×#4111yescharacteristic
222C2×C4×C11C2×#4111yescharacteristic
223C2×C4×C11H(5,𝔽2)11yescharacteristic
224C2×C4×C112−1+411yescharacteristic
225C2×Dic11—22no—
226C2×Dic11—22no—
227C2×Dic11—24no—
228C2×Dic11—22no—
229C2×Dic11—22no—
230C2×Dic11—22no—
231C2×Dic11—22no—
232C2×Dic11—42no—
233C2×Dic11—42no—
234C2×Dic11—42no—
235C2×Dic11—42no—
236C2×Dic11—42no—
237Dic22—24no—
238Dic22—24no—
239Dic22—42no—
240Dic22—42no—
241Dic22—44no—
242C8×C11—22no—
243C8×C11—41no—
244C11⋊C8—42no—
245C22×Q16—112no—
246C22×SD16C2×D1111yescharacteristic
247#556—224no—
248#556—224no—
249C2×C4⋊Q8—111no—
250C2×#126—222no—
251C2×#124—112no—
252C2×C22⋊Q8—221no—
253C2×C4×Q8—111no—
254C2×C4×Q8—112no—
255C2×C4×Q8C2×D1111yescharacteristic
256C2×C4×D4C2×D1111yescharacteristic
257C2×#117—221no—
258#462—224no—
259C4×SD16—24no—
260C2×#106C2×D1111yescharacteristic
261C2×C4⋊C8—111no—
262C2×#103—112no—
263C2×#103—112no—
264C2×#103—112no—
265C2×#103C2×D1111yescharacteristic
266C2×#102—112no—
267C2×#102C2×D1111yescharacteristic
268C2×C4×C8C2×D1111yescharacteristic
269#398—222no—
270#391—112no—
271#391—221no—
272#389—112no—
273#389—221no—
274#384—111no—
275#340—112no—
276#336—112no—
277#336—112no—
278C22×C11—42no—
279C22×C11—42no—
280C22×C11—42no—
281C22×C11—81no—
282C22×C11—81no—
283C22×C11—81no—
284C22×C11—81no—
285C22×C11C2×#12011yescharacteristic
286C22×C11#54811yescharacteristic
287C22×C11#55011yescharacteristic
288C22×C11C2×#13511yescharacteristic
289C22×C11C2×#13711yescharacteristic
290C22×C11#57711yescharacteristic
291C22×C11#57811yescharacteristic
292C4×C11—22no—
293C4×C11—42no—
294C4×C11—42no—
295C4×C11—81no—
296C4×C11#45214yes—
297C4×C11#52511yescharacteristic
298C4×C11#53711yescharacteristic
299C4×C11#54611yescharacteristic
300C4×C11#54711yescharacteristic
301Dic11—44no—
302Dic11—42no—
303Dic11—42no—
304Dic11—82no—
305C22×Q8—111no—
306C22×Q8—112no—
307C22×Q8C22×D1111yescharacteristic
308C22×D4C22×D1111yescharacteristic
309C23×C4—21no—
310C2×Q16—224no—
311C2×Q16—224no—
312C2×Q16—442no—
313C2×SD16—22no—
314C2×SD16—22no—
315C2×SD16—22no—
316C22×C8—112no—
317C22×C8C22×D1111yescharacteristic
318C22×C8C22×D1111yescharacteristic
319C4⋊Q8—114no—
320C4⋊Q8—222no—
321#126—2216no—
322#124—224no—
323#124—224no—
324C22⋊Q8—228no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—224no—
328C4×Q8—224no—
329C4×Q8—222no—
330C4×Q8—222no—
331C4×D4—22no—
332C4×D4C22⋊D1114yes—
333#117—228no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—111no—
336C2×C4⋊C4—112no—
337C2×C4⋊C4—112no—
338C2×C4⋊C4—222no—
339C2×C4⋊C4—221no—
340C2×C4⋊C4—221no—
341C2×C4⋊C4—221no—
342C2×C4⋊C4—222no—
343C2×C4⋊C4—221no—
344C2×C4⋊C4C22×D1111yescharacteristic
345C2×C4⋊C4C22×D1111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—222no—
348C2×C22⋊C4—222no—
349C2×C42—21no—
350C2×C42—112no—
351C2×C42—222no—
352C2×C42C22×D1111yescharacteristic
353#106—22no—
354C4⋊C8—222no—
355#103—22no—
356#103—224no—
357#103—224no—
358#103—442no—
359#102—22no—
360#102—442no—
361C4×C8—22no—
362#95—222no—
363#95—222no—
364#95—222no—
365C2×C11—82no—
366C2×C11#240711yescharacteristic
367C2×C11C2×#45211yescharacteristic
368C2×C11#301311yescharacteristic
369C2×C11#336411yescharacteristic
370C2×C11#336911yescharacteristic
371C2×C11#339411yescharacteristic
372C2×C11#339811yescharacteristic
373C24—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—114no—
380C2×Q8—222no—
381C2×Q8—222no—
382C2×Q8—222no—
383C2×Q8—222no—
384C2×Q8—222no—
385C2×Q8—222no—
386C2×Q8—222no—
387C2×Q8—222no—
388C2×Q8#433912yes—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4C2×C22⋊D1112yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—112no—
402C22×C4—112no—
403C22×C4—222no—
404C22×C4—222no—
405C22×C4D4×D1111yescharacteristic
406C22×C4D4×D1111yescharacteristic
407C22×C4C23×D1111yescharacteristic
408Q16—448no—
409SD16—28no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—222no—
416C2×C8—222no—
417C2×C8—222no—
418C2×C8#433912yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—228no—
424C4⋊C4—222no—
425C4⋊C4—224no—
426C4⋊C4—224no—
427C4⋊C4—224no—
428C4⋊C4—222no—
429C4⋊C4—222no—
430C4⋊C4—442no—
431C4⋊C4—881no—
432C4⋊C4—882no—
433C4⋊C4C2×C22⋊D1112yes—
434C22⋊C4—24no—
435C22⋊C4—228no—
436C22⋊C4—228no—
437C42—42no—
438C42—228no—
439C42—222no—
440C42—222no—
441C42D4×D1112yes—
442C42C2×C22⋊D1112yes—
443C11#2418911yescharacteristic, complemented, Sylow
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C2×D4×D1111yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—224no—
457Q8—224no—
458Q8—444no—
459Q8—442no—
460Q8—442no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—48no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—222no—
478C2×C4—222no—
479C2×C4—222no—
480C2×C4—222no—
481C2×C4—222no—
482C2×C4—222no—
483C2×C4—224no—
484C2×C4—442no—
485C2×C4—442no—
486C2×C4—442no—
487C2×C4—442no—
488C2×C4—442no—
489C2×C4#6696812yes—
490C2×C4#6696912yes—
491C2×C4#6699012yes—
492C2×C4#6699012yes—
493C2×C4C2×#433911yescharacteristic
494C2×C4C2×#433911yescharacteristic
495C2×C4C2×D4×D1111yescharacteristic
496C2×C4#6704211yescharacteristic
497C2×C4#6704311yescharacteristic
498C2×C4C22×C22⋊D1111yescharacteristic
499C8—22no—
500C8—41no—
501C8—442no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22#1062308311yescharacteristic
510C22#1062308411yescharacteristic
511C22C2×#6696811yescharacteristic
512C22C2×#6696911yescharacteristic
513C22#1062330111yescharacteristic
514C22#1062330211yescharacteristic
515C22C2×#6699011yescharacteristic
516C4—22no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—442no—
521C4—444no—
522C4—442no—
523C4—882no—
524C4#1062278614yes—
525C4#1062308111yescharacteristic
526C4#1062308211yescharacteristic
527C4#1062332211yescharacteristic
528C4#1062336411yescharacteristic
529C2—82no—
530C2#1307196111yescharacteristic
531C2C2×#1062278611yescharacteristic
532C2#1307792711yescharacteristic
533C2#1308129911yescharacteristic
534C2#1308135411yescharacteristic
535C2#1308161011yescharacteristic
536C2#1308164511yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C4×C11×SD16C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
hypercenterC2×C4×SD16D11
hypocenterC11#24189
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#10622644C2semidirect
3. #13073986C2semidirect
6. #13069552C2semidirect
7. #13069418C2semidirect
8. #13068954C2semidirect
9. #13068881C2semidirect
443. C11#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#106230732
C2×#106226442
#130739862
#130698552
#130697082
#130695522
#130694182
#130689542
#130688812
#130686162
C2×C4×C11×SD162
#2418911