Finite Group Databasebeta

← #14783652

Subgroups — #14783652

Order 2816.

1098 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622608C211yesmaximal, characteristic, complemented
2C22×#66872C211yesmaximal, characteristic, complemented
3C2×#10622579C211yesmaximal, characteristic
4C2×#10622615C211yesmaximal, characteristic
5#13071204C211yesmaximal, characteristic
6#13070167C212yesmaximal, complemented
7#13070160C212yesmaximal, complemented
8#13070150C212yesmaximal, complemented
9#13070140C212yesmaximal, complemented
10#13068313C211yesmaximal, characteristic, complemented
11C11×#2317C211yesmaximal, characteristic, complemented
12C22×#4339C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic11C2211yescharacteristic
17C22×#4318C2211yescharacteristic
18C23×C11⋊C8C2211yescharacteristic
19C11×#389—22no—
20C11×#385—22no—
21C11×#384C2212yes—
22C11×#382C2212yes—
23#10622615—24no—
24#10622608—24no—
25#10622600—22no—
26C2×#66874C2211yescharacteristic
27C2×#66874C2212yes—
28C2×#66874C2211yescharacteristic
29#10622579—24no—
30C2×#66873C2211yescharacteristic
31C2×#66873C2212yes—
32C2×#66873C2211yescharacteristic
33C2×#66872—24no—
34C2×#66872C2212yes—
35C2×#66872C2212yes—
36C2×#66872C2212yes—
37C2×#66871C2212yes—
38#10622571—22no—
39#10622552C2212yes—
40#10622542C2212yes—
41#10622162C2212yes—
42#10622162C2212yes—
43C23×C4×C11—21no—
44C23×C4×C11C2311yescharacteristic
45C22×C22⋊D11—21no—
46C23×Dic11—21no—
47C2×#4339—22no—
48C2×#4339—22no—
49C2×#4339—22no—
50C2×#4339—22no—
51C2×#4339—22no—
52C2×#4339Q814yes—
53C22×D44C2311yescharacteristic
54C22×C4×D11—21no—
55C22×Dic22C2311yescharacteristic
56C11×#117—22no—
57C11×#117—22no—
58C11×#117—22no—
59C11×#117—22no—
60C2×C11×C4⋊C4—22no—
61C2×C11×C4⋊C4—22no—
62C2×C11×C4⋊C4—21no—
63C2×C11×C4⋊C4—24no—
64C2×C11×C4⋊C4D412yes—
65C2×C11×C4⋊C4C2311yescharacteristic
66C2×C11×C4⋊C4C2312yes—
67C2×C11×C4⋊C4C2311yescharacteristic
68C2×C11×C22⋊C4—21no—
69C2×C11×C22⋊C4D412yes—
70C2×C42×C11—21no—
71C2×C42×C11C2312yes—
72C2×(C2×D11)⋊C4—24no—
73C2×C4⋊Dic11—22no—
74C2×C4⋊Dic11—22no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11C2312yes—
77C2×#4318—22no—
78C2×#4318—22no—
79C2×#4318D414yes—
80C22×C11⋊C8—22no—
81C22×C11⋊C8—22no—
82C22×C11⋊C8D414yes—
83C22×C11⋊C8C2311yescharacteristic
84C22×C11⋊C8C2311yescharacteristic
85C22×C11⋊C8C2312yes—
86C11×#95—24no—
87C11⋊#95—24no—
88#66874—24no—
89#66874—22no—
90#66874—22no—
91#66873—24no—
92#66873—22no—
93#66873—22no—
94#66872—24no—
95#66872—24no—
96#66872—24no—
97#66871—24no—
98#23738—111nomaximal, Sylow
99C24×C11C2×D411yescharacteristic
100C23×D11—21no—
101C22×C4×C11—21no—
102C22×C4×C11—22no—
103C22×C4×C11—21no—
104C22×C4×C11—42no—
105C22×C4×C11—82no—
106C22×C4×C11C2×D414yes—
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11C2×D412yes—
109C22×C4×C11C2×D412yes—
110C22×C4×C11C2×Q812yes—
111C22×C4×C11#4112yes—
112C22×C4×C11#4112yes—
113C22×C4×C11#4112yes—
114C22×C4×C11C2411yescharacteristic
115C2×C22⋊D11—24no—
116C2×C22⋊D11—42no—
117C2×C22⋊D11—42no—
118C22×Dic11—21no—
119C22×Dic11—22no—
120C22×Dic11—42no—
121C22×Dic11—82no—
122#4339—28no—
123#4339—28no—
124#4339—44no—
125#4339—44no—
126C2×D44—21no—
127C2×D44—21no—
128C2×D44—21no—
129C2×D44—21no—
130C2×D44—21no—
131C2×D44C2×Q812yes—
132C2×C4×D11—24no—
133C2×C4×D11—42no—
134C2×C4×D11—42no—
135C2×Dic22—21no—
136C2×Dic22—21no—
137C2×Dic22—21no—
138C2×Dic22—21no—
139C2×Dic22—21no—
140C2×Dic22C2×Q812yes—
141C11×C4⋊C4—24no—
142C11×C4⋊C4—21no—
143C11×C4⋊C4—22no—
144C11×C4⋊C4—21no—
145C11×C4⋊C4—22no—
146C11×C4⋊C4—21no—
147C11×C4⋊C4—21no—
148C11×C4⋊C4—42no—
149C11×C4⋊C4—84no—
150C11×C22⋊C4—24no—
151C11×C22⋊C4—42no—
152C42×C11—22no—
153C42×C11—22no—
154C42×C11—42no—
155(C2×D11)⋊C4—84no—
156C4⋊Dic11—22no—
157C4⋊Dic11—22no—
158C4⋊Dic11—42no—
159#4318—28no—
160C2×C11⋊C8—28no—
161C2×C11⋊C8—21no—
162C2×C11⋊C8—21no—
163C2×C11⋊C8—21no—
164C2×C11⋊C8—21no—
165C2×C11⋊C8—22no—
166C2×C11⋊C8—22no—
167C2×C11⋊C8C2×D414yes—
168C2×C11⋊C8C2×D412yes—
169C2×C11⋊C8C2×D412yes—
170C2×#428—111no—
171C22×#106—111no—
172C2×#417—111no—
173C2×#416—111no—
174#2317D1111yescharacteristic, complemented
175#2311—111no—
176#2097—112no—
177#2096—112no—
178#2094—112no—
179#2092—112no—
180#1729—111no—
181C23×C11—22no—
182C23×C11—22no—
183C23×C11C22⋊Q812yes—
184C23×C11#12312yes—
185C23×C11#12712yes—
186C23×C11C22×D411yescharacteristic
187C22×D11—22no—
188C22×D11—41no—
189C22×D11—41no—
190C22×D11—42no—
191C22×D11—42no—
192C2×C4×C11—24no—
193C2×C4×C11—24no—
194C2×C4×C11—24no—
195C2×C4×C11—41no—
196C2×C4×C11—41no—
197C2×C4×C11—42no—
198C2×C4×C11—42no—
199C2×C4×C11—44no—
200C2×C4×C11—42no—
201C2×C4×C11—84no—
202C2×C4×C11C22⋊Q814yes—
203C2×C4×C11C22⋊Q814yes—
204C2×C4×C11C22⋊Q812yes—
205C2×C4×C11#12314yes—
206C2×C4×C11#12314yes—
207C2×C4×C11#12312yes—
208C2×C4×C11#12712yes—
209C2×C4×C11C22×D411yescharacteristic
210C2×C4×C11C22×D411yescharacteristic
211C2×C4×C11C22×Q811yescharacteristic
212C2×C4×C11C2×#4111yescharacteristic
213C2×C4×C11C2×#4111yescharacteristic
214C2×C4×C11C2×#4111yescharacteristic
215C22⋊D11—48no—
216C2×Dic11—22no—
217C2×Dic11—41no—
218C2×Dic11—41no—
219C2×Dic11—42no—
220C2×Dic11—44no—
221C2×Dic11—84no—
222D44—22no—
223D44—22no—
224D44—41no—
225D44—41no—
226C4×D11—48no—
227Dic22—22no—
228Dic22—22no—
229Dic22—41no—
230Dic22—41no—
231C11⋊C8—22no—
232C11⋊C8—22no—
233C11⋊C8—24no—
234C22×#41—111no—
235C22×M16—111no—
236C23×C8—111no—
237C2×#117C2×D1111yescharacteristic
238C2×#117C2×D1111yescharacteristic
239C22×C4⋊C4—111no—
240C22×C4⋊C4C2×D1111yescharacteristic
241#428—224no—
242C2×#107—112no—
243C2×#106—112no—
244C2×#106—112no—
245C2×#106—112no—
246C2×#106—224no—
247#417—224no—
248#416—224no—
249C2×#103—111no—
250C2×#103—112no—
251C2×#103—111no—
252C2×#102—111no—
253C2×#102—112no—
254C2×#102—111no—
255#391—112no—
256#390—112no—
257#389—22no—
258#386—222no—
259#385—22no—
260#384C2×D1112yes—
261#382C2×D1112yes—
262#380—222no—
263#336—112no—
264#336—112no—
265C22×C11—24no—
266C22×C11—24no—
267C22×C11—22no—
268C22×C11—22no—
269C22×C11#39914yes—
270C22×C11C2×C22⋊Q811yescharacteristic
271C22×C11C2×#12311yescharacteristic
272C22×C11C2×#12711yescharacteristic
273C22×C11#57712yes—
274C22×C11#57712yes—
275C2×D11—44no—
276C2×D11—42no—
277C2×D11—82no—
278C2×D11—81no—
279C2×D11—81no—
280C4×C11—44no—
281C4×C11—81no—
282C4×C11—81no—
283C4×C11—82no—
284C4×C11#39914yes—
285C4×C11C2×C22⋊Q811yescharacteristic
286C4×C11C2×C22⋊Q811yescharacteristic
287C4×C11C2×#12311yescharacteristic
288C4×C11C2×#12311yescharacteristic
289Dic11—42no—
290Dic11—82no—
291C2×#41—114no—
292C2×#41—222no—
293C2×#41—222no—
294C2×#41—222no—
295C2×#41—222no—
296C2×#41—222no—
297C22×Q8—111no—
298C22×D4—111no—
299C22×D4—221no—
300C23×C4—21no—
301C23×C4—221no—
302C23×C4—221no—
303C23×C4C22×D1111yescharacteristic
304C2×M16—114no—
305C2×M16—222no—
306C2×M16—222no—
307C22×C8—114no—
308C22×C8—111no—
309C22×C8—111no—
310C22×C8—112no—
311C22×C8—222no—
312C22×C8—222no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—112no—
322C2×C4⋊C4—222no—
323C2×C4⋊C4—222no—
324C2×C4⋊C4—221no—
325C2×C4⋊C4C22⋊D1112yes—
326C2×C4⋊C4C22×D1112yes—
327C2×C4⋊C4C22×D1111yescharacteristic
328C2×C4⋊C4C22×D1111yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—224no—
331C2×C22⋊C4C22⋊D1112yes—
332C2×C42—21no—
333C2×C42C22×D1112yes—
334#107—224no—
335#106—224no—
336#106—224no—
337#106—224no—
338#103—222no—
339#103—222no—
340#103—224no—
341#102—222no—
342#102—222no—
343#102—224no—
344#95—24no—
345#95—224no—
346C2×C11—24no—
347C2×C11C2×#39911yescharacteristic
348C2×C11#308412yes—
349C2×C11#309912yes—
350C2×C11#315411yescharacteristic
351C2×C11#315411yescharacteristic
352D11—82no—
353C24—221no—
354C24C2×C22⋊D1111yescharacteristic
355#41—228no—
356#41—228no—
357#41—444no—
358#41—444no—
359C2×Q8—112no—
360C2×Q8—221no—
361C2×Q8—221no—
362C2×Q8—221no—
363C2×Q8—221no—
364C2×Q8—221no—
365C2×D4—112no—
366C2×D4—224no—
367C2×D4—221no—
368C2×D4—221no—
369C2×D4—221no—
370C2×D4—221no—
371C2×D4—221no—
372C2×D4—442no—
373C2×D4—442no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—82no—
379C22×C4—221no—
380C22×C4—224no—
381C22×C4—222no—
382C22×C4—442no—
383C22×C4—442no—
384C22×C4—442no—
385C22×C4—882no—
386C22×C4#433912yes—
387C22×C4#433912yes—
388C22×C4D4×D1112yes—
389C22×C4D4×D1112yes—
390C22×C4Q8×D1112yes—
391C22×C4#434312yes—
392C22×C4C2×C22⋊D1114yes—
393C22×C4C2×C22⋊D1111yescharacteristic
394C22×C4C23×D1111yescharacteristic
395M16—228no—
396C2×C8—112no—
397C2×C8—112no—
398C2×C8—114no—
399C2×C8—228no—
400C2×C8—221no—
401C2×C8—221no—
402C2×C8—222no—
403C2×C8—221no—
404C2×C8—221no—
405C2×C8—222no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—222no—
416C4⋊C4—222no—
417C4⋊C4—442no—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—884no—
421C42—22no—
422C42—22no—
423C42—42no—
424C11#2373811yescharacteristic, complemented, Sylow
425C23—22no—
426C23—22no—
427C23—222no—
428C23—441no—
429C23—441no—
430C23—442no—
431C23—442no—
432C23#6698212yes—
433C23#6699312yes—
434C23#6699912yes—
435C23C22×C22⋊D1111yescharacteristic
436Q8—222no—
437Q8—222no—
438Q8—441no—
439Q8—441no—
440D4—222no—
441D4—222no—
442D4—448no—
443D4—441no—
444D4—441no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—222no—
456C2×C4—441no—
457C2×C4—441no—
458C2×C4—448no—
459C2×C4—442no—
460C2×C4—444no—
461C2×C4—884no—
462C2×C4#6694714yes—
463C2×C4#6694714yes—
464C2×C4#6694914yes—
465C2×C4#6694914yes—
466C2×C4#6698212yes—
467C2×C4#6699312yes—
468C2×C4#6699912yes—
469C2×C4C2×#433911yescharacteristic
470C2×C4C2×#433911yescharacteristic
471C2×C4C2×D4×D1111yescharacteristic
472C2×C4C2×D4×D1111yescharacteristic
473C2×C4C2×Q8×D1111yescharacteristic
474C2×C4C2×#434311yescharacteristic
475C8—222no—
476C8—222no—
477C8—224no—
478C22—22no—
479C22—22no—
480C22—24no—
481C22—24no—
482C22—444no—
483C22—442no—
484C22—881no—
485C22—881no—
486C22—882no—
487C22#1062260514yes—
488C22C2×#6698211yescharacteristic
489C22C2×#6699311yescharacteristic
490C22C2×#6699911yescharacteristic
491C22#1062336712yes—
492C22#1062336812yes—
493C4—44no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—442no—
498C4—882no—
499C4#1062260514yes—
500C4C2×#6694711yescharacteristic
501C4C2×#6694711yescharacteristic
502C4C2×#6694911yescharacteristic
503C4C2×#6694911yescharacteristic
504C2—24no—
505C2—882no—
506C2C2×#1062260511yescharacteristic
507C2#1307847412yes—
508C2#1307860512yes—
509C2#1307908811yescharacteristic
510C2#1307909411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2317C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
hypercenter#2317D11
hypocenterC11#23738
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#10622608C2semidirect
2. C22×#66872C2semidirect
6. #13070167C2semidirect
7. #13070160C2semidirect
8. #13070150C2semidirect
9. #13070140C2semidirect
10. #13068313C2semidirect
11. C11×#2317C2semidirect
174. #2317D11semidirect
424. C11#23738semidirect

Maximal subgroups

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

subgroupindex
C2×#106226082
C22×#668722
C2×#106225792
C2×#106226152
#130712042
#130701672
#130701602
#130701502
#130701402
#130683132
C11×#23172
#2373811