Finite Group Databasebeta

← #14767630

Subgroups — #14767630

Order 2816.

874 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622328C211yesmaximal, characteristic
2#13071335C211yesmaximal, characteristic, complemented
3#13071327C211yesmaximal, characteristic
4#13069427C211yesmaximal, characteristic
5#13069418C211yesmaximal, characteristic, complemented
6#13068970C212yesmaximal
7#13068956C212yesmaximal, complemented
8#13068956C212yesmaximal, complemented
9#13068949C212yesmaximal
10#13068750C211yesmaximal, characteristic, complemented
11C11×#1746C211yesmaximal, characteristic, complemented
12C22×Dic44C2212yes—
13C22×#4314C2212yes—
14C2×#66928C2211yescharacteristic
15C2×C4×D44C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C2×C4×Dic22C2211yescharacteristic
18C2×C4×Dic22C2212yes—
19C2×C11×#97C2211yescharacteristic
20C2×C4×C8×C11C2211yescharacteristic
21C11×#376C2211yescharacteristic
22C2×#66885C2212yes—
23C2×#66880C2212yes—
24C2×#66880C2212yes—
25C2×#66880C2212yes—
26#10622552C2211yescharacteristic
27#10622549C2211yescharacteristic
28#10622548C2211yescharacteristic
29#10622542C2212yes—
30#10622542C2211yescharacteristic
31#10622328C418yescomplemented
32#10622283—22no—
33#10622281—22no—
34#10622266—22no—
35#10622266—22no—
36C11×#336C2212yes—
37C11×#336C2212yes—
38#10622189C2212yes—
39#10622189C2212yes—
40C22×D44C2311yescharacteristic
41C22×C4×D11—21no—
42C22×Dic22C2312yes—
43C22×Dic22C2311yescharacteristic
44C22×C8×C11C2312yes—
45C22×C8×C11C2312yes—
46C2×C42×C11C2311yescharacteristic
47C2×C42×C11C2311yescharacteristic
48C2×C42×C11C2311yescharacteristic
49C2×(C2×D11)⋊C4—22no—
50C2×(C2×D11)⋊C4—21no—
51C2×(C2×D11)⋊C4—22no—
52C2×C4⋊Dic11—22no—
53C2×C4⋊Dic11—22no—
54C2×C4⋊Dic11C2312yes—
55C2×C4⋊Dic11C2311yescharacteristic
56C2×C4⋊Dic11C2311yescharacteristic
57C2×Dic11⋊C4—22no—
58C2×Dic11⋊C4—21no—
59C2×C4×Dic11—22no—
60C2×C4×Dic11—21no—
61C2×Dic44—42no—
62C2×Dic44—42no—
63C2×Dic44C2×C418yescomplemented
64C2×#4314—42no—
65C2×#4314—42no—
66C2×#4314C2×C418yes—
67#66928—22no—
68#66928C2×C414yes—
69C4×D44—42no—
70#66922—22no—
71#66922C2×C414yescomplemented
72C4×Dic22—42no—
73C4×Dic22—42no—
74C4×Dic22—42no—
75C11×#97C2×C414yescomplemented
76C4×C8×C11—22no—
77C11×#95D412yes—
78C11×#95D412yes—
79C11⋊#95—22no—
80C11⋊#95—22no—
81C11⋊#95—22no—
82C11⋊#95—22no—
83#66885—42no—
84#66880—42no—
85#66880—42no—
86#66880—42no—
87#22278—111nomaximal, Sylow
88C23×D11—21no—
89C22×C4×C11C2×D411yescharacteristic
90C22×C4×C11C2×D411yescharacteristic
91C22×C4×C11C2×D412yes—
92C22×C4×C11C2×D412yes—
93C22×C4×C11C2411yescharacteristic
94C22×Dic11—22no—
95C22×Dic11—21no—
96C22×Dic11—22no—
97C22×Dic11—21no—
98C22×Dic11—21no—
99C2×D44—21no—
100C2×D44—22no—
101C2×D44—22no—
102C2×D44C22×C412yes—
103C2×C4×D11—42no—
104C2×C4×D11—82no—
105C2×Dic22—22no—
106C2×Dic22—21no—
107C2×Dic22—22no—
108C2×Dic22—22no—
109C2×Dic22—22no—
110C2×Dic22—22no—
111C2×Dic22—22no—
112C2×Dic22—22no—
113C2×Dic22C22×C414yes—
114C2×Dic22C22×C412yes—
115C2×C8×C11—22no—
116C2×C8×C11—22no—
117C2×C8×C11—22no—
118C2×C8×C11—22no—
119C2×C8×C11—22no—
120C2×C8×C11C22×C414yes—
121C42×C11—22no—
122C42×C11—22no—
123C42×C11C22×C412yes—
124C42×C11#4112yes—
125(C2×D11)⋊C4—28no—
126(C2×D11)⋊C4—42no—
127(C2×D11)⋊C4—82no—
128C4⋊Dic11—28no—
129C4⋊Dic11—22no—
130C4⋊Dic11—22no—
131C4⋊Dic11—22no—
132C4⋊Dic11—22no—
133C4⋊Dic11—82no—
134Dic11⋊C4—44no—
135Dic11⋊C4—42no—
136C4×Dic11—44no—
137C4×Dic11—42no—
138Dic44—48no—
139#4314—48no—
140C2×#497—111no—
141#2327—111no—
142#2326—111no—
143#1958—111no—
144#1956—111no—
145#1880—112no—
146#1878—112no—
147#1878—112no—
148#1877—112no—
149#1836—111no—
150#1746D1111yescharacteristic, complemented
151C23×C11C22×D411yescharacteristic
152C22×D11—22no—
153C22×D11—41no—
154C22×D11—41no—
155C22×D11—42no—
156C22×D11—42no—
157C2×C4×C11—22no—
158C2×C4×C11—22no—
159C2×C4×C11—22no—
160C2×C4×C11—21no—
161C2×C4×C11—21no—
162C2×C4×C11—22no—
163C2×C4×C11—22no—
164C2×C4×C11—22no—
165C2×C4×C11—22no—
166C2×C4×C11C4×D414yes—
167C2×C4×C11C23×C411yescharacteristic
168C2×C4×C11C2×#4111yescharacteristic
169C2×C4×C11H(5,𝔽2)12yes—
170C2×C4×C11H(5,𝔽2)12yes—
171C2×Dic11—24no—
172C2×Dic11—22no—
173C2×Dic11—41no—
174C2×Dic11—42no—
175C2×Dic11—42no—
176C2×Dic11—41no—
177C2×Dic11—41no—
178C2×Dic11—42no—
179C2×Dic11—41no—
180C2×Dic11—42no—
181C2×Dic11—42no—
182C2×Dic11—42no—
183C2×Dic11—42no—
184D44—24no—
185D44—42no—
186C4×D11—84no—
187Dic22—24no—
188Dic22—24no—
189Dic22—24no—
190Dic22—42no—
191Dic22—42no—
192Dic22—42no—
193C8×C11—24no—
194C8×C11—42no—
195C22×Q16—112no—
196C22×SD16—112no—
197C2×C4⋊Q8—111no—
198C2×#124—111no—
199C2×C4×Q8—111no—
200C2×C4×Q8—112no—
201C2×C4×D4—111no—
202#497—118no—
203C2×#103—112no—
204C2×#103—112no—
205C2×#103—112no—
206C2×#102—112no—
207C2×#97C2×D1111yescharacteristic
208C2×C4×C8C2×D1111yescharacteristic
209#391—111no—
210#391—112no—
211#390—111no—
212#388—222no—
213#387—222no—
214#387—222no—
215#385—222no—
216#384—111no—
217#382—111no—
218#376C2×D1111yescharacteristic
219#340—112no—
220#340—112no—
221#336C2×D1112yes—
222#336C2×D1112yes—
223C22×C11C2×C4×D411yescharacteristic
224C22×C11#53511yescharacteristic
225C22×C11#53511yescharacteristic
226C22×C11#57712yes—
227C22×C11#57812yes—
228C2×D11—42no—
229C2×D11—44no—
230C2×D11—81no—
231C2×D11—81no—
232C2×D11—82no—
233C4×C11—24no—
234C4×C11—41no—
235C4×C11—41no—
236C4×C11—42no—
237C4×C11#51812yes—
238C4×C11#54012yes—
239Dic11—44no—
240Dic11—42no—
241Dic11—81no—
242Dic11—81no—
243Dic11—82no—
244C22×Q8—111no—
245C22×Q8—112no—
246C22×D4—111no—
247C23×C4—221no—
248C2×Q16—118no—
249C2×Q16—442no—
250C2×Q16—442no—
251C2×SD16—118no—
252C2×SD16—442no—
253C2×SD16—442no—
254C22×C8C22×D1112yes—
255C22×C8C22×D1112yes—
256C4⋊Q8—114no—
257C4⋊Q8—222no—
258#124—114no—
259#124—222no—
260C4×Q8—442no—
261C4×Q8—442no—
262C4×Q8—442no—
263C4×D4—442no—
264C2×C4⋊C4—111no—
265C2×C4⋊C4—112no—
266C2×C4⋊C4—111no—
267C2×C4⋊C4—222no—
268C2×C4⋊C4—221no—
269C2×C4⋊C4—222no—
270C2×C4⋊C4—222no—
271C2×C22⋊C4—222no—
272C2×C22⋊C4—221no—
273C2×C22⋊C4—222no—
274C2×C42—222no—
275C2×C42—221no—
276C2×C42C22×D1111yescharacteristic
277C2×C42C22×D1111yescharacteristic
278C2×C42C22×D1111yescharacteristic
279#103—442no—
280#103—442no—
281#103—442no—
282#102—442no—
283#97C4×D1114yescomplemented
284C4×C8—22no—
285#95—222no—
286#95—222no—
287#95—222no—
288#95—222no—
289#95D4412yes—
290#95D4412yes—
291C2×C11#232811yescharacteristic
292C2×C11#294712yes—
293C2×C11#322211yescharacteristic
294C2×C11#322311yescharacteristic
295C2×C11#322512yes—
296D11—82no—
297C24—221no—
298C2×Q8—112no—
299C2×Q8—114no—
300C2×Q8—221no—
301C2×Q8—222no—
302C2×Q8—222no—
303C2×Q8—222no—
304C2×Q8—222no—
305C2×Q8—222no—
306C2×Q8—222no—
307C2×Q8—222no—
308C2×D4—112no—
309C2×D4—221no—
310C2×D4—222no—
311C2×D4—222no—
312C22×C4—221no—
313C22×C4—222no—
314C22×C4—222no—
315C22×C4—221no—
316C22×C4—221no—
317C22×C4—442no—
318C22×C4—882no—
319C22×C4C2×D4411yescharacteristic
320C22×C4C2×D4411yescharacteristic
321C22×C4C2×D4412yes—
322C22×C4C2×D4412yes—
323C22×C4C23×D1111yescharacteristic
324Q16—448no—
325SD16—448no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8C2×C4×D1114yes—
332C4⋊C4—228no—
333C4⋊C4—222no—
334C4⋊C4—222no—
335C4⋊C4—222no—
336C4⋊C4—222no—
337C4⋊C4—444no—
338C4⋊C4—442no—
339C4⋊C4—882no—
340C22⋊C4—228no—
341C22⋊C4—442no—
342C22⋊C4—882no—
343C42—22no—
344C42—22no—
345C42—442no—
346C42—444no—
347C42C2×C4×D1112yes—
348C42#433912yes—
349C11#2227811yescharacteristic, complemented, Sylow
350C23—222no—
351C23—442no—
352C23—442no—
353C23—441no—
354C23—441no—
355C23C22×D4411yescharacteristic
356Q8—224no—
357Q8—224no—
358Q8—224no—
359Q8—442no—
360Q8—442no—
361Q8—442no—
362D4—224no—
363D4—442no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—222no—
374C2×C4—224no—
375C2×C4—441no—
376C2×C4—442no—
377C2×C4—442no—
378C2×C4—441no—
379C2×C4—442no—
380C2×C4—442no—
381C2×C4—442no—
382C2×C4—442no—
383C2×C4—441no—
384C2×C4—442no—
385C2×C4—441no—
386C2×C4—884no—
387C2×C4C4×D4414yes—
388C2×C4C22×C4×D1111yescharacteristic
389C2×C4C2×#433911yescharacteristic
390C2×C4#6704212yes—
391C2×C4#6704212yes—
392C8—24no—
393C8—42no—
394C22—444no—
395C22—442no—
396C22—881no—
397C22—881no—
398C22—882no—
399C22C2×C4×D4411yescharacteristic
400C22#1062305611yescharacteristic
401C22#1062305611yescharacteristic
402C22#1062328212yes—
403C22#1062328412yes—
404C4—24no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—442no—
409C4—444no—
410C4—881no—
411C4—881no—
412C4—882no—
413C4#1062305212yes—
414C4#1062306612yes—
415C2—882no—
416C2#1307134111yescharacteristic
417C2#1307746112yes—
418C2#1307969011yescharacteristic
419C2#1307970111yescharacteristic
420C2#1307972712yes—
subgroupG/H
centerC23C22×D44
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1746C2
socleC23×C11C22×D4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenter#1746D11
hypocenterC11#22278
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #13071335C2semidirect
5. #13069418C2semidirect
7. #13068956C2semidirect
8. #13068956C2semidirect
10. #13068750C2semidirect
11. C11×#1746C2semidirect
31. #10622328C4semidirect
63. C2×Dic44C2×C4semidirect
71. #66922C2×C4semidirect
75. C11×#97C2×C4semidirect
150. #1746D11semidirect
283. #97C4×D11semidirect
349. C11#22278semidirect

Maximal subgroups

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

subgroupindex
C2×#106223282
#130713352
#130713272
#130694272
#130694182
#130689702
#130689562
#130689492
#130687502
C11×#17462
#2227811