Finite Group Databasebeta

← #14764290

Subgroups — #14764290

Order 2816.

966 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622298C211yesmaximal, characteristic
2#13071021C211yesmaximal, characteristic
3D11×#384C211yesmaximal, characteristic
4#13069378C214yesmaximal, complemented
5#13068810C214yesmaximal
6#13068773C211yesmaximal, characteristic, complemented
7#13068772C211yesmaximal, characteristic, complemented
8#13068743C211yesmaximal, characteristic, complemented
9C11×#1834C211yesmaximal, characteristic, complemented
10C2×#66945—21no—
11C2×C4⋊C4×D11—21no—
12C2×C4⋊C4×D11—21no—
13C2×C4⋊C4×D11C2212yes—
14C2×C42×D11C2211yescharacteristic
15C2×C4×C8×C11C2211yescharacteristic
16C11×#384C2211yescharacteristic
17C11×#384C2211yescharacteristic
18C2×(C2×D11)⋊C8C2212yes—
19C2×#66881C2214yes—
20C2×Dic11⋊C8C2212yes—
21#10622600—21no—
22#10622600C2212yes—
23#10622590—21no—
24#10622589—21no—
25#10622589—21no—
26#10622589C2212yes—
27C4⋊C4×Dic11—21no—
28C4⋊C4×Dic11C2212yes—
29C2×#66872C2214yes—
30#10622548C2211yescharacteristic
31#10622548C2211yescharacteristic
32C2×C4×C11⋊C8C2211yescharacteristic
33#10622298C418yescomplemented
34C11×#340C2214yes—
35#10622167C2214yes—
36C22×C4×D11—22no—
37C22×C4×D11D411yescharacteristic
38C22×C4×D11D411yescharacteristic
39C22×C4×D11C2311yescharacteristic
40C22×C8×C11C2312yes—
41C2×C11×C4⋊C4—21no—
42C2×C11×C4⋊C4—21no—
43C2×C11×C4⋊C4—21no—
44C2×C11×C4⋊C4—21no—
45C2×C11×C4⋊C4C2312yes—
46C2×C11×C4⋊C4C2312yes—
47C2×C42×C11C2311yescharacteristic
48C2×(C2×D11)⋊C4—22no—
49C2×C4⋊Dic11—21no—
50C2×C4⋊Dic11—21no—
51C2×C4⋊Dic11—21no—
52C2×C4⋊Dic11—21no—
53C2×C4⋊Dic11C2312yes—
54C2×C4⋊Dic11C2312yes—
55C2×Dic11⋊C4—22no—
56C2×Dic11⋊C4—22no—
57C2×Dic11⋊C4—22no—
58C2×C4×Dic11—22no—
59C2×C4×Dic11D411yescharacteristic
60C2×C4×Dic11D411yescharacteristic
61C2×C4×Dic11C2311yescharacteristic
62C22×C11⋊C8C2312yes—
63#66945—44no—
64C4⋊C4×D11—44no—
65C4⋊C4×D11—44no—
66C4⋊C4×D11—44no—
67C42×D11C2×C414yes—
68C42×D11Q814yes—
69C4×C8×C11C2×C414yescomplemented
70C11⋊#95—22no—
71C11⋊#95—22no—
72(C2×D11)⋊C8C2×C418yes—
73#66881—44no—
74Dic11⋊C8C2×C418yescomplemented
75#66872—44no—
76C4×C11⋊C8C2×C414yescomplemented
77#21981—111nomaximal, Sylow
78C23×D11C2×D411yescharacteristic
79C22×C4×C11—22no—
80C22×C4×C11—22no—
81C22×C4×C11C2×D411yescharacteristic
82C22×C4×C11C2×D411yescharacteristic
83C22×C4×C11C2411yescharacteristic
84C22×Dic11—22no—
85C22×Dic11—22no—
86C22×Dic11C2×D411yescharacteristic
87C22×Dic11C2×D411yescharacteristic
88C22×Dic11C2×D411yescharacteristic
89C2×C4×D11—22no—
90C2×C4×D11—22no—
91C2×C4×D11—22no—
92C2×C4×D11—22no—
93C2×C4×D11—44no—
94C2×C4×D11—44no—
95C2×C4×D11—44no—
96C2×C4×D11C4⋊C414yes—
97C2×C4×D11C4⋊C414yes—
98C2×C4×D11SD1614yes—
99C2×C4×D11SD1614yes—
100C2×C4×D11C22×C412yes—
101C2×C4×D11C2×Q812yes—
102C2×C8×C11—22no—
103C2×C8×C11—22no—
104C2×C8×C11C22×C414yes—
105C11×C4⋊C4—24no—
106C11×C4⋊C4—24no—
107C11×C4⋊C4—42no—
108C11×C4⋊C4—42no—
109C11×C4⋊C4—42no—
110C11×C4⋊C4—42no—
111C42×C11C22×C412yes—
112C42×C11C2×Q812yes—
113(C2×D11)⋊C4—44no—
114C4⋊Dic11—24no—
115C4⋊Dic11—24no—
116C4⋊Dic11—42no—
117C4⋊Dic11—42no—
118C4⋊Dic11—42no—
119C4⋊Dic11—42no—
120Dic11⋊C4—44no—
121Dic11⋊C4—44no—
122Dic11⋊C4—44no—
123C4×Dic11—44no—
124C4×Dic11C4⋊C414yes—
125C4×Dic11C4⋊C414yes—
126C4×Dic11C22×C412yes—
127C4×Dic11C2×Q812yes—
128C2×C11⋊C8—22no—
129C2×C11⋊C8—22no—
130C2×C11⋊C8C22×C414yes—
131C2×#431—111no—
132#2285—111no—
133C2×#384—111no—
134#1948—114no—
135#1853—114no—
136#1844—111no—
137#1844—111no—
138#1834—111no—
139#1834D1111yescharacteristic, complemented
140C23×C11C22×D411yescharacteristic
141C22×D11—21no—
142C22×D11—21no—
143C22×D11—21no—
144C22×D11—21no—
145C22×D11C2×C4⋊C412yes—
146C22×D11C2×SD1612yes—
147C22×D11C2×SD1612yes—
148C2×C4×C11—42no—
149C2×C4×C11—42no—
150C2×C4×C11—42no—
151C2×C4×C11—42no—
152C2×C4×C11—42no—
153C2×C4×C11—42no—
154C2×C4×C11C2×C4⋊C412yes—
155C2×C4×C11C2×C4⋊C412yes—
156C2×C4×C11C2×SD1612yes—
157C2×C4×C11C2×SD1612yes—
158C2×C4×C11AGL(1,ℤ/8)12yes—
159C2×C4×C11#13712yes—
160C2×C4×C11C23×C411yescharacteristic
161C2×C4×C11C22×Q811yescharacteristic
162C2×C4×C11H(5,𝔽2)12yes—
163C2×C4×C112−1+412yes—
164C2×Dic11—21no—
165C2×Dic11—21no—
166C2×Dic11—21no—
167C2×Dic11—21no—
168C2×Dic11—42no—
169C2×Dic11—42no—
170C2×Dic11—42no—
171C2×Dic11—42no—
172C2×Dic11—42no—
173C2×Dic11—42no—
174C2×Dic11C2×C4⋊C412yes—
175C2×Dic11C2×C4⋊C412yes—
176C2×Dic11C2×C4⋊C412yes—
177C2×Dic11C2×SD1612yes—
178C2×Dic11C2×SD1612yes—
179C4×D11—24no—
180C4×D11—24no—
181C4×D11—28no—
182C4×D11—88no—
183C4×D11#106116yescomplemented
184C8×C11—24no—
185C11⋊C8—24no—
186C2×#117—221no—
187C22×C4⋊C4—112no—
188C22×C4⋊C4—221no—
189C22×C4⋊C4—221no—
190C22×C42—111no—
191#431—118no—
192C2×#106—114no—
193C2×#106—114no—
194C2×C4⋊C8—112no—
195C2×C22⋊C8—112no—
196C2×C4×C8—111no—
197C2×C4×C8C2×D1111yescharacteristic
198#384—112no—
199#384—111no—
200#384—111no—
201#384—221no—
202#384—221no—
203#384C2×D1111yescharacteristic
204#384C2×D1111yescharacteristic
205#382—221no—
206#380—112no—
207#380—221no—
208C4×C4⋊C4—112no—
209C4×C4⋊C4—221no—
210#340—114no—
211#340C2×D1114yes—
212C22×C11C22×C4⋊C411yescharacteristic
213C22×C11#53511yescharacteristic
214C22×C11#53611yescharacteristic
215C22×C11C22×SD1611yescharacteristic
216C22×C11C22×SD1611yescharacteristic
217C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C11C2×#13711yescharacteristic
219C2×D11—22no—
220C2×D11—22no—
221C2×D11—22no—
222C2×D11—22no—
223C2×D11—22no—
224C2×D11—22no—
225C2×D11C2×#10614yes—
226C4×C11—82no—
227C4×C11—82no—
228C4×C11C2×#10614yes—
229C4×C11#42814yes—
230C4×C11#52012yes—
231C4×C11#54412yes—
232Dic11—22no—
233Dic11—22no—
234Dic11—82no—
235Dic11—82no—
236Dic11C2×#10614yes—
237C23×C4—111no—
238C23×C4—111no—
239C23×C4—111no—
240C23×C4—222no—
241C22×C8—112no—
242C22×C8C22×D1112yes—
243#117—444no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—112no—
249C2×C4⋊C4—112no—
250C2×C4⋊C4—222no—
251C2×C4⋊C4—222no—
252C2×C4⋊C4—222no—
253C2×C4⋊C4—221no—
254C2×C4⋊C4—221no—
255C2×C4⋊C4—221no—
256C2×C4⋊C4—221no—
257C2×C4⋊C4—444no—
258C2×C4⋊C4—444no—
259C2×C4⋊C4—444no—
260C2×C4⋊C4C22×D1112yes—
261C2×C4⋊C4C22×D1112yes—
262C2×C22⋊C4—222no—
263C2×C42—111no—
264C2×C42—111no—
265C2×C42—111no—
266C2×C42—114no—
267C2×C42—114no—
268C2×C42—222no—
269C2×C42C22×D1111yescharacteristic
270#106—444no—
271#106—444no—
272C4⋊C8—118no—
273C22⋊C8—118no—
274C4×C8—114no—
275C4×C8C4×D1114yescomplemented
276#95—222no—
277#95—222no—
278C2×C11#231711yescharacteristic
279C2×C11C22×#10611yescharacteristic
280C2×C11C2×#42811yescharacteristic
281C2×C11#320611yescharacteristic
282C2×C11#320811yescharacteristic
283C2×C11#322711yescharacteristic
284C2×C11#322911yescharacteristic
285D11—24no—
286C24—111no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—111no—
290C22×C4—114no—
291C22×C4—114no—
292C22×C4—114no—
293C22×C4—111no—
294C22×C4—111no—
295C22×C4—114no—
296C22×C4—112no—
297C22×C4—112no—
298C22×C4—222no—
299C22×C4—222no—
300C22×C4—222no—
301C22×C4—222no—
302C22×C4—222no—
303C22×C4—222no—
304C22×C4—444no—
305C22×C4—444no—
306C22×C4—444no—
307C22×C4D4×D1111yescharacteristic
308C22×C4D4×D1111yescharacteristic
309C22×C4C23×D1111yescharacteristic
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—114no—
313C2×C8—222no—
314C2×C8—222no—
315C2×C8C2×C4×D1114yes—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—224no—
323C4⋊C4—224no—
324C4⋊C4—444no—
325C4⋊C4—444no—
326C4⋊C4—444no—
327C4⋊C4—442no—
328C4⋊C4—442no—
329C4⋊C4—442no—
330C4⋊C4—442no—
331C22⋊C4—444no—
332C42—114no—
333C42—114no—
334C42—112no—
335C42—112no—
336C42—444no—
337C42C2×C4×D1112yes—
338C42Q8×D1112yes—
339C11#2198111yescharacteristic, complemented, Sylow
340C23—112no—
341C23—112no—
342C23—112no—
343C23—221no—
344C23—221no—
345C23—221no—
346C23—221no—
347C23C2×D4×D1111yescharacteristic
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—112no—
355C2×C4—112no—
356C2×C4—112no—
357C2×C4—1116no—
358C2×C4—112no—
359C2×C4—112no—
360C2×C4—221no—
361C2×C4—221no—
362C2×C4—221no—
363C2×C4—221no—
364C2×C4—228no—
365C2×C4—224no—
366C2×C4—224no—
367C2×C4—442no—
368C2×C4—442no—
369C2×C4—442no—
370C2×C4—442no—
371C2×C4—442no—
372C2×C4—442no—
373C2×C4—888no—
374C2×C4C4⋊C4×D1112yes—
375C2×C4C4⋊C4×D1112yes—
376C2×C4#6696412yes—
377C2×C4SD16×D1112yes—
378C2×C4SD16×D1112yes—
379C2×C4#6697112yes—
380C2×C4C22×C4×D1111yescharacteristic
381C2×C4C2×Q8×D1111yescharacteristic
382C2×C4#6704212yes—
383C2×C4#6704312yes—
384C8—24no—
385C8—224no—
386C22—114no—
387C22—222no—
388C22—222no—
389C22—222no—
390C22—222no—
391C22—222no—
392C22—222no—
393C22C2×C4⋊C4×D1111yescharacteristic
394C22#1062320511yescharacteristic
395C22#1062320611yescharacteristic
396C22C2×#6696411yescharacteristic
397C22C2×SD16×D1111yescharacteristic
398C22C2×SD16×D1111yescharacteristic
399C22C2×#6697111yescharacteristic
400C4—82no—
401C4—82no—
402C4—114no—
403C4—222no—
404C4—222no—
405C4—882no—
406C4—882no—
407C4D11×#10614yes—
408C4#1062249414yes—
409C4#1062305412yes—
410C4#1062322012yes—
411C2—224no—
412C2#1307124111yescharacteristic
413C2C2×D11×#10611yescharacteristic
414C2C2×#1062249411yescharacteristic
415C2#1307954211yescharacteristic
416C2#1307955811yescharacteristic
417C2#1307974011yescharacteristic
418C2#1307975611yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1834C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1834D11
hypocenterC11#21981
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
4. #13069378C2semidirect
6. #13068773C2semidirect
7. #13068772C2semidirect
8. #13068743C2semidirect
9. C11×#1834C2semidirect
33. #10622298C4semidirect
69. C4×C8×C11C2×C4semidirect
74. Dic11⋊C8C2×C4semidirect
76. C4×C11⋊C8C2×C4semidirect
139. #1834D11semidirect
183. C4×D11#106semidirect
275. C4×C8C4×D11semidirect
339. C11#21981semidirect

Maximal subgroups

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

subgroupindex
C2×#106222982
#130710212
D11×#3842
#130693782
#130688102
#130687732
#130687722
#130687432
C11×#18342
#2198111