Finite Group Databasebeta

← #14757439

Subgroups — #14757439

Order 2816.

1726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623008C211yesmaximal, characteristic
2C2×#10622579C214yesmaximal, complemented
3C2×C4×#4318C211yesmaximal, characteristic, complemented
4#13069430C214yesmaximal, complemented
5#13069421C212yesmaximal, complemented
6#13069417C212yesmaximal, complemented
7C2×C11×#384C211yesmaximal, characteristic, complemented
8C22×#4339C2212yes—
9C22×C11×C4⋊C4—22no—
10C22×C11×C4⋊C4C2212yes—
11C22×C42×C11C2211yescharacteristic
12C2×#66983—22no—
13C22×#4318C2212yes—
14#10623008—24no—
15#10623008C418yescomplemented
16C2×#66928C2212yes—
17C2×#66927C2211yescharacteristic
18C2×C4×D44—22no—
19C2×#66922C2211yescharacteristic
20C11×#384—22no—
21C11×#384C2212yes—
22C11×#384C2212yes—
23C2×#66874C2218yes—
24#10622579—216no—
25C2×#66873C2218yes—
26C4×#4318—24no—
27C2×#66867C2212yes—
28C2×C4×C11⋊C8C2212yes—
29C23×C4×C11—22no—
30C23×C4×C11D412yes—
31C23×C4×C11C2311yescharacteristic
32C22×C22⋊D11—22no—
33C2×#4339—24no—
34C2×#4339—28no—
35C2×#4339—28no—
36C2×#4339C2×C418yes—
37C22×D44—22no—
38C22×D44C2312yes—
39C22×C4×D11—22no—
40C22×Dic22C2312yes—
41C2×C11×C4⋊C4—28no—
42C2×C11×C4⋊C4—24no—
43C2×C11×C4⋊C4—24no—
44C2×C11×C4⋊C4—22no—
45C2×C11×C4⋊C4—48no—
46C2×C11×C4⋊C4C2314yes—
47C2×C42×C11C2×C414yes—
48C2×C42×C11D414yes—
49C2×C42×C11D412yes—
50C2×C42×C11C2311yescharacteristic
51C2×C42×C11C2311yescharacteristic
52#66983—216no—
53C2×(C2×D11)⋊C4—24no—
54C2×C4⋊Dic11—22no—
55C2×#4318—24no—
56C2×#4318—24no—
57C2×#4318D418yescomplemented
58C22×C11⋊C8C2314yes—
59#66928—24no—
60#66928C2×C418yescomplemented
61#66927—22no—
62#66927C2×C414yescomplemented
63C4×D44—216no—
64#66922—22no—
65#66922C2×C414yescomplemented
66#66874—216no—
67#66873—216no—
68#66867—24no—
69C4×C11⋊C8—24no—
70#21200—111nomaximal, Sylow
71C24×C11C2×D411yescharacteristic
72C23×D11—22no—
73C22×C4×C11—24no—
74C22×C4×C11—24no—
75C22×C4×C11—24no—
76C22×C4×C11—48no—
77C22×C4×C11—44no—
78C22×C4×C11C22⋊C418yes—
79C22×C4×C11C22×C412yes—
80C22×C4×C11C2×D412yes—
81C22×C4×C11C2×D411yescharacteristic
82C22×C4×C11C2×D412yes—
83C22×C4×C11C2×D412yes—
84C22×C4×C11C2×D414yes—
85C22×C4×C11#4114yes—
86C22×C4×C11C2411yescharacteristic
87C2×C22⋊D11—28no—
88C2×C22⋊D11—44no—
89C2×C22⋊D11—44no—
90C22×Dic11—22no—
91#4339—232no—
92#4339—416no—
93C2×D44—28no—
94C2×D44—22no—
95C2×D44—24no—
96C2×D44—24no—
97C2×D44—44no—
98C2×D44—44no—
99C2×D44C22×C414yes—
100C2×C4×D11—28no—
101C2×C4×D11—44no—
102C2×C4×D11—44no—
103C2×Dic22—24no—
104C2×Dic22—22no—
105C2×Dic22—24no—
106C2×Dic22C22×C414yes—
107C11×C4⋊C4—28no—
108C11×C4⋊C4—48no—
109C11×C4⋊C4—44no—
110C11×C4⋊C4—48no—
111C42×C11C22⋊C418yes—
112C42×C11C22×C412yes—
113C42×C11C22×C412yes—
114C42×C11C2×D412yes—
115C42×C11C2×D412yes—
116(C2×D11)⋊C4—216no—
117C4⋊Dic11—28no—
118#4318—216no—
119C2×C11⋊C8—24no—
120C2×C11⋊C8—24no—
121C2×C11⋊C8C2×D418yes—
122C2×#532—111no—
123C2×#417—114no—
124C2×C4×M16—111no—
125C2×#384D1111yescharacteristic, complemented
126#1959—114no—
127#1957—112no—
128#1956—112no—
129C23×C11—22no—
130C23×C11—22no—
131C23×C11C2×C22⋊C412yes—
132C23×C11#12412yes—
133C23×C11#12712yes—
134C23×C11C22×D411yescharacteristic
135C22×D11—24no—
136C22×D11—42no—
137C22×D11—42no—
138C22×D11—44no—
139C22×D11—44no—
140C2×C4×C11—22no—
141C2×C4×C11—22no—
142C2×C4×C11—216no—
143C2×C4×C11—44no—
144C2×C4×C11—48no—
145C2×C4×C11—88no—
146C2×C4×C11C2×C22⋊C414yes—
147C2×C4×C11C2×C22⋊C412yes—
148C2×C4×C11C2×C22⋊C414yes—
149C2×C4×C11C4×D418yes—
150C2×C4×C11#12118yes—
151C2×C4×C11#12412yes—
152C2×C4×C11#12712yes—
153C2×C4×C11AGL(1,ℤ/8)14yes—
154C2×C4×C11#13714yes—
155C2×C4×C11C23×C411yescharacteristic
156C2×C4×C11C22×D411yescharacteristic
157C2×C4×C11C22×D412yes—
158C2×C4×C11C2×#4112yes—
159C22⋊D11—416no—
160C2×Dic11—24no—
161C2×Dic11—42no—
162C2×Dic11—42no—
163D44—28no—
164D44—44no—
165D44—416no—
166C4×D11—416no—
167Dic22—28no—
168Dic22—44no—
169C11⋊C8—28no—
170C22×#41—112no—
171C22×M16—112no—
172#532—118no—
173#532—224no—
174C2×C4⋊Q8—111no—
175C2×#127—111no—
176C2×#124—112no—
177C2×#121—222no—
178C2×C4×D4—222no—
179C22×C4⋊C4—22no—
180C22×C4⋊C4C2×D1112yes—
181C22×C42C2×D1111yescharacteristic
182#417—2216no—
183C2×#103—118no—
184C2×#102—118no—
185C4×M16—224no—
186C2×#97—112no—
187C2×C4×C8—112no—
188#384—22no—
189#384C2×D1112yes—
190#384C2×D1112yes—
191C22×C11—24no—
192C22×C11—24no—
193C22×C11—22no—
194C22×C11—22no—
195C22×C11#39014yes—
196C22×C11C22×C22⋊C411yescharacteristic
197C22×C11C2×#12411yescharacteristic
198C22×C11C2×#12711yescharacteristic
199C22×C11C2×AGL(1,ℤ/8)12yes—
200C22×C11C2×#13712yes—
201C2×D11—48no—
202C2×D11—44no—
203C2×D11—84no—
204C2×D11—82no—
205C2×D11—82no—
206C4×C11—24no—
207C4×C11—84no—
208C4×C11#39014yes—
209C4×C11#41718yescomplemented
210C4×C11C2×C4×D412yes—
211C4×C11C2×#12112yes—
212Dic11—44no—
213C2×#41—118no—
214C2×#41—224no—
215C2×#41—228no—
216C2×#41—228no—
217C22×Q8—112no—
218C22×D4—112no—
219C22×D4—222no—
220C22×D4—222no—
221C23×C4—22no—
222C23×C4—222no—
223C23×C4C22⋊D1112yes—
224C23×C4C22×D1111yescharacteristic
225C2×M16—118no—
226C2×M16—224no—
227C2×M16—224no—
228C22×C8—114no—
229C4⋊Q8—114no—
230C4⋊Q8—222no—
231#127—114no—
232#127—222no—
233#124—118no—
234#124—224no—
235#121—2216no—
236C4×D4—2216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—222no—
243C2×C4⋊C4C22×D1114yes—
244C2×C22⋊C4—224no—
245C2×C42C4×D1114yes—
246C2×C42D4414yes—
247C2×C42C22⋊D1112yes—
248C2×C42C22×D1111yescharacteristic
249C2×C42C22×D1111yescharacteristic
250#103—2216no—
251#102—2216no—
252#97—224no—
253C4×C8—224no—
254C2×C11—24no—
255C2×C11C2×#39011yescharacteristic
256C2×C11C2×#41712yes—
257C2×C11#314112yes—
258C2×C11#315111yescharacteristic
259C2×C11#315211yescharacteristic
260D11—84no—
261C24—222no—
262C24C2×C22⋊D1111yescharacteristic
263#41—2232no—
264#41—4416no—
265C2×Q8—114no—
266C2×Q8—222no—
267C2×Q8—224no—
268C2×Q8—224no—
269C2×D4—114no—
270C2×D4—228no—
271C2×D4—228no—
272C2×D4—222no—
273C2×D4—224no—
274C2×D4—224no—
275C2×D4—444no—
276C2×D4—444no—
277C2×D4—444no—
278C2×D4—444no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—48no—
283C22×C4—44no—
284C22×C4—222no—
285C22×C4—228no—
286C22×C4—444no—
287C22×C4—444no—
288C22×C4(C2×D11)⋊C418yes—
289C22×C4C2×C4×D1112yes—
290C22×C4C2×D4412yes—
291C22×C4D4×D1114yes—
292C22×C4#434314yes—
293C22×C4C2×C22⋊D1112yes—
294C22×C4C2×C22⋊D1112yes—
295C22×C4C2×C22⋊D1111yescharacteristic
296C22×C4C23×D1111yescharacteristic
297M16—2216no—
298C2×C8—118no—
299C2×C8—224no—
300C2×C8—224no—
301C4⋊C4—28no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—228no—
306C22⋊C4—2216no—
307C42(C2×D11)⋊C418yes—
308C42C2×C4×D1112yes—
309C42C2×C4×D1112yes—
310C42C2×D4412yes—
311C42C2×D4412yes—
312C11#2120011yescharacteristic, complemented, Sylow
313C23—22no—
314C23—22no—
315C23—224no—
316C23—444no—
317C23—442no—
318C23—444no—
319C23—442no—
320C23C2×(C2×D11)⋊C412yes—
321C23#6699312yes—
322C23#6700012yes—
323C23C22×C22⋊D1111yescharacteristic
324Q8—228no—
325Q8—444no—
326D4—228no—
327D4—4416no—
328D4—444no—
329D4—4416no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—216no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—88no—
336C2×C4—224no—
337C2×C4—442no—
338C2×C4—442no—
339C2×C4—4416no—
340C2×C4#6694618yes—
341C2×C4#6694818yes—
342C2×C4C2×(C2×D11)⋊C414yes—
343C2×C4C2×(C2×D11)⋊C414yes—
344C2×C4C2×(C2×D11)⋊C412yes—
345C2×C4#6698514yes—
346C2×C4#6699312yes—
347C2×C4#6699514yes—
348C2×C4#6700012yes—
349C2×C4C22×C4×D1111yescharacteristic
350C2×C4C22×D4411yescharacteristic
351C2×C4C2×D4×D1112yes—
352C2×C4C2×#434312yes—
353C8—228no—
354C22—22no—
355C22—22no—
356C22—24no—
357C22—24no—
358C22—448no—
359C22—444no—
360C22—882no—
361C22—882no—
362C22—884no—
363C22#1062260114yes—
364C22C22×(C2×D11)⋊C411yescharacteristic
365C22C2×#6698512yes—
366C22C2×#6699311yescharacteristic
367C22C2×#6699512yes—
368C22C2×#6700011yescharacteristic
369C4—24no—
370C4—84no—
371C4—444no—
372C4#1062257918yescomplemented
373C4#1062260114yes—
374C4C2×#6694612yes—
375C4C2×#6694812yes—
376C2—24no—
377C2—884no—
378C2C2×#1062260111yescharacteristic
379C2C2×#1062257912yes—
380C2#1307901212yes—
381C2#1307906611yescharacteristic
382C2#1307907311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#384C2
socleC23×C11C22×D4
prime radicalC2×#10623008C2
cosocleC22×C4×C11C24
hypercenterC2×#384D11
hypocenterC11#21200
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#10622579C2semidirect
3. C2×C4×#4318C2semidirect
4. #13069430C2semidirect
5. #13069421C2semidirect
6. #13069417C2semidirect
7. C2×C11×#384C2semidirect
15. #10623008C4semidirect
57. C2×#4318D4semidirect
60. #66928C2×C4semidirect
62. #66927C2×C4semidirect
65. #66922C2×C4semidirect
125. C2×#384D11semidirect
209. C4×C11#417semidirect
312. C11#21200semidirect
372. C4#10622579semidirect

Maximal subgroups

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

subgroupindex
C2×#106230082
C2×#106225792
C2×C4×#43182
#130694302
#130694212
#130694172
C2×C11×#3842
#2120011