Finite Group Databasebeta

← #14764296

Subgroups — #14764296

Order 2816.

966 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622538C211yesmaximal, characteristic
2#13071023C211yesmaximal, characteristic
3#13069372C214yesmaximal, complemented
4#13068804C214yesmaximal
5#13068775C212yesmaximal, complemented
6#13068743C212yesmaximal, complemented
7C2×C11×#384C211yesmaximal, characteristic
8C22×C11×C4⋊C4—21no—
9C22×C11×C4⋊C4—21no—
10C22×C11×C4⋊C4C2212yes—
11C22×C42×C11C2211yescharacteristic
12C2×#66979—21no—
13C11×#384—21no—
14C11×#384—21no—
15C11×#384C2212yes—
16C11×#384C2212yes—
17#10622825—21no—
18#10622825C2212yes—
19C2×C22⋊(C11⋊C8)C2212yes—
20C2×#66871C2218yes—
21#10622549—21no—
22#10622548C2211yescharacteristic
23#10622548C2211yescharacteristic
24C4×C4⋊Dic11—21no—
25C4×C4⋊Dic11C2212yes—
26#10622538C418yescomplemented
27C2×C4⋊(C11⋊C8)C2212yes—
28C2×C4×C11⋊C8C2212yes—
29#10622167C2218yes—
30C23×C4×C11—22no—
31C23×C4×C11D411yescharacteristic
32C23×C4×C11D411yescharacteristic
33C23×C4×C11C2311yescharacteristic
34C2×C11×C4⋊C4—22no—
35C2×C11×C4⋊C4—22no—
36C2×C11×C4⋊C4—22no—
37C2×C11×C4⋊C4—22no—
38C2×C11×C4⋊C4—22no—
39C2×C11×C4⋊C4—44no—
40C2×C11×C4⋊C4—44no—
41C2×C11×C4⋊C4—44no—
42C2×C11×C4⋊C4C2314yes—
43C2×C42×C11C2×C414yes—
44C2×C42×C11D411yescharacteristic
45C2×C42×C11D411yescharacteristic
46C2×C42×C11Q814yes—
47C2×C42×C11C2311yescharacteristic
48C2×C42×C11C2311yescharacteristic
49C2×C22⋊Dic11—22no—
50#66979—44no—
51C2×C4⋊Dic11—21no—
52C2×C4⋊Dic11—21no—
53C2×C4⋊Dic11—21no—
54C2×C4⋊Dic11—21no—
55C2×C4⋊Dic11C2312yes—
56C2×C4⋊Dic11C2312yes—
57C2×C4×Dic11—22no—
58C22×C11⋊C8C2314yes—
59C11⋊#95—22no—
60C11⋊#95—22no—
61C22⋊(C11⋊C8)C2×C418yes—
62#66871—48no—
63C4⋊(C11⋊C8)C2×C418yescomplemented
64C4×C11⋊C8C2×C418yescomplemented
65#21982—111nomaximal, Sylow
66C24×C11C2×D411yescharacteristic
67C22×C4×C11—22no—
68C22×C4×C11—22no—
69C22×C4×C11—22no—
70C22×C4×C11—22no—
71C22×C4×C11—24no—
72C22×C4×C11—44no—
73C22×C4×C11—44no—
74C22×C4×C11—44no—
75C22×C4×C11C4⋊C414yes—
76C22×C4×C11C4⋊C414yes—
77C22×C4×C11D814yes—
78C22×C4×C11Q1614yes—
79C22×C4×C11C22×C412yes—
80C22×C4×C11C2×D411yescharacteristic
81C22×C4×C11C2×D411yescharacteristic
82C22×C4×C11C2×D411yescharacteristic
83C22×C4×C11C2×D411yescharacteristic
84C22×C4×C11C2×D411yescharacteristic
85C22×C4×C11C2×Q812yes—
86C22×C4×C11C2411yescharacteristic
87C22×Dic11—22no—
88C22×Dic11—22no—
89C11×C4⋊C4—28no—
90C11×C4⋊C4—44no—
91C11×C4⋊C4—44no—
92C11×C4⋊C4—44no—
93C11×C4⋊C4—44no—
94C11×C4⋊C4—44no—
95C42×C11C4⋊C414yes—
96C42×C11C4⋊C414yes—
97C42×C11C22×C412yes—
98C42×C11C22×C412yes—
99C42×C11C2×Q812yes—
100C42×C11C2×Q812yes—
101C22⋊Dic11—44no—
102C4⋊Dic11—24no—
103C4⋊Dic11—24no—
104C4⋊Dic11—42no—
105C4⋊Dic11—42no—
106C4⋊Dic11—42no—
107C4⋊Dic11—42no—
108C4×Dic11—44no—
109C2×C11⋊C8—24no—
110C2×C11⋊C8—24no—
111C2×C11⋊C8C22×C418yes—
112C2×#431—111no—
113#2285—111no—
114C2×#384D1111yescharacteristic
115#1947—114no—
116#1852—114no—
117#1845—112no—
118#1834—112no—
119C23×C11—21no—
120C23×C11—21no—
121C23×C11—21no—
122C23×C11—21no—
123C23×C11C2×C4⋊C412yes—
124C23×C11C2×D812yes—
125C23×C11C2×Q1612yes—
126C23×C11C22×D411yescharacteristic
127C2×C4×C11—21no—
128C2×C4×C11—21no—
129C2×C4×C11—21no—
130C2×C4×C11—21no—
131C2×C4×C11—24no—
132C2×C4×C11—24no—
133C2×C4×C11—28no—
134C2×C4×C11—44no—
135C2×C4×C11—44no—
136C2×C4×C11—44no—
137C2×C4×C11—88no—
138C2×C4×C11#107116yescomplemented
139C2×C4×C11C2×C4⋊C412yes—
140C2×C4×C11C2×C4⋊C412yes—
141C2×C4×C11C2×C4⋊C412yes—
142C2×C4×C11C2×C4⋊C412yes—
143C2×C4×C11C2×C4⋊C412yes—
144C2×C4×C11C2×D812yes—
145C2×C4×C11C2×D812yes—
146C2×C4×C11C2×Q1612yes—
147C2×C4×C11C2×Q1612yes—
148C2×C4×C11AGL(1,ℤ/8)12yes—
149C2×C4×C11#13712yes—
150C2×C4×C11C23×C411yescharacteristic
151C2×C4×C11C22×Q811yescharacteristic
152C2×C4×C11H(5,𝔽2)12yes—
153C2×C4×C112−1+412yes—
154C2×Dic11—42no—
155C2×Dic11—42no—
156C2×Dic11—42no—
157C2×Dic11—42no—
158C2×Dic11—42no—
159C2×Dic11—42no—
160C11⋊C8—28no—
161C2×#117—221no—
162C22×C4⋊C4—21no—
163C22×C4⋊C4—21no—
164C22×C4⋊C4C2×D1112yes—
165C22×C42C2×D1111yescharacteristic
166#431—118no—
167C2×#107—118no—
168C2×C4⋊C8—112no—
169C2×C22⋊C8—112no—
170C2×C4×C8—112no—
171#384—21no—
172#384—21no—
173#384—111no—
174#384—111no—
175#384C2×D1112yes—
176#384C2×D1112yes—
177#382—221no—
178#380—112no—
179#380—221no—
180C4×C4⋊C4—112no—
181C4×C4⋊C4—221no—
182#340—118no—
183C22×C11—22no—
184C22×C11—22no—
185C22×C11—22no—
186C22×C11—22no—
187C22×C11—22no—
188C22×C11—22no—
189C22×C11C2×#10714yes—
190C22×C11C22×C4⋊C411yescharacteristic
191C22×C11#53511yescharacteristic
192C22×C11#53611yescharacteristic
193C22×C11C22×D811yescharacteristic
194C22×C11C22×Q1611yescharacteristic
195C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
196C22×C11C2×#13711yescharacteristic
197C4×C11—22no—
198C4×C11—22no—
199C4×C11—84no—
200C4×C11C2×#10714yes—
201C4×C11C2×#10714yes—
202C4×C11#42814yes—
203C4×C11#52012yes—
204C4×C11#54412yes—
205Dic11—82no—
206Dic11—82no—
207C23×C4—22no—
208C23×C4C22⋊D1111yescharacteristic
209C23×C4C22⋊D1111yescharacteristic
210C23×C4C22×D1111yescharacteristic
211C22×C8—114no—
212#117—444no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—112no—
222C2×C4⋊C4—112no—
223C2×C4⋊C4—221no—
224C2×C4⋊C4—221no—
225C2×C4⋊C4—221no—
226C2×C4⋊C4—221no—
227C2×C4⋊C4C22×D1114yes—
228C2×C22⋊C4—222no—
229C2×C42—222no—
230C2×C42Dic2214yes—
231C2×C42C4×D1114yes—
232C2×C42C22⋊D1111yescharacteristic
233C2×C42C22⋊D1111yescharacteristic
234C2×C42C22×D1111yescharacteristic
235C2×C42C22×D1111yescharacteristic
236#107—448no—
237C4⋊C8—118no—
238C22⋊C8—118no—
239C4×C8—118no—
240#95—222no—
241#95—222no—
242C2×C11—24no—
243C2×C11#231711yescharacteristic
244C2×C11C22×#10711yescharacteristic
245C2×C11C2×#42811yescharacteristic
246C2×C11#320511yescharacteristic
247C2×C11#320911yescharacteristic
248C2×C11#322611yescharacteristic
249C2×C11#323011yescharacteristic
250C24C2×C22⋊D1111yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—222no—
260C22×C4—222no—
261C22×C4Dic11⋊C414yes—
262C22×C4Dic11⋊C414yes—
263C22×C4#432314yes—
264C22×C4#432614yes—
265C22×C4C2×Dic2212yes—
266C22×C4C2×C4×D1112yes—
267C22×C4C2×C22⋊D1111yescharacteristic
268C22×C4C2×C22⋊D1111yescharacteristic
269C22×C4C2×C22⋊D1111yescharacteristic
270C22×C4C2×C22⋊D1111yescharacteristic
271C22×C4C2×C22⋊D1111yescharacteristic
272C22×C4C23×D1111yescharacteristic
273C2×C8—118no—
274C2×C8—224no—
275C2×C8—224no—
276C4⋊C4—28no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—224no—
283C4⋊C4—224no—
284C4⋊C4—442no—
285C4⋊C4—442no—
286C4⋊C4—442no—
287C4⋊C4—442no—
288C22⋊C4—444no—
289C42—444no—
290C42Dic11⋊C414yes—
291C42Dic11⋊C414yes—
292C42C2×Dic2212yes—
293C42C2×Dic2212yes—
294C42C2×C4×D1112yes—
295C42C2×C4×D1112yes—
296C11#2198211yescharacteristic, complemented, Sylow
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C2×Dic11⋊C412yes—
302C23C2×#432312yes—
303C23C2×#432612yes—
304C23C22×C22⋊D1111yescharacteristic
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—28no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—88no—
316C2×C4—442no—
317C2×C4—442no—
318C2×C4—442no—
319C2×C4—442no—
320C2×C4—442no—
321C2×C4—442no—
322C2×C4#66871116yescomplemented
323C2×C4C2×Dic11⋊C412yes—
324C2×C4C2×Dic11⋊C412yes—
325C2×C4C2×Dic11⋊C412yes—
326C2×C4C2×Dic11⋊C412yes—
327C2×C4C2×Dic11⋊C412yes—
328C2×C4C2×#432312yes—
329C2×C4C2×#432312yes—
330C2×C4#6698512yes—
331C2×C4#6699512yes—
332C2×C4C2×#432612yes—
333C2×C4C2×#432612yes—
334C2×C4C22×Dic2211yescharacteristic
335C2×C4C22×C4×D1111yescharacteristic
336C2×C4#6703712yes—
337C2×C4#6704012yes—
338C8—228no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22C2×#6687114yes—
346C22C22×Dic11⋊C411yescharacteristic
347C22C22×#432311yescharacteristic
348C22C2×#6698511yescharacteristic
349C22#1062333611yescharacteristic
350C22C2×#6699511yescharacteristic
351C22C22×#432611yescharacteristic
352C22#1062334711yescharacteristic
353C4—22no—
354C4—22no—
355C4—84no—
356C4—882no—
357C4—882no—
358C4C2×#6687114yes—
359C4C2×#6687114yes—
360C4#1062257714yes—
361C4#1062303112yes—
362C4#1062303512yes—
363C2—24no—
364C2#1307124011yescharacteristic
365C2C22×#6687111yescharacteristic
366C2C2×#1062257711yescharacteristic
367C2#1307953411yescharacteristic
368C2#1307956411yescharacteristic
369C2#1307973711yescharacteristic
370C2#1307976711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#384C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
cosocleC22×C4×C11C24
hypercenterC2×#384D11
hypocenterC11#21982
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
3. #13069372C2semidirect
5. #13068775C2semidirect
6. #13068743C2semidirect
26. #10622538C4semidirect
63. C4⋊(C11⋊C8)C2×C4semidirect
64. C4×C11⋊C8C2×C4semidirect
138. C2×C4×C11#107semidirect
296. C11#21982semidirect
322. C2×C4#66871semidirect

Maximal subgroups

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

subgroupindex
C2×#106225382
#130710232
#130693722
#130688042
#130687752
#130687432
C2×C11×#3842
#2198211