Finite Group Databasebeta

← #13073617

Subgroups — #13073617

Order 1408.

676 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66993C211yesmaximal, characteristic
2C2×#66938C212yesmaximal, complemented
3C2×C4×Dic22C211yesmaximal, characteristic, complemented
4C11×#390C211yesmaximal, characteristic, complemented
5#10622834C211yesmaximal, characteristic, complemented
6#10622601C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
8#10622574C212yesmaximal, complemented
9#10622572C212yesmaximal, complemented
10#10622570C212yesmaximal, complemented
11#10622552C211yesmaximal, characteristic, complemented
12C22×D4×C11C2211yescharacteristic
13C22×C22⋊D11C2212yes—
14C22×C22⋊D11C2212yes—
15C22×D44C2211yescharacteristic
16C22×Dic22C2211yescharacteristic
17C2×C11×C4⋊C4C2211yescharacteristic
18C2×C11×C22⋊C4C2212yes—
19C2×C11×C22⋊C4C2212yes—
20C2×C42×C11C2211yescharacteristic
21C2×C22⋊Dic11C2212yes—
22C2×C22⋊Dic11C2212yes—
23#66993—24no—
24C2×(C2×D11)⋊C4C2212yes—
25C2×(C2×D11)⋊C4C2212yes—
26C2×(C2×D11)⋊C4C2212yes—
27C2×(C2×D11)⋊C4C2212yes—
28C2×C4⋊Dic11C2211yescharacteristic
29C2×C4⋊Dic11C2211yescharacteristic
30C2×Dic11⋊C4C2211yescharacteristic
31C2×Dic11⋊C4C2211yescharacteristic
32C2×C4×Dic11C2211yescharacteristic
33C2×C4×Dic11C2212yes—
34C2×C4×Dic11C2211yescharacteristic
35#66938—28no—
36C4×Dic22—24no—
37C11⋊#95C2212yes—
38C24×C11C2312yes—
39C23×D11C2312yes—
40C2×D4×C11—22no—
41C2×D4×C11—22no—
42C2×D4×C11—22no—
43C22×C4×C11C2312yes—
44C22×C4×C11C2311yescharacteristic
45C22×C4×C11C2311yescharacteristic
46C22×C4×C11C2311yescharacteristic
47C2×C22⋊D11—24no—
48C2×C22⋊D11—24no—
49C2×C22⋊D11—24no—
50C2×C22⋊D11—24no—
51C2×C22⋊D11—42no—
52C2×C22⋊D11—42no—
53C22×Dic11C2312yes—
54C22×Dic11C2311yescharacteristic
55C22×Dic11C2311yescharacteristic
56C22×Dic11C2311yescharacteristic
57C22×Dic11C2311yescharacteristic
58C2×D44—22no—
59C2×D44—22no—
60C2×D44—22no—
61C2×Dic22—22no—
62C2×Dic22—22no—
63C2×Dic22D414yescomplemented
64C11×C4⋊C4—22no—
65C11×C22⋊C4—24no—
66C11×C22⋊C4—42no—
67C42×C11—22no—
68C22⋊Dic11—24no—
69C22⋊Dic11—42no—
70(C2×D11)⋊C4—24no—
71(C2×D11)⋊C4—24no—
72(C2×D11)⋊C4—42no—
73(C2×D11)⋊C4—42no—
74C4⋊Dic11—22no—
75C4⋊Dic11—22no—
76Dic11⋊C4—22no—
77Dic11⋊C4—22no—
78C4×Dic11—24no—
79C4×Dic11—22no—
80C4×Dic11D414yes—
81#2548—111nomaximal, Sylow
82C23×C11—22no—
83C23×C11—22no—
84C23×C11—22no—
85C23×C11—22no—
86C23×C11—22no—
87C23×C11—22no—
88C23×C11—22no—
89C23×C11C2411yescharacteristic
90C22×D11—22no—
91C22×D11—22no—
92C22×D11—22no—
93C22×D11—22no—
94C22×D11—22no—
95C22×D11—22no—
96C22×D11—22no—
97D4×C11—44no—
98C2×C4×C11—21no—
99C2×C4×C11—21no—
100C2×C4×C11—22no—
101C2×C4×C11—21no—
102C2×C4×C11—22no—
103C2×C4×C11—21no—
104C2×C4×C11—22no—
105C2×C4×C11—21no—
106C2×C4×C11C2×D412yes—
107C2×C4×C11C2×D412yes—
108C2×C4×C11#4112yes—
109C2×C4×C11#4112yes—
110C22⋊D11—48no—
111C22⋊D11—48no—
112C2×Dic11—21no—
113C2×Dic11—21no—
114C2×Dic11—21no—
115C2×Dic11—21no—
116C2×Dic11—21no—
117C2×Dic11—22no—
118C2×Dic11—22no—
119C2×Dic11—21no—
120C2×Dic11—22no—
121C2×Dic11—21no—
122C2×Dic11C2×D412yes—
123C2×Dic11C2×D412yes—
124C2×Dic11C2×D412yes—
125C2×Dic11C2×D412yes—
126C2×Dic11#4112yes—
127D44—44no—
128Dic22—28no—
129C2×#127—111no—
130C2×#124—112no—
131C2×C4×Q8—111no—
132#394—112no—
133#394—112no—
134#390—111no—
135#390—111no—
136#390—112no—
137#390—111no—
138#390D1111yescharacteristic, complemented
139C4×C4⋊C4—111no—
140C22×C11—42no—
141C22×C11—42no—
142C22×C11—42no—
143C22×C11—42no—
144C22×C11—42no—
145C22×C11—42no—
146C22×C11—42no—
147C22×C11C22×D411yescharacteristic
148C22×C11C22×D411yescharacteristic
149C22×C11C2×#4111yescharacteristic
150C22×C11C2×#4111yescharacteristic
151C22×C11C2×#4111yescharacteristic
152C22×C11H(5,𝔽2)11yescharacteristic
153C22×C11H(5,𝔽2)11yescharacteristic
154C2×D11—42no—
155C2×D11—42no—
156C2×D11—42no—
157C2×D11—42no—
158C2×D11—42no—
159C2×D11—42no—
160C2×D11—42no—
161C4×C11—22no—
162C4×C11—41no—
163C4×C11—42no—
164C4×C11#12414yescomplemented
165Dic11—22no—
166Dic11—22no—
167Dic11—41no—
168Dic11—42no—
169Dic11#12114yes—
170C22×Q8—111no—
171C22×D4—112no—
172C22×D4—112no—
173C22×D4—111no—
174C22×D4C2×D1111yescharacteristic
175#127—224no—
176#124—228no—
177C4×Q8—224no—
178C2×C4⋊C4—111no—
179C2×C4⋊C4—111no—
180C2×C4⋊C4—111no—
181C2×C4⋊C4—111no—
182C2×C4⋊C4C2×D1111yescharacteristic
183C2×C22⋊C4—112no—
184C2×C22⋊C4—112no—
185C2×C22⋊C4—112no—
186C2×C22⋊C4—112no—
187C2×C22⋊C4—112no—
188C2×C22⋊C4—112no—
189C2×C22⋊C4C2×D1112yes—
190C2×C22⋊C4C2×D1112yes—
191C2×C42—111no—
192C2×C42—111no—
193C2×C42—112no—
194C2×C42C2×D1111yescharacteristic
195#95—112no—
196C2×C11—82no—
197C2×C11C2×#12111yescharacteristic
198C2×C11C2×#12411yescharacteristic
199C2×C11#53511yescharacteristic
200C2×C11#55011yescharacteristic
201C2×C11#55011yescharacteristic
202C2×C11#55911yescharacteristic
203C2×C11#55911yescharacteristic
204D11—82no—
205C24—112no—
206C24C22×D1112yes—
207C2×Q8—114no—
208C2×Q8—222no—
209C2×Q8—222no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—224no—
214C2×D4—224no—
215C2×D4—224no—
216C2×D4—224no—
217C2×D4—222no—
218C2×D4—222no—
219C2×D4—222no—
220C2×D4—442no—
221C2×D4—442no—
222C22×C4—111no—
223C22×C4—111no—
224C22×C4—111no—
225C22×C4—112no—
226C22×C4—111no—
227C22×C4C22×D1112yes—
228C22×C4C22×D1111yescharacteristic
229C22×C4C22×D1111yescharacteristic
230C22×C4C22×D1111yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—222no—
233C4⋊C4—222no—
234C4⋊C4—222no—
235C4⋊C4—222no—
236C22⋊C4—24no—
237C22⋊C4—42no—
238C22⋊C4—224no—
239C22⋊C4—224no—
240C22⋊C4—224no—
241C22⋊C4—442no—
242C22⋊C4—442no—
243C22⋊C4—442no—
244C42—22no—
245C42—114no—
246C42—224no—
247C42—222no—
248C11#254811yescharacteristic, complemented, Sylow
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—222no—
257C23—222no—
258C23—222no—
259C23—222no—
260C23—222no—
261C23—222no—
262C23—222no—
263C23C23×D1111yescharacteristic
264Q8—228no—
265D4—44no—
266D4—448no—
267D4—448no—
268D4—444no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—112no—
278C2×C4—112no—
279C2×C4—112no—
280C2×C4—112no—
281C2×C4—112no—
282C2×C4—221no—
283C2×C4—221no—
284C2×C4—221no—
285C2×C4—221no—
286C2×C4—222no—
287C2×C4—222no—
288C2×C4—221no—
289C2×C4—221no—
290C2×C4—222no—
291C2×C4—221no—
292C2×C4#433912yes—
293C2×C4D4×D1112yes—
294C2×C4D4×D1112yes—
295C2×C4#434112yes—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—442no—
304C22—442no—
305C22—442no—
306C22—442no—
307C22—442no—
308C22—442no—
309C22—442no—
310C22C2×#433911yescharacteristic
311C22C2×D4×D1111yescharacteristic
312C22C2×D4×D1111yescharacteristic
313C22C2×#434111yescharacteristic
314C22#6703711yescharacteristic
315C22#41×D1111yescharacteristic
316C22#6704211yescharacteristic
317C4—22no—
318C4—42no—
319C4—41no—
320C4—114no—
321C4—222no—
322C4—222no—
323C4—442no—
324C4—441no—
325C4#6693814yescomplemented
326C2—82no—
327C2—882no—
328C2C2×#6693811yescharacteristic
329C2#1062308411yescharacteristic
330C2#1062309011yescharacteristic
331C2D11×#12111yescharacteristic
332C2#1062313811yescharacteristic
333C2#1062321211yescharacteristic
334C2#1062324911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#390C2
prime radicalC2×#66993C2
hypercenter#390D11
hypocenterC11#2548
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#66938C2semidirect
3. C2×C4×Dic22C2semidirect
4. C11×#390C2semidirect
5. #10622834C2semidirect
6. #10622601C2semidirect
7. C4⋊C4×Dic11C2semidirect
8. #10622574C2semidirect
9. #10622572C2semidirect
10. #10622570C2semidirect
11. #10622552C2semidirect
63. C2×Dic22D4semidirect
138. #390D11semidirect
164. C4×C11#124semidirect
248. C11#2548semidirect
325. C4#66938semidirect

Maximal subgroups

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

subgroupindex
C2×#669932
C2×#669382
C2×C4×Dic222
C11×#3902
#106228342
#106226012
C4⋊C4×Dic112
#106225742
#106225722
#106225702
#106225522
#254811