Finite Group Databasebeta

← #14719702

Subgroups — #14719702

Order 2816.

1098 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623047C211yesmaximal, characteristic
2#13077699C212yesmaximal, complemented
3#13066958C214yesmaximal, complemented
4#13066947C214yesmaximal
5#13066939C212yesmaximal, complemented
6#13066930C211yesmaximal, characteristic
7C11×#1458C211yesmaximal, characteristic
8C23×Dic22C411yescharacteristic
9C2×C11×#117C2211yescharacteristic
10C2×#66979C411yescharacteristic
11#10623047—24no—
12#10623047—24no—
13C2×#66940C412yes—
14C2×#66940C2212yes—
15C2×#66930C412yes—
16C2×#66930C2212yes—
17C2×C4×Dic22C411yescharacteristic
18C2×C4×Dic22C411yescharacteristic
19C2×C4×Dic22C2211yescharacteristic
20C2×C4×Dic22C2211yescharacteristic
21C11×#431C2212yes—
22#10622538C2212yes—
23#10622444—24no—
24#10622332C2218yes—
25#10622292—24no—
26C11×#324C2212yes—
27C11×#323C2212yes—
28#10622166C2212yes—
29#10622163C2212yes—
30#10622107C2218yes—
31C23×C4×C11C2×C411yescharacteristic
32C23×Dic11C2×C412yes—
33C22×Dic22—24no—
34C22×Dic22—22no—
35C22×Dic22—22no—
36C22×Dic22—24no—
37C22×Dic22C2×C412yes—
38C22×Dic22C2×C411yescharacteristic
39C22×Dic22C2×C411yescharacteristic
40C11×#117—22no—
41C11×#117—22no—
42C2×C11×C4⋊C4C2×C411yescharacteristic
43C2×C11×C4⋊C4C2311yescharacteristic
44C2×C11×C22⋊C4C2×C411yescharacteristic
45C2×C11×C22⋊C4C2311yescharacteristic
46C2×C42×C11C2×C411yescharacteristic
47C2×C42×C11C2311yescharacteristic
48C2×C22⋊Dic11C2×C412yes—
49#66979—22no—
50#66979—22no—
51C2×C4⋊Dic11C2×C411yescharacteristic
52C2×C4⋊Dic11C2×C411yescharacteristic
53C2×Dic11⋊C4C2×C412yes—
54C2×Dic11⋊C4C2×C412yes—
55C2×Dic11⋊C4C2×C412yes—
56C2×Dic11⋊C4C2312yes—
57C2×C4×Dic11C2×C412yes—
58C2×C4×Dic11C2×C412yes—
59C2×C4×Dic11C2×C412yes—
60C2×C4×Dic11C2×C412yes—
61C2×C4×Dic11C2312yes—
62#66940—28no—
63#66940—28no—
64#66930—28no—
65#66930—28no—
66C4×Dic22—22no—
67C4×Dic22—22no—
68C4×Dic22—22no—
69C4×Dic22—22no—
70C4×Dic22—22no—
71C4×Dic22—22no—
72C4×Dic22D414yes—
73C4×Dic22D414yes—
74C11×C4⋊C8—22no—
75C11×C22⋊C8C2314yes—
76C4×C8×C11—22no—
77C22⋊(C11⋊C8)C2314yes—
78#66879—28no—
79Dic11⋊C8—28no—
80C4⋊(C11⋊C8)—22no—
81C4×C11⋊C8—22no—
82#17371—111nomaximal, Sylow
83C24×C11C22×C411yescharacteristic
84C22×C4×C11—21no—
85C22×C4×C11—21no—
86C22×C4×C11—22no—
87C22×C4×C11M1614yes—
88C22×C4×C11C22×C411yescharacteristic
89C22×C4×C11C22×C411yescharacteristic
90C22×C4×C11C22×C411yescharacteristic
91C22×C4×C11C22×C411yescharacteristic
92C22×C4×C11C22×C411yescharacteristic
93C22×C4×C11C2411yescharacteristic
94C22×Dic11—24no—
95C22×Dic11—24no—
96C22×Dic11—24no—
97C22×Dic11C22×C412yes—
98C22×Dic11C22×C412yes—
99C22×Dic11C22×C412yes—
100C22×Dic11C22×C412yes—
101C2×Dic22—216no—
102C2×Dic22—216no—
103C2×Dic22—24no—
104C2×Dic22—24no—
105C2×Dic22—21no—
106C2×Dic22—21no—
107C2×Dic22—24no—
108C2×Dic22—22no—
109C2×Dic22—22no—
110C2×Dic22—22no—
111C2×Dic22C22⋊C412yes—
112C2×Dic22C22⋊C412yes—
113C2×Dic22M1614yes—
114C2×C8×C11—24no—
115C11×C4⋊C4—22no—
116C11×C4⋊C4—22no—
117C11×C22⋊C4—22no—
118C11×C22⋊C4—22no—
119C42×C11—21no—
120C42×C11—21no—
121C42×C11C2×D412yes—
122C42×C11C2×D412yes—
123C22⋊Dic11—24no—
124C4⋊Dic11—21no—
125C4⋊Dic11—21no—
126C4⋊Dic11C22⋊C412yes—
127C4⋊Dic11C22⋊C412yes—
128Dic11⋊C4—24no—
129Dic11⋊C4—24no—
130Dic11⋊C4—24no—
131Dic11⋊C4C22⋊C418yes—
132C4×Dic11—24no—
133C4×Dic11—24no—
134C4×Dic11—24no—
135C4×Dic11—24no—
136C4×Dic11C2×D418yes—
137C2×C11⋊C8—24no—
138C2×#519—111no—
139#2983—112no—
140#1467—114no—
141#1464—114no—
142#1461—112no—
143#1458—111no—
144#1458D1111yescharacteristic
145C23×C11—21no—
146C23×C11—21no—
147C23×C11—41no—
148C23×C11—41no—
149C23×C11C2×M1612yes—
150C23×C11C23×C411yescharacteristic
151C2×C4×C11—21no—
152C2×C4×C11—21no—
153C2×C4×C11—24no—
154C2×C4×C11—24no—
155C2×C4×C11—21no—
156C2×C4×C11—21no—
157C2×C4×C11—41no—
158C2×C4×C11—41no—
159C2×C4×C11—41no—
160C2×C4×C11—41no—
161C2×C4×C11C2×C22⋊C412yes—
162C2×C4×C11C2×C22⋊C411yescharacteristic
163C2×C4×C11C2×C22⋊C411yescharacteristic
164C2×C4×C11C2×M1612yes—
165C2×C4×C11C2×M1612yes—
166C2×C4×C11C2×M1612yes—
167C2×C4×C11C22×D412yes—
168C2×Dic11—216no—
169C2×Dic11—22no—
170C2×Dic11—22no—
171C2×Dic11—22no—
172C2×Dic11—22no—
173C2×Dic11—22no—
174C2×Dic11—42no—
175C2×Dic11—42no—
176C2×Dic11C2×C22⋊C414yes—
177C2×Dic11C2×C22⋊C414yes—
178C2×Dic11C2×M1614yes—
179Dic22—216no—
180Dic22—24no—
181Dic22—24no—
182Dic22—24no—
183Dic22—24no—
184C8×C11—44no—
185C11⋊C8—44no—
186C23×Q8—111no—
187#519—224no—
188#519—224no—
189C2×C4×Q8—112no—
190C2×C4×Q8—112no—
191C2×C4×Q8—111no—
192C2×C4×Q8—111no—
193C2×C4×Q8—111no—
194C2×C4×Q8—111no—
195C2×#117—111no—
196C2×#117—112no—
197C2×#117—112no—
198C2×#117C2×D1111yescharacteristic
199#446—224no—
200#446—224no—
201#432—118no—
202#431—112no—
203#431C2×D1112yes—
204#324—112no—
205#324—118no—
206#324C2×D1112yes—
207#323—112no—
208#323C2×D1112yes—
209C22×C11—21no—
210C22×C11—21no—
211C22×C11—22no—
212C22×C11—42no—
213C22×C11—42no—
214C22×C11—41no—
215C22×C11—41no—
216C22×C11C22×C22⋊C411yescharacteristic
217C22×C11C22×M1611yescharacteristic
218C22×C11C22×M1611yescharacteristic
219C4×C11—22no—
220C4×C11—22no—
221C4×C11—22no—
222C4×C11—41no—
223C4×C11—41no—
224C4×C11#40712yes—
225C4×C11#40712yes—
226Dic11—24no—
227Dic11—24no—
228Dic11—24no—
229Dic11—42no—
230C22×Q8—111no—
231C22×Q8—111no—
232C22×Q8—112no—
233C22×Q8—222no—
234C22×Q8—222no—
235C22×Q8—224no—
236C22×Q8—224no—
237C23×C4—112no—
238C23×C4C4×D1111yescharacteristic
239C4×Q8—114no—
240C4×Q8—114no—
241C4×Q8—222no—
242C4×Q8—222no—
243C4×Q8—222no—
244C4×Q8—222no—
245C4×Q8—228no—
246C4×Q8—228no—
247C4×Q8—222no—
248C4×Q8—222no—
249#117—22no—
250#117—22no—
251#117—228no—
252#117—228no—
253#117—222no—
254#117—222no—
255C2×C4⋊C4—112no—
256C2×C4⋊C4—111no—
257C2×C4⋊C4—112no—
258C2×C4⋊C4—112no—
259C2×C4⋊C4—111no—
260C2×C4⋊C4—112no—
261C2×C4⋊C4C4×D1111yescharacteristic
262C2×C4⋊C4C22×D1111yescharacteristic
263C2×C22⋊C4—112no—
264C2×C22⋊C4C4×D1111yescharacteristic
265C2×C22⋊C4C22×D1111yescharacteristic
266C2×C42—112no—
267C2×C42—112no—
268C2×C42—112no—
269C2×C42—112no—
270C2×C42—112no—
271C2×C42C4×D1111yescharacteristic
272C2×C42C22×D1111yescharacteristic
273C4⋊C8—22no—
274C4⋊C8—228no—
275C4⋊C8—222no—
276C22⋊C8—114no—
277C22⋊C8C22×D1114yes—
278#97—228no—
279C4×C8—22no—
280C4×C8—222no—
281C2×C11—21no—
282C2×C11—21no—
283C2×C11—42no—
284C2×C11C2×#40711yescharacteristic
285C2×C11#288411yescharacteristic
286C2×C11#288911yescharacteristic
287C24C2×C4×D1111yescharacteristic
288C2×Q8—112no—
289C2×Q8—112no—
290C2×Q8—114no—
291C2×Q8—224no—
292C2×Q8—224no—
293C2×Q8—2216no—
294C2×Q8—2216no—
295C2×Q8—221no—
296C2×Q8—221no—
297C2×Q8—224no—
298C2×Q8—222no—
299C2×Q8—222no—
300C2×Q8—222no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—112no—
305C22×C4—112no—
306C22×C4—112no—
307C22×C4—112no—
308C22×C4—224no—
309C22×C4—224no—
310C22×C4—224no—
311C22×C4#431314yes—
312C22×C4C2×C4×D1111yescharacteristic
313C22×C4C2×C4×D1111yescharacteristic
314C22×C4C2×C4×D1111yescharacteristic
315C22×C4C2×C4×D1111yescharacteristic
316C22×C4C2×C4×D1111yescharacteristic
317C22×C4C23×D1111yescharacteristic
318C2×C8—24no—
319C2×C8—224no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—118no—
323C4⋊C4—112no—
324C4⋊C4—112no—
325C4⋊C4—224no—
326C4⋊C4—224no—
327C4⋊C4—221no—
328C4⋊C4—221no—
329C4⋊C4—224no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—224no—
333C42—21no—
334C42—21no—
335C42—118no—
336C42—224no—
337C42—224no—
338C42—224no—
339C42—224no—
340C42D4×D1112yes—
341C42D4×D1112yes—
342C11#1737111yescharacteristic, complemented, Sylow
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23C2×#431312yes—
348C23C22×C4×D1111yescharacteristic
349Q8—2216no—
350Q8—224no—
351Q8—224no—
352Q8—224no—
353Q8—224no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—114no—
365C2×C4—114no—
366C2×C4—114no—
367C2×C4—2216no—
368C2×C4—222no—
369C2×C4—222no—
370C2×C4—222no—
371C2×C4—222no—
372C2×C4—222no—
373C2×C4—442no—
374C2×C4—442no—
375C2×C4C22⋊C4×D1112yes—
376C2×C4C22⋊C4×D1111yescharacteristic
377C2×C4C22⋊C4×D1111yescharacteristic
378C2×C4C2×#431312yes—
379C2×C4C2×#431312yes—
380C2×C4M16×D1112yes—
381C2×C4C2×D4×D1112yes—
382C8—44no—
383C8—444no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22C2×C22⋊C4×D1111yescharacteristic
392C22C22×#431311yescharacteristic
393C22C2×M16×D1111yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—41no—
398C4—41no—
399C4—224no—
400C4—224no—
401C4—224no—
402C4—442no—
403C4#1062233912yes—
404C4#1062233912yes—
405C2—21no—
406C2—21no—
407C2—42no—
408C2C2×#1062233911yescharacteristic
409C2#1307707811yescharacteristic
410C2#1307709711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D11
commutatorC23×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1458C2
socleC22×C11C22×C22⋊C4
prime radicalC24×C11C22×C4
cosocleC22×C4×C11C24
hypercenter#1458D11
hypocenterC11#17371
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #13077699C2semidirect
3. #13066958C2semidirect
5. #13066939C2semidirect
342. C11#17371semidirect

Maximal subgroups

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

subgroupindex
C2×#106230472
#130776992
#130669582
#130669472
#130669392
#130669302
C11×#14582
#1737111