Finite Group Databasebeta

← #14648713

Subgroups — #14648713

Order 2816.

578 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13071790C211yesmaximal, characteristic
2#13069590C212yesmaximal, complemented
3C2×#10622178C211yesmaximal, characteristic
4#13066498C212yesmaximal, complemented
5C2×C11×#497C211yesmaximal, characteristic
6C22×C11×Q16—21no—
7C22×C11×SD16—21no—
8C2×C11×C4⋊Q8C2211yescharacteristic
9C2×C11×#124C2211yescharacteristic
10C11×#497—22no—
11C11×#497C414yescomplemented
12C2×C11×#97C2211yescharacteristic
13#10622843—21no—
14#10622833—21no—
15#10622564—22no—
16#10622563—22no—
17#10622549C2212yes—
18C2×#66867C2212yes—
19#10622178—22no—
20#10622167—22no—
21#10622167—22no—
22C22×Q8×C11—21no—
23C22×Q8×C11D411yescharacteristic
24C22×D4×C11D411yescharacteristic
25C23×Dic11—21no—
26C2×C11×Q16—24no—
27C2×C11×Q16—42no—
28C2×C11×Q16—42no—
29C2×C11×SD16—24no—
30C2×C11×SD16—42no—
31C2×C11×SD16—42no—
32C22×C8×C11—21no—
33C11×C4⋊Q8—21no—
34C11×C4⋊Q8—22no—
35C11×C4⋊Q8C2×C412yescomplemented
36C11×#124—22no—
37C11×#124C2×C412yes—
38C11×#124D412yes—
39C2×C11×C4⋊C4—22no—
40C2×C11×C22⋊C4—22no—
41C2×C42×C11C2311yescharacteristic
42C2×C22⋊Dic11—22no—
43C2×C22⋊Dic11—21no—
44C2×C4⋊Dic11—21no—
45C2×C4⋊Dic11D412yes—
46C2×C4⋊Dic11D412yes—
47C2×C4×Dic11—21no—
48C22×C11⋊C8—22no—
49C11×#97—21no—
50C11×#97C2×C412yescomplemented
51C11⋊#95—22no—
52C11⋊#95—22no—
53C11⋊#95—22no—
54#66867—22no—
55#66867—22no—
56#9003—111nomaximal, Sylow
57C24×C11—21no—
58C2×Q8×C11—21no—
59C2×Q8×C11—22no—
60C2×Q8×C11—41no—
61C2×Q8×C11—41no—
62C2×Q8×C11—41no—
63C2×Q8×C11—41no—
64C2×Q8×C11—41no—
65C2×Q8×C11—41no—
66C2×Q8×C11—41no—
67C2×Q8×C11C22⋊C412yes—
68C2×D4×C11—21no—
69C2×D4×C11—41no—
70C2×D4×C11—41no—
71C2×D4×C11C22⋊C412yes—
72C22×C4×C11—21no—
73C22×C4×C11—41no—
74C22×C4×C11C2×D411yescharacteristic
75C22×C4×C11C2×D412yes—
76C22×Dic11—22no—
77C22×Dic11—22no—
78C22×Dic11—81no—
79C22×Dic11—82no—
80C11×Q16—48no—
81C11×SD16—48no—
82C2×C8×C11—22no—
83C2×C8×C11—22no—
84C2×C8×C11—22no—
85C11×C4⋊C4—24no—
86C11×C4⋊C4—42no—
87C11×C22⋊C4—24no—
88C11×C22⋊C4—24no—
89C42×C11C22×C411yescharacteristic
90C42×C11C2×D411yescharacteristic
91C42×C11#4112yes—
92C22⋊Dic11—81no—
93C22⋊Dic11—82no—
94C4⋊Dic11—42no—
95C4⋊Dic11—42no—
96C4⋊Dic11—81no—
97C4×Dic11—81no—
98C2×C11⋊C8—24no—
99C2×C11⋊C8—42no—
100C2×C11⋊C8—42no—
101C2×#497D1111yescharacteristic
102#2379—111no—
103#1993—112no—
104C2×#329—111no—
105#1295—112no—
106C23×C11—22no—
107C23×C11—41no—
108C23×C11—41no—
109C23×C11—42no—
110C23×C11—42no—
111C23×C11#12011yescharacteristic
112Q8×C11—42no—
113Q8×C11—42no—
114Q8×C11—42no—
115Q8×C11—81no—
116Q8×C11—81no—
117Q8×C11—81no—
118D4×C11—42no—
119D4×C11—81no—
120C2×C4×C11—21no—
121C2×C4×C11—21no—
122C2×C4×C11—22no—
123C2×C4×C11—22no—
124C2×C4×C11—22no—
125C2×C4×C11—41no—
126C2×C4×C11—41no—
127C2×C4×C11—42no—
128C2×C4×C11—81no—
129C2×C4×C11—81no—
130C2×C4×C11C2×C22⋊C411yescharacteristic
131C2×C4×C11C4×D412yes—
132C2×C4×C11#12112yes—
133C2×C4×C11#12311yescharacteristic
134C2×Dic11—42no—
135C2×Dic11—42no—
136C2×Dic11—42no—
137C2×Dic11—42no—
138C2×Dic11—42no—
139C2×Dic11—42no—
140C2×Dic11—84no—
141C8×C11—24no—
142C11⋊C8—44no—
143C22×Q16—21no—
144C22×SD16—21no—
145C2×C4⋊Q8C2×D1111yescharacteristic
146C2×#124C2×D1111yescharacteristic
147#497—22no—
148#497Dic1114yescomplemented
149C2×#97—112no—
150C2×#97C2×D1111yescharacteristic
151#391—221no—
152#388—222no—
153#386—221no—
154#385—222no—
155#382—112no—
156#340—222no—
157#340—222no—
158#329—222no—
159C22×C11—44no—
160C22×C11—42no—
161C22×C11—81no—
162C22×C11—81no—
163C22×C11—82no—
164C22×C11#38611yescharacteristic
165C22×C11#45311yescharacteristic
166C22×C11#45311yescharacteristic
167C22×C11#45512yes—
168C22×C11#45612yes—
169C4×C11—22no—
170C4×C11—22no—
171C4×C11—21no—
172C4×C11—21no—
173C4×C11—42no—
174C4×C11—82no—
175Dic11—82no—
176Dic11—82no—
177C22×Q8—21no—
178C22×Q8C22⋊D1111yescharacteristic
179C22×D4C22⋊D1111yescharacteristic
180C23×C4—221no—
181C2×Q16—24no—
182C2×Q16—42no—
183C2×Q16—42no—
184C2×SD16—24no—
185C2×SD16—42no—
186C2×SD16—42no—
187C22×C8—21no—
188C22×C8—222no—
189C4⋊Q8—22no—
190C4⋊Q8—21no—
191C4⋊Q8C2×Dic1112yescomplemented
192#124—22no—
193#124C2×Dic1112yes—
194#124C22⋊D1112yes—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—112no—
197C2×C4⋊C4—112no—
198C2×C4⋊C4—221no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—222no—
201C2×C22⋊C4—221no—
202C2×C42—221no—
203C2×C42C22×D1111yescharacteristic
204#97—21no—
205#97—222no—
206#97—222no—
207#97C2×Dic1112yescomplemented
208#95—222no—
209#95—222no—
210#95—222no—
211C2×C11—82no—
212C2×C11#165911yescharacteristic
213C2×C11#166011yescharacteristic
214C2×C11#166112yes—
215C2×C11#188212yes—
216C2×C11#188711yescharacteristic
217C24—21no—
218C2×Q8—21no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8C22⋊Dic1112yes—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4C22⋊Dic1112yes—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—222no—
235C22×C4—222no—
236C22×C4—882no—
237C22×C4—881no—
238C22×C4D4×D1112yes—
239C22×C4C2×C22⋊D1111yescharacteristic
240Q16—48no—
241SD16—48no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—224no—
246C2×C8—442no—
247C2×C8—442no—
248C4⋊C4—24no—
249C4⋊C4—42no—
250C4⋊C4—442no—
251C4⋊C4—442no—
252C4⋊C4—881no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—882no—
256C22⋊C4—881no—
257C42—881no—
258C42#434112yes—
259C42C22×Dic1111yescharacteristic
260C42C2×C22⋊D1111yescharacteristic
261C11#900311yescharacteristic, complemented, Sylow
262C23—22no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23#6699011yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—42no—
271Q8—81no—
272Q8—81no—
273Q8—81no—
274D4—42no—
275D4—81no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—442no—
287C2×C4—442no—
288C2×C4—442no—
289C2×C4—442no—
290C2×C4—442no—
291C2×C4—442no—
292C2×C4—884no—
293C2×C4D4×Dic1112yes—
294C2×C4#6698811yescharacteristic
295C2×C4#6699212yes—
296C2×C4C2×C22⋊Dic1111yescharacteristic
297C8—24no—
298C8—444no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1062269011yescharacteristic
305C22#1062269011yescharacteristic
306C22#1062281512yes—
307C22#1062281812yes—
308C22#1062283311yescharacteristic
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—882no—
316C4—882no—
317C2—82no—
318C2#1306792011yescharacteristic
319C2#1306792511yescharacteristic
320C2#1306793012yes—
321C2#1306898712yes—
322C2#1306901511yescharacteristic
subgroupG/H
centerC23#66990
commutatorC42×C11C22×C4
FrattiniC2×C42C22×D11
FittingC2×C11×#497C2
socleC23×C11#120
prime radicalC22×D4×C11D4
cosocleC2×C42×C11C23
hypercenterC2×#497D11
hypocenterC11#9003
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. #13069590C2semidirect
4. #13066498C2semidirect
11. C11×#497C4semidirect
35. C11×C4⋊Q8C2×C4semidirect
50. C11×#97C2×C4semidirect
148. #497Dic11semidirect
191. C4⋊Q8C2×Dic11semidirect
207. #97C2×Dic11semidirect
261. C11#9003semidirect

Maximal subgroups

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

subgroupindex
#130717902
#130695902
C2×#106221782
#130664982
C2×C11×#4972
#900311