Finite Group Databasebeta

← #14664896

Subgroups — #14664896

Order 2816.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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#13071981C211yesmaximal, characteristic
2#13066937C211yesmaximal, characteristic
3#13066636C214yesmaximal, complemented
4C11×#1799C211yesmaximal, characteristic
5C23×Dic22C411yescharacteristic, complemented
6C2×C4×D4×C11C2211yescharacteristic
7C2×#66977—21no—
8C2×C11×#99—21no—
9C11×#380—21no—
10#10622833—21no—
11#10622825C411yescharacteristic, complemented
12#10622592C412yescomplemented
13#10622560C2212yes—
14#10622559—21no—
15#10622542—21no—
16#10622538—21no—
17C11×#342C2212yes—
18#10622163—21no—
19#10622163C2212yes—
20C22×D4×C11D411yescharacteristic
21C23×C4×C11C2×C411yescharacteristic
22C23×C4×C11D411yescharacteristic
23C23×Dic11—21no—
24C22×Dic22—21no—
25C22×Dic22—44no—
26C22×Dic22—41no—
27C22×Dic22—41no—
28C22×Dic22C2×C412yes—
29C4×D4×C11—41no—
30C4×D4×C11—41no—
31C2×C11×C4⋊C4—41no—
32C2×C11×C4⋊C4C2×C411yescharacteristic
33C2×C11×C22⋊C4—22no—
34C2×C11×C22⋊C4D411yescharacteristic
35C2×C11×C22⋊C4C2311yescharacteristic
36C2×C42×C11D411yescharacteristic
37C2×C22⋊Dic11—21no—
38C2×C22⋊Dic11—22no—
39C2×C4⋊Dic11—21no—
40#66977—44no—
41C2×Dic11⋊C4—22no—
42C11×#99—42no—
43C11×#95—41no—
44C11⋊#95—22no—
45C11⋊#95—22no—
46C11⋊#95C2×C412yes—
47C22⋊(C11⋊C8)—22no—
48C4⋊(C11⋊C8)—41no—
49C4×C11⋊C8—41no—
50#11898—111nomaximal, Sylow
51C24×C11C22⋊C411yescharacteristic
52C24×C11C2×D411yescharacteristic
53C2×D4×C11—41no—
54C2×D4×C11—41no—
55C2×D4×C11SD1612yes—
56C2×D4×C11Q1612yes—
57C22×C4×C11—21no—
58C22×C4×C11—21no—
59C22×C4×C11—21no—
60C22×C4×C11—21no—
61C22×C4×C11—42no—
62C22×C4×C11—42no—
63C22×C4×C11—42no—
64C22×C4×C11C22⋊C411yescharacteristic
65C22×C4×C11C22⋊C411yescharacteristic
66C22×C4×C11C22⋊C411yescharacteristic
67C22×C4×C11C22×C411yescharacteristic
68C22×C4×C11C2×D411yescharacteristic
69C22×Dic11—22no—
70C22×Dic11—22no—
71C22×Dic11—42no—
72C22×Dic11—44no—
73C2×Dic22—24no—
74C2×Dic22—42no—
75C2×Dic22—42no—
76C2×Dic22—48no—
77C2×Dic22—44no—
78C2×Dic22—44no—
79C2×Dic22—44no—
80C2×Dic22—41no—
81C2×Dic22—41no—
82C11×C4⋊C4—41no—
83C11×C4⋊C4—82no—
84C11×C22⋊C4—41no—
85C11×C22⋊C4—41no—
86C11×C22⋊C4—42no—
87C11×C22⋊C4—42no—
88C42×C11—21no—
89C42×C11—21no—
90C22⋊Dic11—42no—
91C22⋊Dic11—44no—
92C4⋊Dic11—42no—
93Dic11⋊C4—44no—
94C2×C11⋊C8—42no—
95#2409—111no—
96#1799D1111yescharacteristic
97#1460—111no—
98#1351—114no—
99C23×C11—21no—
100C23×C11—21no—
101C23×C11—21no—
102C23×C11—21no—
103C23×C11—41no—
104C23×C11—41no—
105C23×C11—41no—
106C23×C11—41no—
107C23×C11#10312yes—
108C23×C11C2×C22⋊C411yescharacteristic
109C23×C11C2×SD1611yescharacteristic
110C23×C11C2×Q1611yescharacteristic
111D4×C11—42no—
112D4×C11—42no—
113C2×C4×C11—21no—
114C2×C4×C11—21no—
115C2×C4×C11—22no—
116C2×C4×C11—21no—
117C2×C4×C11—21no—
118C2×C4×C11—21no—
119C2×C4×C11—42no—
120C2×C4×C11—42no—
121C2×C4×C11—42no—
122C2×C4×C11—41no—
123C2×C4×C11—41no—
124C2×C4×C11—41no—
125C2×C4×C11—81no—
126C2×C4×C11—82no—
127C2×C4×C11—82no—
128C2×C4×C11—82no—
129C2×C4×C11#9911yescharacteristic
130C2×C4×C11#9911yescharacteristic
131C2×C4×C11#10312yes—
132C2×Dic11—24no—
133C2×Dic11—42no—
134C2×Dic11—48no—
135C2×Dic11—42no—
136C2×Dic11—42no—
137C2×Dic11—42no—
138C2×Dic11—42no—
139Dic22—44no—
140Dic22—44no—
141Dic22—44no—
142Dic22—44no—
143C11⋊C8—82no—
144C23×Q8—111no—
145C2×C22⋊Q8—221no—
146C2×C4×D4C2×D1111yescharacteristic
147#431—221no—
148C2×#99—21no—
149#395—112no—
150#393—112no—
151#391—221no—
152#386—221no—
153#385—221no—
154#380—21no—
155#380—111no—
156#342C2×D1112yes—
157#323—112no—
158#323—221no—
159C22×C11—21no—
160C22×C11—21no—
161C22×C11—21no—
162C22×C11—22no—
163C22×C11—21no—
164C22×C11—42no—
165C22×C11—41no—
166C22×C11—41no—
167C22×C11—42no—
168C22×C11—41no—
169C22×C11—41no—
170C22×C11—41no—
171C22×C11—41no—
172C22×C11—81no—
173C22×C11C2×#9911yescharacteristic
174C22×C11C2×#10311yescharacteristic
175C22×C11#42111yescharacteristic
176C4×C11—21no—
177C4×C11—22no—
178C4×C11—41no—
179C4×C11—41no—
180C4×C11—41no—
181C4×C11—162no—
182C4×C11AΣL(1,𝔽16)12yes—
183Dic11—44no—
184Dic11—82no—
185C22×Q8—112no—
186C22×Q8—221no—
187C22×Q8—441no—
188C22×Q8—441no—
189C22×Q8—444no—
190C22×D4C22⋊D1111yescharacteristic
191C23×C4—221no—
192C23×C4C4×D1111yescharacteristic
193C23×C4C22⋊D1111yescharacteristic
194C22⋊Q8—444no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—222no—
199C2×C4⋊C4—221no—
200C2×C4⋊C4C4×D1111yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—221no—
203C2×C22⋊C4—222no—
204C2×C22⋊C4D4411yescharacteristic
205C2×C22⋊C4C22×D1111yescharacteristic
206C2×C42D4411yescharacteristic
207C4⋊C8—441no—
208#99—42no—
209C22⋊C8—222no—
210C4×C8—441no—
211#95—41no—
212#95—112no—
213#95—222no—
214#95—222no—
215C2×C11—21no—
216C2×C11—21no—
217C2×C11—41no—
218C2×C11—42no—
219C2×C11—41no—
220C2×C11#152011yescharacteristic
221C2×C11C2×AΣL(1,𝔽16)11yescharacteristic
222C2×C11#212911yescharacteristic
223C24(C2×D11)⋊C411yescharacteristic
224C24C2×C22⋊D1111yescharacteristic
225C2×Q8—224no—
226C2×Q8—442no—
227C2×Q8—442no—
228C2×Q8—444no—
229C2×Q8—444no—
230C2×Q8—448no—
231C2×Q8—444no—
232C2×Q8—441no—
233C2×Q8—441no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#432412yes—
237C2×D4#432612yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—222no—
246C22×C4—222no—
247C22×C4—444no—
248C22×C4—442no—
249C22×C4(C2×D11)⋊C411yescharacteristic
250C22×C4(C2×D11)⋊C411yescharacteristic
251C22×C4(C2×D11)⋊C411yescharacteristic
252C22×C4C2×C4×D1111yescharacteristic
253C22×C4C2×D4411yescharacteristic
254C2×C8—442no—
255C4⋊C4—41no—
256C4⋊C4—82no—
257C4⋊C4—442no—
258C4⋊C4—444no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—444no—
264C22⋊C4—442no—
265C42—21no—
266C42—21no—
267C11#1189811yescharacteristic, complemented, Sylow
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#6687412yes—
277C23C2×(C2×D11)⋊C411yescharacteristic
278C23C2×#432411yescharacteristic
279C23C2×#432611yescharacteristic
280Q8—444no—
281Q8—444no—
282Q8—444no—
283Q8—444no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—224no—
303C2×C4—442no—
304C2×C4—448no—
305C2×C4—442no—
306C2×C4—442no—
307C2×C4—442no—
308C2×C4—442no—
309C2×C4#6687011yescharacteristic
310C2×C4#6687011yescharacteristic
311C2×C4#6687412yes—
312C8—882no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#6687011yescharacteristic
328C22C2×#6687411yescharacteristic
329C22#1062261811yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—162no—
336C4—444no—
337C4—882no—
338C4#1062210912yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#1306718711yescharacteristic
345C2C2×#1062210911yescharacteristic
346C2#1307029011yescharacteristic
subgroupG/H
centerC22C2×#66870
commutatorC22×C4×C11C22×C4
FrattiniC2×C22⋊C4C22×D11
FittingC11×#1799C2
socleC22×C11C2×#99
prime radicalC22×D4×C11D4
cosocleC2×C11×C22⋊C4C23
hypercenter#1799D11
hypocenterC11#11898
LCSG ▷ C22×C4×C11 ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #13066636C2semidirect
5. C23×Dic22C4semidirect
11. #10622825C4semidirect
12. #10622592C4semidirect
267. C11#11898semidirect

Maximal subgroups

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

subgroupindex
#130719812
#130669372
#130666362
C11×#17992
#1189811