Finite Group Databasebeta

← #14669408

Subgroups — #14669408

Order 2816.

690 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 127 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#13070305C212yesmaximal, complemented
2C2×#10622180C212yesmaximal, complemented
3#13068730C212yesmaximal, complemented
4C11×#3510C211yesmaximal, characteristic
5C11×#559—22no—
6C11×#556—22no—
7C11×#551—24no—
8C11×#542—22no—
9C11×#536C2211yescharacteristic
10C11×#535C411yescharacteristic
11C11×#533C411yescharacteristic
12C11×#529—22no—
13C2×C11×#124C2211yescharacteristic
14C11×#519C2211yescharacteristic
15C11×#518C411yescharacteristic
16C4×C11×#41C411yescharacteristic
17C2×#66900C2214yes—
18#10622186—22no—
19#10622180—22no—
20#10622180—22no—
21#10622180—22no—
22#10622171—24no—
23C2×C11×#41C2×C411yescharacteristic
24C22×Q8×C11C2311yescharacteristic
25C22×D4×C11C2×C411yescharacteristic
26C11×C4⋊Q8—21no—
27C11×C4⋊Q8D412yes—
28C11×#127—21no—
29C11×#126—24no—
30C11×#126—24no—
31C11×#125—21no—
32C11×#124—24no—
33C11×#124—21no—
34C11×#124—22no—
35C11×#124D412yes—
36C11×#124D412yes—
37C11×#124D412yes—
38C11×#123—24no—
39C11×#123—24no—
40C11×#123—22no—
41C11×C22⋊Q8—24no—
42C11×C22⋊Q8—22no—
43C11×C22⋊Q8—22no—
44C11×#121—22no—
45C11×#120—22no—
46C4×Q8×C11—21no—
47C4×Q8×C11—22no—
48C4×Q8×C11—22no—
49C4×Q8×C11—22no—
50C4×D4×C11—24no—
51C4×D4×C11—21no—
52C4×D4×C11—22no—
53C4×D4×C11—22no—
54C4×D4×C11—22no—
55C11×#117—24no—
56C11×#117—21no—
57C11×#117C2×C411yescharacteristic
58C11×#117C2×C411yescharacteristic
59C11×#117C2×C411yescharacteristic
60C11×#117D414yes—
61C2×C11×C22⋊C4—22no—
62C2×C11×C22⋊C4—22no—
63C2×C42×C11—21no—
64C2×C42×C11C2×C411yescharacteristic
65C2×#4318—24no—
66#66900—24no—
67#66900—24no—
68#12729—111nomaximal, Sylow
69C24×C11—21no—
70C11×#41—42no—
71C2×Q8×C11—22no—
72C2×Q8×C11—42no—
73C2×Q8×C11C22⋊C411yescharacteristic
74C2×Q8×C11C2×D412yes—
75C2×Q8×C11C2×D412yes—
76C2×D4×C11—21no—
77C2×D4×C11—24no—
78C2×D4×C11—21no—
79C2×D4×C11—42no—
80C2×D4×C11C22⋊C412yes—
81C2×D4×C11C22⋊C411yescharacteristic
82C22×C4×C11—22no—
83C22×C4×C11—24no—
84C22×C4×C11—21no—
85C22×C4×C11—21no—
86C22×C4×C11C22×C411yescharacteristic
87C22×C4×C11C2×D412yes—
88C11×C4⋊C4—22no—
89C11×C4⋊C4—22no—
90C11×C4⋊C4—22no—
91C11×C4⋊C4—22no—
92C11×C4⋊C4—21no—
93C11×C4⋊C4—21no—
94C11×C4⋊C4—21no—
95C11×C4⋊C4—24no—
96C11×C4⋊C4—24no—
97C11×C22⋊C4—24no—
98C11×C22⋊C4—22no—
99C11×C22⋊C4—24no—
100C11×C22⋊C4—21no—
101C11×C22⋊C4—24no—
102C11×C22⋊C4—24no—
103C11×C22⋊C4—24no—
104C11×C22⋊C4C22⋊C412yes—
105C11×C22⋊C4C22⋊C412yes—
106C42×C11—22no—
107C42×C11—21no—
108C42×C11—21no—
109C42×C11—21no—
110C42×C11—24no—
111C42×C11C22⋊C411yescharacteristic
112C42×C11C22⋊C411yescharacteristic
113C42×C11C22⋊C412yes—
114#4318—44no—
115#4318—44no—
116C2×C11⋊C8—44no—
117#3510D1111yescharacteristic
118#2133—112no—
119C2×#355—112no—
120#1830—112no—
121C23×C11—22no—
122C23×C11—21no—
123C23×C11—42no—
124C23×C11—82no—
125C23×C11C2×C22⋊C411yescharacteristic
126Q8×C11—42no—
127Q8×C11—42no—
128Q8×C11—41no—
129Q8×C11—82no—
130D4×C11—42no—
131D4×C11—41no—
132D4×C11—42no—
133D4×C11—42no—
134D4×C11—84no—
135C2×C4×C11—24no—
136C2×C4×C11—22no—
137C2×C4×C11—22no—
138C2×C4×C11—21no—
139C2×C4×C11—22no—
140C2×C4×C11—21no—
141C2×C4×C11—21no—
142C2×C4×C11—22no—
143C2×C4×C11—42no—
144C2×C4×C11—82no—
145C2×C4×C11—84no—
146C2×C4×C11—81no—
147C2×C4×C11C2×C22⋊C411yescharacteristic
148C2×C4×C11C2×C22⋊C411yescharacteristic
149C2×C4×C11#12014yes—
150C11⋊C8—84no—
151#559—22no—
152#556—22no—
153#551—24no—
154#542—22no—
155#536C2×D1111yescharacteristic
156#535Dic1111yescharacteristic
157#533Dic1111yescharacteristic
158#529—22no—
159C2×#124C2×D1111yescharacteristic
160#519C2×D1111yescharacteristic
161#518Dic1111yescharacteristic
162C4×#41Dic1111yescharacteristic
163C2×#101—114no—
164#356—222no—
165#355—222no—
166#355—222no—
167#355—222no—
168#344—224no—
169C22×C11—41no—
170C22×C11—42no—
171C22×C11—44no—
172C22×C11—41no—
173C22×C11—42no—
174C22×C11—162no—
175C22×C11#37911yescharacteristic
176C22×C11#41012yes—
177C4×C11—21no—
178C4×C11—21no—
179C4×C11—24no—
180C4×C11—42no—
181C4×C11—41no—
182C4×C11—42no—
183C4×C11—41no—
184C4×C11—42no—
185C4×C11—41no—
186C2×#41C2×Dic1111yescharacteristic
187C22×Q8C22×D1111yescharacteristic
188C22×D4C2×Dic1111yescharacteristic
189C2×M16—224no—
190C4⋊Q8—21no—
191C4⋊Q8C22⋊D1112yes—
192#127—21no—
193#126—24no—
194#126—24no—
195#125—21no—
196#124—24no—
197#124—21no—
198#124—22no—
199#124C22⋊D1112yes—
200#124C22⋊D1112yes—
201#124C22⋊D1112yes—
202#123—24no—
203#123—24no—
204#123—22no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208#121—22no—
209#120—22no—
210C4×Q8—21no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×D4—24no—
215C4×D4—21no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219#117—24no—
220#117—21no—
221#117C2×Dic1111yescharacteristic
222#117C2×Dic1111yescharacteristic
223#117C2×Dic1111yescharacteristic
224#117C22⋊D1114yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—21no—
228C2×C42C2×Dic1111yescharacteristic
229#101—224no—
230#101—224no—
231C2×C11—21no—
232C2×C11—22no—
233C2×C11—81no—
234C2×C11—82no—
235C2×C11#178211yescharacteristic
236C24—21no—
237#41—42no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8C22⋊Dic1111yescharacteristic
241C2×Q8C2×C22⋊D1112yes—
242C2×Q8C2×C22⋊D1112yes—
243C2×D4—21no—
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—42no—
247C2×D4C22⋊Dic1112yes—
248C2×D4C22⋊Dic1111yescharacteristic
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4C22×Dic1111yescharacteristic
254C22×C4C2×C22⋊D1112yes—
255M16—444no—
256M16—444no—
257C2×C8—444no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4C22⋊Dic1112yes—
275C22⋊C4C22⋊Dic1112yes—
276C42—22no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—24no—
281C42C22⋊Dic1111yescharacteristic
282C42C22⋊Dic1111yescharacteristic
283C42C22⋊Dic1112yes—
284C11#1272911yescharacteristic, complemented, Sylow
285C23—22no—
286C23—21no—
287C23—42no—
288C23—82no—
289C23C2×C22⋊Dic1111yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—41no—
293Q8—82no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—84no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—81no—
311C2×C4C2×C22⋊Dic1111yescharacteristic
312C2×C4C2×C22⋊Dic1111yescharacteristic
313C2×C4#6700614yes—
314C8—884no—
315C22—41no—
316C22—42no—
317C22—44no—
318C22—41no—
319C22—42no—
320C22—162no—
321C22#1062285312yes—
322C22#1062286011yescharacteristic
323C4—24no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C2—21no—
333C2—22no—
334C2—82no—
335C2—81no—
336C2#1306853411yescharacteristic
subgroupG/H
centerC2#13068534
commutatorC22×C4×C11C22×C4
FrattiniC22×Q8C22×D11
FittingC11×#3510C2
socleC2×C11#1782
prime radicalC11×#535C4
cosocleC22×Q8×C11C23
hypercenter#3510D11
hypocenterC11#12729
LCSG ▷ C22×C4×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C23 ◁ #519 ◁ #3510

Complements

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

NH ≅ G/Ntype
1. #13070305C2semidirect
2. C2×#10622180C2semidirect
3. #13068730C2semidirect
284. C11#12729semidirect

Maximal subgroups

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

subgroupindex
#130703052
C2×#106221802
#130687302
C11×#35102
#1272911