Finite Group Databasebeta

← #13074101

Subgroups — #13074101

Order 1408.

520 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66989C211yesmaximal, characteristic
2C2×C4×Dic22C211yesmaximal, characteristic, complemented
3C11×#390C211yesmaximal, characteristic
4#10622834C211yesmaximal, characteristic
5C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
6#10622587C211yesmaximal, characteristic, complemented
7#10622565C212yesmaximal, complemented
8#10622564C212yesmaximal, complemented
9#10622560C212yesmaximal, complemented
10#10622557C212yesmaximal, complemented
11#10622542C211yesmaximal, characteristic, complemented
12C22×D4×C11C2211yescharacteristic
13C22×Dic22C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C22⋊C4C2212yes—
17C2×C11×C22⋊C4C2212yes—
18C2×C42×C11C2211yescharacteristic
19C2×C22⋊Dic11C2212yes—
20C2×C22⋊Dic11C2212yes—
21C2×C22⋊Dic11C2212yes—
22C2×C22⋊Dic11C2212yes—
23#66989—24no—
24C2×C4⋊Dic11C2211yescharacteristic
25C2×C4⋊Dic11C2211yescharacteristic
26C2×Dic11⋊C4C2211yescharacteristic
27C2×Dic11⋊C4C2211yescharacteristic
28C2×C4×Dic11C2211yescharacteristic
29C2×C4×Dic11C2212yes—
30C2×C4×Dic11C2211yescharacteristic
31C4×Dic22—24no—
32C11⋊#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C24×C11C2312yes—
38C2×D4×C11—22no—
39C2×D4×C11—22no—
40C2×D4×C11—22no—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×C4×C11C2311yescharacteristic
44C22×C4×C11C2311yescharacteristic
45C22×Dic11C2312yes—
46C22×Dic11C2312yes—
47C22×Dic11C2311yescharacteristic
48C22×Dic11C2311yescharacteristic
49C22×Dic11C2311yescharacteristic
50C22×Dic11C2311yescharacteristic
51C2×Dic22—22no—
52C2×Dic22—22no—
53C2×Dic22—22no—
54C2×Dic22—22no—
55C2×Dic22—22no—
56C2×Dic22—22no—
57C11×C4⋊C4—22no—
58C11×C22⋊C4—24no—
59C11×C22⋊C4—42no—
60C42×C11—22no—
61C22⋊Dic11—24no—
62C22⋊Dic11—24no—
63C22⋊Dic11—24no—
64C22⋊Dic11—42no—
65C4⋊Dic11—22no—
66C4⋊Dic11—22no—
67Dic11⋊C4—22no—
68Dic11⋊C4—22no—
69C4×Dic11—22no—
70C4×Dic11—42no—
71C4×Dic11D414yes—
72#2592—111nomaximal, Sylow
73C23×C11—22no—
74C23×C11—22no—
75C23×C11—22no—
76C23×C11—22no—
77C23×C11—22no—
78C23×C11—22no—
79C23×C11—22no—
80C23×C11C2411yescharacteristic
81D4×C11—44no—
82C2×C4×C11—21no—
83C2×C4×C11—21no—
84C2×C4×C11—21no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11—22no—
88C2×C4×C11—21no—
89C2×C4×C11—21no—
90C2×C4×C11C2×D412yes—
91C2×C4×C11#4112yes—
92C2×C4×C11#4112yes—
93C2×C4×C11#4112yes—
94C2×Dic11—21no—
95C2×Dic11—21no—
96C2×Dic11—21no—
97C2×Dic11—21no—
98C2×Dic11—21no—
99C2×Dic11—21no—
100C2×Dic11—21no—
101C2×Dic11—22no—
102C2×Dic11—22no—
103C2×Dic11—22no—
104C2×Dic11—22no—
105C2×Dic11—22no—
106C2×Dic11—22no—
107C2×Dic11—21no—
108C2×Dic11—21no—
109C2×Dic11C2×D412yes—
110C2×Dic11C2×D412yes—
111C2×Dic11#4112yes—
112Dic22—28no—
113Dic22—44no—
114C2×#124—111no—
115C2×C4×Q8—111no—
116#397—112no—
117#393—112no—
118#391—111no—
119#391—111no—
120#390—111no—
121#390D1111yescharacteristic
122#388—112no—
123#388—112no—
124C4×C4⋊C4—111no—
125C22×C11—42no—
126C22×C11—42no—
127C22×C11—42no—
128C22×C11—42no—
129C22×C11—42no—
130C22×C11—42no—
131C22×C11—42no—
132C22×C11C22×D411yescharacteristic
133C22×C11C2×#4111yescharacteristic
134C22×C11C2×#4111yescharacteristic
135C22×C11C2×#4111yescharacteristic
136C22×C11C2×#4111yescharacteristic
137C22×C11H(5,𝔽2)11yescharacteristic
138C22×C112−1+411yescharacteristic
139C4×C11—22no—
140C4×C11—41no—
141C4×C11—42no—
142C4×C11#12414yes—
143Dic11—22no—
144Dic11—22no—
145Dic11—22no—
146Dic11—42no—
147Dic11—41no—
148Dic11—42no—
149C22×Q8—111no—
150C22×Q8—111no—
151C22×D4C2×D1111yescharacteristic
152#124—224no—
153C4×Q8—224no—
154C2×C4⋊C4—111no—
155C2×C4⋊C4—111no—
156C2×C4⋊C4—111no—
157C2×C4⋊C4—111no—
158C2×C4⋊C4C2×D1111yescharacteristic
159C2×C22⋊C4—112no—
160C2×C22⋊C4—112no—
161C2×C22⋊C4—112no—
162C2×C22⋊C4—112no—
163C2×C22⋊C4C2×D1112yes—
164C2×C22⋊C4C2×D1112yes—
165C2×C42—111no—
166C2×C42—111no—
167C2×C42—112no—
168C2×C42C2×D1111yescharacteristic
169#95—112no—
170#95—112no—
171#95—112no—
172#95—112no—
173#95—112no—
174C2×C11—82no—
175C2×C11#52511yescharacteristic
176C2×C11C2×#12411yescharacteristic
177C2×C11#53711yescharacteristic
178C2×C11#55611yescharacteristic
179C2×C11#55611yescharacteristic
180C2×C11#55911yescharacteristic
181C2×C11#55911yescharacteristic
182C24C22×D1112yes—
183C2×Q8—222no—
184C2×Q8—222no—
185C2×Q8—222no—
186C2×Q8—222no—
187C2×Q8—222no—
188C2×Q8—222no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—111no—
193C22×C4—111no—
194C22×C4—111no—
195C22×C4—112no—
196C22×C4—112no—
197C22×C4—111no—
198C22×C4C22×D1112yes—
199C22×C4C22×D1111yescharacteristic
200C22×C4C22×D1111yescharacteristic
201C22×C4C22×D1111yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—222no—
204C4⋊C4—222no—
205C4⋊C4—222no—
206C4⋊C4—222no—
207C22⋊C4—24no—
208C22⋊C4—42no—
209C22⋊C4—224no—
210C22⋊C4—224no—
211C22⋊C4—224no—
212C22⋊C4—442no—
213C42—22no—
214C42—114no—
215C42—222no—
216C42—442no—
217C11#259211yescharacteristic, complemented, Sylow
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23C23×D1111yescharacteristic
226Q8—228no—
227Q8—444no—
228D4—44no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—112no—
238C2×C4—112no—
239C2×C4—112no—
240C2×C4—221no—
241C2×C4—221no—
242C2×C4—221no—
243C2×C4—221no—
244C2×C4—221no—
245C2×C4—221no—
246C2×C4—222no—
247C2×C4—222no—
248C2×C4—221no—
249C2×C4—222no—
250C2×C4—222no—
251C2×C4—222no—
252C2×C4—221no—
253C2×C4—222no—
254C2×C4—221no—
255C2×C4#433912yes—
256C2×C4D4×D1112yes—
257C2×C4#434112yes—
258C2×C4#434112yes—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22C2×#433911yescharacteristic
267C22C2×D4×D1111yescharacteristic
268C22C2×#434111yescharacteristic
269C22C2×#434111yescharacteristic
270C22#6703711yescharacteristic
271C22#41×D1111yescharacteristic
272C22#6704311yescharacteristic
273C4—22no—
274C4—42no—
275C4—41no—
276C4—222no—
277C4—222no—
278C4—222no—
279C4—442no—
280C4—442no—
281C4—441no—
282C4#6693814yes—
283C2—82no—
284C2C2×#6693811yescharacteristic
285C2#1062307311yescharacteristic
286C2#1062309011yescharacteristic
287C2#1062313711yescharacteristic
288C2#1062313911yescharacteristic
289C2#1062320211yescharacteristic
290C2#1062324411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#390C2
prime radicalC22×D4×C11C22
hypercenter#390D11
hypocenterC11#2592
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic22C2semidirect
5. C4⋊C4×Dic11C2semidirect
6. #10622587C2semidirect
7. #10622565C2semidirect
8. #10622564C2semidirect
9. #10622560C2semidirect
10. #10622557C2semidirect
11. #10622542C2semidirect
217. C11#2592semidirect

Maximal subgroups

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

subgroupindex
C2×#669892
C2×C4×Dic222
C11×#3902
#106228342
C4⋊C4×Dic112
#106225872
#106225652
#106225642
#106225602
#106225572
#106225422
#259211