Finite Group Databasebeta

← #11773784

Subgroups — #11773784

Order 1152.

996 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×#63657C212yesmaximal, complemented
3#10589394C212yesmaximal, complemented
4#10589391C212yesmaximal, complemented
5#71600—34nomaximal
6C22×C32×D4C411yescharacteristic
7C23×C32×C4C411yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C32×C4×D4—22no—
10C32×C4×D4—22no—
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×C32×C22⋊C4C411yescharacteristic
13C2×C32×C22⋊C4C2211yescharacteristic
14C2×C32×C42C411yescharacteristic
15C22×C3⋊(C3⋊C8)—22no—
16#63657—22no—
17#63657C2214yes—
18C2×C3×C4×D4D314yes—
19C2×#830—38no—
20#4528—38no—
21#4525—38no—
22C24×C32C2×C411yescharacteristic
23C24×C32C2×C411yescharacteristic
24C2×C32×D4—22no—
25C2×C32×D4—22no—
26C2×C32×D4C814yescomplemented
27C22×C32×C4—21no—
28C22×C32×C4—21no—
29C22×C32×C4—42no—
30C22×C32×C4—42no—
31C22×C32×C4C2×C411yescharacteristic
32C22×C32×C4C2×C411yescharacteristic
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C32×C4⋊C4—22no—
39C32×C22⋊C4—22no—
40C32×C22⋊C4—22no—
41C32×C42—21no—
42C32×C42—21no—
43C2×C3⋊(C3⋊C8)—24no—
44C2×C3⋊(C3⋊C8)—44no—
45#1459—91noSylow
46C22×C3×D4Dic314yes—
47C23×C3×C4Dic314yes—
48C23×C3×C4C2×D314yes—
49C3×C4×D4—28no—
50C3×C4×D4—28no—
51C2×C3×C4⋊C4C2×D314yes—
52C2×C3×C22⋊C4Dic314yes—
53C2×C3×C22⋊C4C2×D314yes—
54C2×C3×C42Dic314yes—
55C22×C3⋊C8—68no—
56#830—316no—
57#830—68no—
58C23×C32—21no—
59C23×C32—21no—
60C23×C32—41no—
61C23×C32—41no—
62C23×C32—41no—
63C23×C32—41no—
64C23×C32C22⋊C412yes—
65C23×C32C2×C812yes—
66C23×C32C2×C812yescomplemented
67C23×C32M1612yes—
68C23×C32C22×C411yescharacteristic
69C32×D4—28no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—24no—
73C2×C32×C4—22no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—42no—
77C2×C32×C4—41no—
78C2×C32×C4—42no—
79C2×C32×C4—42no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4C22⋊C411yescharacteristic
83C2×C32×C4C22⋊C411yescharacteristic
84C2×C32×C4C2×C812yescomplemented
85C2×C32×C4C2×D412yes—
86C3⋊(C3⋊C8)—44no—
87C2×C4×D4C3⋊D311yescharacteristic
88C2×C22⋊C8—92no—
89#324—92no—
90#323—92no—
91C24×C3C2×Dic314yes—
92C24×C3C2×Dic314yes—
93C2×C3×D4—28no—
94C2×C3×D4—28no—
95C2×C3×D4C3⋊C8116yescomplemented
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—48no—
99C22×C3×C4—48no—
100C22×C3×C4C2×Dic314yes—
101C22×C3×C4C2×Dic314yes—
102C22×C3×C4C2×Dic314yes—
103C22×C3×C4C2×Dic314yes—
104C22×C3×C4#6718yes—
105C22×C3×C4#6718yes—
106C22×C3×C4C22×D314yes—
107C3×C4⋊C4—28no—
108C3×C22⋊C4—28no—
109C3×C22⋊C4—28no—
110C3×C42—24no—
111C3×C42—24no—
112C2×C3⋊C8—616no—
113C2×C3⋊C8—1216no—
114C22×C32—21no—
115C22×C32—22no—
116C22×C32—21no—
117C22×C32—22no—
118C22×C32—22no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32—42no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32C22⋊C814yes—
126C22×C32C2×C22⋊C411yescharacteristic
127C22×C32C22×C811yescharacteristic
128C22×C32C2×M1611yescharacteristic
129C32×C4—22no—
130C32×C4—22no—
131C32×C4—21no—
132C32×C4—41no—
133C32×C4—41no—
134C32×C4#9912yes—
135C22×D4C3⋊Dic311yescharacteristic
136C23×C4C3⋊Dic311yescharacteristic
137C23×C4C2×C3⋊D311yescharacteristic
138C22×C8—182no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4C2×C3⋊D311yescharacteristic
142C2×C22⋊C4C3⋊Dic311yescharacteristic
143C2×C22⋊C4C2×C3⋊D311yescharacteristic
144C2×C42C3⋊Dic311yescharacteristic
145C22⋊C8—94no—
146C22⋊C8—182no—
147C23×C3—24no—
148C23×C3—24no—
149C23×C3—44no—
150C23×C3—44no—
151C23×C3—44no—
152C23×C3—44no—
153C23×C3C2×C3⋊C818yes—
154C23×C3C2×C3⋊C818yescomplemented
155C23×C3#20818yes—
156C23×C3#21718yes—
157C23×C3C22×Dic314yes—
158C3×D4—232no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—216no—
162C2×C3×C4—28no—
163C2×C3×C4—24no—
164C2×C3×C4—24no—
165C2×C3×C4—48no—
166C2×C3×C4—44no—
167C2×C3×C4—48no—
168C2×C3×C4—48no—
169C2×C3×C4—44no—
170C2×C3×C4—44no—
171C2×C3×C4C2×C3⋊C818yescomplemented
172C2×C3×C4#21714yes—
173C2×C3×C4#21714yes—
174C2×C3×C4C2×#6718yes—
175C3⋊C8—1216no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32—24no—
179C2×C32—42no—
180C2×C32C2×C22⋊C811yescharacteristic
181C2×C32C2×#9911yescharacteristic
182C2×C32#41311yescharacteristic
183C24C2×C3⋊Dic311yescharacteristic
184C24C2×C3⋊Dic311yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4C3⋊(C3⋊C8)14yescomplemented
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4C2×C3⋊Dic311yescharacteristic
193C22×C4C2×C3⋊Dic311yescharacteristic
194C22×C4C2×C3⋊Dic311yescharacteristic
195C22×C4C2×C3⋊Dic311yescharacteristic
196C22×C4#63812yes—
197C22×C4#63812yes—
198C22×C4C22×C3⋊D311yescharacteristic
199C2×C8—184no—
200C2×C8—364no—
201C4⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C42—21no—
205C42—21no—
206C22×C3—24no—
207C22×C3—28no—
208C22×C3—24no—
209C22×C3—28no—
210C22×C3—28no—
211C22×C3—48no—
212C22×C3—48no—
213C22×C3—48no—
214C22×C3—48no—
215C22×C3—44no—
216C22×C3—44no—
217C22×C3#830116yes—
218C22×C3C22×C3⋊C814yes—
219C22×C3C2×#20814yes—
220C22×C3C2×#21714yes—
221C3×C4—28no—
222C3×C4—28no—
223C3×C4—24no—
224C3×C4—44no—
225C3×C4—44no—
226C3×C4#83418yes—
227C32#145911yescharacteristic, complemented, Sylow
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23C2×C3⋊(C3⋊C8)12yes—
235C23C2×C3⋊(C3⋊C8)12yescomplemented
236C23#374812yes—
237C23#375712yes—
238C23C22×C3⋊Dic311yescharacteristic
239D4—28no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4C2×C3⋊(C3⋊C8)12yescomplemented
253C2×C4#375711yescharacteristic
254C2×C4#375711yescharacteristic
255C2×C4C2×#63812yes—
256C8—364no—
257C2×C3—24no—
258C2×C3—24no—
259C2×C3—216no—
260C2×C3—48no—
261C2×C3C2×#83014yes—
262C2×C3C2×#83414yes—
263C2×C3#523714yes—
264C22—21no—
265C22—22no—
266C22—21no—
267C22—22no—
268C22—22no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22#6365714yes—
276C22C22×C3⋊(C3⋊C8)11yescharacteristic
277C22C2×#374811yescharacteristic
278C22C2×#375711yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4#6366112yes—
285C3#7160014yescomplemented
286C2—21no—
287C2—21no—
288C2—24no—
289C2—42no—
290C2C2×#6365711yescharacteristic
291C2C2×#6366111yescharacteristic
292C2#1059293611yescharacteristic
subgroupG/H
centerC2×C4#3757
commutatorC22×C32C22×C8
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×C4×D4C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C32×D4C4
cosocleC22×C32×C4C23
hypercenterC2×C4×D4C3⋊D3
hypocenterC32#1459
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#63657C2semidirect
3. #10589394C2semidirect
4. #10589391C2semidirect
26. C2×C32×D4C8semidirect
66. C23×C32C2×C8semidirect
84. C2×C32×C4C2×C8semidirect
95. C2×C3×D4C3⋊C8semidirect
154. C23×C3C2×C3⋊C8semidirect
171. C2×C3×C4C2×C3⋊C8semidirect
187. C2×D4C3⋊(C3⋊C8)semidirect
227. C32#1459semidirect
235. C23C2×C3⋊(C3⋊C8)semidirect
252. C2×C4C2×C3⋊(C3⋊C8)semidirect
285. C3#71600semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×#636572
#105893942
#105893912
#716003