Finite Group Databasebeta

← #11774608

Subgroups — #11774608

Order 1152.

738 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
1C32×C4×#41C211yesmaximal, characteristic
2#10592625C211yesmaximal, characteristic, complemented
3#10592624C211yesmaximal, characteristic, complemented
4#10589401C212yesmaximal, complemented
5D4⋊(C3⋊(C3⋊C8))C212yesmaximal, complemented
6#71640—34nomaximal
7C2×C32×#41C411yescharacteristic
8C32×C4×Q8C411yescharacteristic
9C32×C4×Q8C2211yescharacteristic
10C32×C4×D4—21no—
11C32×C4×D4—21no—
12C32×C4×D4C411yescharacteristic
13C32×C4×D4C2211yescharacteristic
14C32×#117—21no—
15C32×#117C411yescharacteristic
16C2×C32×C42—21no—
17C2×C32×C42C2211yescharacteristic
18#63657—21no—
19#63657—21no—
20C4⋊(C3⋊(C3⋊C8))—21no—
21C4⋊(C3⋊(C3⋊C8))C2212yes—
22#63630—21no—
23C4×C3⋊(C3⋊C8)C2212yes—
24C3×C4×#41D314yes—
25#4926—34no—
26#4925—34no—
27#4535—38no—
28D4⋊(C3⋊C8)—38no—
29C32×#41—22no—
30C32×#41—22no—
31C2×C32×Q8C2×C411yescharacteristic
32C2×C32×D4—21no—
33C2×C32×D4C2×C411yescharacteristic
34C22×C32×C4—21no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D411yescharacteristic
40C32×C4⋊C4—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4C2×C411yescharacteristic
43C32×C4⋊C4C2×C411yescharacteristic
44C32×C22⋊C4—21no—
45C32×C22⋊C4—21no—
46C32×C22⋊C4—21no—
47C32×C42—21no—
48C32×C42—21no—
49C32×C42—21no—
50C32×C42C2×C411yescharacteristic
51C32×C42D411yescharacteristic
52C32×C42D411yescharacteristic
53C32×C42C2311yescharacteristic
54C2×C3⋊(C3⋊C8)—22no—
55C2×C3⋊(C3⋊C8)—22no—
56#1488—91noSylow
57C2×C3×#41Dic314yes—
58C3×C4×Q8Dic314yes—
59C3×C4×Q8C2×D314yes—
60C3×C4×D4—24no—
61C3×C4×D4—24no—
62C3×C4×D4Dic314yes—
63C3×C4×D4C2×D314yes—
64C3×#117—24no—
65C3×#117Dic314yes—
66C2×C3×C42—24no—
67C2×C3×C42C2×D314yes—
68#830—64no—
69#830—64no—
70C4⋊(C3⋊C8)—38no—
71C4⋊(C3⋊C8)—64no—
72#803—64no—
73C4×C3⋊C8—38no—
74C23×C32—21no—
75C23×C32C22⋊C411yescharacteristic
76C32×Q8—21no—
77C32×Q8M1612yes—
78C32×D4—24no—
79C32×D4—21no—
80C32×D4M1612yes—
81C2×C32×C4—24no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C32×C4—42no—
91C2×C32×C4C22⋊C411yescharacteristic
92C2×C32×C4C22⋊C411yescharacteristic
93C2×C32×C4C22⋊C411yescharacteristic
94C2×C32×C4C22×C411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C3⋊(C3⋊C8)—42no—
98C3⋊(C3⋊C8)—42no—
99C4×#41C3⋊D311yescharacteristic
100#432—91no—
101#431—91no—
102Q8⋊C8—92no—
103D4⋊C8—92no—
104C3×#41—28no—
105C3×#41—28no—
106C2×C3×Q8C2×Dic314yes—
107C2×C3×D4—24no—
108C2×C3×D4C2×Dic314yes—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—24no—
112C22×C3×C4C2×Dic314yes—
113C22×C3×C4#6714yes—
114C22×C3×C4#6714yes—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4C2×Dic314yes—
118C3×C4⋊C4C2×Dic314yes—
119C3×C22⋊C4—24no—
120C3×C22⋊C4—24no—
121C3×C22⋊C4—24no—
122C3×C42—24no—
123C3×C42—24no—
124C3×C42—24no—
125C3×C42C2×Dic314yes—
126C3×C42#6714yes—
127C3×C42#6714yes—
128C3×C42C22×D314yes—
129C2×C3⋊C8—68no—
130C2×C3⋊C8—68no—
131C22×C32—21no—
132C22×C32—22no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32C2×C22⋊C411yescharacteristic
138C32×C4—21no—
139C32×C4—22no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—41no—
144C32×C4—41no—
145C32×C4C2×M1611yescharacteristic
146C32×C4C2×M1611yescharacteristic
147C32×C4#13512yes—
148C2×#41C3⋊Dic311yescharacteristic
149C4×Q8C3⋊Dic311yescharacteristic
150C4×Q8C2×C3⋊D311yescharacteristic
151C4×D4—21no—
152C4×D4—21no—
153C4×D4C3⋊Dic311yescharacteristic
154C4×D4C2×C3⋊D311yescharacteristic
155#117—21no—
156#117C3⋊Dic311yescharacteristic
157C2×C42—21no—
158C2×C42C2×C3⋊D311yescharacteristic
159C4⋊C8—92no—
160C4⋊C8—181no—
161C22⋊C8—181no—
162C22⋊C8—181no—
163#97—181no—
164C4×C8—92no—
165C23×C3—24no—
166C23×C3#21714yes—
167C3×Q8—24no—
168C3×Q8#20818yes—
169C3×D4—216no—
170C3×D4—24no—
171C3×D4#20818yes—
172C2×C3×C4—216no—
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—28no—
177C2×C3×C4—24no—
178C2×C3×C4—24no—
179C2×C3×C4—44no—
180C2×C3×C4—48no—
181C2×C3×C4—48no—
182C2×C3×C4#21714yes—
183C2×C3×C4#21714yes—
184C2×C3×C4#21714yes—
185C2×C3×C4C22×Dic314yes—
186C2×C3×C4C2×#6714yes—
187C2×C3×C4C2×#6714yes—
188C3⋊C8—128no—
189C3⋊C8—128no—
190C2×C32—41no—
191C2×C32—42no—
192C2×C32#40711yescharacteristic
193C2×C32#41611yescharacteristic
194C2×C32#42111yescharacteristic
195#41—22no—
196#41—22no—
197C2×Q8C2×C3⋊Dic311yescharacteristic
198C2×D4—21no—
199C2×D4C2×C3⋊Dic311yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4C2×C3⋊Dic311yescharacteristic
204C22×C4#63811yescharacteristic
205C22×C4#63811yescharacteristic
206C2×C8—182no—
207C2×C8—182no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4C2×C3⋊Dic311yescharacteristic
211C4⋊C4C2×C3⋊Dic311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42C2×C3⋊Dic311yescharacteristic
219C42#63811yescharacteristic
220C42#63811yescharacteristic
221C42C22×C3⋊D311yescharacteristic
222C22×C3—24no—
223C22×C3—28no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—44no—
227C22×C3—44no—
228C22×C3C2×#21714yes—
229C3×C4—24no—
230C3×C4—28no—
231C3×C4—24no—
232C3×C4—24no—
233C3×C4—24no—
234C3×C4—44no—
235C3×C4—44no—
236C3×C4C2×#20814yes—
237C3×C4C2×#20814yes—
238C3×C4#95018yes—
239C32#148811yescharacteristic, complemented, Sylow
240C23—21no—
241C23#375711yescharacteristic
242Q8—21no—
243Q8#374812yes—
244D4—24no—
245D4—21no—
246D4#374812yes—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4#375711yescharacteristic
258C2×C4#375711yescharacteristic
259C2×C4#375711yescharacteristic
260C2×C4C22×C3⋊Dic311yescharacteristic
261C2×C4C2×#63811yescharacteristic
262C2×C4C2×#63811yescharacteristic
263C8—362no—
264C8—362no—
265C2×C3—44no—
266C2×C3—48no—
267C2×C3#520714yes—
268C2×C3#523314yes—
269C2×C3#523614yes—
270C22—21no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22C2×#375711yescharacteristic
277C4—21no—
278C4—22no—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4C2×#374811yescharacteristic
285C4C2×#374811yescharacteristic
286C4#6415912yes—
287C3#7164014yescomplemented
288C2—41no—
289C2—42no—
290C2#1059290611yescharacteristic
291C2#1059293211yescharacteristic
292C2#1059293511yescharacteristic
subgroupG/H
centerC2×C4C2×#638
commutatorC2×C32×C4C22×C4
FrattiniC42C22×C3⋊D3
FittingC32×C4×#41C2
socleC22×C32C2×C22⋊C4
prime radicalC2×C32×#41C4
cosocleC32×C42C23
hypercenterC4×#41C3⋊D3
hypocenterC32#1488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #10592625C2semidirect
3. #10592624C2semidirect
4. #10589401C2semidirect
5. D4⋊(C3⋊(C3⋊C8))C2semidirect
239. C32#1488semidirect
287. C3#71640semidirect

Maximal subgroups

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

subgroupindex
C32×C4×#412
#105926252
#105926242
#105894012
D4⋊(C3⋊(C3⋊C8))2
#716403