Finite Group Databasebeta

← #11786182

Subgroups — #11786182

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63641C211yesmaximal, characteristic
2C2×C3×Dic3⋊C8C212yesmaximal
3C4×Dic32C211yesmaximal, characteristic
4C2×#63572C211yesmaximal, characteristic
5C2×Dic3⋊(C3⋊C8)C212yesmaximal
6#70445—32nomaximal
7C22×C32×C8C2211yescharacteristic
8C2×C4×C3⋊Dic3C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×C3×C4×Dic3C2212yes—
11C22×C3×C3⋊C8C2212yes—
12C2×Dic32C411yescharacteristic
13C2×Dic32C412yes—
14C2×Dic32C411yescharacteristic
15#63641—22no—
16C3×Dic3⋊C8—24no—
17#63572—22no—
18Dic3⋊(C3⋊C8)—24no—
19C2×C3×C4⋊C8—32no—
20C2×Dic3⋊C8—32no—
21C2×Dic3⋊C8—32no—
22C2×Dic3⋊C8—32no—
23C2×Dic3⋊C8—61no—
24C2×Dic3⋊C8D312yes—
25C42×Dic3—32no—
26C2×C4⋊(C3⋊C8)—32no—
27C22×C32×C4C2311yescharacteristic
28C22×C3⋊Dic3C2×C411yescharacteristic
29C22×C3⋊Dic3C2×C411yescharacteristic
30C22×C3×Dic3C2×C412yes—
31C22×C3×Dic3C2×C412yes—
32C2×C32×C8—21no—
33C2×C32×C8—22no—
34C4×C3⋊Dic3D412yes—
35C4×C3⋊Dic3Q812yes—
36C2×C3⋊(C3⋊C8)—21no—
37C2×C3⋊(C3⋊C8)—22no—
38C3×C4×Dic3D414yes—
39C3×C4×Dic3Q814yes—
40C2×C3×C3⋊C8—22no—
41C2×C3×C3⋊C8—22no—
42C2×C3×C3⋊C8—22no—
43Dic32—22no—
44Dic32—24no—
45Dic32C814yescomplemented
46#1842—91noSylow
47C22×C3×C8—21no—
48C22×C3×C8—32no—
49C22×C3×C8C2×D312yes—
50C2×C3×C42—32no—
51C2×C4×Dic3—32no—
52C2×C4×Dic3—32no—
53C2×C4×Dic3—32no—
54C2×C4×Dic3—32no—
55C2×C4×Dic3—32no—
56C2×C4×Dic3—61no—
57C2×C4×Dic3C2×D312yes—
58C22×C3⋊C8—32no—
59C22×C3⋊C8—61no—
60C22×C3⋊C8C2×D312yes—
61C3×C4⋊C8—64no—
62Dic3⋊C8—24no—
63Dic3⋊C8—64no—
64Dic3⋊C8—64no—
65Dic3⋊C8—64no—
66Dic3⋊C8—64no—
67C4⋊(C3⋊C8)—64no—
68C23×C32C22×C411yescharacteristic
69C2×C32×C4C2×D411yescharacteristic
70C2×C32×C4C2×D412yes—
71C2×C32×C4C2×Q812yes—
72C2×C32×C4C2×Q811yescharacteristic
73C2×C3⋊Dic3—22no—
74C2×C3⋊Dic3C4⋊C412yes—
75C2×C3⋊Dic3C4⋊C412yes—
76C2×C3⋊Dic3C2×C812yes—
77C2×C3⋊Dic3M1612yes—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—22no—
80C2×C3×Dic3C4⋊C414yes—
81C2×C3×Dic3C4⋊C414yes—
82C2×C3×Dic3C2×C814yes—
83C2×C3×Dic3M1614yes—
84C32×C8—41no—
85C3⋊(C3⋊C8)—41no—
86C3×C3⋊C8—42no—
87C2×C4⋊C8—32no—
88C2×C4⋊C8—92no—
89C2×C4⋊C8—91no—
90C2×C4⋊C8—91no—
91C43—91no—
92C22×C3×C4—21no—
93C22×C3×C4—32no—
94C22×C3×C4—32no—
95C22×C3×C4C22×D312yes—
96C22×Dic3—32no—
97C22×Dic3—32no—
98C22×Dic3—61no—
99C22×Dic3—61no—
100C22×Dic3C4×D312yes—
101C22×Dic3C4×D312yes—
102C2×C3×C8—22no—
103C2×C3×C8—22no—
104C2×C3×C8—22no—
105C2×C3×C8—22no—
106C2×C3×C8—42no—
107C2×C3×C8—62no—
108C2×C3×C8—62no—
109C2×C3×C8—62no—
110C3×C42—34no—
111C3×C42—34no—
112C4×Dic3—34no—
113C4×Dic3—34no—
114C4×Dic3—38no—
115C4×Dic3—62no—
116C4×Dic3—62no—
117C4×Dic3—64no—
118C4×Dic3—64no—
119C4×Dic3—64no—
120C4×Dic3Dic614yes—
121C4×Dic3#6714yes—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—22no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—62no—
126C2×C3⋊C8—62no—
127C2×C3⋊C8—62no—
128C2×C3⋊C8—62no—
129C2×C3⋊C8—122no—
130C22×C32C2×C4⋊C411yescharacteristic
131C22×C32C2×C4⋊C412yes—
132C22×C32C22×C811yescharacteristic
133C22×C32C2×M1612yes—
134C22×C32C2×M1611yescharacteristic
135C32×C4#12711yescharacteristic
136C32×C4C4⋊Q811yescharacteristic
137C32×C4C4⋊Q812yes—
138C3⋊Dic3—22no—
139C3⋊Dic3C4⋊C814yescomplemented
140C3×Dic3—24no—
141C3×Dic3C4⋊C818yescomplemented
142C22×C8—32no—
143C22×C8—91no—
144C22×C8D3211yescharacteristic
145C2×C42—32no—
146C2×C42—91no—
147C2×C42—91no—
148C2×C42—92no—
149C2×C42—91no—
150C4⋊C8—64no—
151C4⋊C8—184no—
152C4⋊C8—182no—
153C4⋊C8—182no—
154C23×C3—21no—
155C23×C3C2×C4×D312yes—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—34no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—34no—
164C2×C3×C4—62no—
165C2×C3×C4—62no—
166C2×C3×C4C2×Dic612yes—
167C2×C3×C4D3×D412yes—
168C2×C3×C4D3×D412yes—
169C2×C3×C4D3×Q812yes—
170C2×C3×C4D3×Q812yes—
171C2×C3×C4C2×#6712yes—
172C2×Dic3—22no—
173C2×Dic3—22no—
174C2×Dic3—34no—
175C2×Dic3—34no—
176C2×Dic3—34no—
177C2×Dic3—34no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—122no—
185C2×Dic3D3×C814yes—
186C2×Dic3ΓL(1,𝔽25)14yes—
187C2×Dic3Dic3⋊C414yes—
188C2×Dic3Dic3⋊C414yes—
189C3×C8—42no—
190C3×C8—42no—
191C3×C8—122no—
192C3⋊C8—42no—
193C3⋊C8—122no—
194C3⋊C8—122no—
195C2×C32#38411yescharacteristic
196C2×C32C2×C4⋊C811yescharacteristic
197C2×C32C2×C4⋊C812yes—
198C2×C32#42311yescharacteristic
199C2×C32#42312yes—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—91no—
203C22×C4—91no—
204C22×C4C2×D3211yescharacteristic
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—62no—
210C2×C8—182no—
211C2×C8—181no—
212C42—34no—
213C42—34no—
214C42—94no—
215C42—92no—
216C42—92no—
217C42—184no—
218C42—182no—
219C22×C3—21no—
220C22×C3—22no—
221C22×C3—21no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3D3×C4⋊C412yes—
225C22×C3D3×C4⋊C412yes—
226C22×C3C2×D3×C812yes—
227C22×C3C2×ΓL(1,𝔽25)12yes—
228C22×C3D3×M1612yes—
229C22×C3D3×M1612yes—
230C22×C3C2×Dic3⋊C412yes—
231C3×C4—22no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—38no—
235C3×C4—64no—
236C3×C4#88812yes—
237C3×C4#88812yes—
238C3×C4#94012yes—
239C3×C4#94412yes—
240Dic3—24no—
241Dic3—38no—
242Dic3—64no—
243Dic3—64no—
244Dic3—122no—
245Dic3Dic3⋊C818yescomplemented
246C32#184211yescharacteristic, complemented, Sylow
247C23C4×D3211yescharacteristic
248C2×C4—34no—
249C2×C4—34no—
250C2×C4—34no—
251C2×C4—34no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—92no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—92no—
258C2×C4—182no—
259C2×C4D3×Dic612yes—
260C2×C4#379711yescharacteristic
261C2×C4D3×#6712yes—
262C2×C4#381111yescharacteristic
263C8—41no—
264C8—122no—
265C8—361no—
266C2×C3—21no—
267C2×C3—22no—
268C2×C3—21no—
269C2×C3—22no—
270C2×C3—21no—
271C2×C3D3×C4⋊C812yes—
272C2×C3D3×C4⋊C812yes—
273C2×C3#483712yes—
274C2×C3#483712yes—
275C2×C3#497512yes—
276C2×C3C2×Dic3⋊C812yes—
277C2×C3#511812yes—
278C22D32×C811yescharacteristic
279C22D3×ΓL(1,𝔽25)12yes—
280C22#6379111yescharacteristic
281C22D3×Dic3⋊C412yes—
282C22#6388311yescharacteristic
283C4—38no—
284C4—64no—
285C4—94no—
286C4—182no—
287C4#6383912yes—
288C4#6384011yescharacteristic
289C4#6397911yescharacteristic
290C3—21no—
291C3#7044512yescomplemented
292C2D3×Dic3⋊C812yes—
293C2#1059065811yescharacteristic
294C2#1059066312yes—
295C2#1059066611yescharacteristic
296C2#1059124411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutatorC22×C32C22×C8
FittingC22×C32×C8C22
socle, prime radicalC23×C32C22×C4
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1842
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
45. Dic32C8semidirect
139. C3⋊Dic3C4⋊C8semidirect
141. C3×Dic3C4⋊C8semidirect
245. Dic3Dic3⋊C8semidirect
246. C32#1842semidirect
291. C3#70445semidirect

Maximal subgroups

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

subgroupindex
C2×#636412
C2×C3×Dic3⋊C82
C4×Dic322
C2×#635722
C2×Dic3⋊(C3⋊C8)2
#704453