Finite Group Databasebeta

← #11787275

Subgroups — #11787275

Order 1152.

1080 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 289 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
1C2×C32×#117C211yesmaximal, characteristic
2C2×#64133C211yesmaximal, characteristic
3C2×#63657C211yesmaximal, characteristic
4C2×#63637C212yesmaximal, complemented
5#10589395C212yesmaximal, complemented
6#71585—34nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×#634C2211yescharacteristic
9C32×#117—24no—
10C2×C32×C4⋊C4C2212yes—
11C2×C32×C22⋊C4—21no—
12C2×C32×C42—21no—
13C2×#3757—21no—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15#64133—22no—
16#64133C414yescomplemented
17C2×#3750—21no—
18C22×C3⋊(C3⋊C8)C2212yes—
19#63657—22no—
20#63637—24no—
21C2×C3×#117D314yes—
22C2×#924—34no—
23C2×#830—34no—
24C2×#810—38no—
25#4529—38no—
26C24×C32D411yescharacteristic
27C22×C32×C4—22no—
28C22×C32×C4—22no—
29C22×C32×C4—22no—
30C22×C32×C4C2×C412yes—
31C22×C32×C4D411yescharacteristic
32C22×C32×C4D412yes—
33C22×C32×C4C2311yescharacteristic
34C22×C3⋊Dic3—21no—
35C22×C3⋊Dic3—21no—
36C2×#634—22no—
37C2×#634—22no—
38C2×#634—21no—
39C2×#634C2×C412yes—
40C32×C4⋊C4—22no—
41C32×C4⋊C4D414yes—
42C32×C22⋊C4—24no—
43C32×C42—24no—
44#3757—24no—
45C4⋊(C3⋊Dic3)—21no—
46C4⋊(C3⋊Dic3)C2×C412yes—
47#3750—24no—
48C2×C3⋊(C3⋊C8)—22no—
49C2×C3⋊(C3⋊C8)—22no—
50C2×C3⋊(C3⋊C8)—22no—
51#1873—91noSylow
52C23×C3×C4C2×D314yes—
53C22×Dic6—34no—
54C3×#117—216no—
55C2×C3×C4⋊C4C2×D318yes—
56C2×C3×C22⋊C4—24no—
57C2×C3×C42—24no—
58C2×#217—64no—
59C2×C4⋊Dic3—34no—
60#924—316no—
61#924—68no—
62C2×Dic3⋊C4—64no—
63C22×C3⋊C8—38no—
64#830—68no—
65#810—616no—
66C23×C32—21no—
67C23×C32—21no—
68C23×C32—22no—
69C23×C32—22no—
70C23×C32C22⋊C412yes—
71C23×C32C2×D411yescharacteristic
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—24no—
75C2×C32×C4—24no—
76C2×C32×C4—24no—
77C2×C32×C4—42no—
78C2×C32×C4—42no—
79C2×C32×C4C22⋊C412yes—
80C2×C32×C4C22×C411yescharacteristic
81C2×C32×C4C2×D411yescharacteristic
82C2×C32×C4C2×D412yes—
83C2×C32×C4#4112yes—
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—42no—
89#634—24no—
90#634—42no—
91C3⋊(C3⋊C8)—42no—
92C2×C22⋊Q8—91no—
93C2×#117C3⋊D311yescharacteristic
94C2×#103—92no—
95C2×C22⋊C8—91no—
96#340—92no—
97C24×C3#6714yes—
98C22×C3×C4—28no—
99C22×C3×C4—28no—
100C22×C3×C4—28no—
101C22×C3×C4C4×D318yes—
102C22×C3×C4D1218yes—
103C22×C3×C4#6714yes—
104C22×C3×C4C22×D314yes—
105C22×Dic3—64no—
106C22×Dic3—64no—
107C2×Dic6—38no—
108C2×Dic6—68no—
109C2×Dic6—68no—
110C2×Dic6—64no—
111C3×C4⋊C4—28no—
112C3×C4⋊C4#67116yes—
113C3×C22⋊C4—216no—
114C3×C42—216no—
115#217—616no—
116C4⋊Dic3—38no—
117C4⋊Dic3—64no—
118Dic3⋊C4—616no—
119C2×C3⋊C8—68no—
120C2×C3⋊C8—68no—
121C2×C3⋊C8—68no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32C2×C22⋊C411yescharacteristic
128C22×C32#12011yescharacteristic
129C22×C32#12311yescharacteristic
130C22×C32#13512yes—
131C22×C32#13712yes—
132C32×C4—22no—
133C32×C4—44no—
134C32×C4C4×D412yes—
135C32×C4#12112yes—
136C3⋊Dic3—42no—
137C3⋊Dic3—42no—
138C22×Q8—91no—
139C23×C4C2×C3⋊D311yescharacteristic
140C22×C8—92no—
141C22⋊Q8—94no—
142C22⋊Q8—182no—
143#117—24no—
144C2×C4⋊C4—91no—
145C2×C4⋊C4—181no—
146C2×C4⋊C4C2×C3⋊D312yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—181no—
149C2×C42—21no—
150#103—184no—
151C22⋊C8—182no—
152C23×C3—24no—
153C23×C3—28no—
154C23×C3—28no—
155C23×C3—24no—
156C23×C3(C2×D3)⋊C418yes—
157C23×C3C2×#6714yes—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—216no—
161C2×C3×C4—216no—
162C2×C3×C4—216no—
163C2×C3×C4—48no—
164C2×C3×C4—48no—
165C2×C3×C4(C2×D3)⋊C418yes—
166C2×C3×C4C2×C4×D314yes—
167C2×C3×C4C2×D1214yes—
168C2×C3×C4#23518yes—
169C2×C3×C4C2×#6718yes—
170C2×Dic3—68no—
171C2×Dic3—68no—
172C2×Dic3—124no—
173C2×Dic3—128no—
174C2×Dic3—124no—
175Dic6—616no—
176Dic6—128no—
177C3⋊C8—128no—
178C2×C32—42no—
179C2×C32#38611yescharacteristic
180C2×C32#41611yescharacteristic
181C2×C32#41911yescharacteristic
182C2×C32#45212yes—
183C2×C32#48512yes—
184C24#63811yescharacteristic
185C2×Q8—92no—
186C2×Q8—181no—
187C2×Q8—182no—
188C2×Q8—182no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—181no—
193C22×C4—181no—
194C22×C4C4×C3⋊D312yes—
195C22×C4C4⋊(C3⋊D3)12yes—
196C22×C4#63811yescharacteristic
197C22×C4C22×C3⋊D311yescharacteristic
198C2×C8—182no—
199C2×C8—182no—
200C2×C8—182no—
201C4⋊C4—22no—
202C4⋊C4—92no—
203C4⋊C4—184no—
204C4⋊C4—181no—
205C4⋊C4#63814yes—
206C22⋊C4—24no—
207C22⋊C4—184no—
208C42—24no—
209C22×C3—28no—
210C22×C3—28no—
211C22×C3—28no—
212C22×C3—48no—
213C22×C3—48no—
214C22×C3C2×(C2×D3)⋊C414yes—
215C22×C3#92914yes—
216C22×C3#95018yes—
217C22×C3#95118yes—
218C22×C3#95314yes—
219C3×C4—28no—
220C3×C4—416no—
221C3×C4C4×#6718yes—
222C3×C4#93018yes—
223Dic3—128no—
224Dic3—128no—
225C32#187311yescharacteristic, complemented, Sylow
226C23—21no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23#375212yes—
231C23C2×#63811yescharacteristic
232Q8—184no—
233Q8—362no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—182no—
242C2×C4—182no—
243C2×C4—361no—
244C2×C4—362no—
245C2×C4—361no—
246C2×C4#375212yes—
247C2×C4C2×C4×C3⋊D311yescharacteristic
248C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
249C2×C4#382812yes—
250C2×C4C2×#63812yes—
251C8—362no—
252C2×C3—48no—
253C2×C3#499618yes—
254C2×C3#500214yes—
255C2×C3#500414yes—
256C2×C3#521314yes—
257C2×C3#523918yes—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—42no—
262C22—42no—
263C22C2×#375211yescharacteristic
264C22#6413811yescharacteristic
265C22#6415912yes—
266C22#6416012yes—
267C22#6416211yescharacteristic
268C4—22no—
269C4—44no—
270C4—362no—
271C4—362no—
272C4C4×#63812yes—
273C4#6413912yes—
274C3#7158514yescomplemented
275C2—42no—
276C2#1059269512yes—
277C2#1059270111yescharacteristic
278C2#1059270311yescharacteristic
279C2#1059291211yescharacteristic
280C2#1059293812yes—
subgroupG/H
centerC23C2×#638
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×#117C2
socleC23×C32C2×D4
prime radicalC24×C32D4
cosocleC22×C32×C4C23
hypercenterC2×#117C3⋊D3
hypocenterC32#1873
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. C2×#63637C2semidirect
5. #10589395C2semidirect
16. #64133C4semidirect
225. C32#1873semidirect
274. C3#71585semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×#641332
C2×#636572
C2×#636372
#105893952
#715853