Finite Group Databasebeta

← #11792401

Subgroups — #11792401

Order 1152.

591 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#10589893C212yesmaximal, complemented
2#10589850C211yesmaximal, characteristic, complemented
3C9×#407C211yesmaximal, characteristic, complemented
4#10588819C212yesmaximal, complemented
5#10588237C211yesmaximal, characteristic, complemented
6#69993—31nomaximal
7C23×C4×C9C2211yescharacteristic
8C2×#3697C2212yes—
9C2×C9×M16C2211yescharacteristic
10C2×C9×M16C2211yescharacteristic
11#63515—22no—
12#63492—22no—
13#63491—22no—
14C4×#611—24no—
15#63488C2212yes—
16#63486—22no—
17#63471—22no—
18#63470—22no—
19#63467—22no—
20C9×C22⋊C8—22no—
21#63384—24no—
22#5791—32no—
23#5748—31no—
24C3×#407D311yescharacteristic
25#5139—32no—
26#4557—31no—
27C24×C9D411yescharacteristic
28C22×C4×C9—22no—
29C22×C4×C9—22no—
30C22×C4×C9—21no—
31C22×C4×C9—21no—
32C22×C4×C9D411yescharacteristic
33C22×C4×C9C2311yescharacteristic
34C2×#611—24no—
35C2×#611—22no—
36#3697—22no—
37#3697—22no—
38#3697—22no—
39#3697—22no—
40C2×D36D412yescomplemented
41C2×C4×D9—22no—
42C2×Dic18D412yescomplemented
43C9×M16—22no—
44C9×M16—22no—
45C2×C8×C9—21no—
46C2×C8×C9D412yescomplemented
47#3676—24no—
48#3676—22no—
49(C2×D9)⋊C4—24no—
50C4⋊Dic9—22no—
51Dic9⋊C4—24no—
52C4×Dic9D414yescomplemented
53D72—22no—
54#3664—24no—
55Dic36—22no—
56#2010—91noSylow
57C23×C3×C4C2×D311yescharacteristic
58C2×#235—32no—
59C2×C3×M16C2×D311yescharacteristic
60C2×C3×M16C2×D311yescharacteristic
61#953—62no—
62#930—62no—
63#929—62no—
64C4×#67—64no—
65#926—32no—
66#924—62no—
67#909—62no—
68#908—62no—
69#904—62no—
70C3×C22⋊C8—22no—
71#825—64no—
72C23×C9—22no—
73C23×C9—21no—
74C23×C9—42no—
75C23×C9C2×D411yescharacteristic
76C22×D9—22no—
77C2×C4×C9—22no—
78C2×C4×C9—21no—
79C2×C4×C9—41no—
80C2×C4×C9—42no—
81C2×C4×C9—41no—
82C2×C4×C9C2×D412yes—
83C2×C4×C9C2×D412yes—
84C2×C4×C9C2×D411yescharacteristic
85C2×C4×C9#4111yescharacteristic
86#611—48no—
87#611—42no—
88#611—42no—
89C2×Dic9—24no—
90C2×Dic9—22no—
91D36—22no—
92D36—22no—
93C4×D9—42no—
94C4×D9—42no—
95Dic18—22no—
96Dic18—22no—
97C8×C9—22no—
98C8×C9—41no—
99#575—91no—
100#525—92no—
101#421—92no—
102#407D911yescharacteristic, complemented
103H(3,ℤ/4)—91no—
104C24×C3D1211yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4D1211yescharacteristic
110C22×C3×C4C22×D311yescharacteristic
111C2×#67—64no—
112C2×#67—62no—
113#235—62no—
114#235—62no—
115#235—62no—
116#235—62no—
117C2×D12—32no—
118C2×C4×D3—62no—
119C2×Dic6—32no—
120C3×M16—22no—
121C3×M16—22no—
122C2×C3×C8—21no—
123C2×C3×C8#6712yes—
124#217—64no—
125#217—62no—
126(C2×D3)⋊C4—64no—
127C4⋊Dic3—62no—
128Dic3⋊C4—64no—
129C4×Dic3—34no—
130Dic12—62no—
131D24—62no—
132#204—64no—
133C22×C9—24no—
134C22×C9—22no—
135C22×C9—41no—
136C22×C9—42no—
137C22×C9—41no—
138C22×C9—41no—
139C22×C9#12012yes—
140C22×C9#12111yescharacteristic
141C2×D9—44no—
142C2×D9—42no—
143C4×C9—41no—
144C4×C9#12011yescharacteristic
145C4×C9#12112yes—
146C4×C9#12711yescharacteristic
147Dic9—44no—
148Dic9—42no—
149C2×#41—92no—
150C23×C4C2×D911yescharacteristic
151#137—182no—
152AGL(1,ℤ/8)—182no—
153#135—182no—
154C2×M16C2×D911yescharacteristic
155C2×M16C2×D911yescharacteristic
156#123—182no—
157C22⋊Q8—182no—
158#121—182no—
159#120—182no—
160C4×D4—184no—
161#117—92no—
162#104—184no—
163C22⋊C8—22no—
164C23×C3—22no—
165C23×C3—21no—
166C23×C3—42no—
167C23×C3C2×D1211yescharacteristic
168C22×D3—62no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4#23511yescharacteristic
175C2×C3×C4D3×D412yes—
176C2×C3×C4D3×D412yes—
177C2×C3×C4C2×#6711yescharacteristic
178#67—128no—
179#67—122no—
180#67—122no—
181C2×Dic3—64no—
182C2×Dic3—62no—
183D12—62no—
184D12—62no—
185C4×D3—122no—
186C4×D3—122no—
187Dic6—62no—
188Dic6—62no—
189C3×C8—22no—
190C3×C8—41no—
191C2×C9—22no—
192C2×C9—21no—
193C2×C9—42no—
194C2×C9#39211yescharacteristic
195D9—82no—
196C24D3611yescharacteristic
197#41—182no—
198#41—182no—
199#41—182no—
200#41—182no—
201C2×Q8—92no—
202C2×D4—92no—
203C2×D4—184no—
204C2×D4—182no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—182no—
210C22×C4D3611yescharacteristic
211C22×C4C22×D911yescharacteristic
212Q16—182no—
213SD16—184no—
214D8—182no—
215M16—22no—
216M16—22no—
217C2×C8—21no—
218C2×C8#61112yescomplemented
219C4⋊C4—184no—
220C4⋊C4—182no—
221C22⋊C4—184no—
222C22⋊C4—184no—
223C22⋊C4—182no—
224C42—94no—
225C22×C3—24no—
226C22×C3—22no—
227C22×C3—41no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3#88212yes—
232C22×C3#93011yescharacteristic
233C2×D3—124no—
234C2×D3—122no—
235C3×C4—41no—
236C3×C4#88412yes—
237C3×C4#93711yescharacteristic
238C3×C4#94011yescharacteristic
239Dic3—124no—
240Dic3—122no—
241C9#201011yescharacteristic, complemented, Sylow
242C23—21no—
243C23—22no—
244C23—42no—
245C23—182no—
246C23C2×D3611yescharacteristic
247Q8—182no—
248Q8—182no—
249D4—182no—
250D4—182no—
251D4—368no—
252D4—362no—
253D4—362no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—184no—
260C2×C4—182no—
261C2×C4—362no—
262C2×C4—362no—
263C2×C4#369711yescharacteristic
264C2×C4D4×D912yes—
265C2×C4D4×D912yes—
266C2×C4C2×#61111yescharacteristic
267C8—22no—
268C8—41no—
269C2×C3—22no—
270C2×C3—21no—
271C2×C3—42no—
272C2×C3#496011yescharacteristic
273D3—242no—
274C22—24no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—364no—
281C22—362no—
282C22#6344412yes—
283C22#6349211yescharacteristic
284C4—41no—
285C4—364no—
286C4—362no—
287C4#6344612yes—
288C4#6349911yescharacteristic
289C4#6350211yescharacteristic
290C3#6999311yescharacteristic
291C2—22no—
292C2—21no—
293C2—42no—
294C2—722no—
295C2#1058864011yescharacteristic
subgroupG/H
centerC4#63502
commutator, cosocleC22×C4×C9C23
FrattiniC22×C3×C4C22×D3
FittingC9×#407C2
socleC2×C3#4960
hypercenter#407D9
hypocenterC9#2010
LCSG ▷ C22×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #10589893C2semidirect
2. #10589850C2semidirect
3. C9×#407C2semidirect
4. #10588819C2semidirect
5. #10588237C2semidirect
40. C2×D36D4semidirect
42. C2×Dic18D4semidirect
46. C2×C8×C9D4semidirect
52. C4×Dic9D4semidirect
102. #407D9semidirect
218. C2×C8#611semidirect
241. C9#2010semidirect

Maximal subgroups

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

subgroupindex
#105898932
#105898502
C9×#4072
#105888192
#105882372
#699933