Finite Group Databasebeta

← #11921409

Subgroups — #11921409

Order 1152.

1328 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 133 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×#63700C211yesmaximal, characteristic, complemented
2#10593100C214yesmaximal, complemented
3#10593099C214yesmaximal, complemented
4#10593098C216yesmaximal, complemented
5#87277—31nomaximal
6#63713—41nomaximal
7#63700C2214yescomplemented
8C2×#41⋊C9C2213yescomplemented
9#63692C2216yescomplemented
10#63691C22112yescomplemented
11#63690C2216yescomplemented
12#63689C2214yescomplemented
13C2×C3×H(5,𝔽2)D311yescharacteristic
14#5964—31no—
15#5836—34no—
16#5836—34no—
17#5835—34no—
18#5835—34no—
19#5822—36no—
20#5819—31no—
21C2×#932—36no—
22C2×Q8×C9—41no—
23#3701—44no—
24Q8×D9—44no—
25#3697—46no—
26#41⋊C9C2316yes—
27C2×Q8⋊C9C2311yescharacteristic
28#3689C2314yescomplemented
29#3688C2314yescomplemented
30#3585—91noSylow
31C3×H(5,𝔽2)—34no—
32C3×H(5,𝔽2)C2×D314yes—
33C2×C3×#41—31no—
34C2×C3×#41C2×D313yes—
35C22×C3×D4—33no—
36#1010—34no—
37#1009—34no—
38D3×#41—64no—
39#1007—61no—
40#1004—63no—
41C2×#235—36no—
42#951—36no—
43#951—36no—
44#950—312no—
45#950—312no—
46#949—36no—
47#949—36no—
48#948—34no—
49#948—34no—
50#942—34no—
51C2×#214—312no—
52#932—312no—
53#932—312no—
54#932—312no—
55C2×#213—312no—
56C2×#208—36no—
57Q8×C9—44no—
58C2×C4×C9—43no—
59#611—44no—
60D36—46no—
61C4×D9—412no—
62Dic18—46no—
63Q8⋊C9C2411yescharacteristic
64#585—91no—
65C2×H(5,𝔽2)D911yescharacteristic, complemented
66#577—94no—
67#577—94no—
68#576—94no—
69#576—94no—
70#575—96no—
71C2×AGL(1,ℤ/8)—96no—
72#568—91no—
73C24×C3—61no—
74C3×#41—34no—
75C3×#41—34no—
76C3×#41—36no—
77C3×#41C22×D316yes—
78C2×C3×Q8S411yescharacteristic
79C2×C3×Q8C22×D311yescharacteristic
80C2×C3×D4—312no—
81C2×C3×D4—33no—
82C2×C3×D4—33no—
83C2×C3×D4—66no—
84C22×C3×C4—33no—
85C2×#67—66no—
86#239—64no—
87D3×Q8—64no—
88#237—612no—
89D3×D4—612no—
90#235—34no—
91#235—312no—
92#235—66no—
93#235—66no—
94C2×D12—36no—
95C2×C4×D3—66no—
96C2×Dic6—36no—
97#216—34no—
98#216—34no—
99#215—34no—
100#215—34no—
101#214—312no—
102#214—312no—
103#213—312no—
104#213—312no—
105#208—34no—
106#208—36no—
107#208—36no—
108C2×C3⋊C8—312no—
109C22×C9—41no—
110C2×D9—44no—
111C4×C9—46no—
112Dic9—44no—
1132−1+4—94no—
1142−1+4—181no—
115H(5,𝔽2)—34no—
116H(5,𝔽2)—94no—
117H(5,𝔽2)—183no—
118H(5,𝔽2)C2×D914yes—
119C2×#41—31no—
120C2×#41—96no—
121C2×#41—184no—
122C2×#41C2×D913yes—
123C22×D4—33no—
124#137—96no—
125#137—96no—
126#137—94no—
127AGL(1,ℤ/8)—912no—
128AGL(1,ℤ/8)—912no—
129AGL(1,ℤ/8)—912no—
130AGL(1,ℤ/8)—96no—
131AGL(1,ℤ/8)—96no—
132#135—912no—
133#135—912no—
134C2×SD16—912no—
135C2×D8—912no—
136#131—94no—
137#131—94no—
138C2×M16—96no—
139C23×C3—33no—
140C23×C3—64no—
141C23×C3—242no—
142C22×D3—66no—
143C3×Q8—31no—
144C3×Q8C2×S414yes—
145C3×Q8C23×D311yescharacteristic
146C3×D4—36no—
147C3×D4—312no—
148C3×D4—63no—
149C2×C3×C4—31no—
150C2×C3×C4—36no—
151C2×C3×C4—36no—
152C2×C3×C4C2×S413yes—
153#67—612no—
154#67—64no—
155C2×Dic3—66no—
156D12—34no—
157D12—66no—
158C4×D3—64no—
159C4×D3—612no—
160Dic6—34no—
161Dic6—66no—
162C3⋊C8—34no—
163C3⋊C8—34no—
164C2×C9—41no—
165C2×C9—81no—
166D9—84no—
167C24—61no—
168#41—34no—
169#41—34no—
170#41—36no—
171#41—912no—
172#41—94no—
173#41—186no—
174#41—1812no—
175#41—184no—
176#41—186no—
177#41C22×D916yes—
178C2×Q8—96no—
179C2×Q8—184no—
180C2×Q8(C22⋊C9)⋊C211yescharacteristic
181C2×Q8C22×D911yescharacteristic
182C2×D4—312no—
183C2×D4—33no—
184C2×D4—33no—
185C2×D4—66no—
186C2×D4—96no—
187C2×D4—1812no—
188C2×D4—186no—
189C22×C4—33no—
190C22×C4—186no—
191Q16—94no—
192Q16—94no—
193SD16—912no—
194SD16—912no—
195SD16—94no—
196SD16—94no—
197D8—912no—
198D8—912no—
199M16—94no—
200M16—96no—
201M16—96no—
202C2×C8—912no—
203C22×C3—36no—
204C22×C3—123no—
205C22×C3—244no—
206C22×C3C22×S411yescharacteristic
207C2×D3—64no—
208C2×D3—246no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4C22×S416yes—
212Dic3—64no—
213C9—41noSylow
214C23—33no—
215C23—64no—
216C23—186no—
217C23—242no—
218Q8—31no—
219Q8—94no—
220Q8—186no—
221Q8C2×(C22⋊C9)⋊C214yes—
222Q8C23×D911yescharacteristic
223D4—36no—
224D4—312no—
225D4—63no—
226D4—94no—
227D4—1812no—
228D4—184no—
229D4—186no—
230C2×C4—31no—
231C2×C4—36no—
232C2×C4—36no—
233C2×C4—184no—
234C2×C4—186no—
235C2×C4—1812no—
236C2×C4C2×(C22⋊C9)⋊C213yes—
237C8—94no—
238C8—94no—
239C2×C3—21no—
240C2×C3—66no—
241C2×C3C23×S411yescharacteristic
242D3—124no—
243C22—36no—
244C22—123no—
245C22—184no—
246C22—244no—
247C22—726no—
248C22C22×(C22⋊C9)⋊C211yescharacteristic
249C4—31no—
250C4—31no—
251C4—184no—
252C4C22×(C22⋊C9)⋊C216yes—
253C3#8749311yescharacteristic
254C2—21no—
255C2—66no—
256C2—364no—
257C2C23×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC2C23×(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C24
Frattini, socleC2×C3C23×S4
FittingC2×C3×H(5,𝔽2)D3
hypercenterC2×Q8(C22⋊C9)⋊C2
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#63700C2semidirect
2. #10593100C2semidirect
3. #10593099C2semidirect
4. #10593098C2semidirect
7. #63700C22semidirect
8. C2×#41⋊C9C22semidirect
9. #63692C22semidirect
10. #63691C22semidirect
11. #63690C22semidirect
12. #63689C22semidirect
28. #3689C23semidirect
29. #3688C23semidirect
65. C2×H(5,𝔽2)D9semidirect

Maximal subgroups

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

subgroupindex
C2×#637002
#105931002
#105930992
#105930982
#872773
#637134