Finite Group Databasebeta

← #11922895

Subgroups — #11922895

Order 1152.

820 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 89 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×#64267C212yesmaximal, complemented
2C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊GL(2,𝔽3))C214yesmaximal, complemented
4#85346—33nomaximal
5#82526—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64267—24no—
8C2×C3⋊GL(2,𝔽3)C2214yescomplemented
9#64083—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C2×#986—36no—
12C3×C4×#41D311yescharacteristic, complemented
13C2×#949—32no—
14C2×#874—31no—
15C4×#41⋊C3D313yescomplemented
16C4⋊GL(2,𝔽3)—312no—
17#5027—34no—
18#5015—34no—
19#5001—32no—
20#4924—31no—
21C2×C4⋊(C3⋊D3)—44no—
22C2×C4⋊(C3⋊D3)—42no—
23C3×#41⋊C3D412yescomplemented
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3⋊GL(2,𝔽3)—24no—
26C32×C42—41no—
27#3038—91noSylow
28C2×C3×#41C2×D311yescharacteristic
29C22×D12—32no—
30C22×D12—62no—
31C2×#41⋊C3C2×D313yes—
32#986—612no—
33C2×GL(2,𝔽3)—312no—
34C3×C4×Q8C2×D312yes—
35C3×C4×D4—32no—
36C3×#117—31no—
37C2×C3×C42—31no—
38#949—64no—
39#949—64no—
40C2×#215—34no—
41C2×#213—34no—
42C2×#208—32no—
43#874—34no—
44#874—61no—
45#874—61no—
46#874—123no—
47C4×SL(2,𝔽3)C2×D316yescomplemented
48#809—38no—
49C4⋊(C3⋊C8)—34no—
50C22×C3⋊D3—44no—
51C2×C32×C4—41no—
52C2×C32×C4—42no—
53C4⋊(C3⋊D3)—84no—
54C4⋊(C3⋊D3)—88no—
55C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
56C2×AGL(1,ℤ/8)—92no—
57C2×#127—91no—
58C4×#41C3⋊D311yescharacteristic, complemented
59#460—94no—
60#459—94no—
61#423—91no—
62#418—92no—
63C23×D3—62no—
64C3×#41—32no—
65C3×#41D1212yescomplemented
66C2×C3×Q8C22×D311yescharacteristic
67C2×C3×D4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—32no—
70C2×D12—34no—
71C2×D12—64no—
72C2×D12—64no—
73C2×D12—64no—
74C2×D12—64no—
75C2×D12—62no—
76C2×D12—1212no—
77C2×D12—126no—
78C2×D12—128no—
79#41⋊C3D1216yescomplemented
80C2×SL(2,𝔽3)C22×D313yes—
81GL(2,𝔽3)—612no—
82C3×C4⋊C4—32no—
83C3×C22⋊C4—61no—
84C3×C42—32no—
85C3×C42—31no—
86C3×C42—43no—
87C3×C42S411yescharacteristic
88#215—64no—
89#215—64no—
90#213—68no—
91#208—64no—
92C2×C3⋊C8—34no—
93C22×C32—41no—
94C2×C3⋊D3—88no—
95C2×C3⋊D3—84no—
96C32×C4—42no—
97C32×C4—44no—
98C2×#41C2×C3⋊D311yescharacteristic
99C22×D4—92no—
100C22×D4—182no—
101AGL(1,ℤ/8)—184no—
102AGL(1,ℤ/8)—184no—
103C2×SD16—94no—
104C2×D8—94no—
105C2×M16—92no—
106#127—94no—
107#127—181no—
108#127—181no—
109C4×Q8C2×C3⋊D312yes—
110C4×D4—32no—
111#117—31no—
112C2×C42—31no—
113C4⋊C8—94no—
114#102—98no—
115C23×C3—31no—
116C22×D3—64no—
117C22×D3—1212no—
118C22×D3—124no—
119C22×D3—128no—
120C3×Q8—31no—
121C3×Q8C2×D1211yescharacteristic
122C3×D4—32no—
123C3×D4—61no—
124C2×C3×C4—31no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—46no—
129C2×C3×C4—43no—
130C2×C3×C4—64no—
131C2×C3×C4C2×S411yescharacteristic
132C2×C3×C4C2×S412yes—
133D12—64no—
134D12—64no—
135D12—128no—
136D12—124no—
137D12—128no—
138D12—2412no—
139D12—2424no—
140SL(2,𝔽3)C2×D1213yescomplemented
141C3⋊C8—64no—
142C2×C32—41no—
143C2×C32—42no—
144C3⋊D3—164no—
145C24—182no—
146#41—32no—
147#41C4⋊(C3⋊D3)12yescomplemented
148C2×Q8C22×C3⋊D311yescharacteristic
149C2×D4—31no—
150C2×D4—94no—
151C2×D4—184no—
152C2×D4—184no—
153C2×D4—184no—
154C2×D4—184no—
155C2×D4—182no—
156C2×D4—368no—
157C22×C4—31no—
158C22×C4—32no—
159SD16—184no—
160SD16—184no—
161D8—188no—
162M16—184no—
163C2×C8—94no—
164C4⋊C4—32no—
165C22⋊C4—61no—
166C42—32no—
167C42—31no—
168C42C3⋊S411yescharacteristic
169C22×C3—32no—
170C22×C3—43no—
171C22×C3—62no—
172C22×C3C22×S411yescharacteristic
173C2×D3—124no—
174C2×D3—128no—
175C2×D3—2424no—
176C2×D3—2412no—
177C2×D3—248no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—46no—
181C3×C4—412no—
182C3×C4—62no—
183C3×C4C4⋊S412yes—
184C3×C4#98614yescomplemented
185C32—41noSylow
186C23—31no—
187C23—184no—
188C23—364no—
189C23—368no—
190Q8—31no—
191Q8C2×C4⋊(C3⋊D3)11yescharacteristic
192D4—32no—
193D4—61no—
194D4—184no—
195D4—184no—
196D4—368no—
197D4—364no—
198D4—368no—
199C2×C4—31no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—64no—
204C2×C4C2×C3⋊S411yescharacteristic
205C2×C4C2×C3⋊S412yes—
206C8—184no—
207C2×C3—43no—
208C2×C3—46no—
209C2×C3—62no—
210C2×C3C2×C4⋊S411yescharacteristic
211C2×C3C2×#98612yes—
212D3—244no—
213D3—4812no—
214C22—32no—
215C22—62no—
216C22—364no—
217C22—368no—
218C22—728no—
219C22C22×C3⋊S411yescharacteristic
220C4—31no—
221C4—31no—
222C4—62no—
223C4C4⋊(C3⋊S4)12yes—
224C4#6426714yescomplemented
225C3—43no—
226C3#8534611yescharacteristic, complemented
227C2—62no—
228C2—724no—
229C2C2×C4⋊(C3⋊S4)11yescharacteristic
230C2C2×#6426712yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×#41D3
socleC22×C3C22×S4
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C3×C4×#41⋊C3C2semidirect
3. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
8. C2×C3⋊GL(2,𝔽3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
12. C3×C4×#41D3semidirect
15. C4×#41⋊C3D3semidirect
23. C3×#41⋊C3D4semidirect
47. C4×SL(2,𝔽3)C2×D3semidirect
55. C3×SL(2,𝔽3)C2×D4semidirect
58. C4×#41C3⋊D3semidirect
65. C3×#41D12semidirect
79. #41⋊C3D12semidirect
140. SL(2,𝔽3)C2×D12semidirect
147. #41C4⋊(C3⋊D3)semidirect
184. C3×C4#986semidirect
224. C4#64267semidirect
226. C3#85346semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C3×C4×#41⋊C32
C4⋊(C3⋊GL(2,𝔽3))2
#853463
#825263
#640834