Finite Group Databasebeta

← #11923290

Subgroups — #11923290

Order 1152.

609 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(#41⋊C3))C213yesmaximal, complemented
3C4×C3⋊(#41⋊C3)C213yesmaximal, complemented
4#87040C311yesmaximal, characteristic, complemented
5#85604—31nomaximal
6C2×C3⋊(#41⋊C3)C2213yes—
7C3×Q8×Dic3—41nomaximal
8C3×C4×SL(2,𝔽3)C2213yes—
9Dic3×SL(2,𝔽3)C2211yescharacteristic
10C3×Q82C2×C311yescharacteristic
11#5734—33no—
12#5725—33no—
13#5577—31no—
14#5577C2×C313yes—
15C4×#239—31no—
16C4×#239C2×C313yes—
17Q8×SL(2,𝔽3)—21no—
18Q8×SL(2,𝔽3)D311yescharacteristic, complemented
19C4⋊(#41⋊C3)—33no—
20C4×#41⋊C3—33no—
21C2×C32×Q8—41no—
22C2×C3×SL(2,𝔽3)C2311yescharacteristic
23C3⋊(#41⋊C3)—23no—
24C3×C4⋊Dic3—43no—
25C3×C4×Dic3—43no—
26#3483—31noSylow
27C2×#239—31no—
28C2×#239—32no—
29C2×#239C22×C313yes—
30C2×#41⋊C3—33no—
31C3×C4⋊Q8—33no—
32C3×C4×Q8—31no—
33C3×C4×Q8—121no—
34C3×C4×Q8C22×C313yes—
35#947—36no—
36Q8×Dic3A411yescharacteristic
37Q8×Dic3C22×C311yescharacteristic
38#895—36no—
39#895—36no—
40#893—33no—
41#893—33no—
42#892—33no—
43#892—33no—
44#874—33no—
45C4×D12—33no—
46C4×D12—33no—
47C42×D3—33no—
48C4×SL(2,𝔽3)—23no—
49C4×SL(2,𝔽3)—31no—
50C4×SL(2,𝔽3)C2×D313yes—
51C32×Q8—44no—
52C2×C32×C4—43no—
53C2×C3×Dic3—41no—
54C2×C3×Dic3—43no—
55C3×SL(2,𝔽3)#4111yescharacteristic, complemented
56Q82C3×D311yescharacteristic
57#555—93no—
58#550—33no—
59#550—91no—
60#532—93no—
61C4×#41—33no—
62C4×#41—91no—
63C2×C3×Q8—32no—
64C2×C3×Q8—41no—
65C2×C3×Q8—81no—
66C2×C3×Q8C2×A411yescharacteristic
67C2×C3×Q8C23×C311yescharacteristic
68#239—23no—
69#239—64no—
70#239—63no—
71#239—124no—
72C2×D12—33no—
73C2×D12—33no—
74C2×D12—36no—
75C2×C4×D3—33no—
76C2×C4×D3—33no—
77C2×C4×D3—33no—
78#41⋊C3—63no—
79C2×SL(2,𝔽3)—21no—
80C2×SL(2,𝔽3)C22×D311yescharacteristic
81C3×C4⋊C4—33no—
82C3×C4⋊C4—33no—
83C3×C4⋊C4—123no—
84C3×C42—33no—
85C3×C42—123no—
86(C2×D3)⋊C4—36no—
87(C2×D3)⋊C4—36no—
88C4⋊Dic3—31no—
89C4⋊Dic3C2×A413yes—
90C4×Dic3—31no—
91C4×Dic3—33no—
92C4×Dic3C2×A413yes—
93C22×C32—41no—
94C32×C4—46no—
95C3×Dic3—42no—
96C3×Dic3—83no—
97C2×#41—33no—
98C2×#41—92no—
99C2×#41—91no—
100C4⋊Q8—33no—
101#127—93no—
102#124—96no—
103#121—96no—
104#121—96no—
105C4×Q8—31no—
106C4×Q8—31no—
107C4×Q8—31no—
108C4×Q8C2×C3×D313yes—
109C4×D4—93no—
110C4×D4—93no—
111C4×D4—93no—
112C4×D4—93no—
113#117—93no—
114#117—93no—
115C2×C42—93no—
116C22×D3—33no—
117C3×Q8—31no—
118C3×Q8—44no—
119C3×Q8—84no—
120C3×Q8—122no—
121C3×Q8#41⋊C314yescomplemented
122C3×Q8C3×#4111yescharacteristic
123C2×C3×C4—31no—
124C2×C3×C4—33no—
125C2×C3×C4—43no—
126C2×C3×C4—83no—
127C2×C3×C4—121no—
128C2×C3×C4—123no—
129C2×C3×C4C22×A413yes—
130C2×Dic3—31no—
131C2×Dic3C22×A413yes—
132C2×Dic3C22×A411yescharacteristic
133D12—63no—
134D12—63no—
135D12—66no—
136D12—126no—
137C4×D3—63no—
138C4×D3—63no—
139C4×D3—66no—
140C4×D3—66no—
141SL(2,𝔽3)—21no—
142SL(2,𝔽3)#23911yescharacteristic, complemented
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146#41—63no—
147#41—184no—
148#41—183no—
149#41—364no—
150C2×Q8—32no—
151C2×Q8D3×A411yescharacteristic
152C2×Q8C22×C3×D311yescharacteristic
153C2×D4—93no—
154C2×D4—96no—
155C2×D4—93no—
156C22×C4—93no—
157C22×C4—93no—
158C22×C4—93no—
159C4⋊C4—33no—
160C4⋊C4—33no—
161C4⋊C4—33no—
162C4⋊C4—91no—
163C22⋊C4—96no—
164C22⋊C4—96no—
165C42—33no—
166C42—33no—
167C42—91no—
168C42—93no—
169C22×C3—41no—
170C22×C3—81no—
171C22×C3C23×A411yescharacteristic
172C2×D3—63no—
173C2×D3—63no—
174C2×D3—63no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—46no—
178C3×C4—63no—
179C3×C4—86no—
180C3×C4—122no—
181C3×C4—243no—
182C3×C4C2×#41⋊C316yes—
183Dic3—23no—
184Dic3—61no—
185Dic3#99412yes—
186C32—41noSylow
187C23—93no—
188Q8—31no—
189Q8—122no—
190Q8C3⋊(#41⋊C3)14yescomplemented
191Q8C3×#23911yescharacteristic
192D4—183no—
193D4—183no—
194D4—186no—
195D4—366no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—33no—
199C2×C4—33no—
200C2×C4—91no—
201C2×C4—183no—
202C2×C4—183no—
203C2×C4—186no—
204C2×C4—186no—
205C2×C4C2×D3×A413yes—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—81no—
210C2×C3—81no—
211C2×C3—81no—
212C2×C3C22×#41⋊C311yescharacteristic
213C2×C3A4×#4111yescharacteristic
214C2×C3C2×#99411yescharacteristic
215D3—123no—
216C22—183no—
217C22—183no—
218C22—183no—
219C22C22×D3×A411yescharacteristic
220C4—31no—
221C4—31no—
222C4—32no—
223C4—63no—
224C4—63no—
225C4—181no—
226C4C2×C3⋊(#41⋊C3)16yes—
227C3—41no—
228C3—81no—
229C3#8560411yescharacteristic, complemented
230C2—363no—
231C2C22×C3⋊(#41⋊C3)11yescharacteristic
232C2A4×#23911yescharacteristic
233C2D3×#99411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C4⋊(C3⋊(#41⋊C3))C2semidirect
3. C4×C3⋊(#41⋊C3)C2semidirect
4. #87040C3semidirect
18. Q8×SL(2,𝔽3)D3semidirect
55. C3×SL(2,𝔽3)#41semidirect
121. C3×Q8#41⋊C3semidirect
142. SL(2,𝔽3)#239semidirect
190. Q8C3⋊(#41⋊C3)semidirect
229. C3#85604semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
C4×C3⋊(#41⋊C3)2
#870403
#856043
C3×Q8×Dic34