Finite Group Databasebeta

← C3×#85604 #11919734

Subgroups — C3×#85604 #11919734

Order 1152.

694 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 166 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
2C3×C4⋊(#41⋊C3)C213yesmaximal, complemented
3C3×C4×#41⋊C3C213yesmaximal, complemented
4C3×#3483C311yesmaximal, characteristic, complemented
5#85604C313yesmaximal, direct product factor
6C2×C3×#41⋊C3C2213yes—
7C32×C4×Q8—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×C4×SL(2,𝔽3)C2213yes—
10C3×Q82C2×C311yescharacteristic
11C3×#555—33no—
12C3×#550—31no—
13C3×#550C2×C313yes—
14C3×#532—33no—
15C3×C4×#41—31no—
16C3×C4×#41C2×C313yes—
17Q8×SL(2,𝔽3)C2×C313yescomplemented
18C4⋊(#41⋊C3)C2×C319yescomplemented
19C4×#41⋊C3C2×C319yescomplemented
20C2×C32×Q8—41no—
21C3×#41⋊C3—23no—
22C2×C3×SL(2,𝔽3)C2311yescharacteristic
23C32×C4⋊C4—43no—
24C32×C42—43no—
25#3483C3211yescharacteristic, complemented, Sylow
26C2×C3×#41—32no—
27C2×C3×#41—31no—
28C2×C3×#41C22×C313yes—
29C2×#41⋊C3C22×C319yes—
30C3×C4⋊Q8—33no—
31C3×#127—33no—
32C3×#124—36no—
33C3×#121—36no—
34C3×#121—36no—
35C3×C4×Q8—31no—
36C3×C4×Q8—43no—
37C3×C4×Q8A411yescharacteristic
38C3×C4×Q8C22×C313yes—
39C3×C4×Q8C22×C311yescharacteristic
40C3×C4×D4—33no—
41C3×C4×D4—33no—
42C3×C4×D4—33no—
43C3×C4×D4—33no—
44C3×#117—33no—
45C3×#117—33no—
46C2×C3×C42—33no—
47C4×SL(2,𝔽3)C22×C313yes—
48C4×SL(2,𝔽3)C22×C319yes—
49C32×Q8—44no—
50C2×C32×C4—43no—
51C2×C32×C4—43no—
52C2×C32×C4—41no—
53C3×SL(2,𝔽3)#4111yescharacteristic, complemented
54Q82C2×C3211yescharacteristic
55#555—33no—
56#550—31no—
57#550C2×C3213yes—
58#532—33no—
59C4×#41—31no—
60C4×#41C2×C3213yes—
61C3×#41—23no—
62C3×#41—64no—
63C3×#41—63no—
64C3×#41—124no—
65C2×C3×Q8—32no—
66C2×C3×Q8—43no—
67C2×C3×Q8C2×A411yescharacteristic
68C2×C3×Q8C23×C311yescharacteristic
69C2×C3×D4—33no—
70C2×C3×D4—36no—
71C2×C3×D4—33no—
72C22×C3×C4—33no—
73C22×C3×C4—33no—
74C22×C3×C4—33no—
75#41⋊C3—29no—
76C2×SL(2,𝔽3)C23×C313yes—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—33no—
79C3×C4⋊C4—33no—
80C3×C4⋊C4—49no—
81C3×C4⋊C4C2×A413yes—
82C3×C22⋊C4—36no—
83C3×C22⋊C4—36no—
84C3×C42—33no—
85C3×C42—33no—
86C3×C42—31no—
87C3×C42—49no—
88C3×C42C2×A413yes—
89C22×C32—41no—
90C32×C4—46no—
91C32×C4—42no—
92C32×C4—83no—
93C2×#41—32no—
94C2×#41—31no—
95C2×#41C22×C3213yes—
96C4⋊Q8—33no—
97#127—33no—
98#124—36no—
99#121—36no—
100#121—36no—
101C4×Q8—31no—
102C4×Q8C3×A411yescharacteristic
103C4×Q8C22×C3211yescharacteristic
104C4×Q8C22×C3213yes—
105C4×D4—33no—
106C4×D4—33no—
107C4×D4—33no—
108C4×D4—33no—
109#117—33no—
110#117—33no—
111C2×C42—33no—
112C23×C3—33no—
113C3×Q8—31no—
114C3×Q8—412no—
115C3×Q8—122no—
116C3×Q8#41⋊C314yescomplemented
117C3×Q8C3×#4111yescharacteristic
118C3×D4—63no—
119C3×D4—63no—
120C3×D4—66no—
121C3×D4—126no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—33no—
125C2×C3×C4—43no—
126C2×C3×C4—49no—
127C2×C3×C4—49no—
128C2×C3×C4—63no—
129C2×C3×C4—63no—
130C2×C3×C4—66no—
131C2×C3×C4—66no—
132C2×C3×C4C22×A413yes—
133C2×C3×C4C22×A413yes—
134C2×C3×C4C22×A411yescharacteristic
135SL(2,𝔽3)C3×#4113yescomplemented
136C2×C32—41no—
137C2×C32—41no—
138C2×C32—41no—
139#41—23no—
140#41—64no—
141#41—63no—
142#41—124no—
143C2×Q8—32no—
144C2×Q8C2×C3×A411yescharacteristic
145C2×Q8C23×C3211yescharacteristic
146C2×D4—33no—
147C2×D4—36no—
148C2×D4—33no—
149C22×C4—33no—
150C22×C4—33no—
151C22×C4—33no—
152C4⋊C4—31no—
153C4⋊C4—33no—
154C4⋊C4—33no—
155C4⋊C4C2×C3×A413yes—
156C22⋊C4—36no—
157C22⋊C4—36no—
158C42—31no—
159C42—33no—
160C42—33no—
161C42C2×C3×A413yes—
162C22×C3—43no—
163C22×C3—63no—
164C22×C3—63no—
165C22×C3—63no—
166C22×C3C23×A411yescharacteristic
167C3×C4—23no—
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—418no—
171C3×C4—46no—
172C3×C4—61no—
173C3×C4—63no—
174C3×C4—89no—
175C3×C4C2×#41⋊C316yes—
176C3×C4#99412yes—
177C32—41noSylow
178C23—33no—
179Q8—31no—
180Q8—122no—
181Q8C3×#41⋊C314yescomplemented
182Q8C32×#4111yescharacteristic
183D4—63no—
184D4—63no—
185D4—66no—
186D4—126no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—33no—
190C2×C4—63no—
191C2×C4—63no—
192C2×C4—66no—
193C2×C4—66no—
194C2×C4C22×C3×A413yes—
195C2×C4C22×C3×A413yes—
196C2×C4C22×C3×A411yescharacteristic
197C2×C3—43no—
198C2×C3—43no—
199C2×C3—43no—
200C2×C3—123no—
201C2×C3C22×#41⋊C311yescharacteristic
202C2×C3A4×#4111yescharacteristic
203C2×C3C2×#99411yescharacteristic
204C22—63no—
205C22—63no—
206C22—63no—
207C22C23×C3×A411yescharacteristic
208C4—23no—
209C4—31no—
210C4—31no—
211C4—61no—
212C4—63no—
213C4C2×C3×#41⋊C316yes—
214C4C3×#99412yes—
215C3—43no—
216C3#8560411yescharacteristic, direct product factor
217C2—123no—
218C2C22×C3×#41⋊C311yescharacteristic
219C2C3×A4×#4111yescharacteristic
220C2C2×C3×#99411yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3483C3
socleC22×C3C23×A4
hypercenterC3×C4×Q8A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. C3×C4×#41⋊C3C2semidirect
4. C3×#3483C3semidirect
5. #85604C3direct (N × H)
17. Q8×SL(2,𝔽3)C2×C3semidirect
18. C4⋊(#41⋊C3)C2×C3semidirect
19. C4×#41⋊C3C2×C3semidirect
25. #3483C32semidirect
53. C3×SL(2,𝔽3)#41semidirect
116. C3×Q8#41⋊C3semidirect
135. SL(2,𝔽3)C3×#41semidirect
181. Q8C3×#41⋊C3semidirect
216. C3#85604direct (N × H)

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
C3×C4⋊(#41⋊C3)2
C3×C4×#41⋊C32
C3×#34833
#856043
C32×C4×Q84