Finite Group Databasebeta

← C3×#3359⋊C3 #11919598

Subgroups — C3×#3359⋊C3 #11919598

Order 1152.

310 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 70 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#3359C311yesmaximal, characteristic, complemented
5#3359⋊C3C313yesmaximal, direct product factor
6C22×C3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C32×#102—41nomaximal
10C3×#557C2×C311yescharacteristic, complemented
11C3×D4×Q8C2×C311yescharacteristic
12#557⋊C3C2×C313yescomplemented
13D4×SL(2,𝔽3)C2×C313yes—
14C3×#487—31no—
15C3×#475—31no—
16C3×#461—31no—
17C3×C8×Q8C2×C311yescharacteristic, complemented
18C3×C4×SD16—31no—
19C8×SL(2,𝔽3)C2×C313yescomplemented
20C2×C32×D4—41no—
21C2×C3×SL(2,𝔽3)—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C2×C32×C8—41no—
24C32×C4⋊C4—41no—
25#3359C3211yescharacteristic, complemented, Sylow
26C22×C3×Q8—21no—
27C22×SL(2,𝔽3)—23no—
28C2×C3×SD16—31no—
29C3×C4⋊Q8—31no—
30C3×C4⋊Q8—31no—
31C3×#125—61no—
32C3×C22⋊Q8—61no—
33C3×C4×Q8—21no—
34C3×C4×Q8—31no—
35C3×C4×Q8C22×C311yescharacteristic
36C3×C4×D4—31no—
37C4×SL(2,𝔽3)—23no—
38C4×SL(2,𝔽3)C22×C313yes—
39C3×#106—31no—
40C3×C4⋊C8—31no—
41C3×#103—32no—
42C3×#103—31no—
43C3×#102—43no—
44C3×#102A411yescharacteristic
45C3×C4×C8—31no—
46C23×C32—81no—
47C32×D4—42no—
48C32×D4—81no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
52C32×C8—81no—
53#557C2×C3211yescharacteristic, complemented
54D4×Q8C2×C3211yescharacteristic
55#487—31no—
56#475—31no—
57#461—31no—
58C8×Q8C2×C3211yescharacteristic, complemented
59C4×SD16—31no—
60C2×C3×Q8—31no—
61C2×C3×Q8—41no—
62C2×C3×Q8—61no—
63C2×C3×Q8—121no—
64C2×C3×Q8C3×D411yescharacteristic
65C2×C3×D4—43no—
66C2×C3×D4C2×A411yescharacteristic
67C22×C3×C4—61no—
68C2×SL(2,𝔽3)—43no—
69C2×SL(2,𝔽3)C3×D413yes—
70C3×SD16—62no—
71C2×C3×C8—31no—
72C2×C3×C8—43no—
73C2×C3×C8C2×A411yescharacteristic
74C3×C4⋊C4—31no—
75C3×C4⋊C4—31no—
76C3×C4⋊C4—43no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4C2×A411yescharacteristic
82C3×C22⋊C4—61no—
83C3×C42—31no—
84C3×C42—61no—
85C22×C32—41no—
86C22×C32—82no—
87C22×C32—161no—
88C22×C32—161no—
89C32×C4—41no—
90C32×C4—41no—
91C32×C4—161no—
92C22×Q8—21no—
93C2×SD16—31no—
94C4⋊Q8—31no—
95C4⋊Q8—31no—
96#125—61no—
97C22⋊Q8—61no—
98C4×Q8—21no—
99C4×Q8—31no—
100C4×Q8C22×C3211yescharacteristic
101C4×D4—31no—
102#106—31no—
103C4⋊C8—31no—
104#103—32no—
105#103—31no—
106#102C3×A411yescharacteristic
107C4×C8—31no—
108C23×C3—21no—
109C23×C3—83no—
110C3×Q8—31no—
111C3×Q8—61no—
112C3×Q8—61no—
113C3×Q8—121no—
114C3×Q8—121no—
115C3×Q8C3×SD1611yescharacteristic
116C3×D4—21no—
117C3×D4—46no—
118C3×D4—83no—
119C3×D4#41⋊C312yes—
120C2×C3×C4—21no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—43no—
124C2×C3×C4—61no—
125C2×C3×C4—61no—
126C2×C3×C4—83no—
127C2×C3×C4—121no—
128C2×C3×C4—121no—
129C2×C3×C4C22×A411yescharacteristic
130SL(2,𝔽3)C3×SD1613yescomplemented
131C3×C8—21no—
132C3×C8—61no—
133C3×C8—83no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—162no—
138C2×Q8—31no—
139C2×Q8—41no—
140C2×Q8—61no—
141C2×Q8—121no—
142C2×Q8C32×D411yescharacteristic
143C2×D4C2×C3×A411yescharacteristic
144C22×C4—61no—
145SD16—62no—
146C2×C8—31no—
147C2×C8C2×C3×A411yescharacteristic
148C4⋊C4—31no—
149C4⋊C4—31no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C4⋊C4—61no—
153C4⋊C4—61no—
154C4⋊C4C2×C3×A411yescharacteristic
155C22⋊C4—61no—
156C42—31no—
157C42—61no—
158C22×C3—22no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3—43no—
162C22×C3—86no—
163C22×C3—163no—
164C22×C3—163no—
165C22×C3D4×A411yescharacteristic
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—41no—
169C3×C4—43no—
170C3×C4—43no—
171C3×C4—61no—
172C3×C4—121no—
173C3×C4—121no—
174C3×C4—163no—
175C3×C4C2×#41⋊C311yescharacteristic
176C3×C42−1+4⋊C311yescharacteristic
177C32—41noSylow
178C23—21no—
179Q8—31no—
180Q8—61no—
181Q8—61no—
182Q8—121no—
183Q8—121no—
184Q8C32×SD1611yescharacteristic
185D4—21no—
186D4C3×#41⋊C312yes—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—121no—
193C2×C4—121no—
194C2×C4C22×C3×A411yescharacteristic
195C8—21no—
196C8—61no—
197C2×C3—42no—
198C2×C3—43no—
199C2×C3—43no—
200C2×C3—43no—
201C2×C3—166no—
202C2×C3#548⋊C311yescharacteristic
203C2×C3A4×SD1611yescharacteristic
204C2×C3#578⋊C311yescharacteristic
205C22—22no—
206C22—41no—
207C22—41no—
208C22C3×D4×A411yescharacteristic
209C4—31no—
210C4—31no—
211C4—41no—
212C4—61no—
213C4—121no—
214C4—121no—
215C4C2×C3×#41⋊C311yescharacteristic
216C4C3×2−1+4⋊C311yescharacteristic
217C3—43no—
218C3#3359⋊C311yescharacteristic, direct product factor
219C2—42no—
220C2C3×#548⋊C311yescharacteristic
221C2C3×A4×SD1611yescharacteristic
222C2C3×#578⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3D4×A4
commutator, cosocleC4×Q8C22×C32
FrattiniC2×C4C22×C3×A4
FittingC3×#3359C3
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC3×#102A4
hypocenterQ8C32×SD16
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
4. C3×#3359C3semidirect
5. #3359⋊C3C3direct (N × H)
10. C3×#557C2×C3semidirect
12. #557⋊C3C2×C3semidirect
17. C3×C8×Q8C2×C3semidirect
19. C8×SL(2,𝔽3)C2×C3semidirect
25. #3359C32semidirect
51. C3×SL(2,𝔽3)SD16semidirect
53. #557C2×C32semidirect
58. C8×Q8C2×C32semidirect
130. SL(2,𝔽3)C3×SD16semidirect
218. C3#3359⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C3×D4×SL(2,𝔽3)2
C3×C8×SL(2,𝔽3)2
C3×#33593
#3359⋊C33
C32×#1024