Finite Group Databasebeta

← C3×#3392⋊C3 #11919612

Subgroups — C3×#3392⋊C3 #11919612

Order 1152.

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

normal subgroups only

This group has 284 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#3392C311yesmaximal, characteristic, complemented
5#3392⋊C3C313yesmaximal, direct product factor
6C2×C3×#41⋊C3—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C32×#106—41nomaximal
10C3×#557C2×C311yescharacteristic
11C3×#550C2×C311yescharacteristic
12#557⋊C3C2×C313yescomplemented
13C4⋊(#41⋊C3)C2×C313yes—
14C3×#492—31no—
15C3×#461—31no—
16C3×#460—31no—
17C3×C8×Q8C2×C311yescharacteristic
18C3×C4×SD16—31no—
19C8×SL(2,𝔽3)C2×C313yescomplemented
20C3×#41⋊C3—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C2×C32×C8—41no—
23C32×C4⋊C4—41no—
24C32×C4⋊C4—41no—
25#3392C3211yescharacteristic, complemented, Sylow
26C2×C3×#41—21no—
27C2×#41⋊C3—23no—
28C2×C3×SD16—32no—
29C2×C3×SD16—31no—
30C3×C4⋊Q8—31no—
31C3×#127—31no—
32C3×#125—61no—
33C3×#121—61no—
34C3×C4×Q8—21no—
35C3×C4×Q8—31no—
36C3×C4×Q8C22×C311yescharacteristic
37C3×C4×D4—31no—
38C4×SL(2,𝔽3)—23no—
39C4×SL(2,𝔽3)C22×C313yes—
40C3×#106—43no—
41C3×#106A411yescharacteristic
42C3×C4⋊C8—31no—
43C3×#103—31no—
44C3×#102—31no—
45C3×C4×C8—31no—
46C2×C32×C4—41no—
47C2×C32×C4—81no—
48C2×C32×C4—81no—
49C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
50C32×C8—42no—
51#557C2×C3211yescharacteristic
52#550C2×C3211yescharacteristic
53#492—31no—
54#461—31no—
55#460—31no—
56C8×Q8C2×C3211yescharacteristic
57C4×SD16—31no—
58C3×#41—41no—
59C3×#41—121no—
60C2×C3×Q8—31no—
61C2×C3×Q8C3×D411yescharacteristic
62C2×C3×D4—31no—
63C2×C3×D4—61no—
64C2×C3×D4—61no—
65C22×C3×C4—61no—
66#41⋊C3—43no—
67C2×SL(2,𝔽3)C3×D413yes—
68C3×SD16—62no—
69C3×SD16—122no—
70C2×C3×C8—31no—
71C2×C3×C8—43no—
72C2×C3×C8C2×A411yescharacteristic
73C3×C4⋊C4—31no—
74C3×C4⋊C4—43no—
75C3×C4⋊C4—43no—
76C3×C4⋊C4—61no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4C2×A411yescharacteristic
80C3×C4⋊C4C2×A411yescharacteristic
81C3×C22⋊C4—61no—
82C3×C42—31no—
83C3×C42—61no—
84C22×C32—41no—
85C32×C4—41no—
86C32×C4—41no—
87C32×C4—161no—
88C32×C4—161no—
89C2×#41—21no—
90C2×SD16—32no—
91C2×SD16—31no—
92C4⋊Q8—31no—
93#127—31no—
94#125—61no—
95#121—61no—
96C4×Q8—21no—
97C4×Q8—31no—
98C4×Q8C22×C3211yescharacteristic
99C4×D4—31no—
100#106C3×A411yescharacteristic
101C4⋊C8—31no—
102#103—31no—
103#102—31no—
104C4×C8—31no—
105C23×C3—61no—
106C3×Q8—31no—
107C3×Q8—61no—
108C3×Q8—61no—
109C3×Q8C3×SD1611yescharacteristic
110C3×D4—61no—
111C3×D4—61no—
112C3×D4—121no—
113C3×D4—121no—
114C3×D4—241no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—43no—
120C2×C3×C4—61no—
121C2×C3×C4—83no—
122C2×C3×C4—83no—
123C2×C3×C4—121no—
124C2×C3×C4—121no—
125C2×C3×C4C22×A411yescharacteristic
126SL(2,𝔽3)C3×SD1613yescomplemented
127C3×C8—46no—
128C3×C8—61no—
129C3×C8#41⋊C312yescomplemented
130C2×C32—41no—
131C2×C32—41no—
132C2×C32—41no—
133#41—41no—
134#41—121no—
135C2×Q8—31no—
136C2×Q8C32×D411yescharacteristic
137C2×D4—31no—
138C2×D4—61no—
139C2×D4—61no—
140C22×C4—61no—
141SD16—62no—
142SD16—122no—
143C2×C8—31no—
144C2×C8C2×C3×A411yescharacteristic
145C4⋊C4—31no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4C2×C3×A411yescharacteristic
150C4⋊C4C2×C3×A411yescharacteristic
151C22⋊C4—61no—
152C42—31no—
153C42—61no—
154C22×C3—43no—
155C22×C3—121no—
156C22×C3—121no—
157C22×C3—121no—
158C22×C3D4×A411yescharacteristic
159C3×C4—31no—
160C3×C4—31no—
161C3×C4—41no—
162C3×C4—41no—
163C3×C4—43no—
164C3×C4—43no—
165C3×C4—61no—
166C3×C4—121no—
167C3×C4—163no—
168C3×C4—163no—
169C3×C4C2×#41⋊C311yescharacteristic
170C3×C4#99411yescharacteristic
171C32—41noSylow
172C23—61no—
173Q8—31no—
174Q8—61no—
175Q8—61no—
176Q8C32×SD1611yescharacteristic
177D4—61no—
178D4—61no—
179D4—121no—
180D4—121no—
181D4—241no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4C22×C3×A411yescharacteristic
190C8—61no—
191C8C3×#41⋊C312yescomplemented
192C2×C3—43no—
193C2×C3—43no—
194C2×C3—43no—
195C2×C3—241no—
196C2×C3C4⋊(#41⋊C3)11yescharacteristic
197C2×C3A4×SD1611yescharacteristic
198C2×C3#577⋊C311yescharacteristic
199C22—121no—
200C22—121no—
201C22—121no—
202C22C3×D4×A411yescharacteristic
203C4—31no—
204C4—31no—
205C4—41no—
206C4—41no—
207C4—61no—
208C4—121no—
209C4C2×C3×#41⋊C311yescharacteristic
210C4C3×#99411yescharacteristic
211C3—43no—
212C3#3392⋊C311yescharacteristic, direct product factor
213C2—241no—
214C2C3×C4⋊(#41⋊C3)11yescharacteristic
215C2C3×A4×SD1611yescharacteristic
216C2C3×#577⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3D4×A4
commutator, cosocleC4×Q8C22×C32
FrattiniC2×C4C22×C3×A4
FittingC3×#3392C3
prime radicalC3×C4⋊(#41⋊C3)C2
hypercenterC3×#106A4
hypocenterQ8C32×SD16
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

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×#3392C3semidirect
5. #3392⋊C3C3direct (N × H)
12. #557⋊C3C2×C3semidirect
19. C8×SL(2,𝔽3)C2×C3semidirect
25. #3392C32semidirect
49. C3×SL(2,𝔽3)SD16semidirect
126. SL(2,𝔽3)C3×SD16semidirect
129. C3×C8#41⋊C3semidirect
191. C8C3×#41⋊C3semidirect
212. C3#3392⋊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×C4⋊(#41⋊C3)2
C3×C8×SL(2,𝔽3)2
C3×#33923
#3392⋊C33
C32×#1064