Finite Group Databasebeta

← C22×(C3⋊#226)⋊C3 #13164414

Subgroups — C22×(C3⋊#226)⋊C3 #13164414

Order 1728.

630 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊#226)⋊C3C216yesmaximal, direct product factor
3C22×C3⋊#226C311yesmaximal, characteristic, complemented
4C22×C3×#216—31nomaximal
5C22×(C3⋊Dic3)⋊C3—41nomaximal
6C2×Q8⋊H(3,𝔽3)C2213yes—
7(C3⋊#226)⋊C3C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic
9C22×C3×Dic6—31no—
10C22×C3×SL(2,𝔽3)—21no—
11C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
12C2×C3⋊#226C2×C316yescomplemented
13C2×C3×#216—36no—
14C2×C3×#216—36no—
15C22×C3×C3⋊C8—31no—
16C23×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—46no—
18Q8⋊H(3,𝔽3)C2311yescharacteristic
19C22×#226—91no—
20C22×C3×Q16—91no—
21C22×#216—31no—
22C2×C32×Q8—33no—
23C2×C32×Q8C2×D313yes—
24C22×C32×C4—31no—
25C22×C3⋊Dic3—41no—
26C22×C3×Dic3—61no—
27C2×C3×Dic6—36no—
28C2×C3×Dic6—63no—
29C2×C3×SL(2,𝔽3)—23no—
30C2×C3×SL(2,𝔽3)C22×C313yes—
31C3⋊#226C22×C314yescomplemented
32C3×#216—312no—
33C3×#216—34no—
34C2×C3×C3⋊C8—36no—
35C22×H(3,𝔽3)—44no—
36C22×H(3,𝔽3)—43no—
37(C3⋊Dic3)⋊C3—44no—
38C22×C3×Q8—31no—
39C22×C3×Q8—91no—
40C22×C3×Q8C3×D311yescharacteristic
41C22×Dic6—31no—
42C22×SL(2,𝔽3)—31no—
43C22×SL(2,𝔽3)—61no—
44C2×#226—96no—
45C2×C3×Q16—96no—
46C2×C3×Q16—96no—
47C22×C3×C8—91no—
48C2×#216—36no—
49C2×#216—36no—
50C22×C3⋊C8—31no—
51C23×C32—41no—
52C23×C32—81no—
53C23×C32S411yescharacteristic
54C32×Q8—33no—
55C32×Q8—61no—
56C32×Q8C22×D311yescharacteristic
57C2×C32×C4—33no—
58C2×C32×C4—33no—
59C2×C3⋊Dic3—46no—
60C2×C3×Dic3—66no—
61C3×Dic6—34no—
62C3×Dic6—66no—
63C3×SL(2,𝔽3)—21no—
64C3×SL(2,𝔽3)C23×C311yescharacteristic
65C3×C3⋊C8—34no—
66C22×Q16—91noSylow
67C2×H(3,𝔽3)—46no—
68C2×H(3,𝔽3)—41no—
69C2×C3×Q8—33no—
70C2×C3×Q8—33no—
71C2×C3×Q8—33no—
72C2×C3×Q8—63no—
73C2×C3×Q8—96no—
74C2×C3×Q8—183no—
75C2×C3×Q8C2×C3×D313yes—
76C22×C3×C4—31no—
77C22×C3×C4—31no—
78C22×C3×C4—61no—
79C22×C3×C4—181no—
80C22×Dic3—61no—
81C22×Dic3—361no—
82C2×Dic6—36no—
83C2×Dic6—63no—
84C2×SL(2,𝔽3)—33no—
85C2×SL(2,𝔽3)—63no—
86#226—94no—
87C3×Q16—912no—
88C3×Q16—94no—
89C2×C3×C8—96no—
90#216—312no—
91#216—34no—
92C2×C3⋊C8—36no—
93C22×C32—43no—
94C22×C32—44no—
95C22×C32—83no—
96C22×C32—84no—
97C22×C32#22614yescomplemented
98C22×C32C2×S413yes—
99C32×C4—31no—
100C32×C4—33no—
101C3⋊Dic3—44no—
102C3×Dic3—64no—
103C22×Q8—91no—
104C22×Q8(C3⋊D3)⋊C311yescharacteristic
105C2×Q16—96no—
106C2×Q16—96no—
107C22×C8—91no—
108H(3,𝔽3)—41noSylow
109C23×C3—31no—
110C23×C3—121no—
111C23×C3—241no—
112C23×C3C3×S411yescharacteristic
113C3×Q8—33no—
114C3×Q8—33no—
115C3×Q8—31no—
116C3×Q8—63no—
117C3×Q8—61no—
118C3×Q8—63no—
119C3×Q8—94no—
120C3×Q8—186no—
121C3×Q8C22×C3×D311yescharacteristic
122C2×C3×C4—33no—
123C2×C3×C4—33no—
124C2×C3×C4—33no—
125C2×C3×C4—33no—
126C2×C3×C4—63no—
127C2×C3×C4—63no—
128C2×C3×C4—186no—
129C2×Dic3—66no—
130C2×Dic3—366no—
131Dic6—34no—
132Dic6—66no—
133SL(2,𝔽3)—31no—
134SL(2,𝔽3)—61no—
135C3×C8—94no—
136C3⋊C8—34no—
137C2×C32—41no—
138C2×C32—46no—
139C2×C32—81no—
140C2×C32—86no—
141C2×C32C2×#22616yescomplemented
142C2×C32C22×S411yescharacteristic
143C2×Q8—33no—
144C2×Q8—96no—
145C2×Q8—183no—
146C2×Q8C2×(C3⋊D3)⋊C313yes—
147C22×C4—31no—
148C22×C4—181no—
149Q16—912no—
150Q16—94no—
151C2×C8—96no—
152C22×C3—34no—
153C22×C3—33no—
154C22×C3—124no—
155C22×C3—123no—
156C22×C3—244no—
157C22×C3—243no—
158C22×C3C3×#22614yes—
159C22×C3C2×C3×S413yes—
160C3×C4—31no—
161C3×C4—33no—
162C3×C4—31no—
163C3×C4—33no—
164C3×C4—61no—
165C3×C4—63no—
166C3×C4—184no—
167Dic3—64no—
168Dic3—364no—
169C32—41no—
170C32—81no—
171C32C22×#22611yescharacteristic, complemented
172C23(C3⋊S4)⋊C311yescharacteristic
173Q8—33no—
174Q8—61no—
175Q8—94no—
176Q8—186no—
177Q8C22×(C3⋊D3)⋊C311yescharacteristic
178C2×C4—33no—
179C2×C4—33no—
180C2×C4—186no—
181C8—94no—
182C2×C3—31no—
183C2×C3—36no—
184C2×C3—121no—
185C2×C3—126no—
186C2×C3—241no—
187C2×C3—246no—
188C2×C3C2×C3×#22616yes—
189C2×C3C22×C3×S411yescharacteristic
190C22(C3⋊#226)⋊C314yesdirect product factor
191C22C2×(C3⋊S4)⋊C313yes—
192C4—31no—
193C4—33no—
194C4—184no—
195C3—31no—
196C3—121no—
197C3—241no—
198C3C22×C3×#22611yescharacteristic
199C2C2×(C3⋊#226)⋊C316yesdirect product factor
200C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×S4
prime radicalC22×Q8⋊H(3,𝔽3)C2
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊#226)⋊C3C2direct (N × H)
3. C22×C3⋊#226C3semidirect
7. (C3⋊#226)⋊C3C22direct (N × H)
12. C2×C3⋊#226C2×C3semidirect
31. C3⋊#226C22×C3semidirect
97. C22×C32#226semidirect
141. C2×C32C2×#226semidirect
171. C32C22×#226semidirect
190. C22(C3⋊#226)⋊C3direct (N × H)
199. C2C2×(C3⋊#226)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×(C3⋊#226)⋊C32
C22×C3⋊#2263
C22×C3×#2163
C22×(C3⋊Dic3)⋊C34