Finite Group Databasebeta

← #10592415⋊C3 #13141527

Subgroups — #10592415⋊C3 #13141527

Order 1728.

418 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#3746⋊C3C211yesmaximal, characteristic
2C4×#634⋊C3C212yesmaximal
3H(3,𝔽3)×#97C211yesmaximal, characteristic
4#63644⋊C3C211yesmaximal, characteristic
5#63643⋊C3C212yesmaximal
6#10592415C311yesmaximal, characteristic, complemented
7C3×#4716—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3750⋊C3—22no—
14C4×(C3⋊Dic3)⋊C3—22no—
15#3746⋊C3C414yescomplemented
16C2×#3746C2×C311yescharacteristic
17C4×#634C2×C312yes—
18C2×C3×Dic12—31no—
19C3×C4×Dic6—32no—
20C32×#97—21no—
21C32×#97D311yescharacteristic
22C32×#97C2×C311yescharacteristic
23#63644C2×C311yescharacteristic
24#63643C2×C312yes—
25C3×#817—31no—
26C3×#816—32no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3C2×C414yes—
33C8×H(3,𝔽3)—21no—
34C8×H(3,𝔽3)C2×C412yes—
35C3×#441—91no—
36#4716—31no—
37#4716—91no—
38C2×#634C22×C312yes—
39C2×C3×Dic6—32no—
40C2×C32×C8—21no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C2×D311yescharacteristic
44C2×C32×C8C22×C311yescharacteristic
45C2×C32×C8C22×C311yescharacteristic
46C32×C42—21no—
47C32×C42C2×D311yescharacteristic
48C32×C42C22×C311yescharacteristic
49C4⋊(C3⋊Dic3)C22×C312yes—
50#3750—22no—
51C4×C3⋊Dic3—22no—
52#3746C3×C414yescomplemented
53C3×C4⋊Dic3—32no—
54C3×Dic3⋊C4—62no—
55C3×C4×Dic3—62no—
56C3×Dic12—34no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)C22×C411yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—42no—
63C2×C3×Q16—91no—
64C3×C4×Q8—92no—
65C2×Dic12—31no—
66C2×Dic12—91no—
67C4×Dic6—32no—
68C4×Dic6—92no—
69C3×#106—91no—
70C3×#103—92no—
71C3×#97—31no—
72C3×#97—31no—
73C3×#97—61no—
74C3×#97C3×D311yescharacteristic
75#817—31no—
76#817—91no—
77#816—32no—
78#816—92no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4D1212yes—
82C2×C32×C4C3×D412yes—
83C2×C32×C4C22×D311yescharacteristic
84C2×C32×C4C23×C311yescharacteristic
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87#634—22no—
88#634C2×C3×C414yes—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C3×Dic6—34no—
92C3×Dic6—62no—
93C32×C8—22no—
94C32×C8—21no—
95C32×C8—21no—
96C32×C8—22no—
97C32×C8C4×D312yes—
98C32×C8C2×C3×C412yes—
99#441—91noSylow
100C2×H(3,𝔽3)C4×D411yescharacteristic
101C2×H(3,𝔽3)#13712yes—
102C2×C3×Q8—92no—
103C2×Dic6—32no—
104C2×Dic6—92no—
105C3×Q16—94no—
106C2×C3×C8—31no—
107C2×C3×C8—31no—
108C2×C3×C8—31no—
109C2×C3×C8—31no—
110C2×C3×C8—61no—
111C2×C3×C8—61no—
112C2×C3×C8C2×C3×D311yescharacteristic
113C2×C3×C8C2×C3×D311yescharacteristic
114C3×C4⋊C4—92no—
115C3×C4⋊C4—182no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—61no—
119C3×C42—182no—
120C3×C42C2×C3×D311yescharacteristic
121C4⋊Dic3—32no—
122C4⋊Dic3—92no—
123Dic3⋊C4—62no—
124Dic3⋊C4—182no—
125C4×Dic3—62no—
126C4×Dic3—182no—
127Dic12—34no—
128Dic12—94no—
129C22×C32—21no—
130C22×C32C2×D1211yescharacteristic
131C22×C32C2×C3×D411yescharacteristic
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—42no—
137C32×C4C2×C4×D311yescharacteristic
138C32×C4#23511yescharacteristic
139C32×C4C22×C3×C411yescharacteristic
140C32×C4C3×#4111yescharacteristic
141C3⋊Dic3—24no—
142C3⋊Dic3—42no—
143C3×Dic3—64no—
144C3×Dic3—122no—
145C2×Q16—91no—
146C4×Q8—92no—
147#106—91no—
148#103—92no—
149#97(C3⋊D3)⋊C311yescharacteristic
150H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
151C3×Q8—94no—
152C3×Q8—182no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—62no—
158C2×C3×C4—61no—
159C2×C3×C4—182no—
160C2×C3×C4—182no—
161C2×C3×C4C3×D1212yes—
162C2×C3×C4C22×C3×D311yescharacteristic
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—182no—
166C2×Dic3—182no—
167Dic6—34no—
168Dic6—62no—
169Dic6—94no—
170Dic6—182no—
171C3×C8—21no—
172C3×C8—32no—
173C3×C8—32no—
174C3×C8—61no—
175C3×C8—61no—
176C3×C8—62no—
177C3×C8—62no—
178C3×C8C3×C4×D312yes—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32C4×D1211yescharacteristic
182C2×C32#90912yes—
183C2×C32C3×C4×D411yescharacteristic
184C2×C32C3×#13712yes—
185C2×Q8—92no—
186Q16—94no—
187C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
188C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
189C4⋊C4—92no—
190C4⋊C4—182no—
191C42—182no—
192C42C2×(C3⋊D3)⋊C311yescharacteristic
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—61no—
196C22×C3C2×C3×D1211yescharacteristic
197C3×C4—22no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—62no—
203C3×C4—62no—
204C3×C4—61no—
205C3×C4—61no—
206C3×C4—122no—
207C3×C4—184no—
208C3×C4—362no—
209C3×C4C2×C3×C4×D311yescharacteristic
210C3×C4C3×#23511yescharacteristic
211Dic3—64no—
212Dic3—122no—
213Dic3—184no—
214Dic3—362no—
215C32—21no—
216C32#471611yescharacteristic, complemented
217C32C3×#44111yescharacteristic, complemented
218Q8—94no—
219Q8—182no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
223C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
224C8—21no—
225C8C4×(C3⋊D3)⋊C312yes—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3C3×C4×D1211yescharacteristic
233C2×C3C3×#90912yes—
234C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
235C4—22no—
236C4—184no—
237C4—362no—
238C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
239C4#3828⋊C311yescharacteristic
240C3—31no—
241C3—31no—
242C3—61no—
243C3C3×#471611yescharacteristic
244C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
245C2#64118⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#441
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
6. #10592415C3semidirect
15. #3746⋊C3C4semidirect
52. #3746C3×C4semidirect
150. H(3,𝔽3)#441semidirect
216. C32#4716semidirect
217. C32C3×#441semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
C4×#634⋊C32
H(3,𝔽3)×#972
#63644⋊C32
#63643⋊C32
#105924153
C3×#47163