Finite Group Databasebeta

← #10592577⋊C3 #13142218

Subgroups — #10592577⋊C3 #13142218

Order 1728.

337 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#64099⋊C3C212yesmaximal
2H(3,𝔽3)×#107C211yesmaximal, characteristic
3#63645⋊C3C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#63635⋊C3C212yesmaximal
6#10592577C311yesmaximal, characteristic, complemented
7C3×#4878—31nomaximal
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4⋊C4×H(3,𝔽3)C2212yes—
10C4⋊((C3⋊Dic3)⋊C3)—22no—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#64099C2×C312yes—
16C3×#890—32no—
17C32×#107—21no—
18C32×#107D311yescharacteristic
19C32×#107C2×C311yescharacteristic
20#63645C2×C311yescharacteristic
21C8×C3⋊Dic3C2×C311yescharacteristic
22#63635C2×C312yes—
23C3×#818—31no—
24C3×C8×Dic3—31no—
25C3×#808—32no—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×(C3⋊Dic3)⋊C3—22no—
29C2×(C3⋊Dic3)⋊C3D411yescharacteristic
30C2×(C3⋊Dic3)⋊C3D411yescharacteristic
31C8×H(3,𝔽3)Q812yes—
32(C3⋊(C3⋊C8))⋊C3Q812yes—
33C3×#499—91no—
34#4878—31no—
35#4878—91no—
36C2×C32×C8—21no—
37C2×C32×C8C2×D311yescharacteristic
38C2×C32×C8C22×C311yescharacteristic
39C32×C4⋊C4—22no—
40C32×C4⋊C4C2×D312yes—
41C32×C4⋊C4C22×C312yes—
42C4⋊(C3⋊Dic3)—22no—
43C4⋊(C3⋊Dic3)C22×C312yes—
44#3750—22no—
45C4×C3⋊Dic3C22×C311yescharacteristic
46C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—62no—
49C3×Dic3⋊C4—62no—
50C3×C4×Dic3—31no—
51C2×C3×C3⋊C8—31no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—42no—
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56(C3⋊Dic3)⋊C3—21no—
57(C3⋊Dic3)⋊C3—21no—
58(C3⋊Dic3)⋊C3—42no—
59C3×#125—92no—
60#890—32no—
61#890—92no—
62C3×#107—31no—
63C3×#107—31no—
64C3×#107—61no—
65C3×#107—91no—
66C3×#107C3×D311yescharacteristic
67C3×#106—92no—
68C3×C4×C8—91no—
69#818—31no—
70#818—91no—
71C8×Dic3—31no—
72C8×Dic3—91no—
73#808—32no—
74#808—92no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4—42no—
79C2×C32×C4C22×D311yescharacteristic
80C2×C32×C4C23×C311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3C3×D411yescharacteristic
83C2×C3⋊Dic3C3×D411yescharacteristic
84C2×C3×Dic3—31no—
85C2×C3×Dic3—31no—
86C2×C3×Dic3—62no—
87C32×C8—22no—
88C32×C8Dic612yes—
89C32×C8C3×Q812yes—
90C3⋊(C3⋊C8)C3×Q812yes—
91C3×C3⋊C8—32no—
92#499—91noSylow
93C2×H(3,𝔽3)C4⋊Q811yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×H(3,𝔽3)#13511yescharacteristic
96C2×C3×C8—31no—
97C2×C3×C8—31no—
98C2×C3×C8—61no—
99C2×C3×C8—91no—
100C2×C3×C8C2×C3×D311yescharacteristic
101C3×C4⋊C4—32no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4—62no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—182no—
106C3×C4⋊C4—182no—
107C3×C4⋊C4C2×C3×D312yes—
108C3×C42—91no—
109C4⋊Dic3—32no—
110C4⋊Dic3—62no—
111C4⋊Dic3—92no—
112C4⋊Dic3—182no—
113Dic3⋊C4—62no—
114Dic3⋊C4—182no—
115C4×Dic3—31no—
116C4×Dic3—91no—
117C2×C3⋊C8—31no—
118C2×C3⋊C8—91no—
119C22×C32—21no—
120C22×C32D3×D411yescharacteristic
121C22×C32C2×C3×D411yescharacteristic
122C32×C4—21no—
123C32×C4—21no—
124C32×C4—42no—
125C32×C4—42no—
126C32×C4—82no—
127C32×C4C2×Dic611yescharacteristic
128C32×C4D3×Q811yescharacteristic
129C32×C4C2×C3×Q811yescharacteristic
130C32×C4C2×C3×Q811yescharacteristic
131C3⋊Dic3—21no—
132C3⋊Dic3—21no—
133C3⋊Dic3—42no—
134C3×Dic3—61no—
135C3×Dic3—61no—
136C3×Dic3—122no—
137#125—92no—
138#107—91no—
139#107(C3⋊D3)⋊C311yescharacteristic
140#106—92no—
141C4×C8—91no—
142H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
143C2×C3×C4—22no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—62no—
147C2×C3×C4—62no—
148C2×C3×C4—61no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—122no—
152C2×C3×C4—182no—
153C2×C3×C4C22×C3×D311yescharacteristic
154C2×Dic3—31no—
155C2×Dic3—31no—
156C2×Dic3—62no—
157C2×Dic3—91no—
158C2×Dic3—91no—
159C2×Dic3—182no—
160C3×C8—32no—
161C3×C8—32no—
162C3×C8—62no—
163C3×C8—92no—
164C3×C8C3×Dic612yes—
165C3⋊C8—32no—
166C3⋊C8—92no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32#88811yescharacteristic
171C2×C32#91211yescharacteristic
172C2×C32#91911yescharacteristic
173C2×C32C3×C4⋊Q811yescharacteristic
174C2×C32C3×#13511yescharacteristic
175C2×C32C3×#13511yescharacteristic
176C2×C8—91no—
177C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
178C4⋊C4—92no—
179C4⋊C4—182no—
180C4⋊C4—182no—
181C4⋊C4C2×(C3⋊D3)⋊C312yes—
182C42—91no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—61no—
186C22×C3C3×D3×D411yescharacteristic
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—31no—
191C3×C4—42no—
192C3×C4—61no—
193C3×C4—61no—
194C3×C4—122no—
195C3×C4—122no—
196C3×C4—181no—
197C3×C4—181no—
198C3×C4—242no—
199C3×C4—362no—
200C3×C4C2×C3×Dic611yescharacteristic
201C3×C4C3×D3×Q811yescharacteristic
202Dic3—61no—
203Dic3—61no—
204Dic3—122no—
205Dic3—181no—
206Dic3—181no—
207Dic3—362no—
208C32—21no—
209C32#487811yescharacteristic, complemented
210C32C3×#49911yescharacteristic, complemented
211C2×C4—22no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—182no—
215C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
216C8—92no—
217C8#634⋊C312yes—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—61no—
225C2×C3—61no—
226C2×C3—61no—
227C2×C3C3×#88811yescharacteristic
228C2×C3C3×#91211yescharacteristic
229C2×C3C3×#91911yescharacteristic
230C22D4×(C3⋊D3)⋊C311yescharacteristic
231C4—42no—
232C4—181no—
233C4—181no—
234C4—362no—
235C4C2×#634⋊C311yescharacteristic
236C4Q8×(C3⋊D3)⋊C311yescharacteristic
237C3—31no—
238C3—31no—
239C3—61no—
240C3C3×#487811yescharacteristic
241C2#64097⋊C311yescharacteristic
242C2#64121⋊C311yescharacteristic
243C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
6. #10592577C3semidirect
142. H(3,𝔽3)#499semidirect
209. C32#4878semidirect
210. C32C3×#499semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
H(3,𝔽3)×#1072
#63645⋊C32
C8×(C3⋊Dic3)⋊C32
#63635⋊C32
#105925773
C3×#48783