Finite Group Databasebeta

← #13164550

Subgroups — #13164550

Order 1728.

366 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 26 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#11722587C211yesmaximal, characteristic, complemented
2C3×#5931—33nomaximal
3C3×#5227—31nomaximal
4C2×#6258—41nomaximal
5#88611—42nomaximal
6C22×C32×Q8D311yescharacteristic, complemented
7C3×C22⋊SL(2,𝔽3)D313yescomplemented
8C2×C3×#215—32no—
9C3×#930—31no—
10C3×#835—32no—
11C3×#830—31no—
12C2×C22⋊H(3,𝔽3)—41no—
13#6258—42no—
14Q8⋊H(3,𝔽3)—42no—
15Q8⋊H(3,𝔽3)—81no—
16#5931—33no—
17C3×#448—91no—
18#5227—31no—
19C2×C3×S4—123no—
20C2×C32×Q8—32no—
21C2×C32×Q8—61no—
22C22×C32×C4—31no—
23C2×C3×#67—61no—
24C2×C3×D12—31no—
25C3×GL(2,𝔽3)—126no—
26C3×#215—62no—
27C3×#215—62no—
28C3×(C2×D3)⋊C4—61no—
29C3×C4⋊Dic3—31no—
30C2×C3×C3⋊C8—32no—
31C2×#272—161no—
32C22⋊H(3,𝔽3)—41no—
33C22×C3×Q8—31no—
34C22×C3×Q8C3⋊D311yescharacteristic
35C22⋊SL(2,𝔽3)—33no—
36C2×C3×SD16—92no—
37C3×#121—91no—
38C2×#215—32no—
39#930—31no—
40C3×#103—92no—
41C3×C22⋊C8—91no—
42#835—32no—
43#830—31no—
44C23×C32S411yescharacteristic
45C22×C3×D3—61no—
46C2×C3×A4—43no—
47C3×S4—126no—
48C32×Q8—21no—
49C32×Q8—32no—
50C32×Q8—61no—
51C32×Q8S412yescomplemented
52C2×C32×C4—31no—
53C2×C32×C4—31no—
54C2×C32×C4—62no—
55C3×#67—121no—
56C3×#67—121no—
57C2×C3×Dic3—61no—
58C3×D12—62no—
59C3×SL(2,𝔽3)—46no—
60C3×SL(2,𝔽3)—83no—
61C3×C3⋊C8—62no—
62#448—91noSylow
63C2×H(3,𝔽3)—161no—
64#272—162no—
65C2×S4—123no—
66C2×C3×Q8—32no—
67C2×C3×Q8—32no—
68C2×C3×Q8—62no—
69C2×C3×Q8—61no—
70C2×C3×Q8—61no—
71C2×C3×Q8—62no—
72C2×C3×D4—91no—
73C2×C3×D4—181no—
74C22×C3×C4—31no—
75C22×C3×C4—31no—
76C22×C3×C4—61no—
77C2×#67—61no—
78C2×D12—31no—
79GL(2,𝔽3)—126no—
80C3×SD16—182no—
81C3×SD16—182no—
82C2×C3×C8—92no—
83C3×C4⋊C4—91no—
84C3×C22⋊C4—181no—
85#215—62no—
86#215—62no—
87(C2×D3)⋊C4—61no—
88C4⋊Dic3—31no—
89C2×C3⋊C8—32no—
90C22×C32—31no—
91C22×C32—31no—
92C22×C32GL(2,𝔽3)11yescharacteristic, complemented
93C2×C3×D3—121no—
94C2×C3×D3—121no—
95C2×C3×D3—121no—
96C2×C3×D3—483no—
97C3×A4—43no—
98C32×C4—32no—
99C32×C4—61no—
100C3×Dic3—121no—
101C22×Q8#27211yescharacteristic, complemented
102C2×SD16—92no—
103#121—91no—
104#103—92no—
105C22⋊C8—91no—
106H(3,𝔽3)—161noSylow
107C23×C3—31no—
108C23×C3—181no—
109C23×C3C3⋊S411yescharacteristic
110C22×D3—61no—
111C2×A4—123no—
112S4—126no—
113C3×Q8—21no—
114C3×Q8—32no—
115C3×Q8—32no—
116C3×Q8—32no—
117C3×Q8—61no—
118C3×Q8—62no—
119C3×Q8—61no—
120C3×Q8—62no—
121C3×Q8—61no—
122C3×Q8C3⋊S412yes—
123C3×D4—182no—
124C3×D4—361no—
125C3×D4—361no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—62no—
136C2×C3×C4—181no—
137#67—121no—
138#67—121no—
139C2×Dic3—61no—
140D12—62no—
141SL(2,𝔽3)—126no—
142SL(2,𝔽3)—243no—
143C3×C8—182no—
144C3⋊C8—62no—
145C2×C32—31no—
146C2×C32—31no—
147C2×C32—163no—
148C2×C32#102011yescharacteristic
149C3×D3—241no—
150C3×D3—486no—
151C2×Q8—32no—
152C2×Q8—61no—
153C2×D4—91no—
154C2×D4—181no—
155C22×C4—31no—
156SD16—182no—
157SD16—182no—
158C2×C8—92no—
159C4⋊C4—91no—
160C22⋊C4—181no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—31no—
165C22×C3—31no—
166C22×C3—61no—
167C22×C3—61no—
168C22×C3—361no—
169C22×C3—361no—
170C22×C3—361no—
171C22×C3C3⋊GL(2,𝔽3)11yescharacteristic
172C2×D3—121no—
173C2×D3—121no—
174C2×D3—121no—
175C2×D3—483no—
176A4—123no—
177C3×C4—32no—
178C3×C4—32no—
179C3×C4—62no—
180C3×C4—61no—
181C3×C4—62no—
182C3×C4—61no—
183C3×C4—361no—
184Dic3—121no—
185C32—163no—
186C32#593111yescharacteristic, complemented
187C23—181no—
188C23#625811yescharacteristic
189Q8—21no—
190Q8—32no—
191Q8—61no—
192Q8#625812yescomplemented
193D4—182no—
194D4—361no—
195D4—361no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—62no—
199C2×C4—181no—
200C8—182no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—483no—
209C2×C3—721no—
210C2×C3C3⋊#102011yescharacteristic
211D3—241no—
212D3—486no—
213C22—31no—
214C22—31no—
215C22—361no—
216C22—361no—
217C22—361no—
218C22#8861111yescharacteristic, complemented
219C4—32no—
220C4—61no—
221C4—361no—
222C3—31no—
223C3—483no—
224C3C3⋊#593111yescharacteristic
225C2—31no—
226C2—31no—
227C2—721no—
228C2#1172405511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3C3⋊#1020
commutator, cosocle, hypocenter#11722587C2
FittingC22×C32×Q8D3
socleC23×C3C3⋊S4
LCSG ▷ #11722587
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. #11722587C2semidirect
6. C22×C32×Q8D3semidirect
7. C3×C22⋊SL(2,𝔽3)D3semidirect
51. C32×Q8S4semidirect
92. C22×C32GL(2,𝔽3)semidirect
101. C22×Q8#272semidirect
186. C32#5931semidirect
192. Q8#6258semidirect
218. C22#88611semidirect

Maximal subgroups

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

subgroupindex
#117225872
C3×#59313
C3×#52273
C2×#62584
#886114