Finite Group Databasebeta

← #13135101

Subgroups — #13135101

Order 1728.

293 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 53 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×#88529C211yesmaximal, characteristic
2#63646⋊C3C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589127—31nomaximal
5#10589126—31nomaximal
6C2×C4⋊#272—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88529—21no—
9#88529C412yes—
10#3748⋊C3—21no—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×C3×D24—31no—
13C2×C3×D24—31no—
14C2×C3×D24—61no—
15#63646D311yescharacteristic, complemented
16#63639D311yescharacteristic, complemented
17C3×#819—31no—
18C3×#812—31no—
19C22×#272—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C4⋊#272—22no—
22C4⋊#272—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26#4563—91no—
27#4516—91no—
28#4495—91no—
29C2×C3×D12—62no—
30C2×C3×D12—61no—
31C2×C3×D12—61no—
32C2×C32×C8—21no—
33C2×C32×C8C2×D311yescharacteristic
34C2×C32×C8C2×D311yescharacteristic
35#3748—21no—
36C3×#208—61no—
37C3×D24—32no—
38C3×D24—32no—
39C3×D24—62no—
40C3×D24—62no—
41C3×D24—61no—
42C3×D24—61no—
43C3⋊(C3⋊C16)—21no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)SD1611yescharacteristic
46C2×#272—42no—
47C2×#272—82no—
48C4×H(3,𝔽3)C22⋊C411yescharacteristic
49C4×H(3,𝔽3)D811yescharacteristic
50C2×C3×D8—91no—
51C2×D24—31no—
52C2×D24—31no—
53C2×D24—61no—
54C3×M32—91no—
55C3×#108—91no—
56#819—31no—
57#819—91no—
58#812—31no—
59#812—91no—
60C22×C3×D3—122no—
61C22×C3×D3—121no—
62C22×C3×D3—121no—
63C2×C32×C4—21no—
64C2×C32×C4D1211yescharacteristic
65C2×C32×C4#6711yescharacteristic
66C3×D12—64no—
67C3×D12—62no—
68C3×D12—62no—
69C3×D12—122no—
70C3×D12—121no—
71C3×D12—121no—
72C32×C8—21no—
73C32×C8—21no—
74C32×C8C4×D311yescharacteristic
75C32×C8C4×D311yescharacteristic
76C32×C8D1211yescharacteristic
77C32×C8#6711yescharacteristic
78C3⋊(C3⋊C8)—41no—
79C3×C3⋊C8—121no—
80#361—271noSylow
81C2×H(3,𝔽3)—21no—
82C2×H(3,𝔽3)#10211yescharacteristic
83#272—82no—
84C2×C3×D4—181no—
85C2×D12—61no—
86C2×D12—61no—
87C2×D12—62no—
88C3×D8—92no—
89C3×D8—181no—
90C3×M16—181no—
91C2×C3×C8—31no—
92C2×C3×C8—31no—
93C2×C3×C8—61no—
94C2×C3×C8D3211yescharacteristic
95#208—61no—
96#208—181no—
97D24—32no—
98D24—32no—
99D24—62no—
100D24—61no—
101D24—61no—
102D24—62no—
103C3×C16—181no—
104C3⋊C16—61no—
105C3⋊C16—181no—
106C22×C32—21no—
107C22×C32#20411yescharacteristic
108C22×C32#21511yescharacteristic
109C2×C3×D3—124no—
110C2×C3×D3—122no—
111C2×C3×D3—122no—
112C2×C3×D3—244no—
113C2×C3×D3—242no—
114C2×C3×D3—242no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4D2411yescharacteristic
118C32×C4(C2×D3)⋊C411yescharacteristic
119C32×C4(C2×D3)⋊C411yescharacteristic
120C32×C4#21311yescharacteristic
121C2×D8—91no—
122M32—91no—
123#108—91no—
124H(3,𝔽3)#36111yescharacteristic, complemented, Sylow
125C23×C3—361no—
126C22×D3—121no—
127C22×D3—121no—
128C22×D3—122no—
129C3×D4—182no—
130C3×D4—361no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—61no—
134C2×C3×C4#62611yescharacteristic
135D12—62no—
136D12—62no—
137D12—64no—
138D12—121no—
139D12—121no—
140D12—122no—
141C3×C8—31no—
142C3×C8—31no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—61no—
146C3×C8—61no—
147C3×C8—361no—
148C3×C8#62411yescharacteristic
149C3×C8#62611yescharacteristic
150C3⋊C8—121no—
151C3⋊C8—361no—
152C2×C32—22no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32#80911yescharacteristic
157C2×C32#82111yescharacteristic
158C3×D3—244no—
159C3×D3—242no—
160C3×D3—242no—
161C2×D4—181no—
162D8—92no—
163D8—181no—
164M16—181no—
165C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
166C16—181no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—362no—
171C22×C3—722no—
172C22×C3#371711yescharacteristic
173C2×D3—122no—
174C2×D3—122no—
175C2×D3—124no—
176C2×D3—242no—
177C2×D3—242no—
178C2×D3—244no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—61no—
184C3×C4—61no—
185C3×C4#371411yescharacteristic
186C3×C4#372211yescharacteristic
187C32—21no—
188C32#449511yescharacteristic, complemented
189C32#451611yescharacteristic, complemented
190C23—361no—
191D4—182no—
192D4—361no—
193C2×C4#620011yescharacteristic
194C8—361no—
195C8#619911yescharacteristic
196C8#620011yescharacteristic
197C2×C3—21no—
198C2×C3—31no—
199C2×C3—31no—
200C2×C3—61no—
201C2×C3—61no—
202C2×C3—62no—
203C2×C3—61no—
204C2×C3—722no—
205C2×C3#6356411yescharacteristic
206D3—242no—
207D3—242no—
208D3—244no—
209C22—362no—
210C22—722no—
211C22#8843711yescharacteristic
212C4#8843611yescharacteristic
213C4#8845111yescharacteristic
214C3—31no—
215C3—31no—
216C3—61no—
217C3#1058912811yescharacteristic
218C2—21no—
219C2—722no—
220C2#1172032411yescharacteristic
subgroupG/H
centerC2#11720324
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63564
prime radicalC2×#88529C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#361
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63646⋊C3C2semidirect
3. #63639⋊C3C2semidirect
15. #63646D3semidirect
16. #63639D3semidirect
124. H(3,𝔽3)#361semidirect
188. C32#4495semidirect
189. C32#4516semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#63646⋊C32
#63639⋊C32
#105891273
#105891263