Finite Group Databasebeta

← #13135104

Subgroups — #13135104

Order 1728.

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

normal subgroups only

This group has 265 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
1#11721779C211yesmaximal, characteristic
2#63644⋊C3C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589124—31nomaximal
5#10589123—31nomaximal
6#88740—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88530C411yescharacteristic, complemented
9#88529C411yescharacteristic, complemented
10#88528—21no—
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12(C3⋊(C3⋊C16))⋊C3—21no—
13C3×#904—31no—
14C3×#904—31no—
15C3×#904—61no—
16#63644D311yescharacteristic, complemented
17#63639D311yescharacteristic, complemented
18C3×#817—31no—
19C3×#812—31no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21#6235—41no—
22C4⋊#272—21no—
23C4×#272—41no—
24#6229—21no—
25C2×(C3⋊Dic3)⋊C3—41no—
26C8×H(3,𝔽3)C2×C411yescharacteristic
27C8×H(3,𝔽3)D411yescharacteristic
28#4566—91no—
29#4518—91no—
30#4488—91no—
31C3×#235—62no—
32C3×#235—61no—
33C3×#235—61no—
34C2×C32×C8—21no—
35C2×C32×C8C2×D311yescharacteristic
36C2×C32×C8C2×D311yescharacteristic
37C4⋊(C3⋊Dic3)—21no—
38C3×C4⋊Dic3—61no—
39C3×Dic12—31no—
40C3×Dic12—31no—
41C3×Dic12—61no—
42C3×D24—31no—
43C3×D24—31no—
44C3×D24—61no—
45C3×#204—62no—
46C3×#204—61no—
47C3×#204—61no—
48C3⋊(C3⋊C16)—21no—
49C3×C3⋊C16—61no—
50C22×H(3,𝔽3)D811yescharacteristic
51C2×#272—41no—
52C4×H(3,𝔽3)C22⋊C411yescharacteristic
53C4×H(3,𝔽3)SD1611yescharacteristic
54#1084—41no—
55(C3⋊Dic3)⋊C3—81no—
56C3×#135—91no—
57#904—31no—
58#904—31no—
59#904—61no—
60C3×M32—91no—
61C3×#106—91no—
62#817—31no—
63#817—91no—
64#812—31no—
65#812—91no—
66C2×C32×C4—21no—
67C2×C32×C4D1211yescharacteristic
68C2×C32×C4#6711yescharacteristic
69C2×C3⋊Dic3—41no—
70C3×#67—122no—
71C3×#67—121no—
72C3×#67—121no—
73C2×C3×Dic3—121no—
74C3×D12—62no—
75C3×D12—61no—
76C3×D12—61no—
77C3×C4×D3—122no—
78C3×C4×D3—121no—
79C3×C4×D3—121no—
80C3×Dic6—62no—
81C3×Dic6—61no—
82C3×Dic6—61no—
83C32×C8—21no—
84C32×C8—21no—
85C32×C8C4×D311yescharacteristic
86C32×C8C4×D311yescharacteristic
87C32×C8D1211yescharacteristic
88C32×C8#6711yescharacteristic
89#360—271noSylow
90C2×H(3,𝔽3)—21no—
91C2×H(3,𝔽3)#10211yescharacteristic
92#272—81no—
93C3×#41—181no—
94#235—61no—
95#235—61no—
96#235—62no—
97C3×Q16—91no—
98C3×SD16—181no—
99C3×D8—91no—
100C2×C3×C8—31no—
101C2×C3×C8—31no—
102C2×C3×C8—61no—
103C2×C3×C8D3211yescharacteristic
104C3×C4⋊C4—181no—
105C4⋊Dic3—61no—
106C4⋊Dic3—181no—
107Dic12—31no—
108Dic12—31no—
109Dic12—61no—
110D24—31no—
111D24—31no—
112D24—61no—
113#204—61no—
114#204—61no—
115#204—62no—
116C3×C16—181no—
117C3⋊C16—61no—
118C3⋊C16—181no—
119C22×C32—21no—
120C22×C32D2411yescharacteristic
121C22×C32#21311yescharacteristic
122C2×C3×D3—122no—
123C2×C3×D3—121no—
124C2×C3×D3—121no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4#20411yescharacteristic
128C32×C4(C2×D3)⋊C411yescharacteristic
129C32×C4(C2×D3)⋊C411yescharacteristic
130C32×C4#21511yescharacteristic
131C3⋊Dic3—81no—
132C3×Dic3—122no—
133C3×Dic3—121no—
134C3×Dic3—121no—
135C3×Dic3—241no—
136#135—91no—
137M32—91no—
138#106—91no—
139H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
140C3×Q8—181no—
141C3×D4—181no—
142C3×D4—361no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—61no—
146C2×C3×C4—361no—
147C2×C3×C4—361no—
148C2×C3×C4#62611yescharacteristic
149#67—121no—
150#67—121no—
151#67—122no—
152C2×Dic3—121no—
153C2×Dic3—361no—
154D12—61no—
155D12—61no—
156D12—62no—
157C4×D3—121no—
158C4×D3—121no—
159C4×D3—122no—
160Dic6—61no—
161Dic6—61no—
162Dic6—62no—
163C3×C8—31no—
164C3×C8—31no—
165C3×C8—31no—
166C3×C8—31no—
167C3×C8—61no—
168C3×C8—61no—
169C3×C8#62411yescharacteristic
170C3×C8#62611yescharacteristic
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—41no—
175C2×C32#80911yescharacteristic
176C2×C32#82111yescharacteristic
177C3×D3—242no—
178C3×D3—241no—
179C3×D3—241no—
180#41—181no—
181Q16—91no—
182SD16—181no—
183D8—91no—
184C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C4⋊C4—181no—
186C16—181no—
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—61no—
190C22×C3—361no—
191C22×C3#371411yescharacteristic
192C2×D3—121no—
193C2×D3—121no—
194C2×D3—122no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—61no—
200C3×C4—61no—
201C3×C4—361no—
202C3×C4—721no—
203C3×C4#371711yescharacteristic
204C3×C4#372211yescharacteristic
205Dic3—121no—
206Dic3—121no—
207Dic3—122no—
208Dic3—241no—
209Dic3—721no—
210C32—21no—
211C32#448811yescharacteristic, complemented
212C32#451811yescharacteristic, complemented
213Q8—181no—
214D4—181no—
215D4—361no—
216C2×C4—361no—
217C2×C4—361no—
218C2×C4#620011yescharacteristic
219C8#619911yescharacteristic
220C8#620011yescharacteristic
221C2×C3—21no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—61no—
225C2×C3—61no—
226C2×C3—61no—
227C2×C3—121no—
228C2×C3—721no—
229C2×C3#6356411yescharacteristic
230D3—241no—
231D3—241no—
232D3—242no—
233C22—361no—
234C22#8843611yescharacteristic
235C4—361no—
236C4—721no—
237C4#8843711yescharacteristic
238C4#8845111yescharacteristic
239C3—31no—
240C3—31no—
241C3—61no—
242C3#1058912511yescharacteristic
243C2—21no—
244C2—721no—
245C2#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 radical#11721779C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#360
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. #63644⋊C3C2semidirect
3. #63639⋊C3C2semidirect
8. #88530C4semidirect
9. #88529C4semidirect
16. #63644D3semidirect
17. #63639D3semidirect
139. H(3,𝔽3)#360semidirect
211. C32#4488semidirect
212. C32#4518semidirect

Maximal subgroups

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

subgroupindex
#117217792
#63644⋊C32
#63639⋊C32
#105891243
#105891233