Finite Group Databasebeta

← #13138736

Subgroups — #13138736

Order 1728.

347 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 65 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×#88530C211yesmaximal, characteristic, complemented
2#64113⋊C3C211yesmaximal, characteristic
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#11720256C212yesmaximal, complemented
5#11720255C212yesmaximal, complemented
6#10590642—31nomaximal
7#10590636—31nomaximal
8C2×#6229—21no—
9#3828⋊C3—21no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—21no—
12#88530C2212yes—
13#3745⋊C3C2211yescharacteristic
14#3744⋊C3—21no—
15#3746⋊C3C2211yescharacteristic
16(C3⋊(C3⋊C16))⋊C3C2212yes—
17#64113D311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C3×#904—31no—
22#63639D311yescharacteristic, complemented
23C3×#812—31no—
24#63551—32no—
25#63550—32no—
26#63549—32no—
27#63547—32no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×#1084—41no—
30#6229—22no—
31#6229—41no—
32#638⋊C3—41no—
33C4⋊((C3⋊D3)⋊C3)—21no—
34C4×(C3⋊D3)⋊C3—41no—
35#634⋊C3—21no—
36C8×H(3,𝔽3)D411yescharacteristic
37C8×H(3,𝔽3)C2311yescharacteristic
38#5195—91no—
39#5179—91no—
40#4909—91no—
41#3828—21no—
42C3×#235—61no—
43C2×C3×Dic6—62no—
44C2×C3×Dic6—61no—
45C2×C3×Dic6—61no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49#3746C2×D311yescharacteristic
50#3745C2×D311yescharacteristic
51#3744—21no—
52C3×Dic12—31no—
53C3×Dic12—32no—
54C3×Dic12—32no—
55C3×Dic12—62no—
56C3×Dic12—62no—
57C3×Dic12—61no—
58C3×Dic12—61no—
59C3×D24—31no—
60C3×#204—61no—
61C3⋊(C3⋊C16)C2×D312yes—
62C3×C3⋊C16—32no—
63C22×H(3,𝔽3)D811yescharacteristic
64C2×(C3⋊D3)⋊C3—41no—
65C4×H(3,𝔽3)D811yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67#1084—42no—
68(C3⋊Dic3)⋊C3—41no—
69C3×#135—91no—
70C2×C3×Q16—91no—
71C2×Dic12—31no—
72C2×Dic12—31no—
73C2×Dic12—61no—
74#904—31no—
75#904—91no—
76C3×M32—91no—
77#829—92no—
78#829—92no—
79#828—92no—
80#827—92no—
81#812—31no—
82#812—91no—
83Dic24—92no—
84#800—92no—
85C2×C32×C4—21no—
86C2×C32×C4D1211yescharacteristic
87C2×C32×C4#6711yescharacteristic
88#638—41no—
89C4⋊(C3⋊D3)—21no—
90C4×C3⋊D3—41no—
91#634—21no—
92C3×#67—121no—
93C2×C3×Dic3—122no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—121no—
96C3×D12—61no—
97C3×C4×D3—121no—
98C3×Dic6—64no—
99C3×Dic6—61no—
100C3×Dic6—62no—
101C3×Dic6—62no—
102C3×Dic6—122no—
103C3×Dic6—121no—
104C3×Dic6—121no—
105C32×C8—21no—
106C32×C8—21no—
107C32×C8D1211yescharacteristic
108C32×C8#6711yescharacteristic
109C32×C8C22×D311yescharacteristic
110C32×C8C22×D311yescharacteristic
111#510—271noSylow
112C2×H(3,𝔽3)—21no—
113C2×H(3,𝔽3)C2×D811yescharacteristic
114(C3⋊D3)⋊C3—81no—
115C3×#41—181no—
116C2×C3×Q8—181no—
117#235—61no—
118#235—181no—
119C2×Dic6—61no—
120C2×Dic6—61no—
121C2×Dic6—62no—
122C3×Q16—91no—
123C3×Q16—92no—
124C3×Q16—181no—
125C3×SD16—181no—
126C3×D8—91no—
127C2×C3×C8—31no—
128C2×C3×C8—31no—
129C2×C3×C8—61no—
130C2×C3×C8D3211yescharacteristic
131Dic12—32no—
132Dic12—32no—
133Dic12—31no—
134Dic12—62no—
135Dic12—61no—
136Dic12—61no—
137Dic12—62no—
138Dic12—91no—
139D24—31no—
140D24—91no—
141#204—61no—
142#204—181no—
143C3×C16—92no—
144C3⋊C16—32no—
145C3⋊C16—92no—
146C22×C32—21no—
147C22×C32D2411yescharacteristic
148C22×C32#21311yescharacteristic
149C2×C3⋊D3—41no—
150C2×C3×D3—121no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4D2411yescharacteristic
154C32×C4#21311yescharacteristic
155C32×C4C2×D1211yescharacteristic
156C32×C4C2×#6711yescharacteristic
157C3⋊Dic3—41no—
158C3×Dic3—124no—
159C3×Dic3—121no—
160C3×Dic3—122no—
161C3×Dic3—122no—
162#135—91no—
163C2×Q16—91no—
164Q32—272no—
165SD32—272no—
166M32—91no—
167H(3,𝔽3)#51011yescharacteristic, complemented, Sylow
168C3×Q8—181no—
169C3×Q8—182no—
170C3×Q8—361no—
171C3×D4—181no—
172C3×D4—361no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—361no—
177C2×C3×C4—361no—
178C2×C3×C4#62611yescharacteristic
179#67—121no—
180#67—361no—
181C2×Dic3—121no—
182C2×Dic3—121no—
183C2×Dic3—122no—
184D12—61no—
185D12—181no—
186C4×D3—121no—
187C4×D3—361no—
188Dic6—62no—
189Dic6—62no—
190Dic6—64no—
191Dic6—61no—
192Dic6—121no—
193Dic6—121no—
194Dic6—122no—
195Dic6—181no—
196C3×C8—31no—
197C3×C8—31no—
198C3×C8—31no—
199C3×C8—31no—
200C3×C8—61no—
201C3×C8—61no—
202C3×C8#62611yescharacteristic
203C3×C8C2×D3211yescharacteristic
204C2×C32—22no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32C2×D2411yescharacteristic
209C2×C32C2×#21311yescharacteristic
210C3⋊D3—81no—
211C3×D3—241no—
212#41—181no—
213C2×Q8—181no—
214Q16—92no—
215Q16—91no—
216Q16—181no—
217SD16—181no—
218D8—91no—
219C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C16—92no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—61no—
224C22×C3—361no—
225C22×C3#371411yescharacteristic
226C2×D3—121no—
227C2×D3—361no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—361no—
235C3×C4—362no—
236C3×C4#371411yescharacteristic
237C3×C4C2×#62611yescharacteristic
238Dic3—122no—
239Dic3—122no—
240Dic3—124no—
241Dic3—121no—
242Dic3—361no—
243C32—21no—
244C32#490911yescharacteristic, complemented
245C32#519511yescharacteristic, complemented
246Q8—181no—
247Q8—182no—
248Q8—361no—
249D4—181no—
250D4—361no—
251C2×C4—361no—
252C2×C4—361no—
253C2×C4#620011yescharacteristic
254C8#620011yescharacteristic
255C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C2×C3—21no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—61no—
260C2×C3—61no—
261C2×C3—62no—
262C2×C3—61no—
263C2×C3—721no—
264C2×C3C2×#371411yescharacteristic
265D3—241no—
266D3—721no—
267C22—361no—
268C22#8843611yescharacteristic
269C4—361no—
270C4—362no—
271C4#8843611yescharacteristic
272C4C2×#620011yescharacteristic
273C3—31no—
274C3—31no—
275C3—61no—
276C3#1059063811yescharacteristic
277C2—21no—
278C2—721no—
279C2C2×#8843611yescharacteristic
subgroupG/H
centerC2C2×#88436
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3714
prime radical#64113⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#510
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
1. C2×#88530C2semidirect
3. #63639⋊C3C2semidirect
4. #11720256C2semidirect
5. #11720255C2semidirect
22. #63639D3semidirect
167. H(3,𝔽3)#510semidirect
244. C32#4909semidirect
245. C32#5195semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#64113⋊C32
#63639⋊C32
#117202562
#117202552
#105906423
#105906363