Finite Group Databasebeta

← #13138792

Subgroups — #13138792

Order 1728.

513 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 95 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×#3745⋊C3C211yesmaximal, characteristic
2#11721136C212yesmaximal, complemented
3C8×#1084C211yesmaximal, characteristic, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590763—31nomaximal
7Dic3×D24—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4×#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3745⋊C3C414yescomplemented
15#88450—22no—
16C2×#3745D311yescharacteristic
17C2×C3×D24—31no—
18#63894—32no—
19Dic3×D12—32no—
20#63645D311yescharacteristic, complemented
21C3×#818—31no—
22C3×C8×Dic3—31no—
23C3×C8×Dic3—31no—
24C3×C8×Dic3—61no—
25#63564—32no—
26#63563—32no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084D411yescharacteristic
30C2×#1084D411yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
34((C3⋊D3)⋊C3)⋊C4—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36#6180—21no—
37Dic3×D8—91no—
38#4872—91no—
39C4×D24—91no—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×D12—32no—
42C2×#624—62no—
43C2×D3×Dic3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C4⋊(C3⋊Dic3)C2×D312yes—
48#3745Dic314yescomplemented
49C3×C4⋊Dic3—32no—
50C3×C4×Dic3—31no—
51C3×C4×Dic3—31no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—61no—
56C3×D24—34no—
57#3722—62no—
58(C2×D3)⋊Dic3—62no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×(C3⋊D3)⋊C3—24no—
61C2×(C3⋊D3)⋊C3—42no—
62C2×(C3⋊D3)⋊C3—42no—
63C4×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65#1084—21no—
66#1084D812yescomplemented
67(C3⋊Dic3)⋊C3—42no—
68C2×C3×D8—91no—
69D4×Dic3—92no—
70C2×D24—31no—
71C2×D24—91no—
72#893—92no—
73C4×D12—92no—
74C3×#107—91no—
75C3×C4×C8—91no—
76#832—92no—
77#821—92no—
78#818—31no—
79#818—91no—
80C8×Dic3—31no—
81C8×Dic3—31no—
82C8×Dic3—61no—
83#809—92no—
84C22×C3⋊D3—22no—
85C22×C3×D3—62no—
86C2×C32×C4—21no—
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C22×D311yescharacteristic
89C2×C3⋊Dic3—22no—
90C4⋊(C3⋊D3)—22no—
91C4⋊(C3⋊D3)C2×Dic314yescomplemented
92C2×C3×Dic3—31no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—31no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C3×D12—34no—
100C3×D12—62no—
101#624—124no—
102D3×Dic3—124no—
103C32×C8—22no—
104C32×C8C4×D312yescomplemented
105C32×C8C2×Dic312yescomplemented
106C3×C3⋊C8—62no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109C4×D8—271noSylow
110C2×H(3,𝔽3)C4×D411yescharacteristic
111C2×H(3,𝔽3)C2×D811yescharacteristic
112C2×H(3,𝔽3)#13511yescharacteristic
113(C3⋊D3)⋊C3—44no—
114C2×C3×D4—92no—
115C22×Dic3—182no—
116C2×D12—32no—
117C2×D12—92no—
118C2×C4×D3—182no—
119C2×C4×D3—182no—
120C3×D8—94no—
121C2×C3×C8—31no—
122C2×C3×C8—31no—
123C2×C3×C8—61no—
124C2×C3×C8—91no—
125C2×C3×C8D3211yescharacteristic
126C3×C4⋊C4—92no—
127C3×C42—91no—
128#217—182no—
129(C2×D3)⋊C4—182no—
130(C2×D3)⋊C4—182no—
131C4⋊Dic3—32no—
132C4⋊Dic3—92no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—61no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—61no—
139D24—34no—
140D24—94no—
141C22×C32—21no—
142C22×C32D3×D411yescharacteristic
143C22×C32D3×D411yescharacteristic
144C2×C3⋊D3—24no—
145C2×C3⋊D3—42no—
146C2×C3⋊D3—42no—
147C2×C3×D3—64no—
148C2×C3×D3—122no—
149C2×C3×D3—122no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4C2×C4×D311yescharacteristic
153C32×C4#23711yescharacteristic
154C32×C4#23911yescharacteristic
155C32×C4C22×Dic311yescharacteristic
156C3⋊Dic3—42no—
157C3×Dic3—32no—
158C3×Dic3—32no—
159C3×Dic3—62no—
160C3×Dic3—62no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—122no—
164C2×D8—91no—
165C4×D4—272no—
166#107—91no—
167#102—272no—
168C4×C8—91no—
169H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
170C23×C3—182no—
171C22×D3—62no—
172C22×D3—182no—
173C3×D4—94no—
174C3×D4—182no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4—182no—
181C2×C3×C4C2×D3211yescharacteristic
182C2×Dic3—31no—
183C2×Dic3—31no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—62no—
189C2×Dic3—182no—
190C2×Dic3—364no—
191D12—34no—
192D12—62no—
193D12—94no—
194D12—182no—
195C4×D3—364no—
196C4×D3—364no—
197C3×C8—32no—
198C3×C8—32no—
199C3×C8—62no—
200C3×C8—181no—
201C3×C8D3×Dic312yes—
202C3⋊C8—61no—
203C3⋊C8—61no—
204C3⋊C8—62no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32#89311yescharacteristic
209C2×C32D3×D811yescharacteristic
210C2×C32D3×D811yescharacteristic
211C2×C32#91211yescharacteristic
212C2×C32#91911yescharacteristic
213C2×C32D4×Dic311yescharacteristic
214C3⋊D3—44no—
215C3×D3—124no—
216C2×D4—92no—
217C22×C4—542no—
218D8—94no—
219C2×C8—91no—
220C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C4⋊C4—92no—
222C22⋊C4—542no—
223C42—91no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—61no—
227C22×C3—184no—
228C22×C3—362no—
229C22×C3—362no—
230C22×C3#380211yescharacteristic
231C2×D3—64no—
232C2×D3—122no—
233C2×D3—122no—
234C2×D3—184no—
235C2×D3—362no—
236C2×D3—362no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—92no—
244C3×C4—181no—
245C3×C4—362no—
246C3×C4#379611yescharacteristic
247C3×C4C2×D3×Dic311yescharacteristic
248Dic3—32no—
249Dic3—32no—
250Dic3—62no—
251Dic3—61no—
252Dic3—61no—
253Dic3—62no—
254Dic3—122no—
255Dic3—362no—
256C32—21no—
257C32#487211yescharacteristic, complemented
258C32Dic3×D811yescharacteristic, complemented
259C23—182no—
260D4—94no—
261D4—182no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—182no—
265C2×C4—1084no—
266C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
267C8—181no—
268C8((C3⋊D3)⋊C3)⋊C412yescomplemented
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—364no—
279C2×C3#6379711yescharacteristic
280C2×C3#6381011yescharacteristic
281C2×C3#6389811yescharacteristic
282D3—124no—
283D3—364no—
284C22—184no—
285C22—362no—
286C22—362no—
287C22#8865511yescharacteristic
288C4—92no—
289C4—181no—
290C4—362no—
291C4#8864911yescharacteristic
292C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
293C3—31no—
294C3—31no—
295C3—61no—
296C3#1059077711yescharacteristic
297C2—364no—
298C2#1172083811yescharacteristic
299C2#1172084411yescharacteristic
300C2#1172113611yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#3745⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721136C2semidirect
3. C8×#1084C2semidirect
4. #63645⋊C3C2semidirect
5. #11720318C2semidirect
14. #3745⋊C3C4semidirect
20. #63645D3semidirect
33. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
48. #3745Dic3semidirect
66. #1084D8semidirect
91. C4⋊(C3⋊D3)C2×Dic3semidirect
104. C32×C8C4×D3semidirect
105. C32×C8C2×Dic3semidirect
169. H(3,𝔽3)C4×D8semidirect
257. C32#4872semidirect
258. C32Dic3×D8semidirect
268. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211362
C8×#10842
#63645⋊C32
#117203182
#105907633
Dic3×D243