Finite Group Databasebeta

← #13138960

Subgroups — #13138960

Order 1728.

545 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 91 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#11721149C212yesmaximal, complemented
3#63645⋊C3C211yesmaximal, characteristic, complemented
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10590777—31nomaximal
7Dic3×D24—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×#6199—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88529C414yescomplemented
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#88451—22no—
16C2×C3×D24—31no—
17C2×C3×D24—31no—
18C2×C3×D24—61no—
19#63898—32no—
20Dic3×D12—32no—
21#63645D311yescharacteristic, complemented
22C8×C3⋊Dic3D311yescharacteristic, complemented
23C3×#818—31no—
24C3×C8×Dic3—31no—
25#63563—32no—
26#63562—32no—
27C22×#272—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C4⋊#272—22no—
30C4⋊#272C2×C414yescomplemented
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C2×(C3⋊Dic3)⋊C3D411yescharacteristic
34#6199—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36(C3⋊(C3⋊C8))⋊C3—21no—
37Dic3×D8—91no—
38#4872—91no—
39C4×D24—91no—
40C2×C3×D12—32no—
41C2×C3×D12—32no—
42C2×C3×D12—62no—
43C2×D3×Dic3—62no—
44C2×D3×Dic3—62no—
45C2×C32×C8—21no—
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C2×D311yescharacteristic
48C4⋊(C3⋊Dic3)C2×D312yes—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
51C3×C4⋊Dic3—32no—
52C3×C4×Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C3×D24—34no—
55C3×D24—34no—
56C3×D24—64no—
57(C2×D3)⋊Dic3—62no—
58(C2×D3)⋊Dic3—62no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—24no—
61C2×#272—42no—
62C2×#272—42no—
63C4×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65(C3⋊Dic3)⋊C3—21no—
66(C3⋊Dic3)⋊C3—42no—
67(C3⋊Dic3)⋊C3D812yescomplemented
68C2×C3×D8—91no—
69D4×Dic3—92no—
70C2×D24—31no—
71C2×D24—31no—
72C2×D24—61no—
73#893—92no—
74C4×D12—92no—
75C3×#107—91no—
76C3×C4×C8—91no—
77#832—92no—
78#821—92no—
79#818—31no—
80#818—91no—
81C8×Dic3—31no—
82C8×Dic3—91no—
83#809—92no—
84C22×C3×D3—62no—
85C22×C3×D3—62no—
86C22×C3×D3—122no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3D1211yescharacteristic
92C2×C3⋊Dic3D1211yescharacteristic
93C2×C3×Dic3—31no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—62no—
96C3×D12—34no—
97C3×D12—34no—
98C3×D12—64no—
99C3×D12—62no—
100C3×D12—62no—
101C3×D12—122no—
102D3×Dic3—124no—
103D3×Dic3—124no—
104C32×C8—22no—
105C32×C8C4×D312yescomplemented
106C32×C8C4×D312yescomplemented
107C3⋊(C3⋊C8)—21no—
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#272—44no—
114C2×C3×D4—92no—
115C22×Dic3—182no—
116C2×D12—32no—
117C2×D12—32no—
118C2×D12—62no—
119C2×C4×D3—182no—
120C2×C4×D3—182no—
121C3×D8—94no—
122C2×C3×C8—31no—
123C2×C3×C8—31no—
124C2×C3×C8—61no—
125C2×C3×C8—91no—
126C2×C3×C8D3211yescharacteristic
127C3×C4⋊C4—92no—
128C3×C42—91no—
129#217—182no—
130(C2×D3)⋊C4—182no—
131(C2×D3)⋊C4—182no—
132C4⋊Dic3—32no—
133C4⋊Dic3—92no—
134C4×Dic3—31no—
135C4×Dic3—91no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—91no—
138D24—34no—
139D24—34no—
140D24—64no—
141C22×C32—21no—
142C22×C32C2×D1211yescharacteristic
143C22×C32D3×D411yescharacteristic
144C2×C3×D3—64no—
145C2×C3×D3—64no—
146C2×C3×D3—124no—
147C2×C3×D3—122no—
148C2×C3×D3—122no—
149C2×C3×D3—122no—
150C2×C3×D3—122no—
151C2×C3×D3—242no—
152C2×C3×D3—242no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4C2×C4×D311yescharacteristic
156C32×C4C2×C4×D311yescharacteristic
157C32×C4#23511yescharacteristic
158C32×C4#23911yescharacteristic
159C3⋊Dic3—21no—
160C3⋊Dic3—42no—
161C3⋊Dic3D2412yescomplemented
162C3×Dic3—32no—
163C3×Dic3—61no—
164C3×Dic3—122no—
165C2×D8—91no—
166C4×D4—272no—
167#107—91no—
168#102—272no—
169C4×C8—91no—
170H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
171C23×C3—182no—
172C22×D3—62no—
173C22×D3—62no—
174C22×D3—122no—
175C3×D4—94no—
176C3×D4—182no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—182no—
183C2×C3×C4C2×D3211yescharacteristic
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—62no—
187C2×Dic3—91no—
188C2×Dic3—91no—
189C2×Dic3—182no—
190C2×Dic3—364no—
191D12—34no—
192D12—34no—
193D12—64no—
194D12—62no—
195D12—62no—
196D12—122no—
197C4×D3—364no—
198C4×D3—364no—
199C3×C8—32no—
200C3×C8—32no—
201C3×C8—62no—
202C3×C8—181no—
203C3×C8#62412yes—
204C3⋊C8—61no—
205C3⋊C8—181no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32C4×D1211yescharacteristic
210C2×C32#89311yescharacteristic
211C2×C32C2×D2411yescharacteristic
212C2×C32#90411yescharacteristic
213C2×C32D3×D811yescharacteristic
214C2×C32#91911yescharacteristic
215C3×D3—124no—
216C3×D3—124no—
217C3×D3—244no—
218C2×D4—92no—
219C22×C4—542no—
220D8—94no—
221C2×C8—91no—
222C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C4⋊C4—92no—
224C22⋊C4—542no—
225C42—91no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—61no—
229C22×C3—184no—
230C22×C3—362no—
231C22×C3—362no—
232C22×C3D3×D1211yescharacteristic
233C2×D3—64no—
234C2×D3—64no—
235C2×D3—124no—
236C2×D3—122no—
237C2×D3—122no—
238C2×D3—122no—
239C2×D3—122no—
240C2×D3—242no—
241C2×D3—242no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—92no—
249C3×C4—181no—
250C3×C4—362no—
251C3×C4#379911yescharacteristic
252C3×C4C2×#62411yescharacteristic
253Dic3—32no—
254Dic3—61no—
255Dic3—92no—
256Dic3—122no—
257Dic3—181no—
258Dic3—362no—
259C32—21no—
260C32C4×D2411yescharacteristic, complemented
261C32#487211yescharacteristic, complemented
262C23—182no—
263D4—94no—
264D4—182no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—182no—
268C2×C4—1084no—
269C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C8—181no—
271C8#619912yescomplemented
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—61no—
279C2×C3—61no—
280C2×C3—61no—
281C2×C3—364no—
282C2×C3D3×D2411yescharacteristic
283C2×C3#6380811yescharacteristic
284C2×C3#6389411yescharacteristic
285D3—124no—
286D3—124no—
287D3—244no—
288C22—184no—
289C22—362no—
290C22—362no—
291C22#8865411yescharacteristic
292C4—92no—
293C4—181no—
294C4—362no—
295C4#8865211yescharacteristic
296C4C2×#619911yescharacteristic
297C3—31no—
298C3—31no—
299C3—61no—
300C3#1059076311yescharacteristic
301C2—364no—
302C2#1172084711yescharacteristic
303C2#1172085011yescharacteristic
304C2#1172114911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88529C2
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. #11721149C2semidirect
3. #63645⋊C3C2semidirect
4. C8×(C3⋊Dic3)⋊C3C2semidirect
5. #11720324C2semidirect
11. #88529C4semidirect
21. #63645D3semidirect
22. C8×C3⋊Dic3D3semidirect
30. C4⋊#272C2×C4semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
67. (C3⋊Dic3)⋊C3D8semidirect
105. C32×C8C4×D3semidirect
106. C32×C8C4×D3semidirect
161. C3⋊Dic3D24semidirect
170. H(3,𝔽3)C4×D8semidirect
260. C32C4×D24semidirect
261. C32#4872semidirect
271. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#117211492
#63645⋊C32
C8×(C3⋊Dic3)⋊C32
#117203242
#105907773
Dic3×D243