Finite Group Databasebeta

← C3×#10592765 #13163104

Subgroups — C3×#10592765 #13163104

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#125C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic
4C3×#63636C211yesmaximal, characteristic
5C3×#63635C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592765C311yesmaximal, characteristic, direct product factor
9C3×#5066—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C33×C4⋊C4—21no—
13C33×C4⋊C4—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C33×C42C2211yescharacteristic
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#125—24no—
22C32×#125D314yescomplemented
23C32×#125C2×C311yescharacteristic, complemented
24C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
25C3×C4×D12—34no—
26#63636C2×C311yescharacteristic
27#63636C2×C311yescharacteristic
28#63635C2×C311yescharacteristic, complemented
29#63634C2×C311yescharacteristic, complemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#809—34no—
32C3×#809—34no—
33C3×#808—34no—
34C3×#807—34no—
35C3×C4⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)Q812yes—
45C3×C4×C3⋊D3—42no—
46C3×C3⋊(C3⋊C8)—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×#478—91no—
49#5066—34no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×C4×C3⋊D3—21no—
52C2×C3×D12—34no—
53C2×C3×C4×D3—64no—
54C32×C4⋊C4—28no—
55C32×C4⋊C4—28no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C22×C311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C42—24no—
67C32×C42C2×D314yes—
68C32×C42C22×C311yescharacteristic
69#3752—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×(C2×D3)⋊C4—64no—
74C3×C4⋊Dic3—34no—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8—34no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C33×C4—21no—
82C33×C4—21no—
83C33×C4—41no—
84C33×C4—41no—
85C33×C4C2×Q811yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—41no—
88C3×#125—24no—
89C3×#125C3×D314yescomplemented
90C3×#125C3⋊D311yescharacteristic, complemented
91C3×C4×D4—91no—
92C4×D12—34no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#102—91no—
97C3×#102—91no—
98#809—34no—
99#809—34no—
100#808—34no—
101#807—34no—
102C4⋊(C3⋊C8)—34no—
103C22×C3⋊D3—21no—
104C22×C3×D3—64no—
105C2×C32×C4—28no—
106C2×C32×C4—28no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4#6714yes—
115C2×C32×C4#6714yes—
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C22×D314yes—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)C3×Q812yes—
123C4×C3⋊D3—42no—
124C2×C3×Dic3—64no—
125C3×D12—38no—
126C3×D12—64no—
127C3×C4×D3—128no—
128C3⋊(C3⋊C8)—21no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—64no—
131C3×C3⋊C8—64no—
132#478—91noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33AGL(1,ℤ/8)11yescharacteristic
136C3×C3⋊D3—42no—
137C2×C3×D4—91no—
138C22×C3×C4—181no—
139C2×D12—34no—
140C2×C4×D3—64no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—28no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4C2×C3×D314yes—
153C3×C4⋊C4C2×C3×D314yes—
154C3×C4⋊C4C2×C3⋊D311yescharacteristic
155C3×C4⋊C4C2×C3⋊D311yescharacteristic
156C3×C22⋊C4—181no—
157C3×C42—24no—
158C3×C42C2×C3×D314yes—
159C3×C42C2×C3⋊D311yescharacteristic
160(C2×D3)⋊C4—64no—
161C4⋊Dic3—34no—
162C2×C3⋊C8—34no—
163C2×C3⋊C8—34no—
164C22×C32—24no—
165C22×C32C2×#6714yes—
166C22×C32C2×C3×D411yescharacteristic
167C2×C3⋊D3—22no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C2×C3×D3—68no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—24no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—44no—
181C32×C4—48no—
182C32×C4—48no—
183C32×C4—44no—
184C32×C4—44no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4D3×Q814yes—
188C32×C4#23914yes—
189C32×C4C2×C3×Q811yescharacteristic
190C32×C4C3×#4111yescharacteristic
191C3⋊Dic3—41no—
192C3×Dic3—124no—
193#125C3×C3⋊D311yescharacteristic, complemented
194C4×D4—91no—
195#107—91no—
196#106—91no—
197C4⋊C8—91no—
198#102—91no—
199#102—91no—
200C33#47811yescharacteristic, complemented, Sylow
201C23×C3—181no—
202C22×D3—64no—
203C3×D4—92no—
204C3×D4—181no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—181no—
215C2×C3×C4—362no—
216C2×C3×C4C3×#6714yes—
217C2×C3×C4C3×#6714yes—
218C2×C3×C4#63811yescharacteristic
219C2×C3×C4#63811yescharacteristic
220C2×C3×C4C22×C3×D314yes—
221C2×C3×C4C22×C3⋊D311yescharacteristic
222C2×Dic3—64no—
223D12—38no—
224D12—64no—
225C4×D3—128no—
226C3×C8—181no—
227C3×C8—181no—
228C3⋊C8—64no—
229C3⋊C8—64no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32#94614yes—
234C2×C32#94914yes—
235C2×C32#95014yes—
236C2×C32C3×C22⋊Q811yescharacteristic
237C2×C32C3×#13511yescharacteristic
238C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
239C3⋊D3—42no—
240C3×D3—128no—
241C2×D4—91no—
242C22×C4—181no—
243C2×C8—91no—
244C2×C8—91no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—91no—
248C4⋊C4C2×C3×C3⋊D311yescharacteristic
249C4⋊C4C2×C3×C3⋊D311yescharacteristic
250C22⋊C4—181no—
251C42C2×C3×C3⋊D311yescharacteristic
252C22×C3—24no—
253C22×C3—182no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3C2×C3×#6714yes—
257C22×C3C2×#63811yescharacteristic
258C2×D3—68no—
259C2×D3—124no—
260C2×D3—124no—
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—44no—
268C3×C4—44no—
269C3×C4—48no—
270C3×C4—48no—
271C3×C4—44no—
272C3×C4—44no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—361no—
276C3×C4C3×D3×Q814yes—
277C3×C4C3×#23914yes—
278C3×C4Q8×C3⋊D311yescharacteristic
279C3×C4#383211yescharacteristic
280Dic3—124no—
281C32—24no—
282C32#506614yescomplemented
283C32C3×#47811yescharacteristic, complemented
284C23—181no—
285D4—92no—
286D4—181no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—181no—
290C2×C4—362no—
291C2×C4C3×#63811yescharacteristic
292C2×C4C3×#63811yescharacteristic
293C2×C4C22×C3×C3⋊D311yescharacteristic
294C8—181no—
295C8—181no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—362no—
300C2×C3C3×#94614yes—
301C2×C3C3×#94914yes—
302C2×C3C3×#95014yes—
303C2×C3#6415511yescharacteristic
304C2×C3#6415811yescharacteristic
305C2×C3#6415911yescharacteristic
306D3—128no—
307C22—182no—
308C22—361no—
309C22—361no—
310C22C2×C3×#63811yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—361no—
316C4C3×Q8×C3⋊D311yescharacteristic
317C4C3×#383211yescharacteristic
318C3—24no—
319C3C3×#506614yescomplemented
320C3#1059276511yescharacteristic, direct product factor
321C2—362no—
322C2C3×#6415511yescharacteristic
323C2C3×#6415811yescharacteristic
324C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#125C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×#125C3⋊D3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C33×#125C2semidirect
5. C3×#63635C2semidirect
6. C3×#63634C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592765C3direct (N × H)
22. C32×#125D3semidirect
23. C32×#125C2×C3semidirect
28. #63635C2×C3semidirect
29. #63634C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
89. C3×#125C3×D3semidirect
90. C3×#125C3⋊D3semidirect
193. #125C3×C3⋊D3semidirect
200. C33#478semidirect
282. C32#5066semidirect
283. C32C3×#478semidirect
319. C3C3×#5066semidirect
320. C3#10592765direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1252
C3×C4×C4⋊(C3⋊D3)2
C3×#636362
C3×#636352
C3×#636342
C3×C4⋊(C3⋊(C3⋊C8))2
#105927653
C3×#50663