Finite Group Databasebeta

← C3×#10592725 #13163064

Subgroups — C3×#10592725 #13163064

Order 1728.

1000 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic
6C3×#63634C211yesmaximal, characteristic, complemented
7C3×#63630C211yesmaximal, characteristic, complemented
8#10592725C311yesmaximal, characteristic, direct product factor
9C3×#5026—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C33×C4⋊C4—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C42—21no—
16C33×C42C2211yescharacteristic
17C3×#3755C414yescomplemented
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4×Q8—24no—
23C32×C4×Q8D314yescomplemented
24C32×C4×Q8C2×C311yescharacteristic, complemented
25C2×#3755C2×C311yescharacteristic
26C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
27C2×C3×#215—34no—
28C3×C4×D12—34no—
29#63662C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic
31#63634C2×C311yescharacteristic, complemented
32#63630C2×C311yescharacteristic, complemented
33C3×#835—34no—
34C3×#809—34no—
35C3×#807—34no—
36C3×#803—34no—
37C22×C3×C3⋊D3—21no—
38C33×Q8—21no—
39C33×Q8C2×C412yescomplemented
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)C2×C412yes—
48C3×C4×C3⋊D3—42no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51C3×#440—91no—
52#5026—34no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D314yes—
55C2×C32×Q8C22×C311yescharacteristic
56C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×C3×D12—34no—
59C2×C3×C4×D3—64no—
60C32×C4⋊C4—28no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C42—28no—
67C32×C42—24no—
68C32×C42—24no—
69C32×C42—21no—
70C32×C42C2×D314yes—
71C32×C42C22×C311yescharacteristic
72#3755C3×C414yescomplemented
73#3752—21no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C3×#215—316no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3—34no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8—34no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3—41no—
86C33×C4—22no—
87C33×C4—21no—
88C33×C4—21no—
89C33×C4—41no—
90C33×C4C22×C411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—41no—
93C2×C3×SD16—91no—
94C3×C4×Q8—24no—
95C3×C4×Q8C3×D314yescomplemented
96C3×C4×Q8C3⋊D311yescharacteristic, complemented
97C3×C4×D4—91no—
98C2×#215—34no—
99C4×D12—34no—
100C3×#107—91no—
101C3×#103—91no—
102C3×#102—91no—
103C3×#97—91no—
104#835—34no—
105#809—34no—
106#807—34no—
107#803—34no—
108C22×C3⋊D3—21no—
109C22×C3×D3—64no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—21no—
113C32×Q8—28no—
114C32×Q8C4×D318yescomplemented
115C32×Q8C2×C3×C412yescomplemented
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—21no—
119C2×C32×C4—28no—
120C2×C32×C4—24no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4#6714yes—
126C2×C32×C4#6714yes—
127C2×C32×C4C3×D411yescharacteristic
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C22×D314yes—
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)C2×C3×C412yes—
134C4×C3⋊D3—42no—
135C2×C3×Dic3—64no—
136C3×D12—38no—
137C3×D12—64no—
138C3×C4×D3—128no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)C2×C3×C412yescomplemented
141C3×C3⋊C8—38no—
142C3×C3⋊C8—64no—
143#440—91noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C3×C3⋊D3—42no—
148C2×C3×Q8—24no—
149C2×C3×Q8C2×C3×D314yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151C2×C3×D4—91no—
152C22×C3×C4—181no—
153C2×D12—34no—
154C2×C4×D3—64no—
155C3×SD16—94no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4C2×C3×D314yes—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C22⋊C4—181no—
166C3×C42—28no—
167C3×C42—24no—
168C3×C42—21no—
169C3×C42—24no—
170C3×C42C2×C3×D314yes—
171C3×C42C2×C3⋊D311yescharacteristic
172#215—316no—
173(C2×D3)⋊C4—64no—
174C4⋊Dic3—34no—
175C2×C3⋊C8—34no—
176C2×C3⋊C8—34no—
177C22×C32—24no—
178C22×C32C2×#6714yes—
179C22×C32C2×C3×D411yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C2×C3×D3—68no—
184C2×C3×D3—124no—
185C2×C3×D3—124no—
186C32×C4—28no—
187C32×C4—216no—
188C32×C4—28no—
189C32×C4—24no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—44no—
197C32×C4—48no—
198C32×C4—44no—
199C32×C4—41no—
200C32×C4C2×C4×D314yes—
201C32×C4#23514yes—
202C32×C4C22×C3×C411yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C3⋊Dic3—41no—
205C3×Dic3—124no—
206C2×SD16—91no—
207C4×Q8C3×C3⋊D311yescharacteristic, complemented
208C4×D4—91no—
209#107—91no—
210#103—91no—
211#102—91no—
212#97—91no—
213C33#44011yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×Q8—28no—
217C3×Q8—24no—
218C3×Q8—21no—
219C3×Q8—28no—
220C3×Q8C3×C4×D318yescomplemented
221C3×Q8C4×C3⋊D312yescomplemented
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—28no—
228C2×C3×C4—24no—
229C2×C3×C4—21no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—181no—
234C2×C3×C4—362no—
235C2×C3×C4C3×#6714yes—
236C2×C3×C4C3×#6714yes—
237C2×C3×C4#63811yescharacteristic
238C2×C3×C4#63811yescharacteristic
239C2×C3×C4C22×C3×D314yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—64no—
242D12—38no—
243D12—64no—
244C4×D3—128no—
245C3×C8—92no—
246C3×C8—181no—
247C3⋊C8—38no—
248C3⋊C8—64no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32C4×#6714yes—
253C2×C32#94214yes—
254C2×C32#94914yes—
255C2×C32C3×C4×D411yescharacteristic
256C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
257C2×C32C3×#13711yescharacteristic
258C3⋊D3—42no—
259C3×D3—128no—
260C2×Q8C2×C3×C3⋊D311yescharacteristic
261C2×D4—91no—
262C22×C4—181no—
263SD16—94no—
264C2×C8—91no—
265C2×C8—91no—
266C4⋊C4—21no—
267C4⋊C4—91no—
268C4⋊C4C2×C3×C3⋊D311yescharacteristic
269C22⋊C4—181no—
270C42—21no—
271C42C2×C3×C3⋊D311yescharacteristic
272C22×C3—24no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3C2×C3×#6714yes—
277C22×C3C2×#63811yescharacteristic
278C2×D3—68no—
279C2×D3—124no—
280C2×D3—124no—
281C3×C4—216no—
282C3×C4—28no—
283C3×C4—28no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—44no—
292C3×C4—48no—
293C3×C4—44no—
294C3×C4—41no—
295C3×C4—361no—
296C3×C4C2×C3×C4×D314yes—
297C3×C4C3×#23514yes—
298C3×C4C2×C4×C3⋊D311yescharacteristic
299C3×C4#382811yescharacteristic
300Dic3—124no—
301C32—24no—
302C32#502614yescomplemented
303C32C3×#44011yescharacteristic, complemented
304C23—181no—
305Q8—21no—
306Q8C3×C4×C3⋊D312yescomplemented
307D4—92no—
308D4—181no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—181no—
312C2×C4—362no—
313C2×C4C3×#63811yescharacteristic
314C2×C4C3×#63811yescharacteristic
315C2×C4C22×C3×C3⋊D311yescharacteristic
316C8—92no—
317C8—181no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—362no—
322C2×C3C3×C4×#6714yes—
323C2×C3C3×#94214yes—
324C2×C3C3×#94914yes—
325C2×C3C4×#63811yescharacteristic
326C2×C3#6415111yescharacteristic
327C2×C3#6415811yescharacteristic
328D3—128no—
329C22—182no—
330C22—361no—
331C22—361no—
332C22C2×C3×#63811yescharacteristic
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—361no—
338C4C2×C3×C4×C3⋊D311yescharacteristic
339C4C3×#382811yescharacteristic
340C3—24no—
341C3C3×#502614yescomplemented
342C3#1059272511yescharacteristic, direct product factor
343C2—362no—
344C2C3×C4×#63811yescharacteristic
345C2C3×#6415111yescharacteristic
346C2C3×#6415811yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4×Q8C2semidirect
4. C3×#63662C2semidirect
6. C3×#63634C2semidirect
7. C3×#63630C2semidirect
8. #10592725C3direct (N × H)
17. C3×#3755C4semidirect
23. C32×C4×Q8D3semidirect
24. C32×C4×Q8C2×C3semidirect
29. #63662C2×C3semidirect
31. #63634C2×C3semidirect
32. #63630C2×C3semidirect
39. C33×Q8C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
72. #3755C3×C4semidirect
95. C3×C4×Q8C3×D3semidirect
96. C3×C4×Q8C3⋊D3semidirect
114. C32×Q8C4×D3semidirect
115. C32×Q8C2×C3×C4semidirect
140. C3⋊(C3⋊C8)C2×C3×C4semidirect
207. C4×Q8C3×C3⋊D3semidirect
213. C33#440semidirect
220. C3×Q8C3×C4×D3semidirect
221. C3×Q8C4×C3⋊D3semidirect
302. C32#5026semidirect
303. C32C3×#440semidirect
306. Q8C3×C4×C3⋊D3semidirect
341. C3C3×#5026semidirect
342. C3#10592725direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×Q82
C2×C3×#37552
C3×C4×C4⋊(C3⋊D3)2
C3×#636622
C3×#636362
C3×#636342
C3×#636302
#105927253
C3×#50263