Finite Group Databasebeta

← C3×#10592502 #13162840

Subgroups — C3×#10592502 #13162840

Order 1728.

1140 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
4C33×#103C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592502C311yesmaximal, characteristic, direct product factor
9C3×#4803—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3743—22no—
24C2×#3832C2×C311yescharacteristic
25C2×C3×#239—34no—
26C2×#3743C2×C311yescharacteristic, complemented
27C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×ΓL(1,𝔽25)—34no—
29C3×D3×C4⋊C4—34no—
30C32×#103—24no—
31C32×#103D314yescomplemented
32C32×#103C2×C311yescharacteristic, complemented
33#63662C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#835—34no—
37C3×#821—34no—
38C3×#809—34no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yescomplemented
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—24no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)C2×C412yescomplemented
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3—42no—
53C3×C4×C3⋊D3C2×C412yes—
54C3×C4×C3⋊D3D412yes—
55C33×C8—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×#417—91no—
58#4803—34no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C22×C311yescharacteristic
62#3832—22no—
63#3832C3×C414yescomplemented
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C3×#239—316no—
70C3×#239—68no—
71C2×C3×D12—34no—
72C2×C3×D12—64no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C2×C3×C4×D3—64no—
76C2×C32×C8—24no—
77C2×C32×C8C2×D314yes—
78C2×C32×C8C22×C311yescharacteristic
79C32×C4⋊C4—24no—
80C32×C4⋊C4C2×D314yes—
81C32×C4⋊C4C22×C311yescharacteristic
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83#3750—21no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85#3743—22no—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—64no—
88C2×C3×C3⋊C8—34no—
89C3×ΓL(1,𝔽25)—68no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3C22⋊C412yes—
97C33×C4—22no—
98C33×C4—41no—
99C33×C4C22×C411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#239—34no—
105C2×C3×M16—91no—
106C2×C3×C4⋊C4—91no—
107C2×ΓL(1,𝔽25)—34no—
108D3×C4⋊C4—34no—
109C3×#103—24no—
110C3×#103—91no—
111C3×#103C3×D314yescomplemented
112C3×#103C3⋊D311yescharacteristic, complemented
113C3×#102—91no—
114C3×#102—91no—
115#835—34no—
116#821—34no—
117#809—34no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3C3×D411yescharacteristic
120C22×C3×D3—34no—
121C22×C3×D3—64no—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×Q8C4×D318yescomplemented
127C32×Q8C2×C3×C412yescomplemented
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—44no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3C3×D411yescharacteristic
139C4⋊(C3⋊D3)—24no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C2×C3×C412yescomplemented
142C4×C3⋊D3—24no—
143C4×C3⋊D3—42no—
144C4×C3⋊D3C2×C3×C412yes—
145C4×C3⋊D3C3×D412yes—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—64no—
148C3×D12—38no—
149C3×D12—616no—
150C3×D12—64no—
151C3×C4×D3—38no—
152C3×C4×D3—38no—
153C3×C4×D3—616no—
154C3×C4×D3—128no—
155C32×C8—21no—
156C32×C8—24no—
157C32×C8—44no—
158C3⋊(C3⋊C8)—21no—
159C3×C3⋊C8—64no—
160#417—91noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3×C3⋊D3—22no—
165C3×C3⋊D3—42no—
166C3×#41—94no—
167C3×#41—182no—
168C2×C3×Q8—24no—
169C2×C3×Q8C2×C3×D314yes—
170C2×C3×Q8C2×C3⋊D311yescharacteristic
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×C3×C4—181no—
176#239—316no—
177#239—68no—
178C2×D12—34no—
179C2×D12—64no—
180C2×C4×D3—34no—
181C2×C4×D3—64no—
182C2×C4×D3—64no—
183C3×M16—182no—
184C2×C3×C8—24no—
185C2×C3×C8—91no—
186C2×C3×C8C2×C3×D314yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—24no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4C2×C3×D314yes—
192C3×C4⋊C4C2×C3⋊D311yescharacteristic
193C4⋊Dic3—34no—
194Dic3⋊C4—64no—
195C2×C3⋊C8—34no—
196ΓL(1,𝔽25)—68no—
197C22×C32—24no—
198C22×C32D3×D414yes—
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—22no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3C3×C22⋊C412yes—
206C2×C3×D3—38no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—68no—
210C2×C3×D3—124no—
211C2×C3×D3—124no—
212C32×C4—22no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—28no—
216C32×C4—48no—
217C32×C4—41no—
218C32×C4—44no—
219C32×C4—84no—
220C32×C4C2×C4×D314yes—
221C32×C4D3×D414yes—
222C32×C4C22×C3×C411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C3⋊Dic3—41no—
225C3⋊Dic3C3×C22⋊C412yes—
226C3×Dic3—38no—
227C3×Dic3—124no—
228C2×#41—91no—
229C2×M16—91no—
230C2×C4⋊C4—91no—
231#103—91no—
232#103C3×C3⋊D311yescharacteristic, complemented
233#102—91no—
234#102—91no—
235C33#41711yescharacteristic, complemented, Sylow
236C23×C3—91no—
237C23×C3—181no—
238C22×D3—34no—
239C22×D3—64no—
240C3×Q8—24no—
241C3×Q8—21no—
242C3×Q8—28no—
243C3×Q8—44no—
244C3×Q8C3×C4×D318yescomplemented
245C3×Q8C4×C3⋊D312yescomplemented
246C3×D4—92no—
247C3×D4—184no—
248C3×D4—181no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—91no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—184no—
260C2×C3×C4—181no—
261C2×C3×C4—362no—
262C2×C3×C4C22×C3×D314yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264C2×Dic3—34no—
265C2×Dic3—64no—
266D12—38no—
267D12—616no—
268D12—64no—
269C4×D3—38no—
270C4×D3—38no—
271C4×D3—616no—
272C4×D3—128no—
273C3×C8—24no—
274C3×C8—21no—
275C3×C8—44no—
276C3×C8—181no—
277C3⋊C8—64no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32D3×C22⋊C414yes—
282C2×C32#91414yes—
283C2×C32#91814yes—
284C2×C32C2×C3×C22⋊C411yescharacteristic
285C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
286C2×C32C3×#13711yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—42no—
289C3×D3—68no—
290C3×D3—128no—
291#41—94no—
292#41—182no—
293C2×Q8C2×C3×C3⋊D311yescharacteristic
294C2×D4—91no—
295C2×D4—181no—
296C22×C4—91no—
297C22×C4—181no—
298C22×C4—181no—
299M16—182no—
300C2×C8—91no—
301C2×C8C2×C3×C3⋊D311yescharacteristic
302C4⋊C4—91no—
303C4⋊C4—181no—
304C4⋊C4C2×C3×C3⋊D311yescharacteristic
305C22×C3—24no—
306C22×C3—92no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3C3×D3×D414yes—
313C22×C3D4×C3⋊D311yescharacteristic
314C2×D3—38no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—68no—
318C2×D3—124no—
319C2×D3—124no—
320C3×C4—28no—
321C3×C4—22no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—44no—
326C3×C4—41no—
327C3×C4—84no—
328C3×C4—92no—
329C3×C4—361no—
330C3×C4C2×C3×C4×D314yes—
331C3×C4C3×D3×D414yes—
332C3×C4C2×C4×C3⋊D311yescharacteristic
333C3×C4D4×C3⋊D311yescharacteristic
334Dic3—38no—
335Dic3—124no—
336C32—24no—
337C32#480314yescomplemented
338C32C3×#41711yescharacteristic, complemented
339C23—91no—
340C23—181no—
341Q8—21no—
342Q8C3×C4×C3⋊D312yescomplemented
343D4—92no—
344D4—184no—
345D4—181no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—184no—
352C2×C4—181no—
353C2×C4—362no—
354C2×C4C22×C3×C3⋊D311yescharacteristic
355C8—21no—
356C8—181no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—182no—
361C2×C3—362no—
362C2×C3C3×D3×C22⋊C414yes—
363C2×C3C3×#91414yes—
364C2×C3C3×#91814yes—
365C2×C3C22⋊C4×C3⋊D311yescharacteristic
366C2×C3#6412311yescharacteristic
367C2×C3#6412711yescharacteristic
368D3—68no—
369D3—128no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—182no—
374C22—361no—
375C22—361no—
376C22C3×D4×C3⋊D311yescharacteristic
377C4—22no—
378C4—41no—
379C4—92no—
380C4—361no—
381C4C2×C3×C4×C3⋊D311yescharacteristic
382C4C3×D4×C3⋊D311yescharacteristic
383C3—24no—
384C3C3×#480314yescomplemented
385C3#1059250211yescharacteristic, direct product factor
386C2—182no—
387C2—362no—
388C2C3×C22⋊C4×C3⋊D311yescharacteristic
389C2C3×#6412311yescharacteristic
390C2C3×#6412711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. C3×C4⋊C4×C3⋊D3C2semidirect
4. C33×#103C2semidirect
5. C3×#63662C2semidirect
6. C3×#63648C2semidirect
7. C3×#63636C2semidirect
8. #10592502C3direct (N × H)
12. C3×#3832C4semidirect
26. C2×#3743C2×C3semidirect
27. C4⋊C4×C3⋊D3C2×C3semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63648C2×C3semidirect
35. #63636C2×C3semidirect
42. C33×Q8C2×C4semidirect
50. C3×C4⋊(C3⋊D3)C2×C4semidirect
63. #3832C3×C4semidirect
111. C3×#103C3×D3semidirect
112. C3×#103C3⋊D3semidirect
126. C32×Q8C4×D3semidirect
127. C32×Q8C2×C3×C4semidirect
141. C4⋊(C3⋊D3)C2×C3×C4semidirect
232. #103C3×C3⋊D3semidirect
235. C33#417semidirect
244. C3×Q8C3×C4×D3semidirect
245. C3×Q8C4×C3⋊D3semidirect
337. C32#4803semidirect
338. C32C3×#417semidirect
342. Q8C3×C4×C3⋊D3semidirect
384. C3C3×#4803semidirect
385. C3#10592502direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×#37432
C3×C4⋊C4×C3⋊D32
C33×#1032
C3×#636622
C3×#636482
C3×#636362
#105925023
C3×#48033