Finite Group Databasebeta

← C3×#10592713 #13163052

Subgroups — C3×#10592713 #13163052

Order 1728.

1168 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C3×#63635C211yesmaximal, characteristic, complemented
7C3×#63630C211yesmaximal, characteristic, complemented
8#10592713C311yesmaximal, characteristic, direct product factor
9C3×#5014—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16C33×C42C2211yescharacteristic
17C3×#3753C414yescomplemented
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4×D4—24no—
23C32×C4×D4D314yescomplemented
24C32×C4×D4C2×C311yescharacteristic, complemented
25C2×#3753C2×C311yescharacteristic
26C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
27C2×C3×#213—34no—
28C3×C4×D12—34no—
29#63659C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32#63630C2×C311yescharacteristic, complemented
33C3×#832—34no—
34C3×#809—34no—
35C3×#808—34no—
36C3×#803—34no—
37C23×C33—21no—
38C22×C3×C3⋊D3—21no—
39C33×D4—21no—
40C33×D4C2×C412yescomplemented
41C2×C33×C4—21no—
42C2×C33×C4—42no—
43C2×C33×C4D411yescharacteristic
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yescomplemented
49C3×C4×C3⋊D3—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52C3×#442—91no—
53#5014—34no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C22×C311yescharacteristic
57C22×C32×C4—28no—
58C22×C32×C4—24no—
59C22×C32×C4—21no—
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×C3×D12—34no—
63C2×C3×C4×D3—64no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4C2×D314yes—
66C32×C4⋊C4C22×C311yescharacteristic
67C32×C22⋊C4—28no—
68C32×C22⋊C4—24no—
69C32×C22⋊C4—21no—
70C32×C42—24no—
71C32×C42C2×D314yes—
72C32×C42C22×C311yescharacteristic
73#3753C3×C414yescomplemented
74#3752—21no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C3×#213—316no—
79C3×(C2×D3)⋊C4—64no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8—34no—
83C22×C33—22no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C33×C4—21no—
91C33×C4—21no—
92C33×C4—41no—
93C33×C4C22×C411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—41no—
96C2×C3×D8—91no—
97C3×C4×D4—24no—
98C3×C4×D4—91no—
99C3×C4×D4C3×D314yescomplemented
100C3×C4×D4C3⋊D311yescharacteristic, complemented
101C2×#213—34no—
102C4×D12—34no—
103C3×#106—91no—
104C3×#102—91no—
105C3×#102—91no—
106C3×#97—91no—
107#832—34no—
108#809—34no—
109#808—34no—
110#803—34no—
111C23×C32—28no—
112C23×C32—24no—
113C23×C32—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C32×D4—28no—
117C32×D4—24no—
118C32×D4—21no—
119C32×D4—28no—
120C32×D4C4×D318yescomplemented
121C32×D4C2×C3×C412yescomplemented
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—416no—
129C2×C32×C4—48no—
130C2×C32×C4—42no—
131C2×C32×C4#6714yes—
132C2×C32×C4#6714yes—
133C2×C32×C4C3×D411yescharacteristic
134C2×C32×C4C3×D411yescharacteristic
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)C2×C3×C412yescomplemented
140C4×C3⋊D3—42no—
141C2×C3×Dic3—64no—
142C3×D12—38no—
143C3×D12—64no—
144C3×C4×D3—128no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)C2×C3×C412yescomplemented
147C3×C3⋊C8—38no—
148C3×C3⋊C8—64no—
149#442—91noSylow
150C2×C33—42no—
151C2×C33C4×D411yescharacteristic
152C2×C33AGL(1,ℤ/8)11yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3×C3⋊D3—42no—
155C2×C3×D4—24no—
156C2×C3×D4—91no—
157C2×C3×D4C2×C3×D314yes—
158C2×C3×D4C2×C3⋊D311yescharacteristic
159C22×C3×C4—28no—
160C22×C3×C4—24no—
161C22×C3×C4—21no—
162C22×C3×C4—181no—
163C2×D12—34no—
164C2×C4×D3—64no—
165C3×D8—94no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4C2×C3×D314yes—
171C3×C4⋊C4C2×C3⋊D311yescharacteristic
172C3×C22⋊C4—28no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—181no—
176C3×C42—24no—
177C3×C42C2×C3×D314yes—
178C3×C42C2×C3⋊D311yescharacteristic
179#213—316no—
180(C2×D3)⋊C4—64no—
181C4⋊Dic3—34no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8—34no—
184C22×C32—216no—
185C22×C32—28no—
186C22×C32—22no—
187C22×C32—24no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32—44no—
191C22×C32—44no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32C2×#6714yes—
195C22×C32C2×C3×D411yescharacteristic
196C2×C3⋊D3—22no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3×D3—68no—
200C2×C3×D3—124no—
201C2×C3×D3—124no—
202C32×C4—28no—
203C32×C4—24no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—44no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4C2×C4×D314yes—
214C32×C4#23514yes—
215C32×C4C22×C3×C411yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—41no—
218C3×Dic3—124no—
219C2×D8—91no—
220C4×D4—91no—
221C4×D4C3×C3⋊D311yescharacteristic, complemented
222#106—91no—
223#102—91no—
224#102—91no—
225#97—91no—
226C33#44211yescharacteristic, complemented, Sylow
227C23×C3—28no—
228C23×C3—24no—
229C23×C3—21no—
230C23×C3—181no—
231C22×D3—64no—
232C3×D4—28no—
233C3×D4—24no—
234C3×D4—21no—
235C3×D4—28no—
236C3×D4—92no—
237C3×D4—181no—
238C3×D4C3×C4×D318yescomplemented
239C3×D4C4×C3⋊D312yescomplemented
240C2×C3×C4—28no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—416no—
247C2×C3×C4—48no—
248C2×C3×C4—42no—
249C2×C3×C4—181no—
250C2×C3×C4—362no—
251C2×C3×C4C3×#6714yes—
252C2×C3×C4C3×#6714yes—
253C2×C3×C4#63811yescharacteristic
254C2×C3×C4#63811yescharacteristic
255C2×C3×C4C22×C3×D314yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×Dic3—64no—
258D12—38no—
259D12—64no—
260C4×D3—128no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—38no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—416no—
269C2×C32—48no—
270C2×C32—42no—
271C2×C32C4×#6714yes—
272C2×C32#93214yes—
273C2×C32#94914yes—
274C2×C32C3×C4×D411yescharacteristic
275C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
276C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
277C3⋊D3—42no—
278C3×D3—128no—
279C2×D4—91no—
280C2×D4C2×C3×C3⋊D311yescharacteristic
281C22×C4—21no—
282C22×C4—181no—
283D8—94no—
284C2×C8—91no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4C2×C3×C3⋊D311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—181no—
290C42C2×C3×C3⋊D311yescharacteristic
291C22×C3—216no—
292C22×C3—28no—
293C22×C3—22no—
294C22×C3—24no—
295C22×C3—48no—
296C22×C3—48no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3C2×C3×#6714yes—
305C22×C3C2×#63811yescharacteristic
306C2×D3—68no—
307C2×D3—124no—
308C2×D3—124no—
309C3×C4—28no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—44no—
317C3×C4—48no—
318C3×C4—44no—
319C3×C4—41no—
320C3×C4—361no—
321C3×C4C2×C3×C4×D314yes—
322C3×C4C3×#23514yes—
323C3×C4C2×C4×C3⋊D311yescharacteristic
324C3×C4#382811yescharacteristic
325Dic3—124no—
326C32—24no—
327C32#501414yescomplemented
328C32C3×#44211yescharacteristic, complemented
329C23—21no—
330C23—181no—
331D4—21no—
332D4—92no—
333D4—181no—
334D4C3×C4×C3⋊D312yescomplemented
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—181no—
338C2×C4—362no—
339C2×C4C3×#63811yescharacteristic
340C2×C4C3×#63811yescharacteristic
341C2×C4C22×C3×C3⋊D311yescharacteristic
342C8—92no—
343C8—181no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—416no—
348C2×C3—48no—
349C2×C3—42no—
350C2×C3—362no—
351C2×C3C3×C4×#6714yes—
352C2×C3C3×#93214yes—
353C2×C3C3×#94914yes—
354C2×C3C4×#63811yescharacteristic
355C2×C3#6414111yescharacteristic
356C2×C3#6415811yescharacteristic
357D3—128no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—182no—
362C22—361no—
363C22—361no—
364C22C2×C3×#63811yescharacteristic
365C4—21no—
366C4—21no—
367C4—41no—
368C4—361no—
369C4C2×C3×C4×C3⋊D311yescharacteristic
370C4C3×#382811yescharacteristic
371C3—24no—
372C3C3×#501414yescomplemented
373C3#1059271311yescharacteristic, direct product factor
374C2—42no—
375C2—362no—
376C2C3×C4×#63811yescharacteristic
377C2C3×#6414111yescharacteristic
378C2C3×#6415811yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C3×#63659C2semidirect
5. C3×#63636C2semidirect
6. C3×#63635C2semidirect
7. C3×#63630C2semidirect
8. #10592713C3direct (N × H)
17. C3×#3753C4semidirect
23. C32×C4×D4D3semidirect
24. C32×C4×D4C2×C3semidirect
26. C4×C4⋊(C3⋊D3)C2×C3semidirect
29. #63659C2×C3semidirect
30. #63636C2×C3semidirect
31. #63635C2×C3semidirect
32. #63630C2×C3semidirect
40. C33×D4C2×C4semidirect
48. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
73. #3753C3×C4semidirect
99. C3×C4×D4C3×D3semidirect
100. C3×C4×D4C3⋊D3semidirect
120. C32×D4C4×D3semidirect
121. C32×D4C2×C3×C4semidirect
139. C4⋊(C3⋊D3)C2×C3×C4semidirect
146. C3⋊(C3⋊C8)C2×C3×C4semidirect
221. C4×D4C3×C3⋊D3semidirect
226. C33#442semidirect
238. C3×D4C3×C4×D3semidirect
239. C3×D4C4×C3⋊D3semidirect
327. C32#5014semidirect
328. C32C3×#442semidirect
334. D4C3×C4×C3⋊D3semidirect
372. C3C3×#5014semidirect
373. C3#10592713direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C2×C3×#37532
C3×C4×C4⋊(C3⋊D3)2
C3×#636592
C3×#636362
C3×#636352
C3×#636302
#105927133
C3×#50143