Finite Group Databasebeta

← C3×#10592717 #13163056

Subgroups — C3×#10592717 #13163056

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic
3C3×C4×#634C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7C3×#63630C211yesmaximal, characteristic, complemented
8#10592717C311yesmaximal, characteristic, direct product factor
9C3×#5018—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4—21no—
12C2×C3×#634C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C22⋊C4—21no—
15C33×C42C2211yescharacteristic
16C3×#3754C414yescomplemented
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4×D4—24no—
23C32×C4×D4D314yes—
24C32×C4×D4C2×C311yescharacteristic
25C2×#3754C2×C311yescharacteristic
26C4×#634C2×C311yescharacteristic, complemented
27C2×C3×#214—34no—
28C3×C4×Dic6—34no—
29#63659C2×C311yescharacteristic
30#63637C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32#63630C2×C311yescharacteristic, complemented
33C3×#832—34no—
34C3×#810—34no—
35C3×#807—34no—
36C3×#803—34no—
37C23×C33—21no—
38C33×D4—21no—
39C33×D4C2×C412yes—
40C2×C33×C4—21no—
41C2×C33×C4—42no—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634C2×C412yescomplemented
49C3×C3⋊(C3⋊C8)—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51C3×#440—91no—
52#5018—34no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C2×C32×D4C22×C311yescharacteristic
56C22×C32×C4—28no—
57C22×C32×C4—24no—
58C22×C32×C4—21no—
59C2×#634C22×C311yescharacteristic
60C2×C3×Dic6—34no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C22⋊C4—28no—
65C32×C22⋊C4—24no—
66C32×C22⋊C4—21no—
67C32×C42—24no—
68C32×C42C2×D314yes—
69C32×C42C22×C311yescharacteristic
70#3754C3×C414yescomplemented
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72#3750—21no—
73C4×C3⋊Dic3—21no—
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C3×#214—316no—
77C3×C4⋊Dic3—34no—
78C3×Dic3⋊C4—64no—
79C3×C4×Dic3—64no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8—34no—
82C22×C33—22no—
83C22×C33—41no—
84C22×C33—41no—
85C22×C33C2×D411yescharacteristic
86C33×C4—21no—
87C33×C4—21no—
88C33×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3—41no—
93C2×C3×SD16—91no—
94C3×C4×Q8—91no—
95C3×C4×D4—24no—
96C3×C4×D4C3×D314yes—
97C3×C4×D4C3⋊D311yescharacteristic
98C2×#214—34no—
99C4×Dic6—34no—
100C3×#107—91no—
101C3×#103—91no—
102C3×#102—91no—
103C3×#97—91no—
104#832—34no—
105#810—34no—
106#807—34no—
107#803—34no—
108C23×C32—28no—
109C23×C32—24no—
110C23×C32—21no—
111C32×D4—28no—
112C32×D4—24no—
113C32×D4—21no—
114C32×D4—28no—
115C32×D4C4×D318yes—
116C32×D4C2×C3×C412yes—
117C2×C32×C4—28no—
118C2×C32×C4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—416no—
124C2×C32×C4—48no—
125C2×C32×C4—42no—
126C2×C32×C4#6714yes—
127C2×C32×C4#6714yes—
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C3×D411yescharacteristic
130C2×C32×C4C22×D314yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134#634—21no—
135#634C2×C3×C412yescomplemented
136C2×C3×Dic3—64no—
137C2×C3×Dic3—64no—
138C3×Dic6—38no—
139C3×Dic6—64no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)C2×C3×C412yescomplemented
142C3×C3⋊C8—38no—
143C3×C3⋊C8—64no—
144#440—91noSylow
145C2×C33—42no—
146C2×C33C4×D411yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C3×Q8—91no—
150C2×C3×D4—24no—
151C2×C3×D4C2×C3×D314yes—
152C2×C3×D4C2×C3⋊D311yescharacteristic
153C22×C3×C4—28no—
154C22×C3×C4—24no—
155C22×C3×C4—21no—
156C2×Dic6—34no—
157C3×SD16—94no—
158C2×C3×C8—91no—
159C2×C3×C8—91no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4C2×C3×D314yes—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C22⋊C4—28no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—21no—
168C3×C42—24no—
169C3×C42—181no—
170C3×C42C2×C3×D314yes—
171C3×C42C2×C3⋊D311yescharacteristic
172#214—316no—
173C4⋊Dic3—34no—
174Dic3⋊C4—64no—
175C4×Dic3—64no—
176C2×C3⋊C8—34no—
177C2×C3⋊C8—34no—
178C22×C32—216no—
179C22×C32—28no—
180C22×C32—22no—
181C22×C32—24no—
182C22×C32—48no—
183C22×C32—48no—
184C22×C32—44no—
185C22×C32—44no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32C2×#6714yes—
189C22×C32C2×C3×D411yescharacteristic
190C32×C4—28no—
191C32×C4—24no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—44no—
198C32×C4—48no—
199C32×C4—44no—
200C32×C4—41no—
201C32×C4C2×C4×D314yes—
202C32×C4#23514yes—
203C32×C4C22×C3×C411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—22no—
206C3⋊Dic3—41no—
207C3×Dic3—68no—
208C3×Dic3—124no—
209C2×SD16—91no—
210C4×Q8—91no—
211C4×D4C3×C3⋊D311yescharacteristic
212#107—91no—
213#103—91no—
214#102—91no—
215#97—91no—
216C33#44011yescharacteristic, complemented, Sylow
217C23×C3—28no—
218C23×C3—24no—
219C23×C3—21no—
220C3×Q8—92no—
221C3×Q8—181no—
222C3×D4—28no—
223C3×D4—24no—
224C3×D4—21no—
225C3×D4—28no—
226C3×D4C3×C4×D318yes—
227C3×D4C4×C3⋊D312yes—
228C2×C3×C4—28no—
229C2×C3×C4—24no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—416no—
235C2×C3×C4—48no—
236C2×C3×C4—42no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C3×#6714yes—
240C2×C3×C4C3×#6714yes—
241C2×C3×C4#63811yescharacteristic
242C2×C3×C4#63811yescharacteristic
243C2×C3×C4C22×C3×D314yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—64no—
246C2×Dic3—64no—
247Dic6—38no—
248Dic6—64no—
249C3×C8—92no—
250C3×C8—181no—
251C3⋊C8—38no—
252C3⋊C8—64no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—416no—
257C2×C32—48no—
258C2×C32—42no—
259C2×C32C4×#6714yes—
260C2×C32#93214yes—
261C2×C32#95114yes—
262C2×C32C3×C4×D411yescharacteristic
263C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
264C2×C32C3×#13711yescharacteristic
265C2×Q8—91no—
266C2×D4C2×C3×C3⋊D311yescharacteristic
267C22×C4—21no—
268SD16—94no—
269C2×C8—91no—
270C2×C8—91no—
271C4⋊C4—91no—
272C4⋊C4—181no—
273C4⋊C4C2×C3×C3⋊D311yescharacteristic
274C22⋊C4—21no—
275C42—181no—
276C42C2×C3×C3⋊D311yescharacteristic
277C22×C3—216no—
278C22×C3—28no—
279C22×C3—22no—
280C22×C3—24no—
281C22×C3—48no—
282C22×C3—48no—
283C22×C3—44no—
284C22×C3—44no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3C2×C3×#6714yes—
288C22×C3C2×#63811yescharacteristic
289C3×C4—28no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—24no—
295C3×C4—24no—
296C3×C4—44no—
297C3×C4—48no—
298C3×C4—44no—
299C3×C4—41no—
300C3×C4—182no—
301C3×C4—361no—
302C3×C4C2×C3×C4×D314yes—
303C3×C4C3×#23514yes—
304C3×C4C2×C4×C3⋊D311yescharacteristic
305C3×C4#382811yescharacteristic
306Dic3—68no—
307Dic3—124no—
308C32—24no—
309C32#501814yescomplemented
310C32C3×#44011yescharacteristic, complemented
311C23—21no—
312Q8—92no—
313Q8—181no—
314D4—21no—
315D4C3×C4×C3⋊D312yes—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4C3×#63811yescharacteristic
321C2×C4C3×#63811yescharacteristic
322C2×C4C22×C3×C3⋊D311yescharacteristic
323C8—92no—
324C8—181no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—416no—
329C2×C3—48no—
330C2×C3—42no—
331C2×C3C3×C4×#6714yes—
332C2×C3C3×#93214yes—
333C2×C3C3×#95114yes—
334C2×C3C4×#63811yescharacteristic
335C2×C3#6414111yescharacteristic
336C2×C3#6416011yescharacteristic
337C22—22no—
338C22—41no—
339C22—41no—
340C22C2×C3×#63811yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—182no—
345C4—361no—
346C4C2×C3×C4×C3⋊D311yescharacteristic
347C4C3×#382811yescharacteristic
348C3—24no—
349C3C3×#501814yescomplemented
350C3#1059271711yescharacteristic, direct product factor
351C2—42no—
352C2C3×C4×#63811yescharacteristic
353C2C3×#6414111yescharacteristic
354C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#440
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
3. C3×C4×#634C2semidirect
5. C3×#63637C2semidirect
6. C3×#63634C2semidirect
7. C3×#63630C2semidirect
8. #10592717C3direct (N × H)
16. C3×#3754C4semidirect
26. C4×#634C2×C3semidirect
30. #63637C2×C3semidirect
31. #63634C2×C3semidirect
32. #63630C2×C3semidirect
48. C3×#634C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
70. #3754C3×C4semidirect
135. #634C2×C3×C4semidirect
141. C3⋊(C3⋊C8)C2×C3×C4semidirect
216. C33#440semidirect
309. C32#5018semidirect
310. C32C3×#440semidirect
349. C3C3×#5018semidirect
350. C3#10592717direct (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×#37542
C3×C4×#6342
C3×#636592
C3×#636372
C3×#636342
C3×#636302
#105927173
C3×#50183