Finite Group Databasebeta

← C3×#10592840 #13163179

Subgroups — C3×#10592840 #13163179

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#131C211yesmaximal, characteristic, complemented
2C3×#64160C211yesmaximal, characteristic, complemented
3C3×#64158C211yesmaximal, characteristic
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C3×#63651C211yesmaximal, characteristic, complemented
6C3×#63650C211yesmaximal, characteristic, complemented
7C3×#63646C211yesmaximal, characteristic, complemented
8#10592840C311yesmaximal, characteristic, direct product factor
9C3×#5141—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×M16—21no—
14C33×M16C2211yescharacteristic
15C2×C33×C8—21no—
16C2×C33×C8C2211yescharacteristic
17C3×#3756—21no—
18C3×#3755—21no—
19C3×#3754—21no—
20C3×#3753—21no—
21C3×#3748C2211yescharacteristic
22C3×#3748C2211yescharacteristic
23C3×#3744—21no—
24C3×#3744—21no—
25C32×#131—24no—
26C32×#131D314yescomplemented
27C32×#131C2×C311yescharacteristic, complemented
28#64160C2×C311yescharacteristic, complemented
29#64158C2×C311yescharacteristic
30C2×#3744C2×C311yescharacteristic, complemented
31C3×#951—34no—
32C3×#949—34no—
33C2×C3×#204—34no—
34#63651C2×C311yescharacteristic, complemented
35#63650C2×C311yescharacteristic, complemented
36#63646C2×C311yescharacteristic, complemented
37C3×#824—34no—
38C3×#823—34no—
39C3×#819—34no—
40C22×C3×C3⋊D3—21no—
41C33×Q8D411yescharacteristic
42C33×D4—21no—
43C33×D4D411yescharacteristic
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×#634—21no—
50C3×#634—21no—
51C33×C8—21no—
52C33×C8D411yescharacteristic
53C33×C8D411yescharacteristic
54C3×C3⋊(C3⋊C8)—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×#471—91no—
57#5141—34no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C22×C311yescharacteristic
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic
63C2×C3×D12—34no—
64C2×C3×Dic6—34no—
65C32×M16—28no—
66C32×M16—24no—
67C32×M16—24no—
68C32×M16—21no—
69C32×M16C2×D314yes—
70C32×M16C22×C311yescharacteristic
71C2×C32×C8—28no—
72C2×C32×C8—24no—
73C2×C32×C8—24no—
74C2×C32×C8—21no—
75C2×C32×C8C2×D314yes—
76C2×C32×C8C22×C311yescharacteristic
77#3756—21no—
78#3755—21no—
79#3754—21no—
80#3753—21no—
81#3748C22×C311yescharacteristic
82#3748C22×C311yescharacteristic
83#3744—21no—
84#3744—21no—
85C3×#216—64no—
86C3×#215—64no—
87C3×#214—64no—
88C3×#213—64no—
89C3×#208—34no—
90C3×#208—34no—
91C3×#204—64no—
92C3×#204—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—81no—
97C33×C4—21no—
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3×C3⋊Dic3—41no—
101C3×#137—91no—
102C3×AGL(1,ℤ/8)—91no—
103C2×C3×SD16—91no—
104C3×#131—24no—
105C3×#131C3×D314yescomplemented
106C3×#131C3⋊D311yescharacteristic, complemented
107#951—34no—
108#949—34no—
109C2×#204—34no—
110C3×#108—91no—
111C3×#101—91no—
112C3×#100—91no—
113#824—34no—
114#823—34no—
115#819—34no—
116C22×C3⋊D3—21no—
117C22×C3×D3—64no—
118C32×Q8—24no—
119C32×Q8D1214yes—
120C32×Q8C3×D411yescharacteristic
121C32×D4—28no—
122C32×D4—24no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4D1214yes—
126C32×D4C3×D411yescharacteristic
127C2×C32×C4—28no—
128C2×C32×C4—24no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C23×C311yescharacteristic
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—64no—
139C3×D12—64no—
140C3×D12—64no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143C32×C8—28no—
144C32×C8—24no—
145C32×C8—21no—
146C32×C8—24no—
147C32×C8—24no—
148C32×C8#6714yes—
149C32×C8#6714yes—
150C32×C8C3×D411yescharacteristic
151C32×C8C3×D411yescharacteristic
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)—21no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8—64no—
156#471—91noSylow
157C2×C33—21no—
158C2×C33—41no—
159C2×C33#12111yescharacteristic
160C3×C3⋊D3—81no—
161C3×#41—24no—
162C3×#41C2×C3×D314yes—
163C3×#41C2×C3⋊D311yescharacteristic
164C2×C3×Q8—91no—
165C2×C3×D4—91no—
166C2×D12—34no—
167C2×Dic6—34no—
168C3×Q16—181no—
169C3×SD16—181no—
170C3×SD16—181no—
171C3×SD16—181no—
172C3×SD16—181no—
173C3×D8—181no—
174C3×M16—28no—
175C3×M16—24no—
176C3×M16—21no—
177C3×M16—24no—
178C3×M16—91no—
179C3×M16—91no—
180C3×M16C2×C3×D314yes—
181C3×M16C2×C3⋊D311yescharacteristic
182C2×C3×C8—28no—
183C2×C3×C8—24no—
184C2×C3×C8—21no—
185C2×C3×C8—24no—
186C2×C3×C8C2×C3×D314yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188#216—64no—
189#215—64no—
190#214—64no—
191#213—64no—
192#208—34no—
193#208—34no—
194#204—64no—
195#204—64no—
196C22×C32—28no—
197C22×C32—24no—
198C22×C32—21no—
199C22×C32—24no—
200C22×C32#23514yes—
201C22×C32C3×#4111yescharacteristic
202C2×C3⋊D3—41no—
203C2×C3⋊D3—81no—
204C2×C3×D3—124no—
205C2×C3×D3—244no—
206C32×C4—28no—
207C32×C4—24no—
208C32×C4—21no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4C2×D1214yes—
212C32×C4C2×#6714yes—
213C32×C4C2×C3×D411yescharacteristic
214C32×C4C2×C3×D411yescharacteristic
215C3⋊Dic3—41no—
216C3×Dic3—124no—
217#137—91no—
218AGL(1,ℤ/8)—91no—
219C2×SD16—91no—
220#131C3×C3⋊D311yescharacteristic, complemented
221#108—91no—
222#101—91no—
223#100—91no—
224C33#47111yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C22×D3—64no—
227C3×Q8—24no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8C3×D1214yes—
231C3×Q8C4⋊(C3⋊D3)11yescharacteristic
232C3×D4—28no—
233C3×D4—24no—
234C3×D4—21no—
235C3×D4—24no—
236C3×D4—181no—
237C3×D4—181no—
238C3×D4C3×D1214yes—
239C3×D4C4⋊(C3⋊D3)11yescharacteristic
240C2×C3×C4—28no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—181no—
245C2×C3×C4C22×C3×D314yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×Dic3—64no—
248D12—64no—
249D12—64no—
250Dic6—64no—
251Dic6—64no—
252C3×C8—28no—
253C3×C8—24no—
254C3×C8—21no—
255C3×C8—24no—
256C3×C8—24no—
257C3×C8—181no—
258C3×C8—181no—
259C3×C8C3×#6714yes—
260C3×C8C3×#6714yes—
261C3×C8#63811yescharacteristic
262C3×C8#63811yescharacteristic
263C3⋊C8—64no—
264C3⋊C8—64no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—48no—
270C2×C32—44no—
271C2×C32—41no—
272C2×C32#93014yes—
273C2×C32C3×#12111yescharacteristic
274C3⋊D3—81no—
275C3×D3—244no—
276#41C2×C3×C3⋊D311yescharacteristic
277C2×Q8—91no—
278C2×D4—91no—
279Q16—181no—
280SD16—181no—
281SD16—181no—
282SD16—181no—
283SD16—181no—
284D8—181no—
285M16—21no—
286M16—91no—
287M16—91no—
288M16C2×C3×C3⋊D311yescharacteristic
289C2×C8—21no—
290C2×C8C2×C3×C3⋊D311yescharacteristic
291C22×C3—28no—
292C22×C3—24no—
293C22×C3—21no—
294C22×C3—24no—
295C22×C3—361no—
296C22×C3—721no—
297C22×C3C3×#23514yes—
298C22×C3#382811yescharacteristic
299C2×D3—124no—
300C2×D3—244no—
301C3×C4—28no—
302C3×C4—24no—
303C3×C4—21no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—361no—
307C3×C4C2×C3×D1214yes—
308C3×C4C2×C3×#6714yes—
309C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
310C3×C4C2×#63811yescharacteristic
311Dic3—124no—
312C32—24no—
313C32#514114yescomplemented
314C32C3×#47111yescharacteristic, complemented
315C23—181no—
316Q8—181no—
317Q8—181no—
318Q8C3×C4⋊(C3⋊D3)11yescharacteristic
319D4—21no—
320D4—181no—
321D4—181no—
322D4C3×C4⋊(C3⋊D3)11yescharacteristic
323C2×C4—21no—
324C2×C4—181no—
325C2×C4C22×C3×C3⋊D311yescharacteristic
326C8—21no—
327C8—181no—
328C8—181no—
329C8C3×#63811yescharacteristic
330C8C3×#63811yescharacteristic
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—48no—
336C2×C3—41no—
337C2×C3—44no—
338C2×C3—721no—
339C2×C3C3×#93014yes—
340C2×C3#6413911yescharacteristic
341D3—244no—
342C22—21no—
343C22—361no—
344C22—721no—
345C22C3×#382811yescharacteristic
346C4—21no—
347C4—361no—
348C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
349C4C2×C3×#63811yescharacteristic
350C3—24no—
351C3C3×#514114yescomplemented
352C3#1059284011yescharacteristic, direct product factor
353C2—21no—
354C2—41no—
355C2—721no—
356C2C3×#6413911yescharacteristic
subgroupG/H
centerC2×C3#64139
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#131C2
socleC2×C33#121
prime radicalC3×#64158C2
hypercenterC3×#131C3⋊D3
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
1. C33×#131C2semidirect
2. C3×#64160C2semidirect
4. C2×C3×#3744C2semidirect
5. C3×#63651C2semidirect
6. C3×#63650C2semidirect
7. C3×#63646C2semidirect
8. #10592840C3direct (N × H)
26. C32×#131D3semidirect
27. C32×#131C2×C3semidirect
28. #64160C2×C3semidirect
30. C2×#3744C2×C3semidirect
34. #63651C2×C3semidirect
35. #63650C2×C3semidirect
36. #63646C2×C3semidirect
105. C3×#131C3×D3semidirect
106. C3×#131C3⋊D3semidirect
220. #131C3×C3⋊D3semidirect
224. C33#471semidirect
313. C32#5141semidirect
314. C32C3×#471semidirect
351. C3C3×#5141semidirect
352. C3#10592840direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1312
C3×#641602
C3×#641582
C2×C3×#37442
C3×#636512
C3×#636502
C3×#636462
#105928403
C3×#51413