Finite Group Databasebeta

← C3×#10592838 #13163177

Subgroups — C3×#10592838 #13163177

Order 1728.

1196 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C33×M16C211yesmaximal, characteristic, complemented
3C3×#64135C211yesmaximal, characteristic, complemented
4C3×#63652C214yesmaximal, complemented
5#10592838C311yesmaximal, characteristic, direct product factor
6C3×#5139—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638—21no—
9C3×#3828—21no—
10C3×#3828—21no—
11C3×#3828C412yescomplemented
12C3×#3828C2212yes—
13C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C411yescharacteristic, complemented
16C33×M16—21no—
17C33×M16C2212yescomplemented
18C2×C33×C8—21no—
19C3×#3757—21no—
20C3×C4⋊(C3⋊Dic3)—21no—
21C3×C4×C3⋊Dic3C2212yescomplemented
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—34no—
24C2×C32×M16—24no—
25C2×C32×M16D314yescomplemented
26C2×C32×M16C2×C311yescharacteristic, complemented
27#64135C2×C311yescharacteristic, complemented
28C3×#926—34no—
29#63652C2×C314yescomplemented
30C3×#825—316no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C3×#638—22no—
40C3×#638—22no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—22no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yes—
45C3×C4×C3⋊D3—22no—
46C3×C4×C3⋊D3—22no—
47C3×#634—21no—
48C3×#634C2×C412yes—
49C33×C8—22no—
50C3×#421—91no—
51#5139—34no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C22×C311yescharacteristic
55C2×#638—21no—
56#3828—21no—
57#3828—21no—
58#3828C3×C412yescomplemented
59#3828C22×C312yes—
60C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
61C2×C4×C3⋊D3—21no—
62C2×#634C3×C411yescharacteristic, complemented
63C2×C3×#67—64no—
64C3×#235—38no—
65C3×#235—38no—
66C3×#235—64no—
67C3×#235—64no—
68C2×C3×D12—34no—
69C2×C3×C4×D3—64no—
70C2×C3×Dic6—34no—
71C32×M16—28no—
72C32×M16—28no—
73C32×M16—24no—
74C32×M16—21no—
75C32×M16C2×D318yescomplemented
76C32×M16C22×C312yescomplemented
77C2×C32×C8—28no—
78C2×C32×C8—24no—
79C2×C32×C8—21no—
80#3757—21no—
81C4⋊(C3⋊Dic3)—21no—
82C4×C3⋊Dic3C22×C312yescomplemented
83C3×#217—64no—
84C3×C4⋊Dic3—64no—
85C3×C4×Dic3—38no—
86C22×C33—41no—
87C22×C33C22⋊C411yescharacteristic
88C22×C33C22⋊C411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—42no—
92C33×C4C22⋊C411yescharacteristic
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—22no—
97C3×C3⋊Dic3—42no—
98C2×C3×#41—91no—
99C2×#235—34no—
100C2×C3×M16—24no—
101C2×C3×M16C3×D314yescomplemented
102C2×C3×M16C3⋊D311yescharacteristic, complemented
103C3×#117—91no—
104#926—34no—
105C3×#104—94no—
106#825—316no—
107C23×C32—24no—
108C23×C32D1214yes—
109C23×C32C3×D411yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4C4×D314yes—
119C2×C32×C4C4×D314yes—
120C2×C32×C4D1214yes—
121C2×C32×C4#6714yes—
122C2×C32×C4#6714yes—
123C2×C32×C4C2×C3×C411yescharacteristic
124C2×C32×C4C2×C3×C411yescharacteristic
125C2×C32×C4C3×D411yescharacteristic
126C2×C32×C4C3×D411yescharacteristic
127C2×C32×C4C3×D411yescharacteristic
128C2×C32×C4C22×D314yes—
129C2×C32×C4C23×C311yescharacteristic
130#638—22no—
131#638—22no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—22no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)C2×C3×C412yes—
136C4×C3⋊D3—22no—
137C4×C3⋊D3—22no—
138#634—21no—
139#634C2×C3×C412yes—
140C3×#67—68no—
141C3×#67—68no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—68no—
144C3×D12—38no—
145C3×D12—64no—
146C3×C4×D3—68no—
147C3×C4×D3—68no—
148C3×Dic6—38no—
149C3×Dic6—64no—
150C32×C8—216no—
151C32×C8—28no—
152C32×C8—22no—
153#421—91noSylow
154C2×C33—21no—
155C2×C33—21no—
156C2×C33—21no—
157C2×C33C2×C22⋊C411yescharacteristic
158C3×C3⋊D3—42no—
159C3×#41—92no—
160C3×#41—92no—
161C3×#41—181no—
162C3×#41—181no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—24no—
167C22×C3×C4—181no—
168C22×C3×C4C2×C3×D314yes—
169C22×C3×C4C2×C3⋊D311yescharacteristic
170C2×#67—64no—
171#235—38no—
172#235—38no—
173#235—64no—
174#235—64no—
175C2×D12—34no—
176C2×C4×D3—64no—
177C2×Dic6—34no—
178C3×M16—28no—
179C3×M16—24no—
180C3×M16—21no—
181C3×M16—28no—
182C3×M16C2×C3×D318yescomplemented
183C3×M16C2×C3⋊D312yescomplemented
184C2×C3×C8—28no—
185C2×C3×C8—24no—
186C2×C3×C8—21no—
187C3×C4⋊C4—181no—
188C3×C22⋊C4—181no—
189C3×C42—92no—
190#217—64no—
191C4⋊Dic3—64no—
192C4×Dic3—38no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—44no—
197C22×C32—41no—
198C22×C32—48no—
199C22×C32(C2×D3)⋊C414yes—
200C22×C32(C2×D3)⋊C414yes—
201C22×C32C3×C22⋊C411yescharacteristic
202C22×C32C3×C22⋊C411yescharacteristic
203C22×C32C2×D1214yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3—42no—
207C2×C3×D3—68no—
208C2×C3×D3—128no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4(C2×D3)⋊C414yes—
214C32×C4(C2×D3)⋊C414yes—
215C32×C4C3×C22⋊C411yescharacteristic
216C32×C4C3×C22⋊C411yescharacteristic
217C32×C4C2×C4×D314yes—
218C32×C4C2×#6714yes—
219C32×C4C22×C3×C411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C3⋊Dic3—22no—
222C3⋊Dic3—42no—
223C3×Dic3—68no—
224C3×Dic3—128no—
225C2×#41—91no—
226C2×M16C3×C3⋊D311yescharacteristic, complemented
227#117—91no—
228#104—94no—
229C33#42111yescharacteristic, complemented, Sylow
230C23×C3—24no—
231C23×C3—181no—
232C23×C3C3×D1214yes—
233C23×C3C4⋊(C3⋊D3)11yescharacteristic
234C22×D3—64no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—92no—
238C3×D4—182no—
239C3×D4—182no—
240C3×D4—181no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—181no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C3×C4×D314yes—
252C2×C3×C4C3×C4×D314yes—
253C2×C3×C4C3×D1214yes—
254C2×C3×C4C3×#6714yes—
255C2×C3×C4C3×#6714yes—
256C2×C3×C4C4×C3⋊D311yescharacteristic
257C2×C3×C4C4×C3⋊D311yescharacteristic
258C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
259C2×C3×C4#63811yescharacteristic
260C2×C3×C4#63811yescharacteristic
261C2×C3×C4C22×C3×D314yes—
262C2×C3×C4C22×C3⋊D311yescharacteristic
263#67—68no—
264#67—68no—
265C2×Dic3—64no—
266C2×Dic3—68no—
267D12—38no—
268D12—64no—
269C4×D3—68no—
270C4×D3—68no—
271Dic6—38no—
272Dic6—64no—
273C3×C8—216no—
274C3×C8—28no—
275C3×C8—22no—
276C2×C32—28no—
277C2×C32—24no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—44no—
285C2×C32—44no—
286C2×C32C2×(C2×D3)⋊C414yes—
287C2×C32C2×C3×C22⋊C411yescharacteristic
288C3⋊D3—42no—
289C3×D3—128no—
290#41—92no—
291#41—92no—
292#41—181no—
293#41—181no—
294C2×Q8—91no—
295C2×D4—91no—
296C2×D4—181no—
297C22×C4—181no—
298C22×C4C2×C3×C3⋊D311yescharacteristic
299M16—21no—
300M16C2×C3×C3⋊D312yescomplemented
301C2×C8—21no—
302C4⋊C4—181no—
303C22⋊C4—181no—
304C42—92no—
305C22×C3—24no—
306C22×C3—24no—
307C22×C3—24no—
308C22×C3—44no—
309C22×C3—41no—
310C22×C3—48no—
311C22×C3—182no—
312C22×C3—362no—
313C22×C3C3×(C2×D3)⋊C414yes—
314C22×C3C3×(C2×D3)⋊C414yes—
315C22×C3#375211yescharacteristic
316C22×C3#375211yescharacteristic
317C22×C3C2×C3×D1214yes—
318C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
319C2×D3—68no—
320C2×D3—128no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—182no—
326C3×C4—362no—
327C3×C4C3×(C2×D3)⋊C414yes—
328C3×C4C3×(C2×D3)⋊C414yes—
329C3×C4#375211yescharacteristic
330C3×C4#375211yescharacteristic
331C3×C4C2×C3×C4×D314yes—
332C3×C4C2×C3×#6714yes—
333C3×C4C2×C4×C3⋊D311yescharacteristic
334C3×C4C2×#63811yescharacteristic
335Dic3—68no—
336Dic3—128no—
337C32—24no—
338C32#513914yescomplemented
339C32C3×#42111yescharacteristic, complemented
340C23—181no—
341C23C3×C4⋊(C3⋊D3)11yescharacteristic
342Q8—92no—
343Q8—181no—
344D4—92no—
345D4—182no—
346D4—182no—
347D4—181no—
348C2×C4—181no—
349C2×C4—182no—
350C2×C4—182no—
351C2×C4—182no—
352C2×C4C3×C4×C3⋊D311yescharacteristic
353C2×C4C3×C4×C3⋊D311yescharacteristic
354C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
355C2×C4C3×#63811yescharacteristic
356C2×C4C3×#63811yescharacteristic
357C2×C4C22×C3×C3⋊D311yescharacteristic
358C8—22no—
359C2×C3—28no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—44no—
368C2×C3—44no—
369C2×C3—362no—
370C2×C3C2×C3×(C2×D3)⋊C414yes—
371C2×C3C2×#375211yescharacteristic
372D3—128no—
373C22—41no—
374C22—182no—
375C22—362no—
376C22C3×#375211yescharacteristic
377C22C3×#375211yescharacteristic
378C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
379C4—182no—
380C4—362no—
381C4C3×#375211yescharacteristic
382C4C3×#375211yescharacteristic
383C4C2×C3×C4×C3⋊D311yescharacteristic
384C4C2×C3×#63811yescharacteristic
385C3—24no—
386C3C3×#513914yescomplemented
387C3#1059283811yescharacteristic, direct product factor
388C2—21no—
389C2—21no—
390C2—21no—
391C2—362no—
392C2C2×C3×#375211yescharacteristic
subgroupG/H
centerC3×C4C2×#638
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×M16C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×M16C3⋊D3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
2. C2×C33×M16C2semidirect
3. C3×#64135C2semidirect
4. C3×#63652C2semidirect
5. #10592838C3direct (N × H)
11. C3×#3828C4semidirect
13. C2×C3×C4⋊(C3⋊D3)C4semidirect
15. C2×C3×#634C4semidirect
17. C33×M16C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
25. C2×C32×M16D3semidirect
26. C2×C32×M16C2×C3semidirect
27. #64135C2×C3semidirect
29. #63652C2×C3semidirect
58. #3828C3×C4semidirect
60. C2×C4⋊(C3⋊D3)C3×C4semidirect
62. C2×#634C3×C4semidirect
75. C32×M16C2×D3semidirect
76. C32×M16C22×C3semidirect
82. C4×C3⋊Dic3C22×C3semidirect
101. C2×C3×M16C3×D3semidirect
102. C2×C3×M16C3⋊D3semidirect
182. C3×M16C2×C3×D3semidirect
183. C3×M16C2×C3⋊D3semidirect
226. C2×M16C3×C3⋊D3semidirect
229. C33#421semidirect
300. M16C2×C3×C3⋊D3semidirect
338. C32#5139semidirect
339. C32C3×#421semidirect
386. C3C3×#5139semidirect
387. C3#10592838direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C33×M162
C3×#641352
C3×#636522
#105928383
C3×#51393