Finite Group Databasebeta

← C3×#10592734 #13163073

Subgroups — C3×#10592734 #13163073

Order 1728.

1012 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic, complemented
6C3×#63635C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592734C311yesmaximal, characteristic, direct product factor
9C3×#5035—34nomaximal
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C22⋊C4—21no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3—21no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#121—24no—
22C32×#121D314yes—
23C32×#121C2×C311yescharacteristic
24#64135C2×C311yescharacteristic, complemented
25C3×#926—34no—
26#63659C2×C311yescharacteristic, complemented
27#63659C2×C311yescharacteristic, complemented
28#63657C2×C311yescharacteristic, complemented
29#63635C2×C311yescharacteristic, complemented
30#63634C2×C311yescharacteristic, complemented
31C3×#832—34no—
32C3×#832—34no—
33C3×#830—34no—
34C3×#808—34no—
35C3×#807—34no—
36C23×C33—21no—
37C23×C33D411yescharacteristic
38C33×D4—21no—
39C33×D4—21no—
40C33×D4—42no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×#482—91no—
51#5035—34no—
52C2×C32×D4—28no—
53C2×C32×D4—24no—
54C2×C32×D4—24no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D314yes—
57C2×C32×D4C22×C311yescharacteristic
58C22×C32×C4—24no—
59C22×C32×C4C2×D314yes—
60C22×C32×C4C22×C311yescharacteristic
61C32×C4⋊C4—24no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C22⋊C4—28no—
65C32×C22⋊C4—24no—
66C32×C22⋊C4—21no—
67#3757—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4×C3⋊Dic3—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×#217—64no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×C4×Dic3—64no—
77C2×C3×C3⋊C8—34no—
78C2×C3×C3⋊C8—34no—
79C22×C33—21no—
80C22×C33—21no—
81C22×C33—21no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33—41no—
85C22×C33C2×D411yescharacteristic
86C33×C4—21no—
87C33×C4—41no—
88C33×C4#4111yescharacteristic
89C33×C4#4111yescharacteristic
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3—41no—
92C3×#121—24no—
93C3×#121C3×D314yes—
94C3×#121C3⋊D311yescharacteristic
95C3×#117—91no—
96#926—34no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#102—91no—
100C3×#102—91no—
101C3×C22⋊C8—91no—
102#832—34no—
103#832—34no—
104#830—34no—
105#808—34no—
106#807—34no—
107C23×C32—28no—
108C23×C32—24no—
109C23×C32—21no—
110C23×C32—24no—
111C23×C32#6714yes—
112C23×C32C3×D411yescharacteristic
113C32×D4—21no—
114C32×D4—21no—
115C32×D4—24no—
116C32×D4—24no—
117C32×D4—44no—
118C32×D4—44no—
119C32×D4—416no—
120C32×D4—48no—
121C32×D4—42no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—28no—
125C2×C32×C4—24no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4—44no—
133C2×C32×C4#6714yes—
134C2×C32×C4C3×D411yescharacteristic
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—64no—
144C3×C3⋊C8—64no—
145#482—91noSylow
146C2×C33—41no—
147C2×C33—81no—
148C2×C33#12311yescharacteristic
149C2×C33#13511yescharacteristic
150C2×C33AGL(1,ℤ/8)11yescharacteristic
151C2×C3×D4—28no—
152C2×C3×D4—24no—
153C2×C3×D4—21no—
154C2×C3×D4—24no—
155C2×C3×D4C2×C3×D314yes—
156C2×C3×D4C2×C3⋊D311yescharacteristic
157C22×C3×C4—24no—
158C22×C3×C4C2×C3×D314yes—
159C22×C3×C4C2×C3⋊D311yescharacteristic
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4C2×C3×D314yes—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167C3×C22⋊C4—28no—
168C3×C22⋊C4—24no—
169C3×C22⋊C4—21no—
170C3×C22⋊C4—181no—
171C3×C42—181no—
172#217—64no—
173C4⋊Dic3—34no—
174C4⋊Dic3—34no—
175C4×Dic3—64no—
176C2×C3⋊C8—34no—
177C2×C3⋊C8—34no—
178C22×C32—28no—
179C22×C32—24no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—24no—
184C22×C32—24no—
185C22×C32—24no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32—48no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32C2×#6714yes—
198C22×C32C2×C3×D411yescharacteristic
199C32×C4—21no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—44no—
204C32×C4—48no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4#23714yes—
208C32×C4#23714yes—
209C32×C4C3×#4111yescharacteristic
210C32×C4C3×#4111yescharacteristic
211C3⋊Dic3—41no—
212C3⋊Dic3—41no—
213C3×Dic3—124no—
214C3×Dic3—124no—
215#121C3×C3⋊D311yescharacteristic
216#117—91no—
217#107—91no—
218#106—91no—
219#102—91no—
220#102—91no—
221C22⋊C8—91no—
222C33#48211yescharacteristic, complemented, Sylow
223C23×C3—28no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—24no—
227C23×C3C3×#6714yes—
228C23×C3#63811yescharacteristic
229C3×D4—24no—
230C3×D4—24no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—44no—
234C3×D4—44no—
235C3×D4—416no—
236C3×D4—48no—
237C3×D4—42no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C3×#6714yes—
252C2×C3×C4#63811yescharacteristic
253C2×C3×C4C22×C3×D314yes—
254C2×C3×C4C22×C3⋊D311yescharacteristic
255C2×Dic3—64no—
256C2×Dic3—64no—
257C3×C8—181no—
258C3×C8—181no—
259C3⋊C8—64no—
260C3⋊C8—64no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—48no—
265C2×C32—44no—
266C2×C32—41no—
267C2×C32—88no—
268C2×C32—84no—
269C2×C32—81no—
270C2×C32#93214yes—
271C2×C32#93514yes—
272C2×C32#95014yes—
273C2×C32C3×#12311yescharacteristic
274C2×C32C3×#13511yescharacteristic
275C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
276C2×D4—21no—
277C2×D4C2×C3×C3⋊D311yescharacteristic
278C22×C4C2×C3×C3⋊D311yescharacteristic
279C2×C8—91no—
280C2×C8—91no—
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4C2×C3×C3⋊D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—181no—
286C42—181no—
287C22×C3—28no—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3—24no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—24no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—48no—
298C22×C3—48no—
299C22×C3—48no—
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3C2×C3×#6714yes—
307C22×C3C2×#63811yescharacteristic
308C3×C4—24no—
309C3×C4—21no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—44no—
313C3×C4—48no—
314C3×C4—44no—
315C3×C4—41no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4C3×#23714yes—
319C3×C4C3×#23714yes—
320C3×C4#383011yescharacteristic
321C3×C4#383011yescharacteristic
322Dic3—124no—
323Dic3—124no—
324C32—24no—
325C32#503514yescomplemented
326C32C3×#48211yescharacteristic, complemented
327C23—21no—
328C23C3×#63811yescharacteristic
329D4—21no—
330D4—21no—
331D4—42no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4C3×#63811yescharacteristic
338C2×C4C22×C3×C3⋊D311yescharacteristic
339C8—181no—
340C8—181no—
341C2×C3—24no—
342C2×C3—24no—
343C2×C3—24no—
344C2×C3—48no—
345C2×C3—44no—
346C2×C3—41no—
347C2×C3—88no—
348C2×C3—84no—
349C2×C3—81no—
350C2×C3C3×#93214yes—
351C2×C3C3×#93514yes—
352C2×C3C3×#95014yes—
353C2×C3#6414111yescharacteristic
354C2×C3#6414411yescharacteristic
355C2×C3#6415911yescharacteristic
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22C2×C3×#63811yescharacteristic
363C4—21no—
364C4—41no—
365C4—361no—
366C4—361no—
367C4C3×#383011yescharacteristic
368C4C3×#383011yescharacteristic
369C3—24no—
370C3C3×#503514yescomplemented
371C3#1059273411yescharacteristic, direct product factor
372C2—41no—
373C2—81no—
374C2C3×#6414111yescharacteristic
375C2C3×#6414411yescharacteristic
376C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenterC3×#121C3⋊D3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C3×#63659C2semidirect
4. C3×#63659C2semidirect
5. C3×#63657C2semidirect
6. C3×#63635C2semidirect
7. C3×#63634C2semidirect
8. #10592734C3direct (N × H)
24. #64135C2×C3semidirect
26. #63659C2×C3semidirect
27. #63659C2×C3semidirect
28. #63657C2×C3semidirect
29. #63635C2×C3semidirect
30. #63634C2×C3semidirect
222. C33#482semidirect
325. C32#5035semidirect
326. C32C3×#482semidirect
370. C3C3×#5035semidirect
371. C3#10592734direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1212
C3×#641352
C3×#636592
C3×#636572
C3×#636352
C3×#636342
#105927343
C3×#50353