Finite Group Databasebeta

← C3×#10592708 #13163047

Subgroups — C3×#10592708 #13163047

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C3×#63659C211yesmaximal, characteristic
5C3×#63635C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592708C311yesmaximal, characteristic, direct product factor
9C3×#5009—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4—21no—
12C33×C4⋊C4C2211yescharacteristic
13C33×C22⋊C4—21no—
14C33×C42C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×#3750—21no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4×D4—24no—
22C32×C4×D4D314yes—
23C32×C4×D4C2×C311yescharacteristic
24#64079C2×C311yescharacteristic, complemented
25C3×#870—34no—
26#63659C2×C311yescharacteristic
27#63659C2×C311yescharacteristic
28#63635C2×C311yescharacteristic, complemented
29#63634C2×C311yescharacteristic, complemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#832—34no—
32C3×#832—34no—
33C3×#808—34no—
34C3×#807—34no—
35C3×C4⋊(C3⋊C8)—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4Q812yes—
39C2×C33×C4—21no—
40C2×C33×C4—42no—
41C2×C33×C4D411yescharacteristic
42C2×C33×C4D411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×C3⋊(C3⋊C8)—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×#478—91no—
49#5009—34no—
50C2×C32×D4—24no—
51C2×C32×D4C2×D314yes—
52C2×C32×D4C22×C311yescharacteristic
53C22×C32×C4—28no—
54C22×C32×C4—24no—
55C22×C32×C4—21no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C22×C311yescharacteristic
59C32×C22⋊C4—28no—
60C32×C22⋊C4—24no—
61C32×C22⋊C4—21no—
62C32×C42—24no—
63C32×C42C2×D314yes—
64C32×C42C22×C311yescharacteristic
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68#3750—21no—
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3—34no—
73C3×Dic3⋊C4—64no—
74C3×Dic3⋊C4—64no—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8—34no—
77C22×C33—22no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33C2×D411yescharacteristic
81C33×C4—21no—
82C33×C4—21no—
83C33×C4—41no—
84C33×C4C2×Q811yescharacteristic
85C33×C4#4111yescharacteristic
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×#125—91no—
89C3×C4×D4—24no—
90C3×C4×D4C3×D314yes—
91C3×C4×D4C3⋊D311yescharacteristic
92#870—34no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#102—91no—
97C3×#102—91no—
98#832—34no—
99#832—34no—
100#808—34no—
101#807—34no—
102C4⋊(C3⋊C8)—34no—
103C23×C32—28no—
104C23×C32—24no—
105C23×C32—21no—
106C32×D4—28no—
107C32×D4—24no—
108C32×D4—21no—
109C32×D4—28no—
110C32×D4Dic618yes—
111C32×D4C3×Q812yes—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—416no—
119C2×C32×C4—48no—
120C2×C32×C4—42no—
121C2×C32×C4#6714yes—
122C2×C32×C4#6714yes—
123C2×C32×C4C3×D411yescharacteristic
124C2×C32×C4C3×D411yescharacteristic
125C2×C32×C4C22×D314yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—64no—
131C3⋊(C3⋊C8)—21no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—64no—
134C3×C3⋊C8—64no—
135#478—91noSylow
136C2×C33—42no—
137C2×C33C22⋊Q811yescharacteristic
138C2×C33#13511yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C3×D4—24no—
141C2×C3×D4C2×C3×D314yes—
142C2×C3×D4C2×C3⋊D311yescharacteristic
143C22×C3×C4—28no—
144C22×C3×C4—24no—
145C22×C3×C4—21no—
146C2×C3×C8—91no—
147C2×C3×C8—91no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4C2×C3×D314yes—
154C3×C4⋊C4C2×C3⋊D311yescharacteristic
155C3×C22⋊C4—28no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—21no—
158C3×C42—24no—
159C3×C42C2×C3×D314yes—
160C3×C42C2×C3⋊D311yescharacteristic
161C4⋊Dic3—34no—
162C4⋊Dic3—34no—
163Dic3⋊C4—64no—
164Dic3⋊C4—64no—
165C2×C3⋊C8—34no—
166C2×C3⋊C8—34no—
167C22×C32—216no—
168C22×C32—28no—
169C22×C32—22no—
170C22×C32—24no—
171C22×C32—48no—
172C22×C32—48no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32C2×#6714yes—
178C22×C32C2×C3×D411yescharacteristic
179C32×C4—28no—
180C32×C4—24no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—44no—
187C32×C4—48no—
188C32×C4—44no—
189C32×C4—41no—
190C32×C4C2×Dic614yes—
191C32×C4#23514yes—
192C32×C4C2×C3×Q811yescharacteristic
193C32×C4C3×#4111yescharacteristic
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3×Dic3—124no—
197C3×Dic3—124no—
198#125—91no—
199C4×D4C3×C3⋊D311yescharacteristic
200#107—91no—
201#106—91no—
202C4⋊C8—91no—
203#102—91no—
204#102—91no—
205C33#47811yescharacteristic, complemented, Sylow
206C23×C3—28no—
207C23×C3—24no—
208C23×C3—21no—
209C3×D4—28no—
210C3×D4—24no—
211C3×D4—21no—
212C3×D4—28no—
213C3×D4C3×Dic618yes—
214C3×D4#63412yes—
215C2×C3×C4—28no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—416no—
222C2×C3×C4—48no—
223C2×C3×C4—42no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C3×#6714yes—
227C2×C3×C4C3×#6714yes—
228C2×C3×C4#63811yescharacteristic
229C2×C3×C4#63811yescharacteristic
230C2×C3×C4C22×C3×D314yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×Dic3—64no—
233C2×Dic3—64no—
234C3×C8—181no—
235C3×C8—181no—
236C3⋊C8—64no—
237C3⋊C8—64no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—416no—
242C2×C32—48no—
243C2×C32—42no—
244C2×C32#92414yes—
245C2×C32#93214yes—
246C2×C32#95014yes—
247C2×C32C3×C22⋊Q811yescharacteristic
248C2×C32C3×#13511yescharacteristic
249C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
250C2×D4C2×C3×C3⋊D311yescharacteristic
251C22×C4—21no—
252C2×C8—91no—
253C2×C8—91no—
254C4⋊C4—91no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4—181no—
258C4⋊C4C2×C3×C3⋊D311yescharacteristic
259C22⋊C4—21no—
260C42C2×C3×C3⋊D311yescharacteristic
261C22×C3—216no—
262C22×C3—28no—
263C22×C3—22no—
264C22×C3—24no—
265C22×C3—48no—
266C22×C3—48no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C2×C3×#6714yes—
272C22×C3C2×#63811yescharacteristic
273C3×C4—28no—
274C3×C4—24no—
275C3×C4—24no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—44no—
281C3×C4—48no—
282C3×C4—44no—
283C3×C4—41no—
284C3×C4—361no—
285C3×C4—361no—
286C3×C4C2×C3×Dic614yes—
287C3×C4C3×#23514yes—
288C3×C4C2×#63411yescharacteristic
289C3×C4#382811yescharacteristic
290Dic3—124no—
291Dic3—124no—
292C32—24no—
293C32#500914yescomplemented
294C32C3×#47811yescharacteristic, complemented
295C23—21no—
296D4—21no—
297D4C3×#63412yes—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4C3×#63811yescharacteristic
303C2×C4C3×#63811yescharacteristic
304C2×C4C22×C3×C3⋊D311yescharacteristic
305C8—181no—
306C8—181no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—416no—
311C2×C3—48no—
312C2×C3—42no—
313C2×C3C3×#92414yes—
314C2×C3C3×#93214yes—
315C2×C3C3×#95014yes—
316C2×C3#6413311yescharacteristic
317C2×C3#6414111yescharacteristic
318C2×C3#6415911yescharacteristic
319C22—22no—
320C22—41no—
321C22—41no—
322C22C2×C3×#63811yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—361no—
327C4—361no—
328C4C2×C3×#63411yescharacteristic
329C4C3×#382811yescharacteristic
330C3—24no—
331C3C3×#500914yescomplemented
332C3#1059270811yescharacteristic, direct product factor
333C2—42no—
334C2C3×#6413311yescharacteristic
335C2C3×#6414111yescharacteristic
336C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#478
LCSG ▷ C2×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
2. C3×#64079C2semidirect
5. C3×#63635C2semidirect
6. C3×#63634C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592708C3direct (N × H)
24. #64079C2×C3semidirect
28. #63635C2×C3semidirect
29. #63634C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
205. C33#478semidirect
293. C32#5009semidirect
294. C32C3×#478semidirect
331. C3C3×#5009semidirect
332. C3#10592708direct (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
C3×#640792
C3×#636592
C3×#636352
C3×#636342
C3×C4⋊(C3⋊(C3⋊C8))2
#105927083
C3×#50093