Finite Group Databasebeta

← C3×#10592422 #13162762

Subgroups — C3×#10592422 #13162762

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C33×C22⋊C8C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C3×#63644C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592422C311yesmaximal, characteristic, direct product factor
9C3×#4723—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C3×#3757—21no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C3×C4×C3⋊Dic3—21no—
21#64135C2×C311yescharacteristic
22#64133C2×C311yescharacteristic
23C3×#926—34no—
24C3×#924—34no—
25C32×C22⋊C8—24no—
26C32×C22⋊C8D314yes—
27C32×C22⋊C8C2×C311yescharacteristic
28#63645C2×C311yescharacteristic, complemented
29#63644C2×C311yescharacteristic, complemented
30#63643C2×C311yescharacteristic, complemented
31#63643C2×C311yescharacteristic, complemented
32C3×#818—34no—
33C3×#817—34no—
34C3×#816—34no—
35C3×#816—34no—
36C23×C33D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C33×C8—21no—
48C33×C8—21no—
49C3×#485—91no—
50#4723—34no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C22×C311yescharacteristic
54C2×#634C22×C311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C32×C8—24no—
57C2×C32×C8—24no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C22×C311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62#3757—21no—
63#3757—21no—
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3—21no—
69C3×#217—64no—
70C3×#217—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—64no—
75C3×C4×Dic3—64no—
76C22×C33—21no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33C2×D411yescharacteristic
80C33×C4—21no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C3×C22⋊Q8—91no—
88C3×#117—91no—
89#926—34no—
90#924—34no—
91C3×#107—91no—
92C3×#106—91no—
93C3×#103—91no—
94C3×#103—91no—
95C3×C22⋊C8—24no—
96C3×C22⋊C8C3×D314yes—
97C3×C22⋊C8C3⋊D311yescharacteristic
98#818—34no—
99#817—34no—
100#816—34no—
101#816—34no—
102C23×C32—24no—
103C23×C32D1214yes—
104C23×C32C3×D411yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—44no—
113C2×C32×C4D1214yes—
114C2×C32×C4C3×D411yescharacteristic
115C2×C32×C4C22×D314yes—
116C2×C32×C4C23×C311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129C32×C8—21no—
130C32×C8—21no—
131C32×C8—24no—
132C32×C8—24no—
133C32×C8—44no—
134C32×C8—44no—
135#485—91noSylow
136C2×C33—41no—
137C2×C33#12311yescharacteristic
138C2×C33#13511yescharacteristic
139C2×C33#13711yescharacteristic
140C2×C3×Q8—91no—
141C22×C3×C4—24no—
142C22×C3×C4C2×C3×D314yes—
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C2×Dic6—34no—
145C2×C3×C8—24no—
146C2×C3×C8—24no—
147C2×C3×C8C2×C3×D314yes—
148C2×C3×C8C2×C3×D314yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C2×C3×C8C2×C3⋊D311yescharacteristic
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—181no—
155C3×C22⋊C4—181no—
156C3×C22⋊C4—181no—
157C3×C42—181no—
158#217—64no—
159#217—64no—
160C4⋊Dic3—34no—
161C4⋊Dic3—34no—
162C4⋊Dic3—34no—
163Dic3⋊C4—64no—
164C4×Dic3—64no—
165C22×C32—28no—
166C22×C32—24no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—24no—
171C22×C32—24no—
172C22×C32—24no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32C2×D1214yes—
176C22×C32C2×C3×D411yescharacteristic
177C32×C4—21no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4#23714yes—
183C32×C4#23714yes—
184C32×C4C3×#4111yescharacteristic
185C32×C4C3×#4111yescharacteristic
186C3⋊Dic3—41no—
187C3⋊Dic3—41no—
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3×Dic3—124no—
191C3×Dic3—124no—
192C3×Dic3—124no—
193C3×Dic3—124no—
194C22⋊Q8—91no—
195#117—91no—
196#107—91no—
197#106—91no—
198#103—91no—
199#103—91no—
200C22⋊C8C3×C3⋊D311yescharacteristic
201C33#48511yescharacteristic, complemented, Sylow
202C23×C3—24no—
203C23×C3C3×D1214yes—
204C23×C3C4⋊(C3⋊D3)11yescharacteristic
205C3×Q8—181no—
206C3×Q8—181no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—44no—
214C2×C3×C4—44no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C3×D1214yes—
220C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
221C2×C3×C4C22×C3×D314yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223C2×Dic3—64no—
224C2×Dic3—64no—
225C2×Dic3—64no—
226C2×Dic3—64no—
227Dic6—64no—
228Dic6—64no—
229C3×C8—24no—
230C3×C8—24no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—44no—
234C3×C8—44no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—48no—
239C2×C32—44no—
240C2×C32—41no—
241C2×C32#88614yes—
242C2×C32#90414yes—
243C2×C32#90914yes—
244C2×C32C3×#12311yescharacteristic
245C2×C32C3×#13511yescharacteristic
246C2×C32C3×#13711yescharacteristic
247C2×Q8—91no—
248C22×C4C2×C3×C3⋊D311yescharacteristic
249C2×C8C2×C3×C3⋊D311yescharacteristic
250C2×C8C2×C3×C3⋊D311yescharacteristic
251C4⋊C4—91no—
252C4⋊C4—91no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C22⋊C4—181no—
256C22⋊C4—181no—
257C42—181no—
258C22×C3—28no—
259C22×C3—24no—
260C22×C3—24no—
261C22×C3—24no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—24no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3C2×C3×D1214yes—
269C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
270C3×C4—24no—
271C3×C4—21no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—44no—
275C3×C4—361no—
276C3×C4—361no—
277C3×C4—361no—
278C3×C4—361no—
279C3×C4C3×#23714yes—
280C3×C4C3×#23714yes—
281C3×C4#383011yescharacteristic
282C3×C4#383011yescharacteristic
283Dic3—124no—
284Dic3—124no—
285Dic3—124no—
286Dic3—124no—
287C32—24no—
288C32#472314yescomplemented
289C32C3×#48511yescharacteristic, complemented
290C23C3×C4⋊(C3⋊D3)11yescharacteristic
291Q8—181no—
292Q8—181no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
300C2×C4C22×C3×C3⋊D311yescharacteristic
301C8—21no—
302C8—21no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—48no—
307C2×C3—44no—
308C2×C3—41no—
309C2×C3C3×#88614yes—
310C2×C3C3×#90414yes—
311C2×C3C3×#90914yes—
312C2×C3#6409511yescharacteristic
313C2×C3#6411311yescharacteristic
314C2×C3#6411811yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
319C4—21no—
320C4—361no—
321C4—361no—
322C4—361no—
323C4—361no—
324C4C3×#383011yescharacteristic
325C4C3×#383011yescharacteristic
326C3—24no—
327C3C3×#472314yescomplemented
328C3#1059242211yescharacteristic, direct product factor
329C2—41no—
330C2C3×#6409511yescharacteristic
331C2C3×#6411311yescharacteristic
332C2C3×#6411811yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C3×#63645C2semidirect
5. C3×#63644C2semidirect
6. C3×#63643C2semidirect
7. C3×#63643C2semidirect
8. #10592422C3direct (N × H)
28. #63645C2×C3semidirect
29. #63644C2×C3semidirect
30. #63643C2×C3semidirect
31. #63643C2×C3semidirect
201. C33#485semidirect
288. C32#4723semidirect
289. C32C3×#485semidirect
327. C3C3×#4723semidirect
328. C3#10592422direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641352
C3×#641332
C33×C22⋊C82
C3×#636452
C3×#636442
C3×#636432
#105924223
C3×#47233