Finite Group Databasebeta

← C3×#10592744 #13163083

Subgroups — C3×#10592744 #13163083

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C22⋊Q8C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic
6C3×#63635C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592744C311yesmaximal, characteristic, direct product factor
9C3×#5045—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C33×C4⋊C4—21no—
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×C22⋊Q8—24no—
22C32×C22⋊Q8D314yes—
23C32×C22⋊Q8C2×C311yescharacteristic
24#64135C2×C311yescharacteristic
25C3×#926—34no—
26#63662C2×C311yescharacteristic, complemented
27#63662C2×C311yescharacteristic, complemented
28#63657C2×C311yescharacteristic
29#63635C2×C311yescharacteristic, complemented
30#63634C2×C311yescharacteristic, complemented
31C3×#835—34no—
32C3×#835—34no—
33C3×#830—34no—
34C3×#808—34no—
35C3×#807—34no—
36C23×C33D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4D411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C3×#485—91no—
50#5045—34no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C22×C311yescharacteristic
54C22×C32×C4—24no—
55C22×C32×C4C2×D314yes—
56C22×C32×C4C22×C311yescharacteristic
57C32×C4⋊C4—28no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C22⋊C4—28no—
64C32×C22⋊C4—24no—
65C32×C22⋊C4—21no—
66#3757—21no—
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×#217—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3—64no—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8—34no—
78C22×C33—21no—
79C22×C33—21no—
80C22×C33—21no—
81C22×C33C2×D411yescharacteristic
82C33×C4—21no—
83C33×C4—41no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3—41no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8C3×D314yes—
91C3×C22⋊Q8C3⋊D311yescharacteristic
92C3×#117—91no—
93#926—34no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#103—91no—
97C3×#103—91no—
98C3×C22⋊C8—91no—
99#835—34no—
100#835—34no—
101#830—34no—
102#808—34no—
103#807—34no—
104C23×C32—24no—
105C23×C32#6714yes—
106C23×C32C3×D411yescharacteristic
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—24no—
111C32×Q8—44no—
112C32×Q8—44no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—28no—
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—44no—
126C2×C32×C4—44no—
127C2×C32×C4#6714yes—
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C22×D314yes—
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—64no—
138C3×C3⋊C8—64no—
139#485—91noSylow
140C2×C33—41no—
141C2×C33#12311yescharacteristic
142C2×C33#13511yescharacteristic
143C2×C33#13711yescharacteristic
144C2×C3×Q8—24no—
145C2×C3×Q8C2×C3×D314yes—
146C2×C3×Q8C2×C3⋊D311yescharacteristic
147C22×C3×C4—24no—
148C22×C3×C4C2×C3×D314yes—
149C22×C3×C4C2×C3⋊D311yescharacteristic
150C2×C3×C8—91no—
151C2×C3×C8—91no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4C2×C3×D314yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160C3×C22⋊C4—28no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—181no—
164C3×C42—181no—
165#217—64no—
166C4⋊Dic3—34no—
167C4⋊Dic3—34no—
168C4×Dic3—64no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8—34no—
171C22×C32—28no—
172C22×C32—24no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—24no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32C2×#6714yes—
182C22×C32C2×C3×D411yescharacteristic
183C32×C4—21no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—44no—
188C32×C4—48no—
189C32×C4—48no—
190C32×C4—44no—
191C32×C4—44no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4#23714yes—
195C32×C4#23714yes—
196C32×C4C3×#4111yescharacteristic
197C32×C4C3×#4111yescharacteristic
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3×Dic3—124no—
201C3×Dic3—124no—
202C22⋊Q8C3×C3⋊D311yescharacteristic
203#117—91no—
204#107—91no—
205#106—91no—
206#103—91no—
207#103—91no—
208C22⋊C8—91no—
209C33#48511yescharacteristic, complemented, Sylow
210C23×C3—24no—
211C23×C3C3×#6714yes—
212C23×C3#63811yescharacteristic
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—44no—
218C3×Q8—44no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—28no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—44no—
232C2×C3×C4—44no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4C3×#6714yes—
236C2×C3×C4#63811yescharacteristic
237C2×C3×C4C22×C3×D314yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—64no—
240C2×Dic3—64no—
241C3×C8—181no—
242C3×C8—181no—
243C3⋊C8—64no—
244C3⋊C8—64no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—48no—
249C2×C32—44no—
250C2×C32—41no—
251C2×C32#93514yes—
252C2×C32#94214yes—
253C2×C32#95014yes—
254C2×C32C3×#12311yescharacteristic
255C2×C32C3×#13511yescharacteristic
256C2×C32C3×#13711yescharacteristic
257C2×Q8C2×C3×C3⋊D311yescharacteristic
258C22×C4C2×C3×C3⋊D311yescharacteristic
259C2×C8—91no—
260C2×C8—91no—
261C4⋊C4—21no—
262C4⋊C4—91no—
263C4⋊C4—91no—
264C4⋊C4C2×C3×C3⋊D311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—181no—
267C42—181no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—24no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3C2×C3×#6714yes—
279C22×C3C2×#63811yescharacteristic
280C3×C4—24no—
281C3×C4—21no—
282C3×C4—24no—
283C3×C4—24no—
284C3×C4—44no—
285C3×C4—48no—
286C3×C4—48no—
287C3×C4—44no—
288C3×C4—44no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—361no—
292C3×C4—361no—
293C3×C4C3×#23714yes—
294C3×C4C3×#23714yes—
295C3×C4#383011yescharacteristic
296C3×C4#383011yescharacteristic
297Dic3—124no—
298Dic3—124no—
299C32—24no—
300C32#504514yescomplemented
301C32C3×#48511yescharacteristic, complemented
302C23C3×#63811yescharacteristic
303Q8—21no—
304Q8—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4C3×#63811yescharacteristic
312C2×C4C22×C3×C3⋊D311yescharacteristic
313C8—181no—
314C8—181no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—48no—
319C2×C3—44no—
320C2×C3—41no—
321C2×C3C3×#93514yes—
322C2×C3C3×#94214yes—
323C2×C3C3×#95014yes—
324C2×C3#6414411yescharacteristic
325C2×C3#6415111yescharacteristic
326C2×C3#6415911yescharacteristic
327C22—21no—
328C22—21no—
329C22—21no—
330C22C2×C3×#63811yescharacteristic
331C4—21no—
332C4—41no—
333C4—41no—
334C4—361no—
335C4—361no—
336C4C3×#383011yescharacteristic
337C4C3×#383011yescharacteristic
338C3—24no—
339C3C3×#504514yescomplemented
340C3#1059274411yescharacteristic, direct product factor
341C2—41no—
342C2C3×#6414411yescharacteristic
343C2C3×#6415111yescharacteristic
344C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
4. C3×#63662C2semidirect
6. C3×#63635C2semidirect
7. C3×#63634C2semidirect
8. #10592744C3direct (N × H)
26. #63662C2×C3semidirect
27. #63662C2×C3semidirect
29. #63635C2×C3semidirect
30. #63634C2×C3semidirect
209. C33#485semidirect
300. C32#5045semidirect
301. C32C3×#485semidirect
339. C3C3×#5045semidirect
340. C3#10592744direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C3×#641352
C3×#636622
C3×#636572
C3×#636352
C3×#636342
#105927443
C3×#50453