Finite Group Databasebeta

← C3×#10590288 #13157303

Subgroups — C3×#10590288 #13157303

Order 1728.

602 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
1C3×#64078C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C32×C4×C3⋊C8C211yesmaximal, characteristic
4C3×#63567C214yesmaximal
5C3×#5093—31nomaximal
6C3×#4681—31nomaximal
7#10590288C311yesmaximal, characteristic, direct product factor
8C2×C3×#634C2212yes—
9C2×C32×Dic6—21no—
10C33×C42C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C32×C4⋊Dic3—21no—
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C32×C4⋊Q8—31no—
17#64078C2×C311yescharacteristic
18C3×#869—21no—
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22C3×#869D311yescharacteristic
23C3×#869C2×C311yescharacteristic
24C32×C4×C8—31no—
25C3×#816—34no—
26C3×#810—34no—
27C3×C4×C3⋊C8—21no—
28C3×C4×C3⋊C8D311yescharacteristic
29C3×C4×C3⋊C8C2×C311yescharacteristic
30#63567C2×C314yes—
31C2×C33×C4D411yescharacteristic
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—22no—
35C3×#634—24no—
36C2×C32×Dic3—22no—
37C32×Dic6—42no—
38C32×C3⋊C8—22no—
39C3×#487—91no—
40#5093—31no—
41#4681—31no—
42C2×C32×Q8—61no—
43C2×#634C22×C312yes—
44C2×C3×Dic6—22no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6—21no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—62no—
50C2×C32×C8—32no—
51C32×C4⋊C4—32no—
52C32×C4⋊C4—61no—
53C32×C42—21no—
54C32×C42—21no—
55C32×C42—21no—
56C32×C42—41no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D311yescharacteristic
59C32×C42C22×C311yescharacteristic
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—62no—
67C3×C4⋊Dic3—62no—
68C3×C4⋊Dic3—61no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3C2×D312yes—
73C3×C4⋊Dic3C22×C312yes—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8C2×D312yes—
76C2×C3×C3⋊C8C22×C312yes—
77C22×C33C2×D411yescharacteristic
78C33×C4SD1612yes—
79C33×C4Q1612yes—
80C33×C4#4112yes—
81C3×C3⋊Dic3—42no—
82C32×Dic3—42no—
83C3×C4⋊Q8—31no—
84C3×C4⋊Q8—31no—
85C3×C4⋊Q8—61no—
86C3×C4⋊Q8—91no—
87#869—31no—
88#869—31no—
89#869—61no—
90#869C3×D311yescharacteristic
91C3×#103—94no—
92C3×C4×C8—31no—
93C3×C4×C8—31no—
94C3×C4×C8—61no—
95#816—34no—
96#810—34no—
97C4×C3⋊C8C3×D311yescharacteristic
98C32×Q8—122no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4D1211yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—22no—
122#634—24no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C3×Dic6—44no—
130C3×Dic6—42no—
131C3×Dic6—42no—
132C3×Dic6—64no—
133C3×Dic6—64no—
134C3×Dic6—124no—
135C32×C8—62no—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139#487—91noSylow
140C2×C33#12411yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C2×C33C2×Q1611yescharacteristic
143C2×C3×Q8—62no—
144C2×C3×Q8—61no—
145C2×C3×Q8—61no—
146C2×C3×Q8—92no—
147C2×Dic6—21no—
148C2×Dic6—32no—
149C2×Dic6—32no—
150C2×Dic6—62no—
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—62no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—32no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—61no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C42—21no—
163C3×C42—21no—
164C3×C42—21no—
165C3×C42—41no—
166C3×C42D3211yescharacteristic
167C3×C42C2×C3×D311yescharacteristic
168C3×C42C2×C3×D311yescharacteristic
169C4⋊Dic3—21no—
170C4⋊Dic3—62no—
171C4⋊Dic3—62no—
172C4⋊Dic3—61no—
173C4⋊Dic3—61no—
174C4⋊Dic3—61no—
175C4⋊Dic3—61no—
176C4⋊Dic3C2×C3×D312yes—
177C2×C3⋊C8C2×C3×D312yes—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32C2×D1211yescharacteristic
183C22×C32C2×#6711yescharacteristic
184C22×C32C2×C3×D411yescharacteristic
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—42no—
195C32×C4—42no—
196C32×C4—42no—
197C32×C4—122no—
198C32×C4#20412yes—
199C32×C4Dic1212yes—
200C32×C4#21412yes—
201C32×C4#21612yes—
202C32×C4C3×SD1612yes—
203C32×C4C3×Q1612yes—
204C32×C4#23512yes—
205C32×C4#23912yes—
206C32×C4C3×#4112yes—
207C3⋊Dic3—42no—
208C3×Dic3—44no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—124no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C4⋊Q8—31no—
215C4⋊Q8—91no—
216#103—94no—
217C4×C8—31no—
218C33#48711yescharacteristic, complemented, Sylow
219C3×Q8—124no—
220C3×Q8—122no—
221C3×Q8—122no—
222C3×Q8—184no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—64no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—182no—
239C2×C3×C4#62611yescharacteristic
240C2×C3×C4#62611yescharacteristic
241C2×C3×C4C3×D1211yescharacteristic
242C2×C3×C4C3×D1211yescharacteristic
243C2×C3×C4C3×#6711yescharacteristic
244C2×C3×C4C3×#6711yescharacteristic
245C2×C3×C4C2×D3211yescharacteristic
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—22no—
249C2×Dic3—64no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252Dic6—42no—
253Dic6—64no—
254Dic6—64no—
255Dic6—124no—
256C3×C8—64no—
257C3×C8—62no—
258C3×C8—62no—
259C3⋊C8—22no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32#87511yescharacteristic
273C2×C32C2×#20411yescharacteristic
274C2×C32C2×Dic1211yescharacteristic
275C2×C32C2×#21411yescharacteristic
276C2×C32C2×#21611yescharacteristic
277C2×C32#94711yescharacteristic
278C2×C32C3×#12411yescharacteristic
279C2×C32C2×C3×SD1611yescharacteristic
280C2×C32C2×C3×Q1611yescharacteristic
281C2×Q8—61no—
282C2×Q8—92no—
283C2×C8—32no—
284C4⋊C4—32no—
285C4⋊C4—61no—
286C4⋊C4—181no—
287C4⋊C4—181no—
288C42C3×D3211yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3C2×#62611yescharacteristic
294C22×C3C2×C3×D1211yescharacteristic
295C22×C3C2×C3×#6711yescharacteristic
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—42no—
308C3×C4—124no—
309C3×C4—122no—
310C3×C4—122no—
311C3×C4—362no—
312C3×C4#371612yes—
313C3×C4#371912yes—
314C3×C4C3×#20412yes—
315C3×C4C3×Dic1212yes—
316C3×C4C3×#21412yes—
317C3×C4C3×#21612yes—
318C3×C4#379912yes—
319C3×C4C3×#23512yes—
320C3×C4C3×#23912yes—
321Dic3—42no—
322Dic3—124no—
323Dic3—122no—
324Dic3—122no—
325C32—21no—
326C32—21no—
327C32—21no—
328C32—41no—
329C32#468111yescharacteristic, complemented
330C32#509311yescharacteristic, complemented
331C32C3×#48711yescharacteristic, complemented
332Q8—122no—
333Q8—184no—
334C2×C4—62no—
335C2×C4—182no—
336C2×C4C3×#62611yescharacteristic
337C2×C4C3×#62611yescharacteristic
338C2×C4C2×C3×D3211yescharacteristic
339C8—62no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3C2×#371611yescharacteristic
353C2×C3C2×#371911yescharacteristic
354C2×C3#6386411yescharacteristic
355C2×C3C3×#87511yescharacteristic
356C2×C3C2×C3×#20411yescharacteristic
357C2×C3C2×C3×Dic1211yescharacteristic
358C2×C3C2×C3×#21411yescharacteristic
359C2×C3C2×C3×#21611yescharacteristic
360C2×C3C3×#94711yescharacteristic
361C22C2×C3×#62611yescharacteristic
362C4—122no—
363C4—362no—
364C4C3×#371612yes—
365C4C3×#371912yes—
366C4C3×#379912yes—
367C3—21no—
368C3—21no—
369C3—21no—
370C3—41no—
371C3#1059028811yescharacteristic, direct product factor
372C3C3×#468111yescharacteristic, complemented
373C3C3×#509311yescharacteristic, complemented
374C2C2×C3×#371611yescharacteristic
375C2C2×C3×#371911yescharacteristic
376C2C3×#6386411yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#487
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
7. #10590288C3direct (N × H)
218. C33#487semidirect
329. C32#4681semidirect
330. C32#5093semidirect
331. C32C3×#487semidirect
371. C3#10590288direct (N × H)
372. C3C3×#4681semidirect
373. C3C3×#5093semidirect

Maximal subgroups

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

subgroupindex
C3×#640782
C32×#8692
C32×C4×C3⋊C82
C3×#635672
C3×#50933
C3×#46813
#105902883