Finite Group Databasebeta

← C3×#10594826 #13174379

Subgroups — C3×#10594826 #13174379

Order 1728.

1046 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C32×#924C212yesmaximal
3C3×#63956C214yesmaximal
4C3×#63873C211yesmaximal, characteristic, complemented
5C3×#63872C211yesmaximal, characteristic, complemented
6C3×#63845C214yesmaximal, complemented
7C3×#63843C212yesmaximal, complemented
8C3×#5625—32nomaximal
9#10594826C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C3×#3757C2212yes—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×#217C2214yes—
17C32×C4⋊Dic3C2212yes—
18C32×Dic3⋊C4C2214yes—
19C3×#3724C2214yes—
20C3×Dic3⋊Dic3C2214yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×Dic32C2214yes—
23C32×C22⋊Q8—32no—
24#64135C2×C311yescharacteristic
25C3×Q8×Dic3—32no—
26C3×#944—32no—
27C3×#926—32no—
28C3×#926—61no—
29C3×#924—22no—
30C3×#924D312yes—
31C3×#924C2×C312yes—
32C3×#890—32no—
33C3×#890—34no—
34C3×#889—34no—
35C3×#887—32no—
36C3×#879—34no—
37C3×#879—34no—
38#63956C2×C314yes—
39#63873C2×C311yescharacteristic, complemented
40#63872C2×C311yescharacteristic, complemented
41#63845C2×C314yescomplemented
42#63843C2×C312yescomplemented
43C23×C33C2311yescharacteristic
44C2×C33×C4—22no—
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3C2312yes—
48C2×C3×C3⋊Dic3C2312yes—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2314yes—
51C32×Dic6—24no—
52C3×#541—91no—
53#5625—32no—
54C2×C32×Q8—32no—
55C22×C32×C4—21no—
56C22×C32×C4—22no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D312yes—
59C22×C32×C4C22×C311yescharacteristic
60C2×C3×Dic6—22no—
61C2×C3×Dic6C2×D312yes—
62C2×C3×Dic6C22×C312yes—
63C32×C4⋊C4—32no—
64C32×C4⋊C4—34no—
65C32×C22⋊C4—34no—
66#3757C22×C312yes—
67C4⋊(C3⋊Dic3)C22×C312yes—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C4×C3⋊Dic3C22×C311yescharacteristic
70C3×#217—24no—
71C3×#217—32no—
72C3×#217—32no—
73C3×#217—62no—
74C3×#217C2×D314yes—
75C3×#217C22×C314yes—
76C3×C4⋊Dic3—22no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—34no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—62no—
82C3×C4⋊Dic3C2×D312yes—
83C3×C4⋊Dic3C22×C312yes—
84C3×Dic3⋊C4—24no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—34no—
88C3×Dic3⋊C4—34no—
89C3×Dic3⋊C4C2×D314yes—
90C3×Dic3⋊C4C22×C314yes—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—34no—
94C3×C4×Dic3—34no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97#3724C22×C314yes—
98Dic3⋊Dic3C22×C314yes—
99Dic3⋊Dic3C22×C314yes—
100Dic32C22×C314yes—
101C22×C33—21no—
102C22×C33—22no—
103C22×C33C2411yescharacteristic
104C33×C4—21no—
105C33×C4#4112yes—
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3—22no—
108C32×Dic3—24no—
109C32×Dic3—24no—
110C3×C4⋊Q8—91no—
111C3×#126—94no—
112C3×#125—94no—
113C3×#125—91no—
114C3×C22⋊Q8—32no—
115C3×C22⋊Q8—32no—
116C3×C22⋊Q8—62no—
117C3×C4×Q8—92no—
118C3×#117—91no—
119Q8×Dic3—32no—
120#944—32no—
121#926—32no—
122#926—61no—
123#924C3×D312yes—
124#890—32no—
125#890—34no—
126#889—34no—
127#887—32no—
128#879—34no—
129#879—34no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—41no—
133C23×C32C22×D312yes—
134C23×C32C23×C311yescharacteristic
135C32×Q8—64no—
136C2×C32×C4—22no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—34no—
144C2×C32×C4—34no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—44no—
148C2×C32×C4—44no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C23×C311yescharacteristic
152C2×C32×C4C23×C311yescharacteristic
153C2×C3⋊Dic3C23×C312yes—
154C2×C3⋊Dic3C23×C312yes—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3C22×D314yes—
164C2×C3×Dic3C22×D314yes—
165C2×C3×Dic3C23×C314yes—
166C2×C3×Dic3C23×C314yes—
167C3×Dic6—24no—
168C3×Dic6—24no—
169C3×Dic6—44no—
170#541—91noSylow
171C2×C33—41no—
172C2×C33C2×#4111yescharacteristic
173C2×C332−1+412yes—
174C2×C3×Q8—32no—
175C2×C3×Q8—32no—
176C2×C3×Q8—62no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—41no—
180C22×C3×C4D3211yescharacteristic
181C22×C3×C4C2×C3×D312yes—
182C2×Dic6C2×C3×D312yes—
183C3×C4⋊C4—34no—
184C3×C4⋊C4—32no—
185C3×C4⋊C4—32no—
186C3×C4⋊C4—34no—
187C3×C4⋊C4—64no—
188C3×C4⋊C4—62no—
189C3×C4⋊C4—94no—
190C3×C4⋊C4—94no—
191C3×C4⋊C4—94no—
192C3×C4⋊C4—92no—
193C3×C22⋊C4—34no—
194C3×C22⋊C4—34no—
195C3×C22⋊C4—64no—
196C3×C22⋊C4—92no—
197C3×C42—94no—
198C3×C42—91no—
199C3×C42—91no—
200#217—32no—
201#217—32no—
202#217—62no—
203#217C2×C3×D314yes—
204C4⋊Dic3—32no—
205C4⋊Dic3—32no—
206C4⋊Dic3—34no—
207C4⋊Dic3—34no—
208C4⋊Dic3—62no—
209C4⋊Dic3C2×C3×D312yes—
210Dic3⋊C4—34no—
211Dic3⋊C4—34no—
212Dic3⋊C4—34no—
213Dic3⋊C4—34no—
214Dic3⋊C4C2×C3×D314yes—
215C4×Dic3—32no—
216C4×Dic3—32no—
217C4×Dic3—34no—
218C4×Dic3—34no—
219C4×Dic3—61no—
220C4×Dic3—61no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—24no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32—42no—
235C22×C32—44no—
236C22×C32C23×D312yes—
237C22×C32C24×C311yescharacteristic
238C32×C4—21no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—24no—
243C32×C4—42no—
244C32×C4—42no—
245C32×C4—42no—
246C32×C4—64no—
247C32×C4—64no—
248C32×C4#23714yes—
249C32×C4C3×#4112yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3×Dic3—24no—
253C3×Dic3—24no—
254C3×Dic3—24no—
255C3×Dic3—24no—
256C3×Dic3—44no—
257C3×Dic3—44no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—122no—
263C3×Dic3—122no—
264C4⋊Q8—91no—
265#126—94no—
266#125—94no—
267#125—91no—
268C22⋊Q8—32no—
269C4×Q8—92no—
270#117—91no—
271C33#54111yescharacteristic, complemented, Sylow
272C23×C3—22no—
273C23×C3—21no—
274C23×C3—41no—
275C23×C3C2×D3211yescharacteristic
276C23×C3C22×C3×D312yes—
277C3×Q8—64no—
278C3×Q8—64no—
279C3×Q8—124no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—34no—
288C2×C3×C4—34no—
289C2×C3×C4—34no—
290C2×C3×C4—34no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—64no—
296C2×C3×C4—64no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C2×D3211yescharacteristic
301C2×C3×C4C22×C3×D312yes—
302C2×C3×C4C22×C3×D312yes—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3C22×C3×D314yes—
310C2×Dic3C22×C3×D314yes—
311Dic6—24no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—41no—
318C2×C32—42no—
319C2×C32—42no—
320C2×C32—44no—
321C2×C32—44no—
322C2×C32—41no—
323C2×C32—42no—
324C2×C32C2×#23712yes—
325C2×C32#100712yes—
326C2×C32#101012yes—
327C2×C32C2×C3×#4111yescharacteristic
328C2×C32C3×2−1+412yes—
329C2×Q8—32no—
330C22×C4C3×D3211yescharacteristic
331C4⋊C4—32no—
332C4⋊C4—34no—
333C4⋊C4—94no—
334C4⋊C4—94no—
335C4⋊C4—94no—
336C4⋊C4—92no—
337C22⋊C4—34no—
338C22⋊C4—92no—
339C42—94no—
340C42—91no—
341C42—91no—
342C22×C3—22no—
343C22×C3—24no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—44no—
356C22×C3—41no—
357C22×C3C22×D3211yescharacteristic
358C22×C3C23×C3×D312yes—
359C3×C4—22no—
360C3×C4—21no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—24no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—64no—
368C3×C4—64no—
369C3×C4—64no—
370C3×C4—64no—
371C3×C4—124no—
372C3×C4—124no—
373C3×C4—182no—
374C3×C4—182no—
375C3×C4#380512yes—
376C3×C4C3×#23714yes—
377Dic3—24no—
378Dic3—24no—
379Dic3—62no—
380Dic3—62no—
381Dic3—62no—
382Dic3—62no—
383Dic3—122no—
384Dic3—122no—
385C32—21no—
386C32—22no—
387C32—41no—
388C32#562512yescomplemented
389C32C3×#54111yescharacteristic, complemented
390C23C2×C3×D3211yescharacteristic
391Q8—64no—
392C2×C4—22no—
393C2×C4—34no—
394C2×C4—34no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4C2×C3×D3211yescharacteristic
398C2×C4C2×C3×D3211yescharacteristic
399C2×C3—21no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—42no—
405C2×C3—44no—
406C2×C3—44no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—42no—
411C2×C3#6429712yes—
412C2×C3C2×#380511yescharacteristic
413C2×C3C2×C3×#23712yes—
414C2×C3C3×#100712yes—
415C2×C3C3×#101012yes—
416C22—21no—
417C22—22no—
418C22C22×C3×D3211yescharacteristic
419C4—21no—
420C4—64no—
421C4—64no—
422C4—182no—
423C4—182no—
424C4C3×#380512yes—
425C3—21no—
426C3—22no—
427C3—41no—
428C3#1059482611yescharacteristic, direct product factor
429C3C3×#562512yescomplemented
430C2—41no—
431C2C3×#6429712yes—
432C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63873C2semidirect
5. C3×#63872C2semidirect
6. C3×#63845C2semidirect
7. C3×#63843C2semidirect
9. #10594826C3direct (N × H)
39. #63873C2×C3semidirect
40. #63872C2×C3semidirect
41. #63845C2×C3semidirect
42. #63843C2×C3semidirect
271. C33#541semidirect
388. C32#5625semidirect
389. C32C3×#541semidirect
428. C3#10594826direct (N × H)
429. C3C3×#5625semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C32×#9242
C3×#639562
C3×#638732
C3×#638722
C3×#638452
C3×#638432
C3×#56253
#105948263