Finite Group Databasebeta

← C3×#10591059 #13158087

Subgroups — C3×#10591059 #13158087

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C2×C32×#208C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63555C212yesmaximal, complemented
5C3×#63553C212yesmaximal, complemented
6C3×#5122—31nomaximal
7C3×#5101—31nomaximal
8#10591059C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C3×#3757C412yes—
11C3×C4⋊(C3⋊Dic3)C412yes—
12C3×C4×C3⋊Dic3C2212yes—
13C32×#208C414yescomplemented
14C2×C32×C3⋊C8C414yescomplemented
15C2×C32×C3⋊C8C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C2×C32×M16—31no—
19C22×C32×C8—31no—
20#64135C2×C311yescharacteristic
21C3×#926—31no—
22C3×#926—31no—
23C3×#926—61no—
24C2×C3×#208—21no—
25C2×C3×#208D311yescharacteristic
26C2×C3×#208C2×C311yescharacteristic
27C22×C3×C3⋊C8—21no—
28C22×C3×C3⋊C8D311yescharacteristic
29C22×C3×C3⋊C8C2×C311yescharacteristic
30C3×#815—32no—
31C3×#815—32no—
32C3×C8×Dic3—32no—
33C3×C8×Dic3—32no—
34#63555C2×C312yescomplemented
35#63553C2×C312yescomplemented
36C23×C33C2×C411yescharacteristic
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4C2×C412yes—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3C2×C414yes—
42C32×C3⋊C8C2×C412yes—
43C32×C3⋊C8C2×C412yes—
44C32×C3⋊C8C2×C414yes—
45C3×#405—91no—
46#5122—31no—
47#5101—31no—
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—41no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C32×M16—34no—
56C2×C32×C8—34no—
57C2×C32×C8—31no—
58C2×C32×C8—32no—
59C2×C32×C8—31no—
60#3757C3×C412yes—
61C4⋊(C3⋊Dic3)C3×C412yes—
62C4×C3⋊Dic3C22×C312yes—
63C3×#217—32no—
64C3×#217—32no—
65C3×#217—62no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—62no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—62no—
72C3×#208—24no—
73C3×#208Dic314yescomplemented
74C3×#208C3×C414yescomplemented
75C2×C3×C3⋊C8—24no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—22no—
79C2×C3×C3⋊C8Dic314yescomplemented
80C2×C3×C3⋊C8C3×C414yescomplemented
81C2×C3×C3⋊C8C2×D311yescharacteristic
82C2×C3×C3⋊C8C2×D311yescharacteristic
83C2×C3×C3⋊C8C2×D312yes—
84C2×C3×C3⋊C8C22×C311yescharacteristic
85C2×C3×C3⋊C8C22×C311yescharacteristic
86C2×C3×C3⋊C8C22×C312yes—
87C22×C33—22no—
88C22×C33C4212yes—
89C22×C33C22×C411yescharacteristic
90C33×C4C4212yes—
91C33×C4C22×C411yescharacteristic
92C33×C4C22×C411yescharacteristic
93C3×C3⋊Dic3—24no—
94C2×C3×M16—31no—
95C2×C3×M16—31no—
96C2×C3×M16—61no—
97C22×C3×C8—31no—
98C22×C3×C8—31no—
99C22×C3×C8—61no—
100C3×#117—91no—
101#926—31no—
102#926—31no—
103#926—61no—
104C2×#208C3×D311yescharacteristic
105C22×C3⋊C8C3×D311yescharacteristic
106C3×#97—92no—
107C3×C4×C8—92no—
108#815—32no—
109#815—32no—
110C8×Dic3—32no—
111C8×Dic3—32no—
112C23×C32—21no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C23×C32C4×D311yescharacteristic
117C23×C32C4×D311yescharacteristic
118C23×C32C2×C3×C411yescharacteristic
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—42no—
132C2×C32×C4—42no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4C4×D312yes—
136C2×C32×C4C4×D312yes—
137C2×C32×C4C4×D311yescharacteristic
138C2×C32×C4C4×D311yescharacteristic
139C2×C32×C4C2×Dic312yes—
140C2×C32×C4C2×Dic312yes—
141C2×C32×C4C2×C3×C411yescharacteristic
142C2×C32×C4C2×C3×C412yes—
143C2×C32×C4C2×C3×C412yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C23×C311yescharacteristic
147C2×C3⋊Dic3C2×C3×C414yes—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—64no—
151C32×C8—32no—
152C32×C8—32no—
153C32×C8—34no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—24no—
157C3×C3⋊C8C2×Dic312yes—
158C3×C3⋊C8C2×Dic312yes—
159C3×C3⋊C8C2×Dic314yes—
160C3×C3⋊C8C2×C3×C412yes—
161C3×C3⋊C8C2×C3×C412yes—
162C3×C3⋊C8C2×C3×C414yes—
163#405—91noSylow
164C2×C33—22no—
165C2×C33C2×C4211yescharacteristic
166C2×C33#13112yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4D3211yescharacteristic
172C22×C3×C4C2×C3×D311yescharacteristic
173C22×C3×C4C2×C3×D311yescharacteristic
174C3×M16—34no—
175C3×M16—34no—
176C3×M16—64no—
177C2×C3×C8—34no—
178C2×C3×C8—34no—
179C2×C3×C8—31no—
180C2×C3×C8—32no—
181C2×C3×C8—31no—
182C2×C3×C8—31no—
183C2×C3×C8—32no—
184C2×C3×C8—31no—
185C2×C3×C8—64no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C2×C3×C8—62no—
189C3×C4⋊C4—92no—
190C3×C22⋊C4—92no—
191C3×C42—92no—
192#217—32no—
193#217—32no—
194#217—62no—
195C4⋊Dic3—32no—
196C4⋊Dic3—32no—
197C4⋊Dic3—62no—
198C4×Dic3—32no—
199C4×Dic3—32no—
200C4×Dic3—62no—
201#208C3×Dic314yescomplemented
202C2×C3⋊C8C3×Dic314yescomplemented
203C2×C3⋊C8C2×C3×D311yescharacteristic
204C2×C3⋊C8C2×C3×D312yes—
205C2×C3⋊C8C2×C3×D311yescharacteristic
206C22×C32—22no—
207C22×C32—24no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—42no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—44no—
221C22×C32C4×Dic312yes—
222C22×C32C4×Dic312yes—
223C22×C32C3×C4212yes—
224C22×C32C2×C4×D311yescharacteristic
225C22×C32C2×C4×D311yescharacteristic
226C22×C32C22×C3×C411yescharacteristic
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—42no—
237C32×C4—41no—
238C32×C4—41no—
239C32×C4C4×Dic312yes—
240C32×C4C4×Dic312yes—
241C32×C4C3×C4212yes—
242C32×C4C2×C4×D311yescharacteristic
243C32×C4C2×C4×D311yescharacteristic
244C32×C4C22×Dic311yescharacteristic
245C32×C4C22×Dic311yescharacteristic
246C32×C4C22×C3×C411yescharacteristic
247C32×C4C22×C3×C411yescharacteristic
248C3⋊Dic3—24no—
249C3×Dic3—64no—
250C3×Dic3—64no—
251C3×Dic3—124no—
252C2×M16—31no—
253C22×C8—31no—
254#117—91no—
255#97—92no—
256C4×C8—92no—
257C33#40511yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3#62411yescharacteristic
263C23×C3C3×C4×D311yescharacteristic
264C23×C3C3×C4×D311yescharacteristic
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—94no—
282C2×C3×C4D3×Dic312yes—
283C2×C3×C4D3×Dic312yes—
284C2×C3×C4#62411yescharacteristic
285C2×C3×C4C3×C4×D312yes—
286C2×C3×C4C3×C4×D312yes—
287C2×C3×C4C3×C4×D311yescharacteristic
288C2×C3×C4C3×C4×D311yescharacteristic
289C2×C3×C4C2×C3×Dic312yes—
290C2×C3×C4C2×C3×Dic312yes—
291C2×C3×C4C2×D3211yescharacteristic
292C2×C3×C4C22×C3×D311yescharacteristic
293C2×C3×C4C22×C3×D311yescharacteristic
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—64no—
297C3×C8—32no—
298C3×C8—32no—
299C3×C8—34no—
300C3×C8—32no—
301C3×C8—32no—
302C3×C8—34no—
303C3×C8—62no—
304C3×C8—62no—
305C3×C8—64no—
306C3⋊C8C2×C3×Dic312yes—
307C3⋊C8C2×C3×Dic312yes—
308C3⋊C8C2×C3×Dic314yes—
309C2×C32—22no—
310C2×C32—24no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—41no—
320C2×C32—42no—
321C2×C32—42no—
322C2×C32—42no—
323C2×C32—44no—
324C2×C32#90112yes—
325C2×C32#90712yes—
326C2×C32C2×C4×Dic311yescharacteristic
327C2×C32C2×C4×Dic311yescharacteristic
328C2×C32C2×C3×C4211yescharacteristic
329C2×C32C3×#13112yes—
330C22×C4C3×D3211yescharacteristic
331M16—34no—
332C2×C8—34no—
333C2×C8—31no—
334C2×C8—32no—
335C2×C8—31no—
336C4⋊C4—92no—
337C22⋊C4—92no—
338C42—92no—
339C22×C3—24no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—44no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3Dic3212yes—
355C22×C3C3×C4×Dic312yes—
356C22×C3C3×C4×Dic312yes—
357C22×C3C2×#62411yescharacteristic
358C22×C3C2×C3×C4×D311yescharacteristic
359C22×C3C2×C3×C4×D311yescharacteristic
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—42no—
372C3×C4—184no—
373C3×C4Dic3212yes—
374C3×C4C3×C4×Dic312yes—
375C3×C4C3×C4×Dic312yes—
376C3×C4C2×D3×Dic311yescharacteristic
377C3×C4C2×D3×Dic311yescharacteristic
378C3×C4C2×C3×C4×D311yescharacteristic
379C3×C4C2×C3×C4×D311yescharacteristic
380C3×C4C22×C3×Dic311yescharacteristic
381C3×C4C22×C3×Dic311yescharacteristic
382Dic3—64no—
383Dic3—64no—
384Dic3—124no—
385C32—21no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32#510111yescharacteristic, complemented
390C32#512211yescharacteristic, complemented
391C32C3×#40511yescharacteristic, complemented
392C23C3×#62411yescharacteristic
393C2×C4—94no—
394C2×C4C3×D3×Dic312yes—
395C2×C4C3×D3×Dic312yes—
396C2×C4C3×#62411yescharacteristic
397C2×C4C2×C3×D3211yescharacteristic
398C8—32no—
399C8—32no—
400C8—34no—
401C2×C3—24no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—42no—
412C2×C3—42no—
413C2×C3—44no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3#6381712yes—
417C2×C3C2×Dic3211yescharacteristic
418C2×C3C3×#90112yes—
419C2×C3C3×#90712yes—
420C2×C3C2×C3×C4×Dic311yescharacteristic
421C2×C3C2×C3×C4×Dic311yescharacteristic
422C22—22no—
423C22C3×Dic3212yes—
424C22C2×C3×#62411yescharacteristic
425C4—184no—
426C4C3×Dic3212yes—
427C4C2×C3×D3×Dic311yescharacteristic
428C4C2×C3×D3×Dic311yescharacteristic
429C3—21no—
430C3—21no—
431C3—21no—
432C3—41no—
433C3#1059105911yescharacteristic, direct product factor
434C3C3×#510111yescharacteristic, complemented
435C3C3×#512211yescharacteristic, complemented
436C2—22no—
437C2C3×#6381712yes—
438C2C2×C3×Dic3211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#405
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63555C2semidirect
5. C3×#63553C2semidirect
8. #10591059C3direct (N × H)
13. C32×#208C4semidirect
14. C2×C32×C3⋊C8C4semidirect
34. #63555C2×C3semidirect
35. #63553C2×C3semidirect
73. C3×#208Dic3semidirect
74. C3×#208C3×C4semidirect
79. C2×C3×C3⋊C8Dic3semidirect
80. C2×C3×C3⋊C8C3×C4semidirect
201. #208C3×Dic3semidirect
202. C2×C3⋊C8C3×Dic3semidirect
257. C33#405semidirect
389. C32#5101semidirect
390. C32#5122semidirect
391. C32C3×#405semidirect
433. C3#10591059direct (N × H)
434. C3C3×#5101semidirect
435. C3C3×#5122semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C2×C32×#2082
C22×C32×C3⋊C82
C3×#635552
C3×#635532
C3×#51223
C3×#51013
#105910593