Finite Group Databasebeta

← C3×#10591215 #13158302

Subgroups — C3×#10591215 #13158302

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C32×Dic3⋊C4C212yesmaximal
3C2×C3×#3724C211yesmaximal, characteristic
4C3×#63579C212yesmaximal
5C3×#63579C211yesmaximal, characteristic
6C3×#4982—32nomaximal
7#10591215C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×C4⋊(C3⋊Dic3)—22no—
14C32×Dic3⋊C4—24no—
15C3×#3724—22no—
16C2×C32×C4⋊C4—32no—
17C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
18C2×C3×C4⋊Dic3—32no—
19C2×C3×C4⋊Dic3—61no—
20C2×C3×Dic3⋊C4—22no—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×Dic3⋊C4D312yes—
23C2×C3×Dic3⋊C4C2×C312yes—
24C2×#3724C2×C311yescharacteristic
25C3×C3⋊#95—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28#63579C2×C312yes—
29#63579C2×C311yescharacteristic
30C23×C33C2311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4D412yes—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D412yes—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3Q814yes—
44C3×#398—91no—
45#4982—32no—
46C22×C32×C4—21no—
47C22×C32×C4—22no—
48C22×C32×C4—32no—
49C22×C32×C4—32no—
50C22×C32×C4—41no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C22×C311yescharacteristic
53C22×C3⋊Dic3C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C311yescharacteristic
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3C2×D312yes—
62C22×C3×Dic3C2×D312yes—
63C22×C3×Dic3C22×C312yes—
64C22×C3×Dic3C22×C312yes—
65C32×C4⋊C4—64no—
66C4⋊(C3⋊Dic3)—22no—
67C3×C4⋊Dic3—64no—
68C3×C4⋊Dic3—64no—
69C3×Dic3⋊C4—28no—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—24no—
72C3×Dic3⋊C4—64no—
73#3724—22no—
74C22×C33C2×D411yescharacteristic
75C22×C33C2×D411yescharacteristic
76C22×C33C2×Q812yes—
77C22×C33#4111yescharacteristic
78C22×C33#4112yes—
79C33×C4—22no—
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—24no—
83C32×Dic3—42no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—62no—
87C2×C3×C4⋊C4—91no—
88C2×C3×C4⋊C4—91no—
89C2×C4⋊Dic3—32no—
90C2×C4⋊Dic3—61no—
91C2×Dic3⋊C4—32no—
92C2×Dic3⋊C4C3×D312yes—
93C3×#95—91no—
94C3×#95—92no—
95C3⋊#95—32no—
96C3⋊#95—32no—
97C3⋊#95—32no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—41no—
101C23×C32C22×D312yes—
102C23×C32C23×C311yescharacteristic
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—24no—
111C2×C32×C4—34no—
112C2×C32×C4—42no—
113C2×C32×C4—42no—
114C2×C32×C4—42no—
115C2×C32×C4—42no—
116C2×C32×C4—42no—
117C2×C32×C4—62no—
118C2×C32×C4—62no—
119C2×C32×C4—62no—
120C2×C32×C4—62no—
121C2×C32×C4—62no—
122C2×C32×C4#6714yes—
123C2×C32×C4C3×D412yes—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3C3×D412yes—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—122no—
153C2×C3×Dic3—122no—
154C2×C3×Dic3Dic614yes—
155C2×C3×Dic3C3×Q814yes—
156#398—91noSylow
157C2×C33#12111yescharacteristic
158C2×C33C22⋊Q812yes—
159C2×C33#12311yescharacteristic
160C2×C33#12512yes—
161C2×C33C4⋊Q811yescharacteristic
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—41no—
169C22×C3×C4—62no—
170C22×C3×C4—62no—
171C22×C3×C4—91no—
172C22×C3×C4—91no—
173C22×C3×C4D3211yescharacteristic
174C22×C3×C4C2×C3×D312yes—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—61no—
179C22×Dic3C2×C3×D312yes—
180C22×Dic3C2×C3×D312yes—
181C3×C4⋊C4—68no—
182C3×C4⋊C4—64no—
183C3×C4⋊C4—64no—
184C3×C4⋊C4—182no—
185C3×C4⋊C4—182no—
186C4⋊Dic3—64no—
187C4⋊Dic3—64no—
188Dic3⋊C4—24no—
189Dic3⋊C4—64no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—42no—
205C22×C32—41no—
206C22×C32—42no—
207C22×C32C2×Dic612yes—
208C22×C32#23512yes—
209C22×C32D3×D412yes—
210C22×C32#23712yes—
211C22×C32D3×Q812yes—
212C22×C32#23912yes—
213C22×C32C2×#6712yes—
214C22×C32C2×C3×D411yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C22×C32C2×C3×Q812yes—
217C22×C32C3×#4112yes—
218C22×C32C3×#4111yescharacteristic
219C32×C4—22no—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—44no—
223C32×C4—44no—
224C32×C4—64no—
225C32×C4—122no—
226C3⋊Dic3—22no—
227C3⋊Dic3—41no—
228C3×Dic3—28no—
229C3×Dic3—24no—
230C3×Dic3—24no—
231C3×Dic3—44no—
232C3×Dic3—42no—
233C3×Dic3—42no—
234C3×Dic3—64no—
235C3×Dic3—122no—
236C3×Dic3—122no—
237C3×Dic3—122no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—91no—
240C2×C4⋊C4—91no—
241#95—91no—
242#95—92no—
243C33#39811yescharacteristic, complemented, Sylow
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—41no—
247C23×C3C2×D3211yescharacteristic
248C23×C3C22×C3×D312yes—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—34no—
258C2×C3×C4—34no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—64no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—64no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—92no—
279C2×C3×C4—122no—
280C2×C3×C4—122no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—181no—
284C2×C3×C4#62512yes—
285C2×C3×C4C3×#6714yes—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—34no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—122no—
300C2×Dic3—122no—
301C2×Dic3C3×Dic614yes—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—41no—
315C2×C32—42no—
316C2×C32—41no—
317C2×C32—42no—
318C2×C32—41no—
319C2×C32#88812yes—
320C2×C32#88912yes—
321C2×C32#89012yes—
322C2×C32#89412yes—
323C2×C32#92412yes—
324C2×C32#93912yes—
325C2×C32#94612yes—
326C2×C32C3×#12111yescharacteristic
327C2×C32C3×C22⋊Q812yes—
328C2×C32C3×#12311yescharacteristic
329C2×C32C3×#12512yes—
330C2×C32C3×C4⋊Q811yescharacteristic
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—91no—
334C22×C4—91no—
335C22×C4C3×D3211yescharacteristic
336C4⋊C4—64no—
337C4⋊C4—182no—
338C4⋊C4—182no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3D3×Dic612yes—
357C22×C3#379912yes—
358C22×C3#380511yescharacteristic
359C22×C3C2×#62511yescharacteristic
360C22×C3#381111yescharacteristic
361C22×C3C2×C3×Dic612yes—
362C22×C3C3×#23512yes—
363C22×C3C3×D3×D412yes—
364C22×C3C3×#23712yes—
365C22×C3C3×D3×Q812yes—
366C22×C3C3×#23912yes—
367C22×C3C2×C3×#6712yes—
368C3×C4—24no—
369C3×C4—24no—
370C3×C4—22no—
371C3×C4—44no—
372C3×C4—44no—
373C3×C4—68no—
374C3×C4—64no—
375C3×C4—64no—
376C3×C4—124no—
377C3×C4—122no—
378C3×C4—122no—
379C3×C4—182no—
380C3×C4—361no—
381Dic3—24no—
382Dic3—42no—
383Dic3—64no—
384Dic3—122no—
385Dic3—122no—
386Dic3—122no—
387C32—22no—
388C32—21no—
389C32—41no—
390C32#498212yescomplemented
391C32C3×#39811yescharacteristic, complemented
392C23C2×C3×D3211yescharacteristic
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—34no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—92no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—181no—
405C2×C4C3×#62512yes—
406C2×C3—21no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—22no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—42no—
421C2×C3—42no—
422C2×C3—41no—
423C2×C3#6384011yescharacteristic
424C2×C3#6384512yes—
425C2×C3#6385612yes—
426C2×C3#6389711yescharacteristic
427C2×C3#6398011yescharacteristic
428C2×C3C3×#88812yes—
429C2×C3C3×#88912yes—
430C2×C3C3×#89012yes—
431C2×C3C3×#89412yes—
432C2×C3C3×#92412yes—
433C2×C3C3×#93912yes—
434C2×C3C3×#94612yes—
435C22C3×D3×Dic612yes—
436C22C3×#379912yes—
437C22C3×#380511yescharacteristic
438C22C2×C3×#62511yescharacteristic
439C22C3×#381111yescharacteristic
440C4—22no—
441C4—64no—
442C4—122no—
443C4—182no—
444C4—361no—
445C3—22no—
446C3—21no—
447C3—41no—
448C3#1059121511yescharacteristic, direct product factor
449C3C3×#498212yescomplemented
450C2C3×#6384011yescharacteristic
451C2C3×#6384512yes—
452C2C3×#6385612yes—
453C2C3×#6389711yescharacteristic
454C2C3×#6398011yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591215C3direct (N × H)
243. C33#398semidirect
390. C32#4982semidirect
391. C32C3×#398semidirect
448. C3#10591215direct (N × H)
449. C3C3×#4982semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×Dic3⋊C42
C2×C3×#37242
C3×#635792
C3×#49823
#105912153