Finite Group Databasebeta

← C3×#10592675 #13163008

Subgroups — C3×#10592675 #13163008

Order 1728.

1388 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic
3C2×C3×#3750C211yesmaximal, characteristic
4C2×C3×#3750C211yesmaximal, characteristic
5C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
6C3×#63658C211yesmaximal, characteristic
7C3×#63658C211yesmaximal, characteristic
8#10592675C311yesmaximal, characteristic, direct product factor
9C3×#4976—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C33×C4⋊C4—22no—
18C3×#3750—22no—
19C3×#3750—22no—
20C3×#3750C414yescomplemented
21C3×C4×C3⋊Dic3—22no—
22C2×C32×C4⋊C4—24no—
23C2×C32×C4⋊C4D314yes—
24C2×C32×C4⋊C4C2×C311yescharacteristic
25C2×#3750C2×C311yescharacteristic
26C2×#3750C2×C311yescharacteristic
27C2×#3750C2×C311yescharacteristic
28C2×C4×C3⋊Dic3C2×C311yescharacteristic
29C2×C3×Dic3⋊C4—34no—
30C2×C3×Dic3⋊C4—34no—
31C2×C3×Dic3⋊C4—34no—
32C2×C3×C4×Dic3—34no—
33#63658C2×C311yescharacteristic
34#63658C2×C311yescharacteristic
35C3×C3⋊#95—34no—
36C3×C3⋊#95—34no—
37C23×C33C2311yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2×C412yes—
46C2×C33×C4D412yes—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3C2×C412yes—
55C2×C3×C3⋊Dic3C2×C412yes—
56C2×C3×C3⋊Dic3D412yes—
57C2×C3×C3⋊Dic3Q812yes—
58C2×C3×C3⋊Dic3Q812yes—
59C3×#389—91no—
60#4976—34no—
61C22×C32×C4—24no—
62C22×C32×C4—24no—
63C22×C32×C4—24no—
64C22×C32×C4C2×D314yes—
65C22×C32×C4C2×D314yes—
66C22×C32×C4C2×D314yes—
67C22×C32×C4C22×C311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3×Dic3—34no—
75C22×C3×Dic3—34no—
76C22×C3×Dic3—34no—
77C22×C3×Dic3—34no—
78C32×C4⋊C4—216no—
79C32×C4⋊C4—28no—
80C32×C4⋊C4—22no—
81#3750—22no—
82#3750—22no—
83#3750C3×C414yescomplemented
84C4×C3⋊Dic3—22no—
85C3×Dic3⋊C4—316no—
86C3×Dic3⋊C4—68no—
87C3×Dic3⋊C4—68no—
88C3×C4×Dic3—68no—
89C22×C33C22×C411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C22×C33C2×D411yescharacteristic
92C22×C33C2×Q811yescharacteristic
93C22×C33C2×Q811yescharacteristic
94C22×C33#4111yescharacteristic
95C22×C33#4111yescharacteristic
96C33×C4—22no—
97C33×C4—22no—
98C33×C4—41no—
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3C4⋊C414yescomplemented
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4—91no—
105C2×C3×C4⋊C4—91no—
106C2×C3×C4⋊C4—91no—
107C2×C3×C4⋊C4C3×D314yes—
108C2×C3×C4⋊C4C3⋊D311yescharacteristic
109C2×C3×C42—91no—
110C2×Dic3⋊C4—34no—
111C2×Dic3⋊C4—34no—
112C2×Dic3⋊C4—34no—
113C2×C4×Dic3—34no—
114C3×#95—91no—
115C3×#95—91no—
116C3⋊#95—34no—
117C3⋊#95—34no—
118C23×C32—24no—
119C23×C32C22×D314yes—
120C23×C32C23×C311yescharacteristic
121C2×C32×C4—28no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—28no—
130C2×C32×C4—24no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—28no—
135C2×C32×C4—28no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—44no—
141C2×C32×C4—44no—
142C2×C32×C4—44no—
143C2×C32×C4—44no—
144C2×C32×C4C4×D318yes—
145C2×C32×C4#6718yes—
146C2×C32×C4C2×C3×C412yes—
147C2×C32×C4C3×D412yes—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3C2×C3×C412yes—
156C2×C3⋊Dic3C2×C3×C412yes—
157C2×C3⋊Dic3C3×D412yes—
158C2×C3⋊Dic3C3×Q812yes—
159C2×C3⋊Dic3C3×Q812yes—
160C2×C3×Dic3—38no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—38no—
163C2×C3×Dic3—38no—
164C2×C3×Dic3—38no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—64no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172#389—91noSylow
173C2×C33C2×C4⋊C411yescharacteristic
174C2×C33C4×D411yescharacteristic
175C2×C33C4×Q811yescharacteristic
176C2×C33#12111yescharacteristic
177C2×C33C22⋊Q811yescharacteristic
178C2×C33#12511yescharacteristic
179C2×C33C4⋊Q811yescharacteristic
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—91no—
184C22×C3×C4—91no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C22×C3×C4C2×C3×D314yes—
188C22×C3×C4C2×C3×D314yes—
189C22×C3×C4C2×C3×D314yes—
190C22×C3×C4C2×C3⋊D311yescharacteristic
191C22×C3×C4C2×C3⋊D311yescharacteristic
192C22×C3×C4C2×C3⋊D311yescharacteristic
193C22×Dic3—34no—
194C22×Dic3—34no—
195C22×Dic3—34no—
196C22×Dic3—34no—
197C3×C4⋊C4—216no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—182no—
202C3×C4⋊C4—182no—
203C3×C42—182no—
204Dic3⋊C4—316no—
205Dic3⋊C4—68no—
206Dic3⋊C4—68no—
207C4×Dic3—68no—
208C22×C32—24no—
209C22×C32—24no—
210C22×C32—24no—
211C22×C32—24no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—24no—
215C22×C32C2×C4×D314yes—
216C22×C32#23514yes—
217C22×C32D3×D414yes—
218C22×C32#23714yes—
219C22×C32D3×Q814yes—
220C22×C32D3×Q814yes—
221C22×C32C2×#6714yes—
222C22×C32C22×C3×C411yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C22×C32C2×C3×Q811yescharacteristic
226C22×C32C2×C3×Q811yescharacteristic
227C22×C32C3×#4111yescharacteristic
228C22×C32C3×#4111yescharacteristic
229C32×C4—216no—
230C32×C4—28no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—28no—
234C32×C4—48no—
235C32×C4—48no—
236C32×C4—44no—
237C32×C4—41no—
238C3⋊Dic3—22no—
239C3⋊Dic3—22no—
240C3⋊Dic3—41no—
241C3⋊Dic3C3×C4⋊C414yescomplemented
242C3×Dic3—316no—
243C3×Dic3—68no—
244C3×Dic3—68no—
245C3×Dic3—124no—
246C2×C4⋊C4—91no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—91no—
249C2×C4⋊C4C3×C3⋊D311yescharacteristic
250C2×C42—91no—
251#95—91no—
252#95—91no—
253C33#38911yescharacteristic, complemented, Sylow
254C23×C3—24no—
255C23×C3C22×C3×D314yes—
256C23×C3C22×C3⋊D311yescharacteristic
257C2×C3×C4—28no—
258C2×C3×C4—28no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—28no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—28no—
275C2×C3×C4—28no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4C3×C4×D318yes—
293C2×C3×C4C3×#6718yes—
294C2×C3×C4C4×C3⋊D312yes—
295C2×C3×C4#63812yes—
296C2×Dic3—38no—
297C2×Dic3—38no—
298C2×Dic3—38no—
299C2×Dic3—38no—
300C2×Dic3—38no—
301C2×Dic3—64no—
302C2×Dic3—64no—
303C2×Dic3—64no—
304C2×Dic3—64no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—64no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—24no—
314C2×C32—24no—
315C2×C32#88714yes—
316C2×C32#88914yes—
317C2×C32D3×C4⋊C414yes—
318C2×C32#89614yes—
319C2×C32C4×#6714yes—
320C2×C32#93914yes—
321C2×C32#94414yes—
322C2×C32C2×C3×C4⋊C411yescharacteristic
323C2×C32C3×C4×D411yescharacteristic
324C2×C32C3×C4×Q811yescharacteristic
325C2×C32C3×#12111yescharacteristic
326C2×C32C3×C22⋊Q811yescharacteristic
327C2×C32C3×#12511yescharacteristic
328C2×C32C3×C4⋊Q811yescharacteristic
329C22×C4—91no—
330C22×C4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4C2×C3×C3⋊D311yescharacteristic
334C22×C4C2×C3×C3⋊D311yescharacteristic
335C22×C4C2×C3×C3⋊D311yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—94no—
338C4⋊C4—182no—
339C4⋊C4—182no—
340C42—182no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—24no—
344C22×C3—24no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3C2×C3×C4×D314yes—
349C22×C3C3×#23514yes—
350C22×C3C3×D3×D414yes—
351C22×C3C3×#23714yes—
352C22×C3C3×D3×Q814yes—
353C22×C3C3×D3×Q814yes—
354C22×C3C2×C3×#6714yes—
355C22×C3C2×C4×C3⋊D311yescharacteristic
356C22×C3#382811yescharacteristic
357C22×C3D4×C3⋊D311yescharacteristic
358C22×C3#383011yescharacteristic
359C22×C3Q8×C3⋊D311yescharacteristic
360C22×C3Q8×C3⋊D311yescharacteristic
361C22×C3C2×#63811yescharacteristic
362C3×C4—216no—
363C3×C4—28no—
364C3×C4—28no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—48no—
368C3×C4—48no—
369C3×C4—44no—
370C3×C4—41no—
371C3×C4—94no—
372C3×C4—182no—
373C3×C4—182no—
374C3×C4—361no—
375Dic3—316no—
376Dic3—68no—
377Dic3—68no—
378Dic3—124no—
379C32—24no—
380C32#497614yescomplemented
381C32C3×#38911yescharacteristic, complemented
382C23C22×C3×C3⋊D311yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—181no—
402C2×C4C3×C4×C3⋊D312yes—
403C2×C4C3×#63812yes—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—24no—
410C2×C3—24no—
411C2×C3C3×#88714yes—
412C2×C3C3×#88914yes—
413C2×C3C3×D3×C4⋊C414yes—
414C2×C3C3×#89614yes—
415C2×C3C3×C4×#6714yes—
416C2×C3C3×#93914yes—
417C2×C3C3×#94414yes—
418C2×C3#6409611yescharacteristic
419C2×C3#6409811yescharacteristic
420C2×C3C4⋊C4×C3⋊D311yescharacteristic
421C2×C3#6410511yescharacteristic
422C2×C3C4×#63811yescharacteristic
423C2×C3#6414811yescharacteristic
424C2×C3#6415311yescharacteristic
425C22C2×C3×C4×C3⋊D311yescharacteristic
426C22C3×#382811yescharacteristic
427C22C3×D4×C3⋊D311yescharacteristic
428C22C3×#383011yescharacteristic
429C22C3×Q8×C3⋊D311yescharacteristic
430C22C3×Q8×C3⋊D311yescharacteristic
431C22C2×C3×#63811yescharacteristic
432C4—22no—
433C4—22no—
434C4—41no—
435C4—94no—
436C4—182no—
437C4—182no—
438C4—361no—
439C3—24no—
440C3C3×#497614yescomplemented
441C3#1059267511yescharacteristic, direct product factor
442C2C3×#6409611yescharacteristic
443C2C3×#6409811yescharacteristic
444C2C3×C4⋊C4×C3⋊D311yescharacteristic
445C2C3×#6410511yescharacteristic
446C2C3×C4×#63811yescharacteristic
447C2C3×#6414811yescharacteristic
448C2C3×#6415311yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#389
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #10592675C3direct (N × H)
20. C3×#3750C4semidirect
83. #3750C3×C4semidirect
102. C3×C3⋊Dic3C4⋊C4semidirect
241. C3⋊Dic3C3×C4⋊C4semidirect
253. C33#389semidirect
380. C32#4976semidirect
381. C32C3×#389semidirect
440. C3C3×#4976semidirect
441. C3#10592675direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C3×#636582
#105926753
C3×#49763