Finite Group Databasebeta

← C3×#10592645 #13162984

Subgroups — C3×#10592645 #13162984

Order 1728.

1932 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C33×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#3757C211yesmaximal, characteristic
4C2×C3×#3750C212yesmaximal, complemented
5C3×#63658C212yesmaximal, complemented
6#10592645C311yesmaximal, characteristic, direct product factor
7C3×#4946—34nomaximal
8C24×C33C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3—22no—
12C22×C3×C3⋊Dic3—24no—
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2212yes—
15C33×C22⋊C4—22no—
16C3×#3757C414yescomplemented
17C3×#3750—24no—
18C23×C3⋊Dic3C2×C311yescharacteristic
19C23×C3×Dic3—34no—
20C2×C32×C22⋊C4—24no—
21C2×C32×C22⋊C4D314yes—
22C2×C32×C22⋊C4C2×C311yescharacteristic
23C2×#3757C2×C311yescharacteristic
24C2×#3750C2×C312yescomplemented
25C2×C3×#217—34no—
26C2×C3×Dic3⋊C4—38no—
27#63658C2×C312yescomplemented
28C3×C3⋊#95—38no—
29C23×C33—22no—
30C23×C33—22no—
31C23×C33C2×C412yes—
32C23×C33D412yes—
33C23×C33Q812yes—
34C23×C33C2311yescharacteristic
35C2×C33×C4—21no—
36C2×C33×C4—21no—
37C2×C33×C4—22no—
38C2×C33×C4—22no—
39C2×C3×C3⋊Dic3—28no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3C2×C414yescomplemented
45C2×C3×C3⋊Dic3D414yes—
46C3×#385—91no—
47#4946—34no—
48C24×C32—24no—
49C24×C32C2×D314yes—
50C24×C32C22×C311yescharacteristic
51C22×C32×C4—24no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C2×D314yes—
55C22×C32×C4C22×C311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C3⋊Dic3—22no—
58C22×C3⋊Dic3—24no—
59C22×C3⋊Dic3C22×C312yes—
60C22×C3⋊Dic3C22×C312yes—
61C22×C3×Dic3—38no—
62C22×C3×Dic3—38no—
63C22×C3×Dic3—616no—
64C22×C3×Dic3—68no—
65C32×C22⋊C4—216no—
66C32×C22⋊C4—28no—
67C32×C22⋊C4—22no—
68#3757C3×C414yescomplemented
69#3750—24no—
70C3×#217—316no—
71C3×Dic3⋊C4—616no—
72C22×C33—24no—
73C22×C33—22no—
74C22×C33—24no—
75C22×C33—22no—
76C22×C33C4⋊C414yescomplemented
77C22×C33C22×C411yescharacteristic
78C22×C33C2×D411yescharacteristic
79C22×C33C2×D412yes—
80C22×C33C2×Q811yescharacteristic
81C22×C33#4112yes—
82C33×C4—41no—
83C33×C4—41no—
84C3×C3⋊Dic3—24no—
85C3×C3⋊Dic3—24no—
86C23×C3×C4—91no—
87C23×Dic3—34no—
88C2×C3×C4⋊C4—92no—
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—91no—
91C2×C3×C22⋊C4C3×D314yes—
92C2×C3×C22⋊C4C3⋊D311yescharacteristic
93C2×#217—34no—
94C2×Dic3⋊C4—38no—
95C3×#95—92no—
96C3⋊#95—38no—
97C23×C32—22no—
98C23×C32—22no—
99C23×C32—28no—
100C23×C32—28no—
101C23×C32—28no—
102C23×C32—24no—
103C23×C32—28no—
104C23×C32—28no—
105C23×C32—48no—
106C23×C32—48no—
107C23×C32Dic618yes—
108C23×C32C4×D318yes—
109C23×C32#6718yes—
110C23×C32C2×C3×C412yes—
111C23×C32C3×D412yes—
112C23×C32C3×Q812yes—
113C23×C32C22×D314yes—
114C23×C32C23×C311yescharacteristic
115C2×C32×C4—28no—
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—28no—
124C2×C32×C4—28no—
125C2×C32×C4—48no—
126C2×C32×C4—48no—
127C2×C3⋊Dic3—28no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3C2×C3×C414yescomplemented
133C2×C3⋊Dic3C3×D414yes—
134C2×C3×Dic3—316no—
135C2×C3×Dic3—316no—
136C2×C3×Dic3—632no—
137C2×C3×Dic3—68no—
138C2×C3×Dic3—68no—
139C2×C3×Dic3—68no—
140C2×C3×Dic3—68no—
141#385—91noSylow
142C2×C33—24no—
143C2×C33C2×C4⋊C411yescharacteristic
144C2×C33C4×D412yes—
145C2×C33#12011yescharacteristic
146C2×C33C22⋊Q812yes—
147C2×C33#12311yescharacteristic
148C24×C3—24no—
149C24×C3C2×C3×D314yes—
150C24×C3C2×C3⋊D311yescharacteristic
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—92no—
154C22×C3×C4—92no—
155C22×C3×C4—184no—
156C22×C3×C4—182no—
157C22×C3×C4C2×C3×D314yes—
158C22×C3×C4C2×C3×D314yes—
159C22×C3×C4C2×C3⋊D311yescharacteristic
160C22×C3×C4C2×C3⋊D311yescharacteristic
161C22×Dic3—38no—
162C22×Dic3—38no—
163C22×Dic3—616no—
164C22×Dic3—68no—
165C3×C4⋊C4—184no—
166C3×C22⋊C4—216no—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—94no—
170#217—316no—
171Dic3⋊C4—616no—
172C22×C32—24no—
173C22×C32—22no—
174C22×C32—24no—
175C22×C32—22no—
176C22×C32—216no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—28no—
180C22×C32—28no—
181C22×C32—24no—
182C22×C32—216no—
183C22×C32—216no—
184C22×C32—28no—
185C22×C32—28no—
186C22×C32—416no—
187C22×C32—416no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32Dic3⋊C4116yescomplemented
191C22×C32C3×C4⋊C414yescomplemented
192C22×C32C2×Dic614yes—
193C22×C32C2×C4×D314yes—
194C22×C32D3×D418yes—
195C22×C32#23718yes—
196C22×C32C2×#6714yes—
197C22×C32C22×C3×C411yescharacteristic
198C22×C32C2×C3×D412yes—
199C22×C32C2×C3×D411yescharacteristic
200C22×C32C2×C3×Q811yescharacteristic
201C22×C32C3×#4112yes—
202C32×C4—48no—
203C32×C4—48no—
204C32×C4—44no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4—41no—
208C3⋊Dic3—24no—
209C3⋊Dic3—24no—
210C3×Dic3—616no—
211C3×Dic3—616no—
212C23×C4—91no—
213C2×C4⋊C4—92no—
214C2×C22⋊C4—91no—
215C2×C22⋊C4C3×C3⋊D311yescharacteristic
216#95—92no—
217C33#38511yescharacteristic, complemented, Sylow
218C23×C3—28no—
219C23×C3—28no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—28no—
223C23×C3—28no—
224C23×C3—28no—
225C23×C3—24no—
226C23×C3—48no—
227C23×C3—48no—
228C23×C3C3×Dic618yes—
229C23×C3C3×C4×D318yes—
230C23×C3C3×#6718yes—
231C23×C3#63412yes—
232C23×C3C4×C3⋊D312yes—
233C23×C3#63812yes—
234C23×C3C22×C3×D314yes—
235C23×C3C22×C3⋊D311yescharacteristic
236C2×C3×C4—28no—
237C2×C3×C4—28no—
238C2×C3×C4—28no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—48no—
247C2×C3×C4—48no—
248C2×C3×C4—94no—
249C2×C3×C4—94no—
250C2×C3×C4—188no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×Dic3—316no—
256C2×Dic3—316no—
257C2×Dic3—632no—
258C2×Dic3—68no—
259C2×Dic3—68no—
260C2×Dic3—68no—
261C2×Dic3—68no—
262C2×C32—24no—
263C2×C32—28no—
264C2×C32—28no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—216no—
269C2×C32—416no—
270C2×C32#87818yes—
271C2×C32#67⋊C418yes—
272C2×C32C2×Dic3⋊C414yes—
273C2×C32#93514yes—
274C2×C32#93714yes—
275C2×C32C2×C3×C4⋊C411yescharacteristic
276C2×C32C3×C4×D412yes—
277C2×C32C3×#12011yescharacteristic
278C2×C32C3×C22⋊Q812yes—
279C2×C32C3×#12311yescharacteristic
280C24C2×C3×C3⋊D311yescharacteristic
281C22×C4—92no—
282C22×C4—92no—
283C22×C4—184no—
284C22×C4—182no—
285C22×C4C2×C3×C3⋊D311yescharacteristic
286C22×C4C2×C3×C3⋊D311yescharacteristic
287C4⋊C4—184no—
288C22⋊C4—22no—
289C22⋊C4—94no—
290C22×C3—216no—
291C22×C3—216no—
292C22×C3—28no—
293C22×C3—28no—
294C22×C3—24no—
295C22×C3—24no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—216no—
299C22×C3—24no—
300C22×C3—24no—
301C22×C3—28no—
302C22×C3—28no—
303C22×C3—24no—
304C22×C3—416no—
305C22×C3—416no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3C3×Dic3⋊C4116yescomplemented
309C22×C3#375014yescomplemented
310C22×C3C2×C3×Dic614yes—
311C22×C3C2×C3×C4×D314yes—
312C22×C3C3×D3×D418yes—
313C22×C3C3×#23718yes—
314C22×C3C2×C3×#6714yes—
315C22×C3C2×#63411yescharacteristic
316C22×C3C2×C4×C3⋊D311yescharacteristic
317C22×C3D4×C3⋊D312yes—
318C22×C3#383012yes—
319C22×C3C2×#63811yescharacteristic
320C3×C4—48no—
321C3×C4—48no—
322C3×C4—44no—
323C3×C4—44no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—184no—
327C3×C4—184no—
328Dic3—616no—
329Dic3—616no—
330C32—24no—
331C32#494614yescomplemented
332C32C3×#38511yescharacteristic, complemented
333C23—22no—
334C23—22no—
335C23C3×#63412yes—
336C23C3×C4×C3⋊D312yes—
337C23C3×#63812yes—
338C23C22×C3×C3⋊D311yescharacteristic
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—94no—
344C2×C4—94no—
345C2×C4—188no—
346C2×C4—182no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4—182no—
350C2×C3—216no—
351C2×C3—24no—
352C2×C3—28no—
353C2×C3—28no—
354C2×C3—24no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—416no—
358C2×C3C3×#87818yes—
359C2×C3C3×#67⋊C418yes—
360C2×C3C2×C3×Dic3⋊C414yes—
361C2×C3C3×#93514yes—
362C2×C3C3×#93714yes—
363C2×C3#6408712yes—
364C2×C3#6409012yes—
365C2×C3C2×#375011yescharacteristic
366C2×C3#6414411yescharacteristic
367C2×C3#6414611yescharacteristic
368C22—24no—
369C22—24no—
370C22—22no—
371C22—22no—
372C22C3×#375014yescomplemented
373C22C2×C3×#63411yescharacteristic
374C22C2×C3×C4×C3⋊D311yescharacteristic
375C22C3×D4×C3⋊D312yes—
376C22C3×#383012yes—
377C22C2×C3×#63811yescharacteristic
378C4—41no—
379C4—41no—
380C4—184no—
381C4—184no—
382C3—24no—
383C3C3×#494614yescomplemented
384C3#1059264511yescharacteristic, direct product factor
385C2—24no—
386C2C3×#6408712yes—
387C2C3×#6409012yes—
388C2C2×C3×#375011yescharacteristic
389C2C3×#6414411yescharacteristic
390C2C3×#6414611yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
cosocleC23×C32C23×C3
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×#385
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3750C2semidirect
5. C3×#63658C2semidirect
6. #10592645C3direct (N × H)
16. C3×#3757C4semidirect
24. C2×#3750C2×C3semidirect
27. #63658C2×C3semidirect
44. C2×C3×C3⋊Dic3C2×C4semidirect
68. #3757C3×C4semidirect
76. C22×C33C4⋊C4semidirect
132. C2×C3⋊Dic3C2×C3×C4semidirect
190. C22×C32Dic3⋊C4semidirect
191. C22×C32C3×C4⋊C4semidirect
217. C33#385semidirect
308. C22×C3C3×Dic3⋊C4semidirect
309. C22×C3#3750semidirect
331. C32#4946semidirect
332. C32C3×#385semidirect
372. C22C3×#3750semidirect
383. C3C3×#4946semidirect
384. C3#10592645direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C33×C22⋊C42
C2×C3×#37572
C2×C3×#37502
C3×#636582
#105926453
C3×#49463