Finite Group Databasebeta

← C3×#10592652 #13162991

Subgroups — C3×#10592652 #13162991

Order 1728.

1548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C22⋊C4C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C3×#63658C212yesmaximal, complemented
5C3×#63658C211yesmaximal, characteristic, complemented
6#10592652C311yesmaximal, characteristic, direct product factor
7C3×#4953—34nomaximal
8C24×C33C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2212yes—
13C33×C22⋊C4—22no—
14C3×#3757—24no—
15C3×C4⋊(C3⋊Dic3)—22no—
16C2×C32×C22⋊C4—24no—
17C2×C32×C22⋊C4D314yes—
18C2×C32×C22⋊C4C2×C311yescharacteristic
19C2×#3757C2×C312yes—
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
21C2×C3×#217—38no—
22C2×C3×C4⋊Dic3—34no—
23#63658C2×C312yescomplemented
24#63658C2×C311yescharacteristic, complemented
25C3×C3⋊#95—38no—
26C3×C3⋊#95—34no—
27C23×C33—21no—
28C23×C33—21no—
29C23×C33—22no—
30C23×C33—22no—
31C23×C33D412yes—
32C23×C33C2311yescharacteristic
33C2×C33×C4—22no—
34C2×C33×C4—21no—
35C2×C33×C4—22no—
36C2×C33×C4D412yes—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C3×#397—91no—
44#4953—34no—
45C24×C32—24no—
46C24×C32C2×D314yes—
47C24×C32C22×C311yescharacteristic
48C22×C32×C4—24no—
49C22×C32×C4—24no—
50C22×C32×C4C2×D314yes—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C22×C311yescharacteristic
53C22×C32×C4C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C312yes—
55C22×C3⋊Dic3C22×C312yes—
56C22×C3×Dic3—38no—
57C22×C3×Dic3—38no—
58C32×C22⋊C4—216no—
59C32×C22⋊C4—28no—
60C32×C22⋊C4—22no—
61#3757—24no—
62C4⋊(C3⋊Dic3)—22no—
63C3×#217—616no—
64C3×C4⋊Dic3—68no—
65C22×C33—22no—
66C22×C33—24no—
67C22×C33—41no—
68C22×C33—41no—
69C22×C33—42no—
70C22×C33C2×D411yescharacteristic
71C22×C33C2×D411yescharacteristic
72C22×C33#4111yescharacteristic
73C22×C33#4112yes—
74C22×C33#4112yes—
75C33×C4—22no—
76C33×C4—41no—
77C3×C3⋊Dic3—42no—
78C3×C3⋊Dic3—42no—
79C2×C3×C4⋊C4—91no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—92no—
82C2×C3×C22⋊C4C3×D314yes—
83C2×C3×C22⋊C4C3⋊D311yescharacteristic
84C2×#217—38no—
85C2×C4⋊Dic3—34no—
86C3×#95—91no—
87C3×#95—92no—
88C3⋊#95—34no—
89C3⋊#95—38no—
90C23×C32—28no—
91C23×C32—28no—
92C23×C32—24no—
93C23×C32—24no—
94C23×C32—21no—
95C23×C32—21no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—28no—
99C23×C32—24no—
100C23×C32—28no—
101C23×C32—28no—
102C23×C32—48no—
103C23×C32—48no—
104C23×C32D1218yes—
105C23×C32C3×D412yes—
106C23×C32C22×D314yes—
107C23×C32C23×C311yescharacteristic
108C2×C32×C4—22no—
109C2×C32×C4—28no—
110C2×C32×C4—24no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4—28no—
114C2×C32×C4—28no—
115C2×C32×C4—28no—
116C2×C32×C4—48no—
117C2×C32×C4—48no—
118C2×C32×C4#6718yes—
119C2×C32×C4C3×D412yes—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3×Dic3—68no—
127C2×C3×Dic3—68no—
128C2×C3×Dic3—68no—
129C2×C3×Dic3—68no—
130C2×C3×Dic3—68no—
131C2×C3×Dic3—68no—
132#397—91noSylow
133C2×C33—42no—
134C2×C33#12111yescharacteristic
135C2×C33#12312yes—
136C2×C33#12311yescharacteristic
137C2×C33#12411yescharacteristic
138C2×C33#12612yes—
139C24×C3—24no—
140C24×C3C2×C3×D314yes—
141C24×C3C2×C3⋊D311yescharacteristic
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—92no—
145C22×C3×C4—92no—
146C22×C3×C4C2×C3×D314yes—
147C22×C3×C4C2×C3×D314yes—
148C22×C3×C4C2×C3⋊D311yescharacteristic
149C22×C3×C4C2×C3⋊D311yescharacteristic
150C22×Dic3—38no—
151C22×Dic3—38no—
152C3×C4⋊C4—182no—
153C3×C22⋊C4—216no—
154C3×C22⋊C4—28no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—184no—
157#217—616no—
158C4⋊Dic3—68no—
159C22×C32—216no—
160C22×C32—28no—
161C22×C32—22no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—24no—
165C22×C32—28no—
166C22×C32—28no—
167C22×C32—24no—
168C22×C32—216no—
169C22×C32—416no—
170C22×C32—416no—
171C22×C32—48no—
172C22×C32—48no—
173C22×C32—48no—
174C22×C32—44no—
175C22×C32—44no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—42no—
179C22×C32C2×D1214yes—
180C22×C32#23514yes—
181C22×C32#23718yes—
182C22×C32#23718yes—
183C22×C32C2×#6714yes—
184C22×C32C2×C3×D411yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C22×C32C3×#4111yescharacteristic
187C22×C32C3×#4112yes—
188C22×C32C3×#4112yes—
189C32×C4—22no—
190C32×C4—28no—
191C32×C4—48no—
192C32×C4—48no—
193C32×C4—44no—
194C32×C4—41no—
195C3⋊Dic3—42no—
196C3⋊Dic3—42no—
197C3×Dic3—128no—
198C3×Dic3—128no—
199C2×C4⋊C4—91no—
200C2×C22⋊C4—92no—
201C2×C22⋊C4C3×C3⋊D311yescharacteristic
202#95—91no—
203#95—92no—
204C33#39711yescharacteristic, complemented, Sylow
205C23×C3—28no—
206C23×C3—28no—
207C23×C3—24no—
208C23×C3—28no—
209C23×C3—28no—
210C23×C3—24no—
211C23×C3—21no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—21no—
215C23×C3—28no—
216C23×C3—24no—
217C23×C3—48no—
218C23×C3—48no—
219C23×C3C3×D1218yes—
220C23×C3C4⋊(C3⋊D3)12yes—
221C23×C3C22×C3×D314yes—
222C23×C3C22×C3⋊D311yescharacteristic
223C2×C3×C4—28no—
224C2×C3×C4—22no—
225C2×C3×C4—28no—
226C2×C3×C4—28no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—28no—
231C2×C3×C4—48no—
232C2×C3×C4—48no—
233C2×C3×C4—182no—
234C2×C3×C4—182no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4C3×#6718yes—
240C2×C3×C4#63812yes—
241C2×Dic3—68no—
242C2×Dic3—68no—
243C2×Dic3—68no—
244C2×Dic3—68no—
245C2×Dic3—68no—
246C2×Dic3—68no—
247C2×C32—28no—
248C2×C32—28no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—416no—
253C2×C32—48no—
254C2×C32—42no—
255C2×C32#87918yes—
256C2×C32#88618yes—
257C2×C32#93014yes—
258C2×C32#93514yes—
259C2×C32#93614yes—
260C2×C32C3×#12111yescharacteristic
261C2×C32C3×#12312yes—
262C2×C32C3×#12311yescharacteristic
263C2×C32C3×#12411yescharacteristic
264C2×C32C3×#12612yes—
265C24C2×C3×C3⋊D311yescharacteristic
266C22×C4—92no—
267C22×C4—92no—
268C22×C4C2×C3×C3⋊D311yescharacteristic
269C22×C4C2×C3×C3⋊D311yescharacteristic
270C4⋊C4—182no—
271C22⋊C4—22no—
272C22⋊C4—184no—
273C22×C3—216no—
274C22×C3—216no—
275C22×C3—28no—
276C22×C3—24no—
277C22×C3—22no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—28no—
281C22×C3—28no—
282C22×C3—24no—
283C22×C3—416no—
284C22×C3—416no—
285C22×C3—48no—
286C22×C3—48no—
287C22×C3—48no—
288C22×C3—44no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3C2×C3×D1214yes—
294C22×C3C3×#23514yes—
295C22×C3C3×#23718yes—
296C22×C3C3×#23718yes—
297C22×C3C2×C3×#6714yes—
298C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
299C22×C3#382811yescharacteristic
300C22×C3#383012yes—
301C22×C3#383012yes—
302C22×C3C2×#63811yescharacteristic
303C3×C4—28no—
304C3×C4—22no—
305C3×C4—48no—
306C3×C4—48no—
307C3×C4—44no—
308C3×C4—41no—
309C3×C4—362no—
310C3×C4—362no—
311Dic3—128no—
312Dic3—128no—
313C32—24no—
314C32#495314yescomplemented
315C32C3×#39711yescharacteristic, complemented
316C23—21no—
317C23—22no—
318C23—22no—
319C23—21no—
320C23C3×C4⋊(C3⋊D3)12yes—
321C23C22×C3×C3⋊D311yescharacteristic
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—182no—
331C2×C4C3×#63812yes—
332C2×C3—28no—
333C2×C3—28no—
334C2×C3—24no—
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—416no—
338C2×C3—48no—
339C2×C3—42no—
340C2×C3C3×#87918yes—
341C2×C3C3×#88618yes—
342C2×C3C3×#93014yes—
343C2×C3C3×#93514yes—
344C2×C3C3×#93614yes—
345C2×C3#6408812yes—
346C2×C3#6409512yes—
347C2×C3#6413911yescharacteristic
348C2×C3#6414411yescharacteristic
349C2×C3#6414511yescharacteristic
350C22—24no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
356C22C3×#382811yescharacteristic
357C22C3×#383012yes—
358C22C3×#383012yes—
359C22C2×C3×#63811yescharacteristic
360C4—22no—
361C4—41no—
362C4—362no—
363C4—362no—
364C3—24no—
365C3C3×#495314yescomplemented
366C3#1059265211yescharacteristic, direct product factor
367C2—42no—
368C2C3×#6408812yes—
369C2C3×#6409512yes—
370C2C3×#6413911yescharacteristic
371C2C3×#6414411yescharacteristic
372C2C3×#6414511yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
4. C3×#63658C2semidirect
5. C3×#63658C2semidirect
6. #10592652C3direct (N × H)
20. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
23. #63658C2×C3semidirect
24. #63658C2×C3semidirect
204. C33#397semidirect
314. C32#4953semidirect
315. C32C3×#397semidirect
365. C3C3×#4953semidirect
366. C3#10592652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C22⋊C42
C2×C3×#37572
C2×C3×C4⋊(C3⋊Dic3)2
C3×#636582
#105926523
C3×#49533