Finite Group Databasebeta

← C3×C22⋊C4×C3⋊Dic3 #13162979

Subgroups — C3×C22⋊C4×C3⋊Dic3 #13162979

Order 1728.

2076 conjugacy classes of subgroups.

The subgroup lattice has 386 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×C4×C3⋊Dic3C212yesmaximal, complemented
5C3×#63658C212yesmaximal, complemented
6C22⋊C4×C3⋊Dic3C311yesmaximal, characteristic, direct product factor
7C3×Dic3×C22⋊C4—34nomaximal
8C24×C33C2211yescharacteristic
9C22×C33×C4C2212yes—
10C22×C3×C3⋊Dic3—24no—
11C22×C3×C3⋊Dic3C414yescomplemented
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C33×C22⋊C4C414yescomplemented
16C3×#3757C414yescomplemented
17C3×C4×C3⋊Dic3—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×C4×C3⋊Dic3C2×C312yescomplemented
25C2×C3×#217—34no—
26C2×C3×C4×Dic3—38no—
27#63658C2×C312yescomplemented
28C3×C3⋊#95—38no—
29C23×C33—22no—
30C23×C33—22no—
31C23×C33C2×C412yes—
32C23×C33C2×C412yes—
33C23×C33C2×C412yes—
34C23×C33C2311yescharacteristic
35C2×C33×C4—22no—
36C2×C33×C4—22no—
37C2×C33×C4C2×C414yescomplemented
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—28no—
42C2×C3×C3⋊Dic3C2×C412yes—
43C2×C3×C3⋊Dic3C2×C412yes—
44C2×C3×C3⋊Dic3C2×C414yescomplemented
45C2×C3×C3⋊Dic3D414yes—
46C3×C4×C22⋊C4—91no—
47Dic3×C22⋊C4—34no—
48C24×C32—24no—
49C24×C32C2×D314yes—
50C24×C32C22×C311yescharacteristic
51C22×C32×C4—28no—
52C22×C32×C4C2×D318yes—
53C22×C32×C4C22×C312yes—
54C22×C3⋊Dic3—24no—
55C22×C3⋊Dic3C3×C414yescomplemented
56C22×C3⋊Dic3C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C312yes—
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3×Dic3—34no—
60C22×C3×Dic3—316no—
61C22×C3×Dic3—38no—
62C22×C3×Dic3—34no—
63C22×C3×Dic3—616no—
64C32×C22⋊C4—216no—
65C32×C22⋊C4Dic3116yescomplemented
66C32×C22⋊C4C3×C414yescomplemented
67#3757C3×C414yescomplemented
68C4×C3⋊Dic3—24no—
69C3×#217—316no—
70C3×C4×Dic3—616no—
71C22×C33—22no—
72C22×C33—22no—
73C22×C33—24no—
74C22×C33—24no—
75C22×C33C4214yescomplemented
76C22×C33C22×C411yescharacteristic
77C22×C33C22×C411yescharacteristic
78C22×C33C22×C411yescharacteristic
79C22×C33C2×D412yes—
80C22×C33#4112yes—
81C33×C4—24no—
82C3×C3⋊Dic3—22no—
83C3×C3⋊Dic3—24no—
84C3×C3⋊Dic3C22⋊C414yesdirect product factor
85C23×C3×C4—91no—
86C23×Dic3—34no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C22⋊C4C3×D314yes—
90C2×C3×C22⋊C4C3⋊D311yescharacteristic
91C2×C3×C42—92no—
92C2×#217—34no—
93C2×C4×Dic3—38no—
94C3×#95—92no—
95C3⋊#95—38no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—28no—
99C23×C32—28no—
100C23×C32—28no—
101C23×C32—24no—
102C23×C32—28no—
103C23×C32—28no—
104C23×C32—48no—
105C23×C32—48no—
106C23×C32C4×D318yes—
107C23×C32C4×D318yes—
108C23×C32C2×Dic318yes—
109C23×C32C2×C3×C412yes—
110C23×C32C2×C3×C412yes—
111C23×C32C2×C3×C412yes—
112C23×C32C22×D314yes—
113C23×C32C23×C311yescharacteristic
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—216no—
117C2×C32×C4—28no—
118C2×C32×C4—28no—
119C2×C32×C4—48no—
120C2×C32×C4—48no—
121C2×C32×C4C2×Dic3116yescomplemented
122C2×C32×C4C2×C3×C414yescomplemented
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—28no—
127C2×C3⋊Dic3C2×C3×C412yes—
128C2×C3⋊Dic3C2×C3×C412yes—
129C2×C3⋊Dic3C2×C3×C414yescomplemented
130C2×C3⋊Dic3C3×D414yes—
131C2×C3×Dic3—38no—
132C2×C3×Dic3—316no—
133C2×C3×Dic3—316no—
134C2×C3×Dic3—38no—
135C2×C3×Dic3—632no—
136C2×C3×Dic3—68no—
137C2×C3×Dic3—68no—
138C2×C3×Dic3—68no—
139C4×C22⋊C4—91noSylow
140C2×C33—24no—
141C2×C33C2×C4211yescharacteristic
142C2×C33C2×C22⋊C411yescharacteristic
143C2×C33#11711yescharacteristic
144C2×C33C4×D412yes—
145C2×C33C4×D412yes—
146C24×C3—24no—
147C24×C3C2×C3×D314yes—
148C24×C3C2×C3⋊D311yescharacteristic
149C22×C3×C4—28no—
150C22×C3×C4—91no—
151C22×C3×C4—94no—
152C22×C3×C4—91no—
153C22×C3×C4—92no—
154C22×C3×C4—184no—
155C22×C3×C4C2×C3×D318yes—
156C22×C3×C4C2×C3⋊D312yes—
157C22×Dic3—34no—
158C22×Dic3—316no—
159C22×Dic3—34no—
160C22×Dic3—38no—
161C22×Dic3—616no—
162C3×C22⋊C4—216no—
163C3×C22⋊C4—94no—
164C3×C22⋊C4C3×Dic3116yescomplemented
165C3×C22⋊C4C3⋊Dic314yesdirect product factor
166C3×C42—184no—
167#217—316no—
168C4×Dic3—616no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—24no—
172C22×C32—24no—
173C22×C32—216no—
174C22×C32—24no—
175C22×C32—24no—
176C22×C32—28no—
177C22×C32—28no—
178C22×C32—24no—
179C22×C32—28no—
180C22×C32—28no—
181C22×C32—216no—
182C22×C32—216no—
183C22×C32—48no—
184C22×C32—48no—
185C22×C32—416no—
186C22×C32—416no—
187C22×C32C4×Dic3116yescomplemented
188C22×C32C3×C4214yescomplemented
189C22×C32C2×C4×D314yes—
190C22×C32C2×C4×D314yes—
191C22×C32D3×D418yes—
192C22×C32#23718yes—
193C22×C32C22×Dic314yes—
194C22×C32C22×C3×C411yescharacteristic
195C22×C32C22×C3×C411yescharacteristic
196C22×C32C22×C3×C411yescharacteristic
197C22×C32C2×C3×D412yes—
198C22×C32C3×#4112yes—
199C32×C4—24no—
200C32×C4—216no—
201C32×C4—416no—
202C3⋊Dic3—22no—
203C3⋊Dic3—24no—
204C3⋊Dic3C3×C22⋊C414yesdirect product factor
205C3×Dic3—316no—
206C3×Dic3—68no—
207C3×Dic3—616no—
208C23×C4—91no—
209C2×C22⋊C4—91no—
210C2×C22⋊C4C3×C3⋊D311yescharacteristic
211C2×C42—92no—
212#95—92no—
213C33C4×C22⋊C411yescharacteristic, complemented, Sylow
214C23×C3—28no—
215C23×C3—28no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—28no—
219C23×C3—28no—
220C23×C3—28no—
221C23×C3—24no—
222C23×C3—48no—
223C23×C3—48no—
224C23×C3C3×C4×D318yes—
225C23×C3C3×C4×D318yes—
226C23×C3C2×C3×Dic318yes—
227C23×C3C4×C3⋊D312yes—
228C23×C3C4×C3⋊D312yes—
229C23×C3C2×C3⋊Dic312yes—
230C23×C3C22×C3×D314yes—
231C23×C3C22×C3⋊D311yescharacteristic
232C2×C3×C4—28no—
233C2×C3×C4—28no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—216no—
237C2×C3×C4—48no—
238C2×C3×C4—48no—
239C2×C3×C4—92no—
240C2×C3×C4—94no—
241C2×C3×C4—94no—
242C2×C3×C4—92no—
243C2×C3×C4—188no—
244C2×C3×C4—182no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4C2×C3×Dic3116yescomplemented
248C2×C3×C4C2×C3⋊Dic314yescomplemented
249C2×Dic3—38no—
250C2×Dic3—316no—
251C2×Dic3—38no—
252C2×Dic3—316no—
253C2×Dic3—632no—
254C2×Dic3—68no—
255C2×Dic3—68no—
256C2×Dic3—68no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32—28no—
261C2×C32—24no—
262C2×C32—28no—
263C2×C32—216no—
264C2×C32—416no—
265C2×C32#87714yes—
266C2×C32D3×C22⋊C414yes—
267C2×C32#67⋊C418yes—
268C2×C32C2×C4×Dic314yes—
269C2×C32D4×Dic318yes—
270C2×C32C2×C3×C4211yescharacteristic
271C2×C32C2×C3×C22⋊C411yescharacteristic
272C2×C32C3×#11711yescharacteristic
273C2×C32C3×C4×D412yes—
274C2×C32C3×C4×D412yes—
275C24C2×C3×C3⋊D311yescharacteristic
276C22×C4—91no—
277C22×C4—94no—
278C22×C4—91no—
279C22×C4—92no—
280C22×C4—184no—
281C22×C4C2×C3×C3⋊D312yes—
282C22⋊C4—94no—
283C22⋊C4C3×C3⋊Dic314yesdirect product factor
284C42—184no—
285C22×C3—28no—
286C22×C3—28no—
287C22×C3—216no—
288C22×C3—216no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—24no—
292C22×C3—24no—
293C22×C3—216no—
294C22×C3—24no—
295C22×C3—24no—
296C22×C3—28no—
297C22×C3—28no—
298C22×C3—24no—
299C22×C3—48no—
300C22×C3—48no—
301C22×C3—416no—
302C22×C3—416no—
303C22×C3C3×C4×Dic3116yescomplemented
304C22×C3C4×C3⋊Dic314yescomplemented
305C22×C3C2×C3×C4×D314yes—
306C22×C3C2×C3×C4×D314yes—
307C22×C3C3×D3×D418yes—
308C22×C3C3×#23718yes—
309C22×C3C22×C3×Dic314yes—
310C22×C3C2×C4×C3⋊D311yescharacteristic
311C22×C3C2×C4×C3⋊D311yescharacteristic
312C22×C3D4×C3⋊D312yes—
313C22×C3#383012yes—
314C22×C3C22×C3⋊Dic311yescharacteristic
315C3×C4—216no—
316C3×C4—24no—
317C3×C4—416no—
318C3×C4—94no—
319C3×C4—182no—
320C3×C4—184no—
321Dic3—316no—
322Dic3—68no—
323Dic3—616no—
324C32—24no—
325C32Dic3×C22⋊C414yescomplemented
326C32C3×C4×C22⋊C411yescharacteristic, complemented
327C23—22no—
328C23—22no—
329C23C3×C4×C3⋊D312yes—
330C23C3×C4×C3⋊D312yes—
331C23C2×C3×C3⋊Dic312yes—
332C23C22×C3×C3⋊D311yescharacteristic
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—92no—
336C2×C4—94no—
337C2×C4—94no—
338C2×C4—92no—
339C2×C4—188no—
340C2×C4—182no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4C2×C3×C3⋊Dic314yescomplemented
344C2×C3—216no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—24no—
348C2×C3—28no—
349C2×C3—24no—
350C2×C3—28no—
351C2×C3—416no—
352C2×C3C3×#87714yes—
353C2×C3C3×D3×C22⋊C414yes—
354C2×C3C3×#67⋊C418yes—
355C2×C3C2×C3×C4×Dic314yes—
356C2×C3C3×D4×Dic318yes—
357C2×C3#6408611yescharacteristic
358C2×C3C22⋊C4×C3⋊D311yescharacteristic
359C2×C3#6409012yes—
360C2×C3C2×C4×C3⋊Dic311yescharacteristic
361C2×C3D4×C3⋊Dic312yes—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—24no—
366C22C3×C4×C3⋊Dic314yescomplemented
367C22C2×C3×C4×C3⋊D311yescharacteristic
368C22C2×C3×C4×C3⋊D311yescharacteristic
369C22C3×D4×C3⋊D312yes—
370C22C3×#383012yes—
371C22C22×C3×C3⋊Dic311yescharacteristic
372C4—24no—
373C4—94no—
374C4—182no—
375C4—184no—
376C3—24no—
377C3C3×Dic3×C22⋊C414yescomplemented
378C3C22⋊C4×C3⋊Dic311yescharacteristic, direct product factor
379C2—24no—
380C2C3×#6408611yescharacteristic
381C2C3×C22⋊C4×C3⋊D311yescharacteristic
382C2C3×#6409012yes—
383C2C2×C3×C4×C3⋊Dic311yescharacteristic
384C2C3×D4×C3⋊Dic312yes—
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC2×C32C2×C3×C42
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
cosocleC23×C32C23×C3
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×C4×C22⋊C4
LCSG ▷ C2×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×C4×C3⋊Dic3C2semidirect
5. C3×#63658C2semidirect
6. C22⋊C4×C3⋊Dic3C3direct (N × H)
11. C22×C3×C3⋊Dic3C4semidirect
15. C33×C22⋊C4C4semidirect
16. C3×#3757C4semidirect
24. C2×C4×C3⋊Dic3C2×C3semidirect
27. #63658C2×C3semidirect
37. C2×C33×C4C2×C4semidirect
44. C2×C3×C3⋊Dic3C2×C4semidirect
55. C22×C3⋊Dic3C3×C4semidirect
65. C32×C22⋊C4Dic3semidirect
66. C32×C22⋊C4C3×C4semidirect
67. #3757C3×C4semidirect
75. C22×C33C42semidirect
84. C3×C3⋊Dic3C22⋊C4direct (N × H)
121. C2×C32×C4C2×Dic3semidirect
122. C2×C32×C4C2×C3×C4semidirect
129. C2×C3⋊Dic3C2×C3×C4semidirect
164. C3×C22⋊C4C3×Dic3semidirect
165. C3×C22⋊C4C3⋊Dic3direct (N × H)
187. C22×C32C4×Dic3semidirect
188. C22×C32C3×C42semidirect
204. C3⋊Dic3C3×C22⋊C4direct (N × H)
213. C33C4×C22⋊C4semidirect
247. C2×C3×C4C2×C3×Dic3semidirect
248. C2×C3×C4C2×C3⋊Dic3semidirect
283. C22⋊C4C3×C3⋊Dic3direct (N × H)
303. C22×C3C3×C4×Dic3semidirect
304. C22×C3C4×C3⋊Dic3semidirect
325. C32Dic3×C22⋊C4semidirect
326. C32C3×C4×C22⋊C4semidirect
343. C2×C4C2×C3×C3⋊Dic3semidirect
366. C22C3×C4×C3⋊Dic3semidirect
377. C3C3×Dic3×C22⋊C4semidirect
378. C3C22⋊C4×C3⋊Dic3direct (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×C4×C3⋊Dic32
C3×#636582
C22⋊C4×C3⋊Dic33
C3×Dic3×C22⋊C43