Finite Group Databasebeta

← C22×C3×Dic3×A4 #13181454

Subgroups — C22×C3×Dic3×A4 #13181454

Order 1728.

1854 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 1854 conjugacy classes of subgroups — too many to draw. Only its 344 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×A4C211yesmaximal, characteristic
2C2×C3×Dic3×A4C216yesmaximal, direct product factor
3C24×C3×Dic3C311yesmaximal, characteristic, complemented
4C22×C3×C4×A4—31nomaximal
5C22×Dic3×A4C313yesmaximal, direct product factor
6C22×C32×A4C414yescomplemented
7C22×C32×A4C2213yes—
8C22×C32×Dic3—41nomaximal
9C3×Dic3×A4C2214yesdirect product factor
10C25×C32C2×C311yescharacteristic
11C23×C3×A4—23no—
12C23×C3×A4—21no—
13C23×C3×A4D311yescharacteristic
14C23×C3×A4C2×C313yes—
15C23×C3×Dic3—31no—
16C23×C3×Dic3—31no—
17C23×C3×Dic3—36no—
18C23×C3×Dic3C2×C316yescomplemented
19C2×C3×C4×A4—36no—
20C2×Dic3×A4C2×C3118yesdirect product factor
21C23×C33—41no—
22C2×C32×A4C2×C416yescomplemented
23C2×C32×A4C2311yescharacteristic
24C2×C32×Dic3—46no—
25C24×C3×C4—31no—
26C24×Dic3C3211yescharacteristic, complemented
27C22×C4×A4—33no—
28C24×C32—31no—
29C24×C32—33no—
30C24×C32—34no—
31C24×C32C3×C414yescomplemented
32C24×C32C22×C313yes—
33C22×C3×A4—24no—
34C22×C3×A4—23no—
35C22×C3×A4—212no—
36C22×C3×A4—29no—
37C22×C3×A4Dic314yesdirect product factor
38C22×C3×A4C3×C4112yescomplemented
39C22×C3×A4C2×D313yes—
40C22×C3×A4C22×C319yes—
41C22×C32×C4—121no—
42C22×C3×Dic3—36no—
43C22×C3×Dic3—38no—
44C22×C3×Dic3—31no—
45C22×C3×Dic3—36no—
46C22×C3×Dic3—36no—
47C22×C3×Dic3—36no—
48C22×C3×Dic3—312no—
49C22×C3×Dic3—43no—
50C22×C3×Dic3A411yescharacteristic, direct product factor
51C22×C3×Dic3C22×C314yescomplemented
52C3×C4×A4—34no—
53Dic3×A4C22×C3112yesdirect product factor
54C22×C33—44no—
55C22×C33—43no—
56C32×A4C22×C411yescharacteristic, complemented
57C32×Dic3—44no—
58C25×C3—21no—
59C25×C3C3×D311yescharacteristic
60C25×C3C2×C3211yescharacteristic
61C23×A4—23no—
62C23×A4C3×D313yes—
63C23×C3×C4—36no—
64C23×C3×C4—91no—
65C23×C3×C4—96no—
66C23×C3×C4—91no—
67C23×Dic3—31no—
68C23×Dic3—31no—
69C23×Dic3—36no—
70C23×Dic3C2×C3216yescomplemented
71C2×C4×A4—318no—
72C23×C32—312no—
73C23×C32—36no—
74C23×C32—33no—
75C23×C32—312no—
76C23×C32—32no—
77C23×C32—34no—
78C23×C32—34no—
79C23×C32—33no—
80C23×C32—33no—
81C23×C32—41no—
82C23×C32—43no—
83C23×C32—83no—
84C23×C32—81no—
85C23×C32C2×C3×C416yescomplemented
86C23×C32C2×A411yescharacteristic
87C23×C32C23×C311yescharacteristic
88C2×C3×A4—26no—
89C2×C3×A4—21no—
90C2×C3×A4—218no—
91C2×C3×A4—23no—
92C2×C3×A4C2×Dic316yesdirect product factor
93C2×C3×A4C2×C3×C4118yescomplemented
94C2×C3×A4C22×D311yescharacteristic
95C2×C3×A4C23×C313yes—
96C2×C32×C4—126no—
97C2×C3×Dic3—312no—
98C2×C3×Dic3—36no—
99C2×C3×Dic3—312no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—418no—
103C2×C3×Dic3C2×A416yesdirect product factor
104C24×C4—31noSylow
105C2×C33—46no—
106C2×C33—41no—
107C24×C3—24no—
108C24×C3—23no—
109C24×C3—34no—
110C24×C3—33no—
111C24×C3—34no—
112C24×C3—33no—
113C24×C3—31no—
114C24×C3—31no—
115C24×C3—61no—
116C24×C3—63no—
117C24×C3—64no—
118C24×C3C3×Dic314yescomplemented
119C24×C3C32×C414yescomplemented
120C24×C3C2×C3×D313yes—
121C24×C3C22×C3213yes—
122C22×A4—212no—
123C22×A4—29no—
124C22×A4C3×Dic3112yesdirect product factor
125C22×A4C2×C3×D319yes—
126C22×C3×C4—31no—
127C22×C3×C4—34no—
128C22×C3×C4—96no—
129C22×C3×C4—96no—
130C22×C3×C4—98no—
131C22×C3×C4—96no—
132C22×C3×C4—91no—
133C22×C3×C4—912no—
134C22×C3×C4—96no—
135C22×C3×C4—123no—
136C22×Dic3—36no—
137C22×Dic3—38no—
138C22×Dic3—31no—
139C22×Dic3—36no—
140C22×Dic3—36no—
141C22×Dic3—36no—
142C22×Dic3—312no—
143C22×Dic3C3×A411yescharacteristic, direct product factor
144C22×Dic3C22×C3214yescomplemented
145C4×A4—312no—
146C22×C32—36no—
147C22×C32—31no—
148C22×C32—38no—
149C22×C32—36no—
150C22×C32—312no—
151C22×C32—36no—
152C22×C32—36no—
153C22×C32—33no—
154C22×C32—31no—
155C22×C32—43no—
156C22×C32—44no—
157C22×C32—49no—
158C22×C32—412no—
159C22×C32—89no—
160C22×C32—812no—
161C22×C32—83no—
162C22×C32—84no—
163C22×C32C4×A414yescomplemented
164C22×C32C22×C3×C411yescharacteristic, complemented
165C22×C32C22×A413yes—
166C3×A4—21no—
167C3×A4—23no—
168C3×A4C22×Dic311yescharacteristic, direct product factor
169C3×A4C22×C3×C413yescomplemented
170C32×C4—124no—
171C3×Dic3—34no—
172C3×Dic3—412no—
173C3×Dic3C22×A414yesdirect product factor
174C25C32×D311yescharacteristic
175C23×C4—36no—
176C23×C4—91no—
177C23×C4—96no—
178C23×C4—91no—
179C33—41noSylow
180C23×C3—26no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—33no—
184C23×C3—33no—
185C23×C3—312no—
186C23×C3—312no—
187C23×C3—36no—
188C23×C3—36no—
189C23×C3—32no—
190C23×C3—32no—
191C23×C3—312no—
192C23×C3—312no—
193C23×C3—33no—
194C23×C3—33no—
195C23×C3—33no—
196C23×C3—33no—
197C23×C3—34no—
198C23×C3—34no—
199C23×C3—34no—
200C23×C3—34no—
201C23×C3—43no—
202C23×C3—63no—
203C23×C3—64no—
204C23×C3—63no—
205C23×C3—64no—
206C23×C3—63no—
207C23×C3—612no—
208C23×C3—66no—
209C23×C3—62no—
210C23×C3—612no—
211C23×C3—83no—
212C23×C3C2×C3×Dic316yescomplemented
213C23×C3C2×C32×C416yescomplemented
214C23×C3D3×A411yescharacteristic
215C23×C3C2×C3×A411yescharacteristic
216C23×C3C22×C3×D311yescharacteristic
217C23×C3C23×C3211yescharacteristic
218C2×A4—218no—
219C2×A4—23no—
220C2×A4C2×C3×Dic3118yesdirect product factor
221C2×A4C22×C3×D313yes—
222C2×C3×C4—36no—
223C2×C3×C4—94no—
224C2×C3×C4—912no—
225C2×C3×C4—912no—
226C2×C3×C4—96no—
227C2×C3×C4—94no—
228C2×C3×C4—1218no—
229C2×Dic3—312no—
230C2×Dic3—36no—
231C2×Dic3—312no—
232C2×Dic3—34no—
233C2×Dic3—34no—
234C2×Dic3C2×C3×A416yesdirect product factor
235C2×C32—36no—
236C2×C32—31no—
237C2×C32—31no—
238C2×C32—46no—
239C2×C32—41no—
240C2×C32—418no—
241C2×C32—43no—
242C2×C32—818no—
243C2×C32—83no—
244C2×C32—86no—
245C2×C32—81no—
246C2×C32C2×C4×A416yescomplemented
247C2×C32C23×A411yescharacteristic
248C24—33no—
249C24—34no—
250C24—31no—
251C24C32×Dic314yescomplemented
252C24C2×C32×D313yes—
253C22×C4—31no—
254C22×C4—34no—
255C22×C4—96no—
256C22×C4—96no—
257C22×C4—98no—
258C22×C4—96no—
259C22×C4—91no—
260C22×C4—912no—
261C22×C4—96no—
262C22×C3—23no—
263C22×C3—24no—
264C22×C3—21no—
265C22×C3—36no—
266C22×C3—36no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—36no—
270C22×C3—38no—
271C22×C3—36no—
272C22×C3—38no—
273C22×C3—36no—
274C22×C3—312no—
275C22×C3—36no—
276C22×C3—36no—
277C22×C3—312no—
278C22×C3—36no—
279C22×C3—31no—
280C22×C3—33no—
281C22×C3—31no—
282C22×C3—33no—
283C22×C3—49no—
284C22×C3—412no—
285C22×C3—66no—
286C22×C3—61no—
287C22×C3—612no—
288C22×C3—66no—
289C22×C3—63no—
290C22×C3—66no—
291C22×C3—61no—
292C22×C3—66no—
293C22×C3—68no—
294C22×C3—89no—
295C22×C3—812no—
296C22×C3Dic3×A414yesdirect product factor
297C22×C3C3×C4×A414yescomplemented
298C22×C3C22×C3×Dic311yescharacteristic, complemented
299C22×C3C22×C32×C411yescharacteristic, complemented
300C22×C3C2×D3×A413yes—
301C22×C3C22×C3×A413yes—
302A4—23no—
303A4C22×C3×Dic313yesdirect product factor
304C3×C4—34no—
305C3×C4—94no—
306C3×C4—1212no—
307Dic3—34no—
308Dic3C22×C3×A414yesdirect product factor
309C32—41no—
310C32—43no—
311C32—83no—
312C32—81no—
313C32C22×C4×A411yescharacteristic, complemented
314C23—33no—
315C23—312no—
316C23—36no—
317C23—32no—
318C23—312no—
319C23—33no—
320C23—33no—
321C23—34no—
322C23—34no—
323C23C2×C32×Dic316yescomplemented
324C23C3×D3×A411yescharacteristic
325C23C22×C32×D311yescharacteristic
326C2×C4—36no—
327C2×C4—94no—
328C2×C4—912no—
329C2×C4—912no—
330C2×C4—96no—
331C2×C4—94no—
332C2×C3—26no—
333C2×C3—21no—
334C2×C3—31no—
335C2×C3—36no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—36no—
339C2×C3—31no—
340C2×C3—418no—
341C2×C3—43no—
342C2×C3—61no—
343C2×C3—66no—
344C2×C3—61no—
345C2×C3—818no—
346C2×C3—83no—
347C2×C3C2×Dic3×A416yesdirect product factor
348C2×C3C2×C3×C4×A416yescomplemented
349C2×C3C22×D3×A411yescharacteristic
350C2×C3C23×C3×A411yescharacteristic
351C22—36no—
352C22—31no—
353C22—36no—
354C22—38no—
355C22—36no—
356C22—312no—
357C22—36no—
358C22—31no—
359C22—33no—
360C22C3×Dic3×A414yesdirect product factor
361C22C22×C32×Dic311yescharacteristic, complemented
362C22C2×C3×D3×A413yes—
363C4—34no—
364C4—94no—
365C3—21no—
366C3—43no—
367C3—83no—
368C3C22×Dic3×A411yescharacteristic, direct product factor
369C3C22×C3×C4×A411yescharacteristic, complemented
370C2—31no—
371C2—36no—
372C2—31no—
373C2C2×C3×Dic3×A416yesdirect product factor
374C2C22×C3×D3×A411yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×A4
commutator, hypocenterC22×C3C22×C32×C4
FrattiniC2C22×C3×D3×A4
Fitting, socleC25×C32C2×C3
prime radicalC23×C32×A4C2
cosocleC23×C3C23×C32
LCSG ▷ C22×C3
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3×A4C2direct (N × H)
3. C24×C3×Dic3C3semidirect
5. C22×Dic3×A4C3direct (N × H)
6. C22×C32×A4C4semidirect
9. C3×Dic3×A4C22direct (N × H)
18. C23×C3×Dic3C2×C3semidirect
20. C2×Dic3×A4C2×C3direct (N × H)
22. C2×C32×A4C2×C4semidirect
26. C24×Dic3C32semidirect
31. C24×C32C3×C4semidirect
37. C22×C3×A4Dic3direct (N × H)
38. C22×C3×A4C3×C4semidirect
50. C22×C3×Dic3A4direct (N × H)
51. C22×C3×Dic3C22×C3semidirect
53. Dic3×A4C22×C3direct (N × H)
56. C32×A4C22×C4semidirect
70. C23×Dic3C2×C32semidirect
85. C23×C32C2×C3×C4semidirect
92. C2×C3×A4C2×Dic3direct (N × H)
93. C2×C3×A4C2×C3×C4semidirect
103. C2×C3×Dic3C2×A4direct (N × H)
118. C24×C3C3×Dic3semidirect
119. C24×C3C32×C4semidirect
124. C22×A4C3×Dic3direct (N × H)
143. C22×Dic3C3×A4direct (N × H)
144. C22×Dic3C22×C32semidirect
163. C22×C32C4×A4semidirect
164. C22×C32C22×C3×C4semidirect
168. C3×A4C22×Dic3direct (N × H)
169. C3×A4C22×C3×C4semidirect
173. C3×Dic3C22×A4direct (N × H)
212. C23×C3C2×C3×Dic3semidirect
213. C23×C3C2×C32×C4semidirect
220. C2×A4C2×C3×Dic3direct (N × H)
234. C2×Dic3C2×C3×A4direct (N × H)
246. C2×C32C2×C4×A4semidirect
251. C24C32×Dic3semidirect
296. C22×C3Dic3×A4direct (N × H)
297. C22×C3C3×C4×A4semidirect
298. C22×C3C22×C3×Dic3semidirect
299. C22×C3C22×C32×C4semidirect
303. A4C22×C3×Dic3direct (N × H)
308. Dic3C22×C3×A4direct (N × H)
313. C32C22×C4×A4semidirect
323. C23C2×C32×Dic3semidirect
347. C2×C3C2×Dic3×A4direct (N × H)
348. C2×C3C2×C3×C4×A4semidirect
360. C22C3×Dic3×A4direct (N × H)
361. C22C22×C32×Dic3semidirect
368. C3C22×Dic3×A4direct (N × H)
369. C3C22×C3×C4×A4semidirect
373. C2C2×C3×Dic3×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C3×Dic3×A42
C24×C3×Dic33
C22×C3×C4×A43
C22×Dic3×A43
C22×C32×Dic34