Finite Group Databasebeta

← C2×S4×Dic9 #13164144

Subgroups — C2×S4×Dic9 #13164144

Order 1728.

731 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×C9×S4C211yesmaximal, characteristic
2C2×A4×Dic9C211yesmaximal, characteristic, complemented
3C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4S4×Dic9C214yesmaximal, direct product factor
5C2×Dic3×S4—31nomaximal
6C2×D4×Dic9—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C2×C9×S4C414yescomplemented
9C2×C9×S4C2212yes—
10C2×D3×Dic9—41nomaximal
11A4×Dic9C2212yescomplemented
12C9⋊SL(2,ℤ/4)C2212yescomplemented
13C22×C3×S4D311yescharacteristic
14C2×Dic3×A4—31no—
15C2×C3⋊SL(2,ℤ/4)—31no—
16Dic3×S4—34no—
17C22×D4×C9—31no—
18C23×Dic9—31no—
19C23×Dic9D311yescharacteristic, complemented
20C2×(C22⋊C9)⋊C4—61no—
21C2×#3676—31no—
22C2×#3676—31no—
23D4×Dic9—34no—
24D4×Dic9—34no—
25C2×C4⋊Dic9—31no—
26C2×C4×Dic9—31no—
27C22×D3×C9—41no—
28C2×C9×A4C2×C412yescomplemented
29C2×C9×A4C2311yescharacteristic
30C9×S4C2×C414yescomplemented
31C2×C3⋊Dic9—41no—
32C2×C3×Dic9—41no—
33D3×Dic9—44no—
34C2×C4×S4—91no—
35C2×D4×Dic3—91no—
36C22×C3×A4C2×D311yescharacteristic
37C2×C3×S4Dic314yes—
38C2×C3×S4C2×D312yes—
39C2×D3×Dic3—121no—
40Dic3×A4—32no—
41C3⋊SL(2,ℤ/4)—32no—
42C24×C9—31no—
43C24×C9C2×D311yescharacteristic
44C22×C22⋊C9—21no—
45C2×D4×C9—32no—
46C2×D4×C9—34no—
47C2×D4×C9—34no—
48C2×D4×C9—32no—
49C22×C4×C9—31no—
50C22×Dic9—31no—
51C22×Dic9—32no—
52C22×Dic9—31no—
53C22×Dic9—34no—
54C22×Dic9—31no—
55C22×Dic9—31no—
56C22×Dic9—61no—
57C22×Dic9—62no—
58C22×Dic9—61no—
59C22×Dic9C2×D312yescomplemented
60(C22⋊C9)⋊C4—62no—
61#3676—34no—
62#3676—32no—
63#3676—32no—
64C4⋊Dic9—34no—
65C4×Dic9—34no—
66C22×C3×C9—41no—
67C2×D3×C9—42no—
68C2×D3×C9—44no—
69C9×A4C22×C411yescharacteristic, complemented
70C3⋊Dic9—42no—
71C3×Dic9—42no—
72C22×S4D911yescharacteristic
73C22×C3×D4—31no—
74C23×Dic3—31no—
75C23×Dic3—91no—
76C2×C4×A4—91no—
77C2×SL(2,ℤ/4)—91no—
78C2×SL(2,ℤ/4)—181no—
79C4×S4—94no—
80C2×#217—91no—
81C2×#217—91no—
82D4×Dic3—94no—
83D4×Dic3—94no—
84C2×C4⋊Dic3—91no—
85C2×C4×Dic3—91no—
86C22×C3×D3—41no—
87C2×C3×A4C2×Dic312yes—
88C2×C3×A4C22×D311yescharacteristic
89C3×S4C2×Dic314yes—
90C2×C3⋊Dic3—121no—
91C2×C3×Dic3—121no—
92D3×Dic3—124no—
93C23×C9—31no—
94C23×C9—31no—
95C23×C9—32no—
96C23×C9—32no—
97C23×C9—34no—
98C23×C9—61no—
99C23×C9—62no—
100C23×C9—61no—
101C23×C9C4×D312yescomplemented
102C23×C9C22×D311yescharacteristic
103C2×C22⋊C9—22no—
104C2×C22⋊C9—21no—
105D4×C9—38no—
106D4×C9—34no—
107D4×C9—34no—
108C2×C4×C9—32no—
109C2×C4×C9—34no—
110C2×Dic9—31no—
111C2×Dic9—32no—
112C2×Dic9—32no—
113C2×Dic9—32no—
114C2×Dic9—32no—
115C2×Dic9—32no—
116C2×Dic9—61no—
117C2×Dic9—64no—
118C2×Dic9—62no—
119C2×Dic9—61no—
120C2×Dic9—61no—
121C2×Dic9—64no—
122C2×Dic9—241no—
123C2×Dic9S411yescharacteristic, direct product factor
124C2×C4×D4—271noSylow
125C2×C3×C9—42no—
126C2×C3×C9—41no—
127D3×C9—44no—
128C24×C3—31no—
129C24×C3D3211yescharacteristic
130C22×A4—21no—
131C22×A4C2×D911yescharacteristic
132C2×S4Dic914yesdirect product factor
133C2×S4C2×D912yes—
134C2×C3×D4—32no—
135C2×C3×D4—34no—
136C2×C3×D4—34no—
137C2×C3×D4—32no—
138C22×C3×C4—31no—
139C22×Dic3—32no—
140C22×Dic3—91no—
141C22×Dic3—91no—
142C22×Dic3—92no—
143C22×Dic3—91no—
144C22×Dic3—91no—
145C22×Dic3—94no—
146C22×Dic3—181no—
147C22×Dic3—182no—
148C22×Dic3—181no—
149C2×C4×D3—361no—
150C4×A4—92no—
151SL(2,ℤ/4)—92no—
152SL(2,ℤ/4)—182no—
153#217—94no—
154#217—92no—
155#217—92no—
156C4⋊Dic3—94no—
157C4×Dic3—94no—
158C22×C32—41no—
159C2×C3×D3—44no—
160C2×C3×D3—42no—
161C3×A4C22×Dic311yescharacteristic
162C3⋊Dic3—122no—
163C3×Dic3—122no—
164C22×C9—32no—
165C22×C9—31no—
166C22×C9—32no—
167C22×C9—32no—
168C22×C9—31no—
169C22×C9—31no—
170C22×C9—34no—
171C22×C9—61no—
172C22×C9—62no—
173C22×C9—64no—
174C22×C9—64no—
175C22×C9—62no—
176C22×C9—81no—
177C22×C9C2×C4×D311yescharacteristic, complemented
178C22×C9C2×S411yescharacteristic
179C22⋊C9—21no—
180C4×C9—34no—
181Dic9—32no—
182Dic9—62no—
183Dic9—62no—
184Dic9—242no—
185Dic9C2×S412yesdirect product factor
186C22×D4—31no—
187C23×C4—91no—
188C23×C4—271no—
189C4×D4—274no—
190C4×D4—274no—
191C2×C4⋊C4—271no—
192C2×C22⋊C4—271no—
193C2×C22⋊C4—271no—
194C2×C42—271no—
195C3×C9—41noSylow
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—32no—
199C23×C3—32no—
200C23×C3—34no—
201C23×C3—61no—
202C23×C3—62no—
203C23×C3—61no—
204C23×C3D3×Dic312yes—
205C23×C3C2×D3211yescharacteristic
206C22×D3—41no—
207C2×A4—22no—
208C2×A4—21no—
209C2×A4C2×Dic912yescomplemented
210C2×A4C22×D911yescharacteristic
211S4C2×Dic914yesdirect product factor
212C3×D4—38no—
213C3×D4—34no—
214C3×D4—34no—
215C2×C3×C4—32no—
216C2×C3×C4—34no—
217C2×C3×C4—361no—
218C2×Dic3—31no—
219C2×Dic3—92no—
220C2×Dic3—92no—
221C2×Dic3—91no—
222C2×Dic3—92no—
223C2×Dic3—92no—
224C2×Dic3—92no—
225C2×Dic3—184no—
226C2×Dic3—181no—
227C2×Dic3—184no—
228C2×Dic3—182no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—361no—
232C2×Dic3—721no—
233C4×D3—364no—
234C2×C32—41no—
235C2×C32—42no—
236C3×D3—44no—
237C2×C9—31no—
238C2×C9—31no—
239C2×C9—32no—
240C2×C9—64no—
241C2×C9—81no—
242C2×C9—82no—
243C2×C9C4×S412yescomplemented
244C2×C9C22×S411yescharacteristic
245C24—31no—
246C24D3×D911yescharacteristic
247C2×D4—32no—
248C2×D4—34no—
249C2×D4—34no—
250C2×D4—32no—
251C22×C4—31no—
252C22×C4—92no—
253C22×C4—271no—
254C22×C4—271no—
255C22×C4—272no—
256C22×C4—271no—
257C22×C4—271no—
258C22×C4—274no—
259C22×C4—541no—
260C22×C4—542no—
261C22×C4—541no—
262C4⋊C4—274no—
263C22⋊C4—274no—
264C22⋊C4—272no—
265C22⋊C4—272no—
266C42—274no—
267C22×C3—32no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—34no—
274C22×C3—41no—
275C22×C3—61no—
276C22×C3—62no—
277C22×C3—64no—
278C22×C3—62no—
279C22×C3—64no—
280C22×C3—81no—
281C22×C3C2×D3×Dic311yescharacteristic
282C22×C3D3×S411yescharacteristic
283C2×D3—42no—
284C2×D3—44no—
285A4—21no—
286A4C22×Dic911yescharacteristic, complemented
287C3×C4—34no—
288C3×C4—362no—
289Dic3—32no—
290Dic3—92no—
291Dic3—182no—
292Dic3—182no—
293Dic3—362no—
294Dic3—722no—
295C32—41no—
296C9—81no—
297C9C2×C4×S411yescharacteristic, complemented
298C23—31no—
299C23—31no—
300C23—32no—
301C23—32no—
302C23—34no—
303C23—61no—
304C23—62no—
305C23—61no—
306C23D3×Dic912yescomplemented
307C23C2×D3×D911yescharacteristic
308D4—38no—
309D4—34no—
310D4—34no—
311C2×C4—32no—
312C2×C4—34no—
313C2×C4—91no—
314C2×C4—272no—
315C2×C4—272no—
316C2×C4—271no—
317C2×C4—272no—
318C2×C4—272no—
319C2×C4—272no—
320C2×C4—544no—
321C2×C4—541no—
322C2×C4—544no—
323C2×C4—541no—
324C2×C4—542no—
325C2×C4—541no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—32no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—64no—
332C2×C3—82no—
333C2×C3—81no—
334C2×C3Dic3×S412yes—
335C2×C3C2×D3×S411yescharacteristic
336D3—44no—
337C22—32no—
338C22—31no—
339C22—32no—
340C22—32no—
341C22—31no—
342C22—31no—
343C22—34no—
344C22—61no—
345C22—62no—
346C22—64no—
347C22—64no—
348C22—62no—
349C22C2×D3×Dic911yescharacteristic, complemented
350C22D9×S411yescharacteristic
351C4—34no—
352C4—92no—
353C4—272no—
354C4—542no—
355C4—542no—
356C3—41no—
357C3—81no—
358C3C2×Dic3×S411yescharacteristic
359C2—31no—
360C2—31no—
361C2—32no—
362C2—64no—
363C2S4×Dic912yesdirect product factor
364C2C2×D9×S411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×A4C22×C4
FrattiniC2×C3C2×D3×S4
FittingC24×C9C2×D3
socleC24×C3D32
prime radicalC22×C3×S4D3
cosocleC2×C9×A4C23
LCSG ▷ C9×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic9C2semidirect
3. C2×C9⋊SL(2,ℤ/4)C2semidirect
4. S4×Dic9C2direct (N × H)
8. C2×C9×S4C4semidirect
11. A4×Dic9C22semidirect
12. C9⋊SL(2,ℤ/4)C22semidirect
19. C23×Dic9D3semidirect
28. C2×C9×A4C2×C4semidirect
30. C9×S4C2×C4semidirect
59. C22×Dic9C2×D3semidirect
69. C9×A4C22×C4semidirect
101. C23×C9C4×D3semidirect
123. C2×Dic9S4direct (N × H)
132. C2×S4Dic9direct (N × H)
177. C22×C9C2×C4×D3semidirect
185. Dic9C2×S4direct (N × H)
209. C2×A4C2×Dic9semidirect
211. S4C2×Dic9direct (N × H)
243. C2×C9C4×S4semidirect
286. A4C22×Dic9semidirect
297. C9C2×C4×S4semidirect
306. C23D3×Dic9semidirect
349. C22C2×D3×Dic9semidirect
363. C2S4×Dic9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C9×S42
C2×A4×Dic92
C2×C9⋊SL(2,ℤ/4)2
S4×Dic92
C2×Dic3×S43
C2×D4×Dic93
C2×D3×Dic94