Finite Group Databasebeta

← C2×C4×C9⋊S4 #13164450

Subgroups — C2×C4×C9⋊S4 #13164450

Order 1728.

915 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 915 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4×C9×A4C211yesmaximal, characteristic, complemented
3C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C4×C9⋊S4C214yesmaximal, direct product factor
5C2×C4×C3⋊S4—31nomaximal
6C2×C4×(C22⋊C9)⋊C2—32nomaximal
7C2×C4×#611—31nomaximal
8C22×C9×A4C2211yescharacteristic
9C2×C9⋊S4C414yesdirect product factor
10C2×C9⋊S4C2212yes—
11C2×C4×C3⋊D9—41nomaximal
12C4×C9×A4C2212yescomplemented
13C9⋊SL(2,ℤ/4)C2212yescomplemented
14C22×C3⋊S4—31no—
15C2×C3×C4×A4D311yescharacteristic
16C2×C3⋊SL(2,ℤ/4)—31no—
17C4×C3⋊S4—34no—
18C22×(C22⋊C9)⋊C2—32no—
19C23×C4×C9D311yescharacteristic, complemented
20C22×#611—31no—
21C22×C4×D9—31no—
22C2×C4×C22⋊C9D312yescomplemented
23C2×(C22⋊C9)⋊C4—32no—
24C4×(C22⋊C9)⋊C2—38no—
25C2×#3676—31no—
26C4×#611—34no—
27C4×#611—34no—
28C2×(C2×D9)⋊C4—31no—
29C2×Dic9⋊C4—31no—
30C2×C4×Dic9—31no—
31C22×C3⋊D9—41no—
32C2×C9×A4C2×C412yescomplemented
33C2×C9×A4C2311yescharacteristic
34C9⋊S4C2×C414yesdirect product factor
35C2×C3×C4×C9—41no—
36C2×C3⋊Dic9—41no—
37C4×C3⋊D9—44no—
38C2×C4×S4—93no—
39C2×C4×#67—91no—
40C22×C3×A4C2×D311yescharacteristic
41C2×C3⋊S4—34no—
42C2×C3⋊S4—32no—
43C2×C4×C3⋊D3—121no—
44C3×C4×A4C2×D312yes—
45C3⋊SL(2,ℤ/4)—32no—
46C24×C9C2×D311yescharacteristic
47C23×D9—31no—
48C22×C22⋊C9C2×D312yes—
49C2×(C22⋊C9)⋊C2—38no—
50C2×(C22⋊C9)⋊C2—34no—
51C22×C4×C9—31no—
52C22×C4×C9—31no—
53C22×C4×C9—32no—
54C22×C4×C9C2×D312yescomplemented
55C2×#611—32no—
56C2×#611—34no—
57C2×#611—34no—
58C2×#611—32no—
59C22×Dic9—31no—
60C22×Dic9—31no—
61C22×Dic9—31no—
62C2×C4×D9—34no—
63C2×C4×D9—61no—
64C2×C4×D9—62no—
65C2×C4×D9—61no—
66C2×C4×D9—122no—
67C4×C22⋊C9C2×D314yescomplemented
68(C22⋊C9)⋊C4—34no—
69#3676—32no—
70#3676—32no—
71(C2×D9)⋊C4—34no—
72Dic9⋊C4—34no—
73C4×Dic9—34no—
74C22×C3×C9—41no—
75C2×C3⋊D9—44no—
76C2×C3⋊D9—42no—
77C9×A4C22×C411yescharacteristic, complemented
78C3×C4×C9—42no—
79C3⋊Dic9—42no—
80C22×S4—93no—
81C23×C3×C4C3⋊D311yescharacteristic
82C22×#67—91no—
83C22×C4×D3—91no—
84C2×C4×A4D913yescomplemented
85C2×SL(2,ℤ/4)—93no—
86C4×S4—912no—
87C2×#217—91no—
88C4×#67—94no—
89C4×#67—94no—
90C2×(C2×D3)⋊C4—91no—
91C2×Dic3⋊C4—91no—
92C2×C4×Dic3—91no—
93C22×C3⋊D3—121no—
94C2×C3×A4C4×D312yes—
95C2×C3×A4C22×D311yescharacteristic
96C3⋊S4—34no—
97C2×C32×C4—41no—
98C2×C3⋊Dic3—121no—
99C4×C3⋊D3—124no—
100C23×C9—31no—
101C23×C9—31no—
102C23×C9—32no—
103C23×C9C4×D312yescomplemented
104C23×C9C22×D311yescharacteristic
105C22×D9—32no—
106C22×D9—34no—
107C22×D9—61no—
108C22×D9—62no—
109C22×D9—61no—
110C22×D9—122no—
111C2×C22⋊C9C4×D314yescomplemented
112C2×C22⋊C9C22×D312yes—
113(C22⋊C9)⋊C2—38no—
114C2×C4×C9—31no—
115C2×C4×C9—32no—
116C2×C4×C9—32no—
117C2×C4×C9—32no—
118C2×C4×C9—42no—
119C2×C4×C9—61no—
120C2×C4×C9S411yescharacteristic, complemented
121#611—38no—
122#611—34no—
123#611—34no—
124C2×Dic9—32no—
125C2×Dic9—34no—
126C2×Dic9—32no—
127C2×Dic9—32no—
128C2×Dic9—61no—
129C2×Dic9—62no—
130C2×Dic9—61no—
131C2×Dic9—122no—
132C4×D9—64no—
133C4×D9—64no—
134C4×D9—128no—
135C2×C4×D4—271noSylow
136C2×C3×C9—42no—
137C2×C3×C9—41no—
138C3⋊D9—44no—
139C24×C3C2×C3⋊D311yescharacteristic
140C23×D3—91no—
141C22×A4C2×D913yes—
142C2×S4—912no—
143C2×S4—96no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—32no—
147C22×C3×C4C2×C3⋊D312yes—
148C2×#67—92no—
149C2×#67—94no—
150C2×#67—94no—
151C2×#67—92no—
152C22×Dic3—91no—
153C22×Dic3—91no—
154C22×Dic3—91no—
155C2×C4×D3—94no—
156C2×C4×D3—181no—
157C2×C4×D3—182no—
158C2×C4×D3—181no—
159C2×C4×D3—363no—
160C4×A4C2×D916yescomplemented
161SL(2,ℤ/4)—96no—
162#217—92no—
163#217—92no—
164(C2×D3)⋊C4—94no—
165Dic3⋊C4—94no—
166C4×Dic3—94no—
167C22×C32—41no—
168C2×C3⋊D3—122no—
169C2×C3⋊D3—124no—
170C3×A4C2×C4×D311yescharacteristic
171C32×C4—42no—
172C3⋊Dic3—122no—
173C22×C9—32no—
174C22×C9—31no—
175C22×C9—32no—
176C22×C9—32no—
177C22×C9—31no—
178C22×C9—31no—
179C22×C9—42no—
180C22×C9—61no—
181C22×C9C2×C4×D311yescharacteristic, complemented
182C22×C9C2×S411yescharacteristic
183C2×D9—34no—
184C2×D9—62no—
185C2×D9—64no—
186C2×D9—64no—
187C2×D9—62no—
188C2×D9—124no—
189C2×D9—128no—
190C22⋊C9C2×C4×D312yescomplemented
191C4×C9—32no—
192C4×C9—44no—
193C4×C9C2×S412yescomplemented
194Dic9—34no—
195Dic9—62no—
196Dic9—62no—
197Dic9—124no—
198C22×D4—271no—
199C23×C4—271no—
200C23×C4C3⋊D911yescharacteristic, complemented
201C4×D4—274no—
202C4×D4—274no—
203C2×C4⋊C4—271no—
204C2×C22⋊C4—271no—
205C2×C22⋊C4—271no—
206C2×C42—271no—
207C3×C9—41noSylow
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—32no—
211C23×C3C4×C3⋊D312yes—
212C23×C3C22×C3⋊D311yescharacteristic
213C22×D3—92no—
214C22×D3—94no—
215C22×D3—181no—
216C22×D3—182no—
217C22×D3—181no—
218C22×D3—363no—
219C2×A4C4×D916yescomplemented
220C2×A4C22×D913yes—
221S4—912no—
222C2×C3×C4—31no—
223C2×C3×C4—32no—
224C2×C3×C4—32no—
225C2×C3×C4—32no—
226C2×C3×C4—43no—
227C2×C3×C4—61no—
228C2×C3×C4C3⋊S411yescharacteristic
229#67—98no—
230#67—94no—
231#67—94no—
232C2×Dic3—92no—
233C2×Dic3—94no—
234C2×Dic3—92no—
235C2×Dic3—92no—
236C2×Dic3—181no—
237C2×Dic3—182no—
238C2×Dic3—181no—
239C2×Dic3—363no—
240C4×D3—184no—
241C4×D3—184no—
242C4×D3—3612no—
243C2×C32—42no—
244C2×C32—41no—
245C3⋊D3—124no—
246C2×C9—31no—
247C2×C9—31no—
248C2×C9—32no—
249C2×C9—44no—
250C2×C9—42no—
251C2×C9C4×S412yescomplemented
252C2×C9C22×S411yescharacteristic
253D9—64no—
254D9—128no—
255C24—271no—
256C24C2×C3⋊D911yescharacteristic
257C2×D4—272no—
258C2×D4—274no—
259C2×D4—274no—
260C2×D4—272no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—274no—
265C22×C4—271no—
266C22×C4—271no—
267C22×C4—271no—
268C22×C4—541no—
269C22×C4—542no—
270C22×C4—541no—
271C22×C4C2×C3⋊D912yescomplemented
272C4⋊C4—274no—
273C22⋊C4—274no—
274C22⋊C4—272no—
275C22⋊C4—272no—
276C42—274no—
277C22×C3—32no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—43no—
284C22×C3—61no—
285C22×C3C2×C4×C3⋊D311yescharacteristic
286C22×C3C2×C3⋊S411yescharacteristic
287C2×D3—94no—
288C2×D3—182no—
289C2×D3—184no—
290C2×D3—184no—
291C2×D3—182no—
292C2×D3—366no—
293C2×D3—3612no—
294A4C2×C4×D913yescomplemented
295C3×C4—32no—
296C3×C4—46no—
297C3×C4C2×C3⋊S412yes—
298Dic3—94no—
299Dic3—182no—
300Dic3—182no—
301Dic3—366no—
302C32—41no—
303C9—42no—
304C9C2×C4×S411yescharacteristic, complemented
305C23—31no—
306C23—31no—
307C23—32no—
308C23—272no—
309C23—274no—
310C23—541no—
311C23—542no—
312C23—541no—
313C23C4×C3⋊D912yescomplemented
314C23C22×C3⋊D911yescharacteristic
315D4—278no—
316D4—274no—
317D4—274no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—61no—
323C2×C4—272no—
324C2×C4—274no—
325C2×C4—272no—
326C2×C4—272no—
327C2×C4—544no—
328C2×C4—541no—
329C2×C4—542no—
330C2×C4—541no—
331C2×C4—544no—
332C2×C4C9⋊S411yescharacteristic, direct product factor
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—32no—
336C2×C3—43no—
337C2×C3—46no—
338C2×C3C4×C3⋊S412yes—
339C2×C3C22×C3⋊S411yescharacteristic
340D3—184no—
341D3—3612no—
342C22—32no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—32no—
347C22—32no—
348C22—61no—
349C22—274no—
350C22—542no—
351C22—544no—
352C22—542no—
353C22—544no—
354C22C2×C4×C3⋊D911yescharacteristic, complemented
355C22C2×C9⋊S411yescharacteristic
356C4—32no—
357C4—274no—
358C4—542no—
359C4—542no—
360C4C2×C9⋊S412yesdirect product factor
361C3—43no—
362C3C2×C4×C3⋊S411yescharacteristic
363C2—31no—
364C2—31no—
365C2—32no—
366C2—544no—
367C2C4×C9⋊S412yesdirect product factor
368C2C22×C9⋊S411yescharacteristic
subgroupG/H
center, hypercenterC2×C4C9⋊S4
commutator, hypocenterC9×A4C22×C4
FrattiniC2×C3C22×C3⋊S4
FittingC23×C4×C9D3
socleC24×C3C2×C3⋊D3
prime radicalC22×C9⋊S4C2
cosocleC2×C9×A4C23
LCSG ▷ C9×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9×A4C2semidirect
3. C2×C9⋊SL(2,ℤ/4)C2semidirect
4. C4×C9⋊S4C2direct (N × H)
9. C2×C9⋊S4C4direct (N × H)
12. C4×C9×A4C22semidirect
13. C9⋊SL(2,ℤ/4)C22semidirect
19. C23×C4×C9D3semidirect
22. C2×C4×C22⋊C9D3semidirect
32. C2×C9×A4C2×C4semidirect
34. C9⋊S4C2×C4direct (N × H)
54. C22×C4×C9C2×D3semidirect
67. C4×C22⋊C9C2×D3semidirect
77. C9×A4C22×C4semidirect
84. C2×C4×A4D9semidirect
103. C23×C9C4×D3semidirect
111. C2×C22⋊C9C4×D3semidirect
120. C2×C4×C9S4semidirect
160. C4×A4C2×D9semidirect
181. C22×C9C2×C4×D3semidirect
190. C22⋊C9C2×C4×D3semidirect
193. C4×C9C2×S4semidirect
200. C23×C4C3⋊D9semidirect
219. C2×A4C4×D9semidirect
251. C2×C9C4×S4semidirect
271. C22×C4C2×C3⋊D9semidirect
294. A4C2×C4×D9semidirect
304. C9C2×C4×S4semidirect
313. C23C4×C3⋊D9semidirect
332. C2×C4C9⋊S4direct (N × H)
354. C22C2×C4×C3⋊D9semidirect
360. C4C2×C9⋊S4direct (N × H)
367. C2C4×C9⋊S4direct (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×C4×C9×A42
C2×C9⋊SL(2,ℤ/4)2
C4×C9⋊S42
C2×C4×C3⋊S43
C2×C4×(C22⋊C9)⋊C23
C2×C4×#6113
C2×C4×C3⋊D94