Finite Group Databasebeta

← C2×C4⋊(D9⋊A4) #13164566

Subgroups — C2×C4⋊(D9⋊A4) #13164566

Order 1728.

967 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 967 conjugacy classes of subgroups — too many to draw. Only its 97 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×D9⋊A4C212yesmaximal, complemented
2C2×C4×C9⋊A4C211yesmaximal, characteristic, complemented
3C4⋊(D9⋊A4)C214yesmaximal, direct product factor
4C2×A4×D12—31nomaximal
5C23×D36C311yesmaximal, characteristic, complemented
6C22×C9⋊A4C2211yescharacteristic
7C2×D9⋊A4—24no—
8C2×D9⋊A4C2214yescomplemented
9C2×C4⋊AGL(1,ℤ/9)—41nomaximal
10C4×C9⋊A4C2212yescomplemented
11C22×D3×A4—32no—
12C2×C3×C4×A4D311yescharacteristic
13A4×D12—34no—
14C24×D9C2×C312yescomplemented
15C23×C4×C9C2×C311yescharacteristic, complemented
16C22×D36—31no—
17C22×D36—32no—
18C22×D36—31no—
19C22×D36—34no—
20C22×D36C2×C314yescomplemented
21C2×C4×C22⋊C9—21no—
22C22×AGL(1,ℤ/9)—42no—
23C2×C9⋊A4D412yescomplemented
24C2×C9⋊A4C2311yescharacteristic
25D9⋊A4—24no—
26C2×C4×3−1+2—41no—
27C4⋊AGL(1,ℤ/9)—44no—
28C23×D12—31no—
29C2×D4×A4—91no—
30C22×C3×A4C2×D311yescharacteristic
31C2×D3×A4—34no—
32C2×D3×A4—64no—
33C2×C3×D12—121no—
34C3×C4×A4C2×D312yes—
35C24×C9C22×C311yescharacteristic
36C23×D9—24no—
37C23×D9—32no—
38C23×D9—32no—
39C23×D9—34no—
40C23×D9—64no—
41C23×D9C22×C314yescomplemented
42C22×C22⋊C9—21no—
43C22×C4×C9—31no—
44C22×C4×C9—31no—
45C22×C4×C9—32no—
46C22×C4×C9C22×C312yescomplemented
47C2×D36—32no—
48C2×D36—38no—
49C2×D36—34no—
50C2×D36—32no—
51C2×D36—31no—
52C2×D36—34no—
53C2×D36—34no—
54C2×D36—38no—
55C2×D36—34no—
56C2×D36A411yescharacteristic, complemented
57C4×C22⋊C9—22no—
58C22×3−1+2—41no—
59C2×AGL(1,ℤ/9)—44no—
60C2×AGL(1,ℤ/9)—84no—
61C9⋊A4C2×D411yescharacteristic, complemented
62C4×3−1+2—42no—
63C24×D3—32no—
64C23×A4—92no—
65C23×C3×C4C3×D311yescharacteristic
66C22×D12—34no—
67C22×D12—91no—
68C22×D12—94no—
69C22×D12—92no—
70C22×D12—91no—
71D4×A4—94no—
72C2×C4×A4—31no—
73C22×C3×D3—122no—
74C2×C3×A4D1212yes—
75C2×C3×A4C22×D311yescharacteristic
76D3×A4—64no—
77C2×C32×C4—41no—
78C3×D12—124no—
79C23×C9—31no—
80C23×C9—32no—
81C23×C9—31no—
82C23×C9C3×D412yescomplemented
83C23×C9C23×C311yescharacteristic
84C22×D9—24no—
85C22×D9—34no—
86C22×D9—32no—
87C22×D9—34no—
88C22×D9—34no—
89C22×D9—34no—
90C22×D9—64no—
91C22×D9—64no—
92C22×D9—68no—
93C22×D9—68no—
94C22×D9—64no—
95C22×D9—64no—
96C22×D9—64no—
97C22×D9C2×A412yescomplemented
98C2×C22⋊C9—22no—
99C2×C22⋊C9—21no—
100C2×C4×C9—32no—
101C2×C4×C9—31no—
102C2×C4×C9—32no—
103C2×C4×C9—32no—
104C2×C4×C9—32no—
105C2×C4×C9—81no—
106C2×C4×C9C2×A411yescharacteristic, complemented
107D36—38no—
108D36—38no—
109D36—34no—
110D36C2×A414yescomplemented
111C23×D4—91noSylow
112C2×3−1+2—42no—
113C2×3−1+2—41no—
114AGL(1,ℤ/9)—84no—
115C24×C3C2×C3×D311yescharacteristic
116C23×D3—34no—
117C23×D3—64no—
118C23×D3—92no—
119C23×D3—94no—
120C23×D3—92no—
121C23×D3—184no—
122C22×A4—31no—
123C22×A4—94no—
124C22×A4—184no—
125C2×C3×D4—361no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—32no—
129C22×C3×C4C2×C3×D312yes—
130C2×D12—31no—
131C2×D12—94no—
132C2×D12—94no—
133C2×D12—92no—
134C2×D12—98no—
135C2×D12—92no—
136C2×D12—94no—
137C2×D12—91no—
138C2×D12—98no—
139C2×D12—94no—
140C4×A4—32no—
141C22×C32—41no—
142C2×C3×D3—124no—
143C2×C3×D3—244no—
144C3×A4C2×D1211yescharacteristic
145C32×C4—42no—
146C22×C9—32no—
147C22×C9—31no—
148C22×C9—32no—
149C22×C9—32no—
150C22×C9—32no—
151C22×C9—31no—
152C22×C9—31no—
153C22×C9—81no—
154C22×C9C2×C3×D411yescharacteristic, complemented
155C22×C9C22×A411yescharacteristic
156C2×D9—24no—
157C2×D9—34no—
158C2×D9—64no—
159C2×D9—68no—
160C2×D9—64no—
161C2×D9—68no—
162C2×D9—64no—
163C2×D9—64no—
164C2×D9—64no—
165C2×D9C22×A414yescomplemented
166C22⋊C9—21no—
167C4×C9—32no—
168C4×C9—82no—
169C4×C9C22×A412yescomplemented
170C25—92no—
171C22×D4—94no—
172C22×D4—271no—
173C22×D4—271no—
174C22×D4—274no—
175C22×D4—272no—
176C23×C4AGL(1,ℤ/9)11yescharacteristic, complemented
1773−1+2—41noSylow
178C23×C3—31no—
179C23×C3—32no—
180C23×C3—31no—
181C23×C3—362no—
182C23×C3C3×D1212yes—
183C23×C3C22×C3×D311yescharacteristic
184C22×D3—32no—
185C22×D3—64no—
186C22×D3—94no—
187C22×D3—94no—
188C22×D3—94no—
189C22×D3—92no—
190C22×D3—94no—
191C22×D3—184no—
192C22×D3—184no—
193C22×D3—188no—
194C22×D3—188no—
195C22×D3—184no—
196C22×D3—184no—
197C22×D3—184no—
198C2×A4—32no—
199C2×A4—31no—
200C2×A4—184no—
201C3×D4—364no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—121no—
208C2×C3×C4D3×A411yescharacteristic
209D12—34no—
210D12—98no—
211D12—98no—
212D12—94no—
213C2×C32—42no—
214C2×C32—41no—
215C3×D3—244no—
216C2×C9—32no—
217C2×C9—31no—
218C2×C9—31no—
219C2×C9—82no—
220C2×C9—81no—
221C2×C9D4×A412yescomplemented
222C2×C9C23×A411yescharacteristic
223D9—24no—
224D9—64no—
225C24—94no—
226C24—184no—
227C24—272no—
228C24—274no—
229C24—272no—
230C24—544no—
231C24C2×AGL(1,ℤ/9)11yescharacteristic
232C2×D4—91no—
233C2×D4—274no—
234C2×D4—274no—
235C2×D4—272no—
236C2×D4—278no—
237C2×D4—272no—
238C2×D4—274no—
239C2×D4—274no—
240C2×D4—278no—
241C2×D4—271no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—32no—
245C22×C4C2×AGL(1,ℤ/9)12yescomplemented
246C22×C3—32no—
247C22×C3—31no—
248C22×C3—32no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—121no—
254C22×C3—364no—
255C22×C3—724no—
256C22×C3C2×C3×D1211yescharacteristic
257C22×C3C2×D3×A411yescharacteristic
258C2×D3—34no—
259C2×D3—64no—
260C2×D3—94no—
261C2×D3—184no—
262C2×D3—184no—
263C2×D3—188no—
264C2×D3—188no—
265C2×D3—184no—
266C2×D3—184no—
267C2×D3—184no—
268A4—31no—
269C3×C4—32no—
270C3×C4—122no—
271C3×C4C2×D3×A412yes—
272C32—41no—
273C9—81no—
274C9C2×D4×A411yescharacteristic, complemented
275C23—31no—
276C23—32no—
277C23—31no—
278C23—92no—
279C23—184no—
280C23—274no—
281C23—274no—
282C23—274no—
283C23—274no—
284C23—272no—
285C23—544no—
286C23—544no—
287C23—544no—
288C23—548no—
289C23—544no—
290C23—544no—
291C23—548no—
292C23C4⋊AGL(1,ℤ/9)12yescomplemented
293C23C22×AGL(1,ℤ/9)11yescharacteristic
294D4—94no—
295D4—278no—
296D4—278no—
297D4—274no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4D9⋊A411yescharacteristic, complemented
304C2×C3—31no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—121no—
308C2×C3—122no—
309C2×C3—724no—
310C2×C3A4×D1212yes—
311C2×C3C22×D3×A411yescharacteristic
312D3—64no—
313D3—184no—
314C22—32no—
315C22—31no—
316C22—32no—
317C22—31no—
318C22—31no—
319C22—32no—
320C22—32no—
321C22—94no—
322C22—184no—
323C22—274no—
324C22—544no—
325C22—544no—
326C22—548no—
327C22—544no—
328C22—544no—
329C22—548no—
330C22—544no—
331C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic, complemented
332C22C2×D9⋊A411yescharacteristic
333C4—32no—
334C4C2×D9⋊A412yescomplemented
335C3—121no—
336C3C2×A4×D1211yescharacteristic
337C2—31no—
338C2—32no—
339C2—31no—
340C2—184no—
341C2—544no—
342C2C4⋊(D9⋊A4)12yesdirect product factor
343C2C22×D9⋊A411yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocleC23×C9C23×C3
FrattiniC2×C3C22×D3×A4
FittingC23×C4×C9C2×C3
socleC24×C3C2×C3×D3
hypercenterC2×C4D9⋊A4
hypocenterC22×C9C2×C3×D4
LCSG ▷ C23×C9 ▷ C22×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D9⋊A4C2semidirect
2. C2×C4×C9⋊A4C2semidirect
3. C4⋊(D9⋊A4)C2direct (N × H)
5. C23×D36C3semidirect
8. C2×D9⋊A4C22semidirect
10. C4×C9⋊A4C22semidirect
14. C24×D9C2×C3semidirect
15. C23×C4×C9C2×C3semidirect
20. C22×D36C2×C3semidirect
23. C2×C9⋊A4D4semidirect
41. C23×D9C22×C3semidirect
46. C22×C4×C9C22×C3semidirect
56. C2×D36A4semidirect
61. C9⋊A4C2×D4semidirect
82. C23×C9C3×D4semidirect
97. C22×D9C2×A4semidirect
106. C2×C4×C9C2×A4semidirect
110. D36C2×A4semidirect
154. C22×C9C2×C3×D4semidirect
165. C2×D9C22×A4semidirect
169. C4×C9C22×A4semidirect
176. C23×C4AGL(1,ℤ/9)semidirect
221. C2×C9D4×A4semidirect
245. C22×C4C2×AGL(1,ℤ/9)semidirect
274. C9C2×D4×A4semidirect
292. C23C4⋊AGL(1,ℤ/9)semidirect
303. C2×C4D9⋊A4semidirect
331. C22C2×C4⋊AGL(1,ℤ/9)semidirect
334. C4C2×D9⋊A4semidirect
342. C2C4⋊(D9⋊A4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D9⋊A42
C2×C4×C9⋊A42
C4⋊(D9⋊A4)2
C2×A4×D123
C23×D363
C2×C4⋊AGL(1,ℤ/9)4