Finite Group Databasebeta

← A4×Dic9⋊C4 #13147152

Subgroups — A4×Dic9⋊C4 #13147152

Order 1728.

371 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×C4×C9×A4C211yesmaximal, characteristic
2C2×A4×Dic9C211yesmaximal, characteristic
3C2×A4×Dic9C211yesmaximal, characteristic
4A4×Dic3⋊C4—31nomaximal
5C22×Dic9⋊C4C311yesmaximal, characteristic, complemented
6C22×C9×A4C2211yescharacteristic
7C4×C9×A4—21no—
8A4×Dic9—21no—
9A4×Dic9C412yescomplemented
10C3×Dic9⋊C4—41nomaximal
11C2×C3×C4×A4D311yescharacteristic
12C2×Dic3×A4—31no—
13C2×Dic3×A4—31no—
14C23×C4×C9C2×C311yescharacteristic
15C23×Dic9C2×C311yescharacteristic
16C23×Dic9C2×C311yescharacteristic
17C2×C4×C22⋊C9—21no—
18C2×Dic9⋊C4—31no—
19C2×Dic9⋊C4—31no—
20C2×Dic9⋊C4—31no—
21C2×Dic9⋊C4—31no—
22C2×C9×A4C2×C411yescharacteristic
23C2×C9×A4D411yescharacteristic
24C2×C9×A4Q811yescharacteristic
25C2×C3×C4×C9—41no—
26C2×C3×Dic9—41no—
27C2×C3×Dic9—41no—
28C22×Dic3⋊C4—31no—
29A4×C4⋊C4—91no—
30C22×C3×A4C2×D311yescharacteristic
31C3×C4×A4—21no—
32Dic3×A4—32no—
33Dic3×A4—61no—
34C24×C9C22×C311yescharacteristic
35C22×C22⋊C9—21no—
36C3×Dic3⋊C4—121no—
37C22×C4×C9—21no—
38C22×C4×C9—31no—
39C22×C4×C9—31no—
40C22×C4×C9—61no—
41C22×Dic9—21no—
42C22×Dic9—31no—
43C22×Dic9—32no—
44C22×Dic9—31no—
45C22×Dic9—31no—
46C22×Dic9—31no—
47C22×Dic9—61no—
48C22×Dic9C3×C412yescomplemented
49C4×C22⋊C9—22no—
50Dic9⋊C4—32no—
51Dic9⋊C4—31no—
52Dic9⋊C4—31no—
53Dic9⋊C4—31no—
54Dic9⋊C4A411yescharacteristic, direct product factor
55C22×C3×C9—41no—
56C9×A4C4⋊C411yescharacteristic, complemented
57C3×C4×C9—81no—
58C3×Dic9—42no—
59C3×Dic9—81no—
60C23×C3×C4C3×D311yescharacteristic
61C23×Dic3—31no—
62C23×Dic3—31no—
63C2×C4×A4—21no—
64C2×C4×A4—91no—
65C2×C4×A4—91no—
66C2×C4×A4D911yescharacteristic
67C2×Dic3⋊C4—91no—
68C2×Dic3⋊C4—91no—
69C2×Dic3⋊C4—91no—
70C2×Dic3⋊C4—91no—
71C2×C3×A4Dic611yescharacteristic
72C2×C3×A4C4×D311yescharacteristic
73C2×C3×A4#6711yescharacteristic
74C2×C32×C4—41no—
75C2×C3×Dic3—121no—
76C2×C3×Dic3—121no—
77C23×C9—31no—
78C23×C9—31no—
79C23×C9—31no—
80C23×C9—31no—
81C23×C9C2×C3×C411yescharacteristic
82C23×C9C3×D411yescharacteristic
83C23×C9C3×Q811yescharacteristic
84C2×C22⋊C9—21no—
85C2×C22⋊C9—21no—
86C2×C22⋊C9—21no—
87C2×C4×C9—31no—
88C2×C4×C9—61no—
89C2×C4×C9—61no—
90C2×C4×C9—61no—
91C2×C4×C9—61no—
92C2×C4×C9—81no—
93C2×C4×C9C2×A411yescharacteristic
94C2×Dic9—32no—
95C2×Dic9—31no—
96C2×Dic9—32no—
97C2×Dic9—32no—
98C2×Dic9—32no—
99C2×Dic9—31no—
100C2×Dic9—61no—
101C2×Dic9—61no—
102C2×Dic9—61no—
103C2×Dic9—61no—
104C2×Dic9C2×A411yescharacteristic
105C2×Dic9C2×A411yescharacteristic
106C22×C4⋊C4—91noSylow
107C2×C3×C9—41no—
108C2×C3×C9—41no—
109C2×C3×C9—41no—
110C24×C3C2×C3×D311yescharacteristic
111C22×A4—21no—
112C22×A4C2×D911yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—61no—
117C22×Dic3—32no—
118C22×Dic3—61no—
119C22×Dic3—91no—
120C22×Dic3—91no—
121C22×Dic3—91no—
122C22×Dic3—92no—
123C22×Dic3—91no—
124C22×Dic3—181no—
125C4×A4—22no—
126C4×A4—21no—
127C4×A4—92no—
128C4×A4—181no—
129C3×C4⋊C4—361no—
130Dic3⋊C4—31no—
131Dic3⋊C4—91no—
132Dic3⋊C4—91no—
133Dic3⋊C4—92no—
134Dic3⋊C4—91no—
135C22×C32—41no—
136C3×A4Dic3⋊C411yescharacteristic
137C32×C4—81no—
138C3×Dic3—122no—
139C3×Dic3—241no—
140C22×C9—31no—
141C22×C9—31no—
142C22×C9—31no—
143C22×C9—32no—
144C22×C9—31no—
145C22×C9—31no—
146C22×C9—31no—
147C22×C9—31no—
148C22×C9—31no—
149C22×C9—31no—
150C22×C9—81no—
151C22×C9C3×C4⋊C411yescharacteristic, complemented
152C22×C9C22×A411yescharacteristic
153C22⋊C9—21no—
154C4×C9—21no—
155C4×C9—61no—
156C4×C9—82no—
157Dic9—21no—
158Dic9—32no—
159Dic9—61no—
160Dic9C4×A412yescomplemented
161C23×C4—91no—
162C23×C4—91no—
163C23×C4C3×D911yescharacteristic
164C2×C4⋊C4—271no—
165C2×C4⋊C4—271no—
166C2×C4⋊C4—271no—
167C2×C4⋊C4—271no—
168C3×C9—41noSylow
169C23×C3—31no—
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—31no—
173C23×C3C3×Dic611yescharacteristic
174C23×C3C3×C4×D311yescharacteristic
175C23×C3C3×#6711yescharacteristic
176C2×A4—21no—
177C2×A4—21no—
178C2×A4—21no—
179C2×A4Dic1811yescharacteristic
180C2×A4C4×D911yescharacteristic
181C2×A4#61111yescharacteristic
182C2×C3×C4—31no—
183C2×C3×C4—41no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—81no—
189C2×C3×C4—361no—
190C2×C3×C4—361no—
191C2×C3×C4D3×A411yescharacteristic
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—92no—
195C2×Dic3—92no—
196C2×Dic3—92no—
197C2×Dic3—91no—
198C2×Dic3—92no—
199C2×Dic3—91no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C9—31no—
208C2×C9—31no—
209C2×C9—31no—
210C2×C9—31no—
211C2×C9—81no—
212C2×C9—81no—
213C2×C9—81no—
214C2×C9C2×C4×A411yescharacteristic
215C2×C9D4×A411yescharacteristic
216C2×C9Q8×A411yescharacteristic
217C24C2×C3×D911yescharacteristic
218C22×C4—21no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—92no—
223C22×C4—181no—
224C22×C4—271no—
225C22×C4—271no—
226C22×C4—271no—
227C22×C4—272no—
228C22×C4—271no—
229C22×C4—541no—
230C4⋊C4—91no—
231C4⋊C4—271no—
232C4⋊C4—271no—
233C4⋊C4—272no—
234C4⋊C4—271no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—32no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—41no—
246C22×C3—81no—
247C22×C3C3×Dic3⋊C411yescharacteristic
248C22×C3C2×D3×A411yescharacteristic
249A4—21no—
250A4Dic9⋊C411yescharacteristic, direct product factor
251C3×C4—21no—
252C3×C4—61no—
253C3×C4—81no—
254C3×C4—82no—
255C3×C4—362no—
256C3×C4—721no—
257Dic3—32no—
258Dic3—61no—
259Dic3—92no—
260Dic3—181no—
261C32—41no—
262C9—81no—
263C9A4×C4⋊C411yescharacteristic, complemented
264C23—31no—
265C23—31no—
266C23—31no—
267C23—31no—
268C23C3×Dic1811yescharacteristic
269C23C3×C4×D911yescharacteristic
270C23C3×#61111yescharacteristic
271C2×C4—31no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—272no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—271no—
282C2×C4—272no—
283C2×C4—271no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4—541no—
287C2×C4—541no—
288C2×C4A4×D911yescharacteristic
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—81no—
297C2×C3—81no—
298C2×C3—81no—
299C2×C3A4×Dic611yescharacteristic
300C2×C3C4×D3×A411yescharacteristic
301C2×C3A4×#6711yescharacteristic
302C22—31no—
303C22—31no—
304C22—31no—
305C22—32no—
306C22—31no—
307C22—31no—
308C22—31no—
309C22—31no—
310C22—31no—
311C22—31no—
312C22C3×Dic9⋊C411yescharacteristic, complemented
313C22C2×A4×D911yescharacteristic
314C4—21no—
315C4—61no—
316C4—92no—
317C4—181no—
318C4—272no—
319C4—541no—
320C3—41no—
321C3—81no—
322C3A4×Dic3⋊C411yescharacteristic
323C2—31no—
324C2—31no—
325C2—31no—
326C2—31no—
327C2A4×Dic1811yescharacteristic
328C2C4×A4×D911yescharacteristic
329C2A4×#61111yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutatorC23×C9C2×C3×C4
FrattiniC22×C3C2×D3×A4
FittingC23×C4×C9C2×C3
socleC24×C3C2×C3×D3
prime radicalC22×C3×A4C2×D3
cosocleC24×C9C22×C3
hypercenterC2×C4A4×D9
hypocenterC22×C9C3×C4⋊C4
LCSG ▷ C23×C9 ▷ C22×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. C22×Dic9⋊C4C3semidirect
9. A4×Dic9C4semidirect
48. C22×Dic9C3×C4semidirect
54. Dic9⋊C4A4direct (N × H)
56. C9×A4C4⋊C4semidirect
151. C22×C9C3×C4⋊C4semidirect
160. Dic9C4×A4semidirect
250. A4Dic9⋊C4direct (N × H)
263. C9A4×C4⋊C4semidirect
312. C22C3×Dic9⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C9×A42
C2×A4×Dic92
A4×Dic3⋊C43
C22×Dic9⋊C43
C3×Dic9⋊C44