Finite Group Databasebeta

← C9×SL(2,ℤ/4)⋊C4 #13146775

Subgroups — C9×SL(2,ℤ/4)⋊C4 #13146775

Order 1728.

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

normal subgroups only

This group has 359 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×C9×SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C9×SL(2,ℤ/4)C211yesmaximal, characteristic
4C3×SL(2,ℤ/4)⋊C4C311yesmaximal, characteristic
5C9×#385—31nomaximal
6C22×C9×A4C2211yescharacteristic
7C4×C9×A4—21no—
8C9×SL(2,ℤ/4)—21no—
9C9×SL(2,ℤ/4)C412yescomplemented
10C9×Dic3⋊C4—41nomaximal
11C2×C3×C4×A4C2×C311yescharacteristic
12C2×C3×SL(2,ℤ/4)C2×C311yescharacteristic
13C2×C3×SL(2,ℤ/4)C2×C311yescharacteristic
14C23×C4×C9D311yescharacteristic
15C2×C4×C22⋊C9—21no—
16C2×C9×C4⋊C4—31no—
17C2×C9×C4⋊C4—31no—
18C2×C9×C22⋊C4—31no—
19C2×C9×C22⋊C4—31no—
20C9×#95—31no—
21C9×#95—31no—
22C2×C9×A4C2×C411yescharacteristic
23C2×C9×A4D411yescharacteristic
24C2×C9×A4Q811yescharacteristic
25C2×C3×C4×C9—41no—
26C2×C9×Dic3—41no—
27C2×C9×Dic3—41no—
28SL(2,ℤ/4)⋊C4C911yescharacteristic, direct product factor
29C3×#385—31no—
30C22×C3×A4C22×C311yescharacteristic
31C3×C4×A4—21no—
32C3×SL(2,ℤ/4)—21no—
33C3×SL(2,ℤ/4)C3×C412yes—
34C24×C9C2×D311yescharacteristic
35C22×C22⋊C9—21no—
36C3×Dic3⋊C4—41no—
37C22×C4×C9—21no—
38C22×C4×C9—31no—
39C22×C4×C9—31no—
40C22×C4×C9—31no—
41C22×C4×C9—31no—
42C22×C4×C9—31no—
43C22×C4×C9—31no—
44C22×C4×C9—61no—
45C4×C22⋊C9—22no—
46C9×C4⋊C4—62no—
47C9×C4⋊C4—61no—
48C9×C4⋊C4—61no—
49C9×C22⋊C4—32no—
50C9×C22⋊C4—32no—
51C9×C22⋊C4—61no—
52C9×C22⋊C4—61no—
53C22×C3×C9—41no—
54C9×A4C4⋊C411yescharacteristic, complemented
55C3×C4×C9—81no—
56C9×Dic3—42no—
57C9×Dic3—81no—
58C23×C3×C4C3×D311yescharacteristic
59C2×C4×A4—21no—
60C2×C4×A4C2×C911yescharacteristic
61C2×SL(2,ℤ/4)C2×C911yescharacteristic
62C2×SL(2,ℤ/4)C2×C911yescharacteristic
63C2×C3×C4⋊C4—31no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C22⋊C4—31no—
66C2×C3×C22⋊C4—31no—
67C3×#95—31no—
68C3×#95—31no—
69C2×C3×A4C2×C3×C411yescharacteristic
70C2×C3×A4C3×D411yescharacteristic
71C2×C3×A4C3×Q811yescharacteristic
72C2×C32×C4—41no—
73C2×C3×Dic3—41no—
74C2×C3×Dic3—41no—
75C23×C9—31no—
76C23×C9—31no—
77C23×C9—31no—
78C23×C9—31no—
79C23×C9Dic611yescharacteristic
80C23×C9C4×D311yescharacteristic
81C23×C9#6711yescharacteristic
82C2×C22⋊C9—21no—
83C2×C22⋊C9—21no—
84C2×C22⋊C9—21no—
85C2×C4×C9—32no—
86C2×C4×C9—32no—
87C2×C4×C9—32no—
88C2×C4×C9—31no—
89C2×C4×C9—61no—
90C2×C4×C9—61no—
91C2×C4×C9—61no—
92C2×C4×C9—61no—
93C2×C4×C9—61no—
94C2×C4×C9—61no—
95C2×C4×C9—61no—
96C2×C4×C9—61no—
97C2×C4×C9—61no—
98C2×C4×C9—61no—
99C2×C4×C9—61no—
100C2×C4×C9—61no—
101C2×C4×C9—61no—
102C2×C4×C9—81no—
103C2×C4×C9S411yescharacteristic
104#385—31noSylow
105C2×C3×C9—41no—
106C2×C3×C9—41no—
107C2×C3×C9—41no—
108C24×C3C2×C3×D311yescharacteristic
109C22×A4—21no—
110C22×A4C22×C911yescharacteristic
111C22×C3×C4—21no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C4×A4—22no—
120C4×A4—21no—
121SL(2,ℤ/4)—21no—
122SL(2,ℤ/4)C4×C912yescomplemented
123C3×C4⋊C4—62no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—61no—
126C3×C22⋊C4—32no—
127C3×C22⋊C4—32no—
128C3×C22⋊C4—61no—
129C3×C22⋊C4—61no—
130Dic3⋊C4—41no—
131C22×C32—41no—
132C3×A4C3×C4⋊C411yescharacteristic
133C32×C4—81no—
134C3×Dic3—42no—
135C3×Dic3—81no—
136C22×C9—31no—
137C22×C9—31no—
138C22×C9—31no—
139C22×C9—31no—
140C22×C9—31no—
141C22×C9—31no—
142C22×C9—31no—
143C22×C9—31no—
144C22×C9—31no—
145C22×C9—61no—
146C22×C9—81no—
147C22×C9Dic3⋊C411yescharacteristic, complemented
148C22×C9C2×S411yescharacteristic
149C22⋊C9—21no—
150C4×C9—21no—
151C4×C9—62no—
152C4×C9—62no—
153C4×C9—61no—
154C4×C9—82no—
155C4×C9—121no—
156C4×C9—121no—
157C23×C4D3×C911yescharacteristic
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162#95—31no—
163#95—31no—
164C3×C9—41noSylow
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—31no—
169C23×C3C3×Dic611yescharacteristic
170C23×C3C3×C4×D311yescharacteristic
171C23×C3C3×#6711yescharacteristic
172C2×A4—21no—
173C2×A4—21no—
174C2×A4—21no—
175C2×A4C2×C4×C911yescharacteristic
176C2×A4D4×C911yescharacteristic
177C2×A4Q8×C911yescharacteristic
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—41no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—81no—
197C2×C3×C4C3×S411yescharacteristic
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C9—31no—
204C2×C9—31no—
205C2×C9—31no—
206C2×C9—31no—
207C2×C9—81no—
208C2×C9—81no—
209C2×C9—81no—
210C2×C9A4⋊Q811yescharacteristic
211C2×C9C4×S411yescharacteristic
212C2×C9GL(2,ℤ/4)11yescharacteristic
213C24C2×D3×C911yescharacteristic
214C22×C4—21no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—61no—
222C4⋊C4—62no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—61no—
228C22⋊C4—61no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—41no—
239C22×C3—61no—
240C22×C3—81no—
241C22×C3C3×Dic3⋊C411yescharacteristic
242C22×C3C2×C3×S411yescharacteristic
243A4—21no—
244A4C9×C4⋊C411yescharacteristic, complemented
245C3×C4—21no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4—62no—
249C3×C4—81no—
250C3×C4—82no—
251C3×C4—121no—
252C3×C4—121no—
253Dic3—42no—
254Dic3—81no—
255C32—41no—
256C9—81no—
257C9SL(2,ℤ/4)⋊C411yescharacteristic, direct product factor
258C23—31no—
259C23—31no—
260C23—31no—
261C23—31no—
262C23C9×Dic611yescharacteristic
263C23C4×D3×C911yescharacteristic
264C23C9×#6711yescharacteristic
265C2×C4—32no—
266C2×C4—32no—
267C2×C4—32no—
268C2×C4—31no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4C9×S411yescharacteristic
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—81no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3C3×A4⋊Q811yescharacteristic
294C2×C3C3×C4×S411yescharacteristic
295C2×C3C3×GL(2,ℤ/4)11yescharacteristic
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—61no—
306C22C9×Dic3⋊C411yescharacteristic, complemented
307C22C2×C9×S411yescharacteristic
308C4—21no—
309C4—62no—
310C4—62no—
311C4—61no—
312C4—121no—
313C4—121no—
314C3—41no—
315C3—81no—
316C3C3×SL(2,ℤ/4)⋊C411yescharacteristic
317C2—31no—
318C2—31no—
319C2—31no—
320C2—31no—
321C2C9×A4⋊Q811yescharacteristic
322C2C4×C9×S411yescharacteristic
323C2C9×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C9C2×S4
commutatorC2×A4C2×C4×C9
FrattiniC22×C3C2×C3×S4
FittingC23×C4×C9D3
socleC24×C3C2×C3×D3
prime radical, cosocleC22×C3×A4C22×C3
hypercenterC2×C4×C9S4
hypocenterA4C9×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C9 ◁ C2×C4×C9

Complements

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

NH ≅ G/Ntype
9. C9×SL(2,ℤ/4)C4semidirect
28. SL(2,ℤ/4)⋊C4C9direct (N × H)
54. C9×A4C4⋊C4semidirect
122. SL(2,ℤ/4)C4×C9semidirect
147. C22×C9Dic3⋊C4semidirect
244. A4C9×C4⋊C4semidirect
257. C9SL(2,ℤ/4)⋊C4direct (N × H)
306. C22C9×Dic3⋊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×C9×SL(2,ℤ/4)2
C3×SL(2,ℤ/4)⋊C43
C9×#3853
C9×Dic3⋊C44