Finite Group Databasebeta

← C9×(C2×S4)⋊C4 #13146777

Subgroups — C9×(C2×S4)⋊C4 #13146777

Order 1728.

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

normal subgroups only

This group has 461 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
1C22×C9×S4C211yesmaximal, characteristic
2C2×C4×C9×A4C211yesmaximal, characteristic, complemented
3C2×C9×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C311yesmaximal, characteristic
5C9×#386—31nomaximal
6C22×C9×A4C2211yescharacteristic
7C2×C9×S4—21no—
8C2×C9×S4—21no—
9C2×C9×S4C412yescomplemented
10C4×C9×A4—21no—
11C9×SL(2,ℤ/4)—21no—
12C9×(C2×D3)⋊C4—41nomaximal
13C22×C3×S4C2×C311yescharacteristic
14C2×C3×C4×A4C2×C311yescharacteristic
15C2×C3×SL(2,ℤ/4)C2×C311yescharacteristic
16C22×D4×C9—31no—
17C23×C4×C9D311yescharacteristic, complemented
18C2×C4×C22⋊C9—21no—
19C2×C9×C22⋊C4—31no—
20C2×C9×C22⋊C4—31no—
21C2×C9×C22⋊C4—31no—
22C9×#95—31no—
23C9×#95—31no—
24C22×D3×C9—41no—
25C2×C9×A4C2×C411yescharacteristic
26C2×C9×A4D411yescharacteristic
27C2×C9×A4D411yescharacteristic
28C9×S4—22no—
29C2×C3×C4×C9—41no—
30C2×C9×Dic3—41no—
31(C2×S4)⋊C4C911yescharacteristic, direct product factor
32C3×#386—31no—
33C22×C3×A4C22×C311yescharacteristic
34C2×C3×S4—21no—
35C2×C3×S4—21no—
36C2×C3×S4C3×C412yes—
37C3×C4×A4—21no—
38C3×SL(2,ℤ/4)—21no—
39C24×C9—31no—
40C24×C9C2×D311yescharacteristic
41C22×C22⋊C9—21no—
42C3×(C2×D3)⋊C4—41no—
43C2×D4×C9—32no—
44C2×D4×C9—32no—
45C2×D4×C9—61no—
46C2×D4×C9—61no—
47C2×D4×C9—61no—
48C2×D4×C9—61no—
49C22×C4×C9—21no—
50C22×C4×C9—31no—
51C22×C4×C9—31no—
52C22×C4×C9—31no—
53C22×C4×C9—31no—
54C22×C4×C9—31no—
55C22×C4×C9—61no—
56C4×C22⋊C9—22no—
57C9×C22⋊C4—62no—
58C9×C22⋊C4—61no—
59C9×C22⋊C4—61no—
60C9×C22⋊C4—61no—
61C9×C22⋊C4—61no—
62C22×C3×C9—41no—
63C2×D3×C9—42no—
64C2×D3×C9—81no—
65C2×D3×C9—81no—
66C9×A4C22⋊C411yescharacteristic, complemented
67C3×C4×C9—81no—
68C9×Dic3—81no—
69C22×S4C2×C911yescharacteristic
70C22×C3×D4—31no—
71C23×C3×C4C3×D311yescharacteristic
72C2×C4×A4—21no—
73C2×C4×A4C2×C911yescharacteristic, complemented
74C2×SL(2,ℤ/4)C2×C911yescharacteristic, complemented
75C2×C3×C22⋊C4—31no—
76C2×C3×C22⋊C4—31no—
77C2×C3×C22⋊C4—31no—
78C3×#95—31no—
79C3×#95—31no—
80C22×C3×D3—41no—
81C2×C3×A4C2×C3×C411yescharacteristic
82C2×C3×A4C3×D411yescharacteristic
83C2×C3×A4C3×D411yescharacteristic
84C3×S4—22no—
85C2×C32×C4—41no—
86C2×C3×Dic3—41no—
87C23×C9—31no—
88C23×C9—31no—
89C23×C9—31no—
90C23×C9—31no—
91C23×C9—32no—
92C23×C9—61no—
93C23×C9—61no—
94C23×C9—61no—
95C23×C9—61no—
96C23×C9—61no—
97C23×C9—61no—
98C23×C9C4×D311yescharacteristic
99C23×C9D1211yescharacteristic
100C23×C9#6711yescharacteristic
101C2×C22⋊C9—21no—
102C2×C22⋊C9—21no—
103C2×C22⋊C9—21no—
104D4×C9—62no—
105D4×C9—62no—
106D4×C9—62no—
107D4×C9—62no—
108C2×C4×C9—32no—
109C2×C4×C9—32no—
110C2×C4×C9—31no—
111C2×C4×C9—61no—
112C2×C4×C9—61no—
113C2×C4×C9—61no—
114C2×C4×C9—61no—
115C2×C4×C9—61no—
116C2×C4×C9—61no—
117C2×C4×C9—61no—
118C2×C4×C9—61no—
119C2×C4×C9—61no—
120C2×C4×C9—61no—
121C2×C4×C9—61no—
122C2×C4×C9—81no—
123C2×C4×C9S411yescharacteristic, complemented
124#386—31noSylow
125C2×C3×C9—41no—
126C2×C3×C9—41no—
127C2×C3×C9—41no—
128D3×C9—82no—
129C24×C3—31no—
130C24×C3C2×C3×D311yescharacteristic
131C22×A4—21no—
132C22×A4C22×C911yescharacteristic
133C2×S4—21no—
134C2×S4—21no—
135C2×S4C4×C912yescomplemented
136C2×C3×D4—32no—
137C2×C3×D4—32no—
138C2×C3×D4—61no—
139C2×C3×D4—61no—
140C2×C3×D4—61no—
141C2×C3×D4—61no—
142C22×C3×C4—21no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—31no—
148C22×C3×C4—61no—
149C4×A4—22no—
150C4×A4—21no—
151SL(2,ℤ/4)—21no—
152C3×C22⋊C4—62no—
153C3×C22⋊C4—61no—
154C3×C22⋊C4—61no—
155C3×C22⋊C4—61no—
156C3×C22⋊C4—61no—
157(C2×D3)⋊C4—41no—
158C22×C32—41no—
159C2×C3×D3—42no—
160C2×C3×D3—81no—
161C2×C3×D3—81no—
162C3×A4C3×C22⋊C411yescharacteristic
163C32×C4—81no—
164C3×Dic3—81no—
165C22×C9—31no—
166C22×C9—31no—
167C22×C9—31no—
168C22×C9—31no—
169C22×C9—31no—
170C22×C9—31no—
171C22×C9—31no—
172C22×C9—31no—
173C22×C9—31no—
174C22×C9—61no—
175C22×C9—62no—
176C22×C9—62no—
177C22×C9—62no—
178C22×C9—81no—
179C22×C9—121no—
180C22×C9—121no—
181C22×C9—121no—
182C22×C9—121no—
183C22×C9(C2×D3)⋊C411yescharacteristic, complemented
184C22×C9C2×S411yescharacteristic
185C22⋊C9—21no—
186C4×C9—21no—
187C4×C9—62no—
188C4×C9—61no—
189C4×C9—82no—
190C4×C9—121no—
191C4×C9—121no—
192C22×D4—31no—
193C23×C4D3×C911yescharacteristic, complemented
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—31no—
197#95—31no—
198#95—31no—
199C3×C9—41noSylow
200C23×C3—31no—
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—31no—
204C23×C3—32no—
205C23×C3—61no—
206C23×C3—61no—
207C23×C3—61no—
208C23×C3—61no—
209C23×C3—61no—
210C23×C3—61no—
211C23×C3C3×C4×D311yescharacteristic
212C23×C3C3×D1211yescharacteristic
213C23×C3C3×#6711yescharacteristic
214C22×D3—41no—
215C2×A4—21no—
216C2×A4—21no—
217C2×A4—21no—
218C2×A4C2×C4×C911yescharacteristic
219C2×A4D4×C911yescharacteristic
220C2×A4D4×C911yescharacteristic
221S4—22no—
222C3×D4—62no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—41no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—81no—
242C2×C3×C4C3×S411yescharacteristic
243C2×Dic3—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C3×D3—82no—
248C2×C9—31no—
249C2×C9—31no—
250C2×C9—31no—
251C2×C9—31no—
252C2×C9—81no—
253C2×C9—81no—
254C2×C9—81no—
255C2×C9—122no—
256C2×C9C4×S411yescharacteristic
257C2×C9C4⋊S411yescharacteristic
258C2×C9GL(2,ℤ/4)11yescharacteristic
259C24—31no—
260C24C2×D3×C911yescharacteristic
261C2×D4—32no—
262C2×D4—32no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—21no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—61no—
274C22⋊C4—62no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—41no—
289C22×C3—61no—
290C22×C3—62no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—81no—
294C22×C3—121no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3—121no—
298C22×C3C3×(C2×D3)⋊C411yescharacteristic
299C22×C3C2×C3×S411yescharacteristic
300C2×D3—42no—
301C2×D3—81no—
302C2×D3—81no—
303A4—21no—
304A4C9×C22⋊C411yescharacteristic, complemented
305C3×C4—21no—
306C3×C4—62no—
307C3×C4—61no—
308C3×C4—81no—
309C3×C4—82no—
310C3×C4—121no—
311C3×C4—121no—
312Dic3—81no—
313C32—41no—
314C9—81no—
315C9(C2×S4)⋊C411yescharacteristic, direct product factor
316C23—31no—
317C23—31no—
318C23—31no—
319C23—31no—
320C23—32no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23C4×D3×C911yescharacteristic
328C23C9×D1211yescharacteristic
329C23C9×#6711yescharacteristic
330D4—62no—
331D4—62no—
332D4—62no—
333D4—62no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4C9×S411yescharacteristic, complemented
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—122no—
360C2×C3C3×C4×S411yescharacteristic
361C2×C3C3×C4⋊S411yescharacteristic
362C2×C3C3×GL(2,ℤ/4)11yescharacteristic
363D3—82no—
364C22—31no—
365C22—31no—
366C22—31no—
367C22—31no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—31no—
372C22—31no—
373C22—61no—
374C22—62no—
375C22—62no—
376C22—62no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22C9×(C2×D3)⋊C411yescharacteristic, complemented
382C22C2×C9×S411yescharacteristic
383C4—21no—
384C4—62no—
385C4—61no—
386C4—121no—
387C4—121no—
388C3—41no—
389C3—81no—
390C3C3×(C2×S4)⋊C411yescharacteristic
391C2—31no—
392C2—31no—
393C2—31no—
394C2—31no—
395C2—122no—
396C2C4×C9×S411yescharacteristic
397C2C9×C4⋊S411yescharacteristic
398C2C9×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 radicalC22×C3×S4C2×C3
cosocleC22×C3×A4C22×C3
hypercenterC2×C4×C9S4
hypocenterA4C9×C22⋊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
2. C2×C4×C9×A4C2semidirect
3. C2×C9×SL(2,ℤ/4)C2semidirect
9. C2×C9×S4C4semidirect
17. C23×C4×C9D3semidirect
31. (C2×S4)⋊C4C9direct (N × H)
66. C9×A4C22⋊C4semidirect
73. C2×C4×A4C2×C9semidirect
74. C2×SL(2,ℤ/4)C2×C9semidirect
123. C2×C4×C9S4semidirect
135. C2×S4C4×C9semidirect
183. C22×C9(C2×D3)⋊C4semidirect
193. C23×C4D3×C9semidirect
304. A4C9×C22⋊C4semidirect
315. C9(C2×S4)⋊C4direct (N × H)
348. C2×C4C9×S4semidirect
381. C22C9×(C2×D3)⋊C4semidirect

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
C3×(C2×S4)⋊C43
C9×#3863
C9×(C2×D3)⋊C44