Finite Group Databasebeta

← C2×(C9⋊S4)⋊C4 #13164146

Subgroups — C2×(C9⋊S4)⋊C4 #13164146

Order 1728.

763 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 79 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×A4×Dic9C211yesmaximal, characteristic, complemented
3C2×C9×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4(C9⋊S4)⋊C4C214yesmaximal, direct product factor
5C2×(C3⋊S4)⋊C4—31nomaximal
6C2×#611⋊C4—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C2×C9⋊S4C414yescomplemented
9C2×C9⋊S4C2212yes—
10C2×(C3⋊D9)⋊C4—41nomaximal
11A4×Dic9C2212yescomplemented
12C9×SL(2,ℤ/4)C2212yescomplemented
13C22×C3⋊S4—31no—
14C2×Dic3×A4—31no—
15C2×C3×SL(2,ℤ/4)D311yescharacteristic
16(C3⋊S4)⋊C4—34no—
17C22×(C22⋊C9)⋊C2—61no—
18C22×#611—31no—
19C23×Dic9D311yescharacteristic, complemented
20C22×C4×D9—31no—
21C2×C9×C22⋊C4—31no—
22C2×(C2×D9)⋊C4—31no—
23C2×Dic9⋊C4—31no—
24C2×C4×Dic9—31no—
25#611⋊C4—34no—
26#611⋊C4—34no—
27C22×C3⋊D9—41no—
28C2×C9×A4C2×C412yescomplemented
29C2×C9×A4C2311yescharacteristic
30C9⋊S4C2×C414yescomplemented
31C2×C9×Dic3—41no—
32C2×C3×Dic9—41no—
33(C3⋊D9)⋊C4—44no—
34C2×C4×S4—91no—
35C2×#67⋊C4—91no—
36C22×C3×A4C2×D311yescharacteristic
37C2×C3⋊S4—34no—
38C2×C3⋊S4—32no—
39C2×#624—121no—
40Dic3×A4—32no—
41C3×SL(2,ℤ/4)C2×D312yes—
42C24×C9C2×D311yescharacteristic
43C23×D9—31no—
44C22×C22⋊C9—21no—
45C2×(C22⋊C9)⋊C2—64no—
46C2×(C22⋊C9)⋊C2—62no—
47C22×C4×C9—31no—
48C22×C4×C9—31no—
49C2×#611—32no—
50C2×#611—34no—
51C2×#611—34no—
52C2×#611—32no—
53C22×Dic9—31no—
54C22×Dic9—31no—
55C22×Dic9—31no—
56C22×Dic9—32no—
57C22×Dic9C2×D312yescomplemented
58C2×C4×D9—34no—
59C2×C4×D9—61no—
60C2×C4×D9—62no—
61C2×C4×D9—61no—
62C9×C22⋊C4—32no—
63C9×C22⋊C4—32no—
64(C2×D9)⋊C4—34no—
65Dic9⋊C4—34no—
66C4×Dic9—34no—
67C22×C3×C9—41no—
68C2×C3⋊D9—42no—
69C2×C3⋊D9—44no—
70C9×A4C22×C411yescharacteristic, complemented
71C9×Dic3—42no—
72C3×Dic9—42no—
73C22×S4—91no—
74C22×S4—181no—
75C22×#67—91no—
76C23×Dic3—31no—
77C22×C4×D3—91no—
78C2×C4×A4—91no—
79C2×SL(2,ℤ/4)D911yescharacteristic, complemented
80C4×S4—94no—
81C2×C3×C22⋊C4—31no—
82C2×(C2×D3)⋊C4—91no—
83C2×Dic3⋊C4—91no—
84C2×C4×Dic3—91no—
85#67⋊C4—94no—
86#67⋊C4—94no—
87C22×C3⋊D3—121no—
88C2×C3×A4C4×D312yes—
89C2×C3×A4C22×D311yescharacteristic
90C3⋊S4—34no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—121no—
93#624—124no—
94C23×C9—31no—
95C23×C9—31no—
96C23×C9—32no—
97C23×C9C4×D312yescomplemented
98C23×C9C22×D311yescharacteristic
99C22×D9—32no—
100C22×D9—34no—
101C22×D9—61no—
102C22×D9—62no—
103C22×D9—61no—
104C22×D9—241no—
105C2×C22⋊C9—22no—
106C2×C22⋊C9—21no—
107(C22⋊C9)⋊C2—64no—
108C2×C4×C9—32no—
109C2×C4×C9—32no—
110C2×C4×C9—61no—
111C2×C4×C9—62no—
112C2×C4×C9—61no—
113#611—38no—
114#611—34no—
115#611—34no—
116C2×Dic9—32no—
117C2×Dic9—34no—
118C2×Dic9—31no—
119C2×Dic9—32no—
120C2×Dic9—32no—
121C2×Dic9—32no—
122C2×Dic9—61no—
123C2×Dic9S411yescharacteristic, complemented
124C4×D9—64no—
125C4×D9—64no—
126C2×C4×D4—271noSylow
127C2×C3×C9—42no—
128C2×C3×C9—41no—
129C3⋊D9—44no—
130C24×C3D3211yescharacteristic
131C23×D3—91no—
132C22×A4—21no—
133C22×A4C2×D911yescharacteristic
134C2×S4—94no—
135C2×S4—92no—
136C2×S4—184no—
137C2×S4—182no—
138C22×C3×C4—31no—
139C22×C3×C4—31no—
140C2×#67—92no—
141C2×#67—94no—
142C2×#67—94no—
143C2×#67—92no—
144C22×Dic3—32no—
145C22×Dic3—91no—
146C22×Dic3—91no—
147C22×Dic3—92no—
148C22×Dic3—91no—
149C2×C4×D3—94no—
150C2×C4×D3—181no—
151C2×C4×D3—182no—
152C2×C4×D3—181no—
153C2×C4×D3—361no—
154C4×A4—92no—
155SL(2,ℤ/4)C2×D912yescomplemented
156C3×C22⋊C4—32no—
157C3×C22⋊C4—32no—
158(C2×D3)⋊C4—94no—
159Dic3⋊C4—94no—
160C4×Dic3—94no—
161C22×C32—41no—
162C2×C3⋊D3—122no—
163C2×C3⋊D3—124no—
164C3×A4C2×C4×D311yescharacteristic
165C3×Dic3—42no—
166C3×Dic3—122no—
167C22×C9—32no—
168C22×C9—31no—
169C22×C9—32no—
170C22×C9—32no—
171C22×C9—31no—
172C22×C9—31no—
173C22×C9—61no—
174C22×C9—81no—
175C22×C9C2×C4×D311yescharacteristic, complemented
176C22×C9C2×S411yescharacteristic
177C2×D9—34no—
178C2×D9—62no—
179C2×D9—64no—
180C2×D9—64no—
181C2×D9—62no—
182C2×D9—242no—
183C2×D9—244no—
184C22⋊C9—21no—
185C4×C9—62no—
186C4×C9—62no—
187Dic9—34no—
188Dic9—32no—
189Dic9C2×S412yescomplemented
190C22×D4—271no—
191C23×C4—91no—
192C23×C4—271no—
193C4×D4—274no—
194C4×D4—274no—
195C2×C4⋊C4—271no—
196C2×C22⋊C4—31no—
197C2×C22⋊C4—271no—
198C2×C42—271no—
199C3×C9—41noSylow
200C23×C3—31no—
201C23×C3—31no—
202C23×C3—32no—
203C23×C3#62412yes—
204C23×C3C2×D3211yescharacteristic
205C22×D3—92no—
206C22×D3—94no—
207C22×D3—181no—
208C22×D3—182no—
209C22×D3—181no—
210C22×D3—361no—
211C22×D3—721no—
212C2×A4—22no—
213C2×A4—21no—
214C2×A4C4×D912yescomplemented
215C2×A4C22×D911yescharacteristic
216S4—94no—
217S4—184no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—61no—
221C2×C3×C4—62no—
222C2×C3×C4—61no—
223C2×C3×C4—361no—
224#67—98no—
225#67—94no—
226#67—94no—
227C2×Dic3—31no—
228C2×Dic3—41no—
229C2×Dic3—92no—
230C2×Dic3—92no—
231C2×Dic3—92no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—92no—
235C2×Dic3—181no—
236C4×D3—184no—
237C4×D3—184no—
238C4×D3—364no—
239C2×C32—41no—
240C2×C32—42no—
241C3⋊D3—124no—
242C2×C9—31no—
243C2×C9—31no—
244C2×C9—32no—
245C2×C9—81no—
246C2×C9—82no—
247C2×C9C4×S412yescomplemented
248C2×C9C22×S411yescharacteristic
249D9—64no—
250D9—244no—
251C24—271no—
252C24D3×D911yescharacteristic
253C2×D4—272no—
254C2×D4—274no—
255C2×D4—274no—
256C2×D4—272no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—92no—
260C22×C4—274no—
261C22×C4—271no—
262C22×C4—271no—
263C22×C4—271no—
264C22×C4—272no—
265C22×C4—541no—
266C22×C4—542no—
267C22×C4—541no—
268C4⋊C4—274no—
269C22⋊C4—32no—
270C22⋊C4—32no—
271C22⋊C4—274no—
272C42—274no—
273C22×C3—32no—
274C22×C3—31no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—41no—
280C22×C3—61no—
281C22×C3—81no—
282C22×C3C2×#62411yescharacteristic
283C22×C3D3×S411yescharacteristic
284C2×D3—94no—
285C2×D3—182no—
286C2×D3—184no—
287C2×D3—182no—
288C2×D3—184no—
289C2×D3—362no—
290C2×D3—364no—
291C2×D3—724no—
292C2×D3—722no—
293A4—21no—
294A4C2×C4×D911yescharacteristic, complemented
295C3×C4—62no—
296C3×C4—62no—
297C3×C4—362no—
298Dic3—32no—
299Dic3—42no—
300Dic3—94no—
301Dic3—92no—
302C32—41no—
303C9—81no—
304C9C2×C4×S411yescharacteristic, complemented
305C23—31no—
306C23—31no—
307C23—32no—
308C23—272no—
309C23—274no—
310C23—541no—
311C23—542no—
312C23—541no—
313C23(C3⋊D9)⋊C412yescomplemented
314C23C2×D3×D911yescharacteristic
315D4—278no—
316D4—274no—
317D4—274no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—91no—
324C2×C4—272no—
325C2×C4—272no—
326C2×C4—272no—
327C2×C4—274no—
328C2×C4—271no—
329C2×C4—272no—
330C2×C4—544no—
331C2×C4—544no—
332C2×C4—541no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—32no—
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—82no—
339C2×C3—81no—
340C2×C3(C3⋊S4)⋊C412yes—
341C2×C3C2×D3×S411yescharacteristic
342D3—184no—
343D3—364no—
344D3—724no—
345C22—32no—
346C22—31no—
347C22—32no—
348C22—32no—
349C22—31no—
350C22—31no—
351C22—61no—
352C22—274no—
353C22—542no—
354C22—544no—
355C22—544no—
356C22—542no—
357C22C2×(C3⋊D9)⋊C411yescharacteristic, complemented
358C22D9×S411yescharacteristic
359C4—62no—
360C4—62no—
361C4—92no—
362C4—274no—
363C4—272no—
364C3—41no—
365C3—81no—
366C3C2×(C3⋊S4)⋊C411yescharacteristic
367C2—31no—
368C2—31no—
369C2—32no—
370C2—544no—
371C2(C9⋊S4)⋊C412yesdirect product factor
372C2C2×D9×S411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×A4C22×C4
FrattiniC2×C3C2×D3×S4
FittingC24×C9C2×D3
socleC24×C3D32
prime radicalC22×C9⋊S4C2
cosocleC2×C9×A4C23
LCSG ▷ C9×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic9C2semidirect
3. C2×C9×SL(2,ℤ/4)C2semidirect
4. (C9⋊S4)⋊C4C2direct (N × H)
8. C2×C9⋊S4C4semidirect
11. A4×Dic9C22semidirect
12. C9×SL(2,ℤ/4)C22semidirect
19. C23×Dic9D3semidirect
28. C2×C9×A4C2×C4semidirect
30. C9⋊S4C2×C4semidirect
57. C22×Dic9C2×D3semidirect
70. C9×A4C22×C4semidirect
79. C2×SL(2,ℤ/4)D9semidirect
97. C23×C9C4×D3semidirect
123. C2×Dic9S4semidirect
155. SL(2,ℤ/4)C2×D9semidirect
175. C22×C9C2×C4×D3semidirect
189. Dic9C2×S4semidirect
214. C2×A4C4×D9semidirect
247. C2×C9C4×S4semidirect
294. A4C2×C4×D9semidirect
304. C9C2×C4×S4semidirect
313. C23(C3⋊D9)⋊C4semidirect
357. C22C2×(C3⋊D9)⋊C4semidirect
371. C2(C9⋊S4)⋊C4direct (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×A4×Dic92
C2×C9×SL(2,ℤ/4)2
(C9⋊S4)⋊C42
C2×(C3⋊S4)⋊C43
C2×#611⋊C43
C2×(C3⋊D9)⋊C44