Finite Group Databasebeta

← C2×S4×C22⋊C9 #13180040

Subgroups — C2×S4×C22⋊C9 #13180040

Order 1728.

650 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 45 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×A4×C22⋊C9C211yesmaximal, characteristic, complemented
2S4×C22⋊C9C212yesmaximal, direct product factor
3C23×C3×S4C311yesmaximal, characteristic
4C2×D4×C22⋊C9—31nomaximal
5C2×D3×C22⋊C9—41nomaximal
6C2×C9×S4—41nomaximal
7A4×C22⋊C9C2211yescharacteristic, complemented
8C23×C3×A4C2×C311yescharacteristic
9C22×C3×S4—31no—
10C22×C3×S4—31no—
11C22×C3×S4—32no—
12C22×C3×S4C2×C312yes—
13C2×C22⋊(C22⋊C9)—21no—
14C23×C22⋊C9—31no—
15C23×C22⋊C9D311yescharacteristic, complemented
16D4×C22⋊C9—32no—
17D4×C22⋊C9—32no—
18C2×C4×C22⋊C9—31no—
19C2×C3×C22⋊C9—41no—
20C2×C9×A4—41no—
21D3×C22⋊C9—42no—
22C9×S4—42no—
23C23×S4C911yescharacteristic, complemented
24C23×C3×D4—31no—
25C23×C3×D3—41no—
26C22×C3×A4—31no—
27C22×C3×A4—31no—
28C22×C3×A4C22×C311yescharacteristic
29C2×C3×S4—32no—
30C2×C3×S4—31no—
31C2×C3×S4—32no—
32C2×C3×S4—32no—
33C2×C3×S4—32no—
34C2×C3×S4A411yescharacteristic
35C22⋊(C22⋊C9)—21no—
36C22×C22⋊C9—31no—
37C22×C22⋊C9—31no—
38C22×C22⋊C9—32no—
39C22×C22⋊C9—61no—
40C22×C22⋊C9—61no—
41C22×C22⋊C9C2×D311yescharacteristic, complemented
42C2×D4×C9—121no—
43C4×C22⋊C9—32no—
44C2×D3×C9—161no—
45C3×C22⋊C9—41no—
46C9×A4—41no—
47C25×C3—31no—
48C25×C3C3×D311yescharacteristic
49C23×A4—21no—
50C23×A4C2×C911yescharacteristic, complemented
51C22×S4—31no—
52C22×S4—31no—
53C22×S4—32no—
54C22×S4C2×C912yescomplemented
55C22×C3×D4—32no—
56C22×C3×D4—32no—
57C22×C3×D4—91no—
58C22×C3×D4—92no—
59C22×C3×D4—91no—
60C22×C3×D4—91no—
61C22×C3×D4—92no—
62C22×C3×D4—91no—
63C23×C3×C4—31no—
64C23×C32—41no—
65C22×C3×D3—42no—
66C22×C3×D3—121no—
67C22×C3×D3—122no—
68C22×C3×D3—121no—
69C2×C3×A4—31no—
70C2×C3×A4—31no—
71C2×C3×A4C2×A411yescharacteristic
72C3×S4—32no—
73C3×S4C2×A412yes—
74C23×C9—41no—
75C23×C9—121no—
76C2×C22⋊C9—31no—
77C2×C22⋊C9—31no—
78C2×C22⋊C9—62no—
79C2×C22⋊C9—81no—
80C2×C22⋊C9—81no—
81C2×C22⋊C9—241no—
82C2×C22⋊C9S411yescharacteristic, direct product factor
83D4×C9—122no—
84D4×C9—122no—
85C2×C4×C9—121no—
86C23×D4—31noSylow
87C2×C3×C9—161no—
88D3×C9—162no—
89C24×C3—31no—
90C24×C3—31no—
91C24×C3—31no—
92C24×C3—31no—
93C24×C3—32no—
94C24×C3—61no—
95C24×C3—61no—
96C24×C3—91no—
97C24×C3—91no—
98C24×C3—91no—
99C24×C3—91no—
100C24×C3—92no—
101C24×C3—181no—
102C24×C3—181no—
103C24×C3C2×C3×D311yescharacteristic
104C23×D3—41no—
105C22×A4—21no—
106C22×A4—31no—
107C22×A4—31no—
108C22×A4—61no—
109C22×A4—61no—
110C22×A4C22×C911yescharacteristic, complemented
111C2×S4—32no—
112C2×S4—31no—
113C2×S4—32no—
114C2×S4—32no—
115C2×S4—32no—
116C2×S4C22⋊C911yescharacteristic, direct product factor
117C2×C3×D4—31no—
118C2×C3×D4—91no—
119C2×C3×D4—92no—
120C2×C3×D4—92no—
121C2×C3×D4—92no—
122C2×C3×D4—92no—
123C2×C3×D4—92no—
124C2×C3×D4—91no—
125C2×C3×D4—94no—
126C2×C3×D4—92no—
127C2×C3×D4—92no—
128C2×C3×D4—94no—
129C2×C3×D4—91no—
130C2×C3×D4—92no—
131C2×C3×D4—92no—
132C2×C3×D4—92no—
133C2×C3×D4—92no—
134C2×C3×D4—92no—
135C2×C3×D4—92no—
136C22×C3×C4—32no—
137C22×C3×C4—91no—
138C22×C3×C4—92no—
139C22×C3×C4—91no—
140C22×C32—41no—
141C22×C32—121no—
142C22×C32—121no—
143C2×C3×D3—41no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C2×C3×D3—121no—
147C2×C3×D3—122no—
148C2×C3×D3—122no—
149C3×A4C22×A411yescharacteristic
150C22×C9—41no—
151C22×C9—121no—
152C22×C9—121no—
153C22×C9—122no—
154C22×C9—241no—
155C22×C9—241no—
156C22⋊C9—81no—
157C22⋊C9—81no—
158C22⋊C9—241no—
159C22⋊C9C2×S411yescharacteristic, direct product factor
160C4×C9—122no—
161C25—31no—
162C25D3×C911yescharacteristic, complemented
163C22×D4—32no—
164C22×D4—32no—
165C22×D4—91no—
166C22×D4—92no—
167C22×D4—91no—
168C22×D4—91no—
169C22×D4—92no—
170C22×D4—91no—
171C23×C4—31no—
172C3×C9—161noSylow
173C23×C3—31no—
174C23×C3—31no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—31no—
178C23×C3—41no—
179C23×C3—61no—
180C23×C3—62no—
181C23×C3—81no—
182C23×C3—91no—
183C23×C3—91no—
184C23×C3—91no—
185C23×C3—91no—
186C23×C3—91no—
187C23×C3—91no—
188C23×C3—91no—
189C23×C3—92no—
190C23×C3—91no—
191C23×C3—92no—
192C23×C3—91no—
193C23×C3—92no—
194C23×C3—91no—
195C23×C3—92no—
196C23×C3—92no—
197C23×C3—181no—
198C23×C3—182no—
199C23×C3—181no—
200C23×C3—181no—
201C23×C3—181no—
202C23×C3—181no—
203C23×C3—181no—
204C23×C3—181no—
205C23×C3—182no—
206C23×C3—181no—
207C23×C3—182no—
208C23×C3—181no—
209C23×C3—181no—
210C23×C3—184no—
211C23×C3C3×S411yescharacteristic
212C23×C3D3×A411yescharacteristic
213C22×D3—42no—
214C22×D3—121no—
215C22×D3—122no—
216C22×D3—121no—
217C2×A4—21no—
218C2×A4—31no—
219C2×A4—31no—
220C2×A4—61no—
221C2×A4—61no—
222C2×A4C2×C22⋊C911yescharacteristic, complemented
223S4—32no—
224S4C2×C22⋊C912yesdirect product factor
225C3×D4—32no—
226C3×D4—32no—
227C3×D4—92no—
228C3×D4—92no—
229C3×D4—92no—
230C3×D4—92no—
231C3×D4—92no—
232C3×D4—94no—
233C3×D4—94no—
234C3×D4—92no—
235C2×C3×C4—31no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—91no—
239C2×C3×C4—92no—
240C2×C3×C4—92no—
241C2×C32—41no—
242C2×C32—121no—
243C2×C32—121no—
244C3×D3—42no—
245C3×D3—122no—
246C2×C9—41no—
247C2×C9—121no—
248C2×C9—121no—
249C2×C9—242no—
250C2×C9—321no—
251C24—31no—
252C24—31no—
253C24—31no—
254C24—31no—
255C24—32no—
256C24—61no—
257C24—61no—
258C24—91no—
259C24—91no—
260C24—91no—
261C24—91no—
262C24—92no—
263C24—181no—
264C24—181no—
265C24C2×D3×C911yescharacteristic, complemented
266C2×D4—31no—
267C2×D4—91no—
268C2×D4—92no—
269C2×D4—92no—
270C2×D4—92no—
271C2×D4—92no—
272C2×D4—92no—
273C2×D4—91no—
274C2×D4—94no—
275C2×D4—92no—
276C2×D4—92no—
277C2×D4—94no—
278C2×D4—91no—
279C2×D4—92no—
280C2×D4—92no—
281C2×D4—92no—
282C2×D4—92no—
283C2×D4—92no—
284C2×D4—92no—
285C22×C4—32no—
286C22×C4—91no—
287C22×C4—92no—
288C22×C4—91no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—32no—
294C22×C3—41no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—81no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3—92no—
308C22×C3—92no—
309C22×C3—91no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—182no—
313C22×C3—182no—
314C22×C3—181no—
315C22×C3—182no—
316C22×C3—181no—
317C22×C3—182no—
318C22×C3—182no—
319C22×C3—182no—
320C22×C3—181no—
321C22×C3—182no—
322C22×C3—182no—
323C22×C3—181no—
324C22×C3—241no—
325C22×C3—241no—
326C22×C3C2×C3×S411yescharacteristic
327C22×C3C2×D3×A411yescharacteristic
328C2×D3—41no—
329C2×D3—122no—
330C2×D3—122no—
331C2×D3—121no—
332C2×D3—122no—
333C2×D3—122no—
334A4—21no—
335A4C22×C22⋊C911yescharacteristic, complemented
336C3×C4—32no—
337C3×C4—92no—
338C32—41no—
339C9—41no—
340C9—321no—
341C23—31no—
342C23—31no—
343C23—31no—
344C23—31no—
345C23—31no—
346C23—61no—
347C23—62no—
348C23—91no—
349C23—91no—
350C23—91no—
351C23—91no—
352C23—91no—
353C23—91no—
354C23—91no—
355C23—92no—
356C23—91no—
357C23—92no—
358C23—91no—
359C23—92no—
360C23—91no—
361C23—92no—
362C23—92no—
363C23—181no—
364C23—182no—
365C23—181no—
366C23—181no—
367C23—181no—
368C23—181no—
369C23—181no—
370C23—181no—
371C23—182no—
372C23—181no—
373C23—182no—
374C23—181no—
375C23—181no—
376C23—184no—
377C23C9×S411yescharacteristic, complemented
378C23D3×C22⋊C911yescharacteristic, complemented
379D4—32no—
380D4—32no—
381D4—92no—
382D4—92no—
383D4—92no—
384D4—92no—
385D4—92no—
386D4—94no—
387D4—94no—
388D4—92no—
389C2×C4—31no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—91no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—41no—
400C2×C3—62no—
401C2×C3—81no—
402C2×C3—91no—
403C2×C3—91no—
404C2×C3—121no—
405C2×C3—121no—
406C2×C3—182no—
407C2×C3—241no—
408C2×C3—241no—
409C2×C3A4×S411yescharacteristic
410D3—42no—
411D3—122no—
412C22—31no—
413C22—31no—
414C22—31no—
415C22—31no—
416C22—32no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—91no—
421C22—91no—
422C22—91no—
423C22—91no—
424C22—91no—
425C22—91no—
426C22—91no—
427C22—91no—
428C22—92no—
429C22—92no—
430C22—91no—
431C22—182no—
432C22—182no—
433C22—181no—
434C22—182no—
435C22—182no—
436C22—181no—
437C22—182no—
438C22—182no—
439C22—181no—
440C22—182no—
441C22—182no—
442C22—181no—
443C22C2×C9×S411yescharacteristic, complemented
444C22C2×D3×C22⋊C911yescharacteristic, complemented
445C4—32no—
446C4—92no—
447C3—41no—
448C3—81no—
449C3C2×A4×S411yescharacteristic
450C2—31no—
451C2—31no—
452C2—31no—
453C2—31no—
454C2—62no—
455C2—91no—
456C2—91no—
457C2—182no—
458C2S4×C22⋊C911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3A4×S4
commutator, hypocenterC22×A4C22×C9
FrattiniC3C2×A4×S4
Fitting, socleC25×C3C3×D3
prime radicalC23×C3×S4C3
cosocleC22×C3×A4C22×C3
LCSG ▷ C22×A4
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×A4×C22⋊C9C2semidirect
2. S4×C22⋊C9C2direct (N × H)
7. A4×C22⋊C9C22semidirect
15. C23×C22⋊C9D3semidirect
23. C23×S4C9semidirect
41. C22×C22⋊C9C2×D3semidirect
50. C23×A4C2×C9semidirect
54. C22×S4C2×C9semidirect
82. C2×C22⋊C9S4direct (N × H)
110. C22×A4C22×C9semidirect
116. C2×S4C22⋊C9direct (N × H)
159. C22⋊C9C2×S4direct (N × H)
162. C25D3×C9semidirect
222. C2×A4C2×C22⋊C9semidirect
224. S4C2×C22⋊C9direct (N × H)
265. C24C2×D3×C9semidirect
335. A4C22×C22⋊C9semidirect
377. C23C9×S4semidirect
378. C23D3×C22⋊C9semidirect
443. C22C2×C9×S4semidirect
444. C22C2×D3×C22⋊C9semidirect
458. C2S4×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×C22⋊C92
S4×C22⋊C92
C23×C3×S43
C2×D4×C22⋊C93
C2×D3×C22⋊C94
C2×C9×S44