Finite Group Databasebeta

← C2×C4×C3⋊((C22⋊C9)⋊C2) #13164471

Subgroups — C2×C4×C3⋊((C22⋊C9)⋊C2) #13164471

Order 1728.

1290 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 147 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×C3⋊((C22⋊C9)⋊C2)C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C9C211yesmaximal, characteristic, complemented
3C2×C3⋊((C22⋊C9)⋊C4)C211yesmaximal, characteristic, complemented
4C4×C3⋊((C22⋊C9)⋊C2)C214yesmaximal, direct product factor
5C2×C4×#638—31nomaximal
6C2×C4×(C22⋊C9)⋊C2—33nomaximal
7C22×C3×C22⋊C9C2211yescharacteristic
8C2×C3⋊((C22⋊C9)⋊C2)C414yesdirect product factor
9C2×C3⋊((C22⋊C9)⋊C2)C2212yes—
10C2×C4×C3⋊D9—41nomaximal
11C3×C4×C22⋊C9C2212yescomplemented
12C3⋊((C22⋊C9)⋊C4)C2212yescomplemented
13C23×C32×C4D311yescharacteristic
14C22×#638—31no—
15C22×C4×C3⋊D3—31no—
16C22×(C22⋊C9)⋊C2—33no—
17C2×#3757—31no—
18C4×#638—34no—
19C4×#638—34no—
20C2×#3752—31no—
21C2×#3750—31no—
22C2×C4×C3⋊Dic3—31no—
23C2×C4×C22⋊C9D313yescomplemented
24C2×(C22⋊C9)⋊C4—33no—
25C4×(C22⋊C9)⋊C2—312no—
26C22×C3⋊D9—41no—
27C2×C3×C22⋊C9C2×C412yescomplemented
28C2×C3×C22⋊C9C2311yescharacteristic
29C3⋊((C22⋊C9)⋊C2)C2×C414yesdirect product factor
30C2×C3×C4×C9—41no—
31C2×C3⋊Dic9—41no—
32C4×C3⋊D9—44no—
33C2×C4×#67—93no—
34C2×C4×#67—91no—
35C24×C32C2×D311yescharacteristic
36C23×C3⋊D3—31no—
37C22×C32×C4—31no—
38C22×C32×C4—31no—
39C22×C32×C4—32no—
40C22×C32×C4C2×D312yes—
41C2×#638—34no—
42C2×#638—32no—
43C2×#638—34no—
44C2×#638—32no—
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—31no—
48C2×C4×C3⋊D3—34no—
49C2×C4×C3⋊D3—61no—
50C2×C4×C3⋊D3—62no—
51C2×C4×C3⋊D3—61no—
52C22×C22⋊C9C2×D313yes—
53C2×(C22⋊C9)⋊C2—312no—
54C2×(C22⋊C9)⋊C2—36no—
55#3757—32no—
56#3757—32no—
57#3752—34no—
58#3750—34no—
59C4×C3⋊Dic3—34no—
60C2×C4×D9—123no—
61C4×C22⋊C9C2×D316yescomplemented
62(C22⋊C9)⋊C4—36no—
63C22×C3×C9—41no—
64C2×C3⋊D9—44no—
65C2×C3⋊D9—42no—
66C3×C22⋊C9C22×C411yescharacteristic, complemented
67C3×C4×C9—42no—
68C3⋊Dic9—42no—
69C23×C3×C4D913yescomplemented
70C23×C3×C4C3⋊D311yescharacteristic
71C22×#67—93no—
72C22×#67—91no—
73C22×C4×D3—93no—
74C22×C4×D3—91no—
75C2×#217—93no—
76C2×#217—91no—
77C4×#67—912no—
78C4×#67—94no—
79C4×#67—912no—
80C4×#67—94no—
81C2×(C2×D3)⋊C4—93no—
82C2×(C2×D3)⋊C4—91no—
83C2×Dic3⋊C4—93no—
84C2×Dic3⋊C4—91no—
85C2×C4×Dic3—93no—
86C2×C4×Dic3—91no—
87C23×C32—31no—
88C23×C32—31no—
89C23×C32—32no—
90C23×C32C4×D312yes—
91C23×C32C22×D311yescharacteristic
92C22×C3⋊D3—32no—
93C22×C3⋊D3—34no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—61no—
97C2×C32×C4—31no—
98C2×C32×C4—32no—
99C2×C32×C4—32no—
100C2×C32×C4—32no—
101C2×C32×C4—61no—
102C2×C32×C4S411yescharacteristic
103#638—38no—
104#638—34no—
105#638—34no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—61no—
113C4×C3⋊D3—64no—
114C4×C3⋊D3—64no—
115C22×D9—123no—
116C2×C22⋊C9C4×D316yescomplemented
117C2×C22⋊C9C22×D313yes—
118(C22⋊C9)⋊C2—312no—
119C2×C4×C9—43no—
120C2×Dic9—123no—
121C4×D9—1212no—
122C2×C4×D4—271noSylow
123C2×C3×C9—42no—
124C2×C3×C9—41no—
125C3⋊D9—44no—
126C24×C3C2×D913yes—
127C24×C3C2×C3⋊D311yescharacteristic
128C23×D3—91no—
129C23×D3—93no—
130C22×C3×C4—33no—
131C22×C3×C4—31no—
132C22×C3×C4—33no—
133C22×C3×C4—31no—
134C22×C3×C4—36no—
135C22×C3×C4—32no—
136C22×C3×C4C2×D916yescomplemented
137C22×C3×C4C2×C3⋊D312yes—
138C2×#67—96no—
139C2×#67—92no—
140C2×#67—912no—
141C2×#67—94no—
142C2×#67—912no—
143C2×#67—96no—
144C2×#67—94no—
145C2×#67—92no—
146C22×Dic3—91no—
147C22×Dic3—93no—
148C22×Dic3—91no—
149C22×Dic3—93no—
150C22×Dic3—91no—
151C22×Dic3—93no—
152C2×C4×D3—94no—
153C2×C4×D3—912no—
154C2×C4×D3—181no—
155C2×C4×D3—183no—
156C2×C4×D3—182no—
157C2×C4×D3—186no—
158C2×C4×D3—183no—
159C2×C4×D3—181no—
160#217—96no—
161#217—92no—
162#217—96no—
163#217—92no—
164(C2×D3)⋊C4—912no—
165(C2×D3)⋊C4—94no—
166Dic3⋊C4—912no—
167Dic3⋊C4—94no—
168C4×Dic3—94no—
169C4×Dic3—912no—
170C22×C32—32no—
171C22×C32—31no—
172C22×C32—32no—
173C22×C32—32no—
174C22×C32—31no—
175C22×C32—31no—
176C22×C32—61no—
177C22×C32C2×C4×D311yescharacteristic
178C22×C32C2×S411yescharacteristic
179C2×C3⋊D3—34no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—64no—
182C2×C3⋊D3—64no—
183C2×C3⋊D3—62no—
184C32×C4—32no—
185C32×C4C2×S412yes—
186C3⋊Dic3—34no—
187C3⋊Dic3—62no—
188C3⋊Dic3—62no—
189C22×C9—43no—
190C2×D9—126no—
191C2×D9—1212no—
192C22⋊C9C2×C4×D313yescomplemented
193C4×C9—46no—
194Dic9—126no—
195C22×D4—271no—
196C23×C4—271no—
197C23×C4C3⋊D911yescharacteristic, complemented
198C4×D4—274no—
199C4×D4—274no—
200C2×C4⋊C4—271no—
201C2×C22⋊C4—271no—
202C2×C22⋊C4—271no—
203C2×C42—271no—
204C3×C9—41noSylow
205C23×C3—31no—
206C23×C3—33no—
207C23×C3—33no—
208C23×C3—31no—
209C23×C3—36no—
210C23×C3—32no—
211C23×C3C4×D916yescomplemented
212C23×C3C22×D913yes—
213C23×C3C4×C3⋊D312yes—
214C23×C3C22×C3⋊D311yescharacteristic
215C22×D3—92no—
216C22×D3—96no—
217C22×D3—94no—
218C22×D3—912no—
219C22×D3—181no—
220C22×D3—183no—
221C22×D3—182no—
222C22×D3—186no—
223C22×D3—181no—
224C22×D3—183no—
225C2×C3×C4—31no—
226C2×C3×C4—33no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—36no—
230C2×C3×C4—36no—
231C2×C3×C4—36no—
232C2×C3×C4—32no—
233C2×C3×C4—61no—
234C2×C3×C4—63no—
235C2×C3×C4(C22⋊C9)⋊C213yescomplemented
236C2×C3×C4C3⋊S411yescharacteristic
237#67—924no—
238#67—912no—
239#67—98no—
240#67—94no—
241#67—912no—
242#67—94no—
243C2×Dic3—92no—
244C2×Dic3—96no—
245C2×Dic3—94no—
246C2×Dic3—912no—
247C2×Dic3—92no—
248C2×Dic3—96no—
249C2×Dic3—92no—
250C2×Dic3—96no—
251C2×Dic3—181no—
252C2×Dic3—183no—
253C2×Dic3—182no—
254C2×Dic3—186no—
255C2×Dic3—181no—
256C2×Dic3—183no—
257C4×D3—184no—
258C4×D3—1812no—
259C4×D3—184no—
260C4×D3—1812no—
261C2×C32—31no—
262C2×C32—31no—
263C2×C32—32no—
264C2×C32C4×S412yes—
265C2×C32C22×S411yescharacteristic
266C3⋊D3—64no—
267C2×C9—46no—
268C2×C9—43no—
269D9—1212no—
270C24—271no—
271C24C2×C3⋊D911yescharacteristic
272C2×D4—272no—
273C2×D4—274no—
274C2×D4—274no—
275C2×D4—272no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—32no—
279C22×C4—274no—
280C22×C4—271no—
281C22×C4—271no—
282C22×C4—271no—
283C22×C4—541no—
284C22×C4—542no—
285C22×C4—541no—
286C22×C4C2×C3⋊D912yescomplemented
287C4⋊C4—274no—
288C22⋊C4—274no—
289C22⋊C4—272no—
290C22⋊C4—272no—
291C42—274no—
292C22×C3—36no—
293C22×C3—33no—
294C22×C3—32no—
295C22×C3—31no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—33no—
301C22×C3—33no—
302C22×C3—36no—
303C22×C3—36no—
304C22×C3—63no—
305C22×C3—61no—
306C22×C3C2×C4×D913yescomplemented
307C22×C3C2×(C22⋊C9)⋊C213yes—
308C22×C3C2×C4×C3⋊D311yescharacteristic
309C22×C3C2×C3⋊S411yescharacteristic
310C2×D3—912no—
311C2×D3—94no—
312C2×D3—186no—
313C2×D3—1812no—
314C2×D3—182no—
315C2×D3—184no—
316C2×D3—184no—
317C2×D3—182no—
318C2×D3—186no—
319C2×D3—1812no—
320C3×C4—32no—
321C3×C4—36no—
322C3×C4C2×(C22⋊C9)⋊C216yescomplemented
323C3×C4C2×C3⋊S412yes—
324Dic3—912no—
325Dic3—94no—
326Dic3—186no—
327Dic3—182no—
328Dic3—182no—
329Dic3—186no—
330C32C2×C4×S411yescharacteristic
331C9—43no—
332C23—31no—
333C23—31no—
334C23—32no—
335C23—272no—
336C23—274no—
337C23—541no—
338C23—542no—
339C23—541no—
340C23C4×C3⋊D912yescomplemented
341C23C22×C3⋊D911yescharacteristic
342D4—278no—
343D4—274no—
344D4—274no—
345C2×C4—31no—
346C2×C4—32no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—61no—
350C2×C4—272no—
351C2×C4—274no—
352C2×C4—272no—
353C2×C4—272no—
354C2×C4—544no—
355C2×C4—541no—
356C2×C4—542no—
357C2×C4—541no—
358C2×C4—544no—
359C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic, direct product factor
360C2×C3—33no—
361C2×C3—33no—
362C2×C3—36no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—32no—
366C2×C3C4×(C22⋊C9)⋊C216yescomplemented
367C2×C3C22×(C22⋊C9)⋊C213yes—
368C2×C3C4×C3⋊S412yes—
369C2×C3C22×C3⋊S411yescharacteristic
370D3—1812no—
371D3—184no—
372C22—32no—
373C22—31no—
374C22—31no—
375C22—31no—
376C22—32no—
377C22—32no—
378C22—61no—
379C22—274no—
380C22—542no—
381C22—544no—
382C22—542no—
383C22—544no—
384C22C2×C4×C3⋊D911yescharacteristic, complemented
385C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
386C4—32no—
387C4—274no—
388C4—542no—
389C4—542no—
390C4C2×C3⋊((C22⋊C9)⋊C2)12yesdirect product factor
391C3C2×C4×(C22⋊C9)⋊C213yescomplemented
392C3C2×C4×C3⋊S411yescharacteristic
393C2—31no—
394C2—31no—
395C2—32no—
396C2—544no—
397C2C4×C3⋊((C22⋊C9)⋊C2)12yesdirect product factor
398C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
commutator, hypocenterC3×C22⋊C9C22×C4
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radicalC22×C3⋊((C22⋊C9)⋊C2)C2
cosocleC2×C3×C22⋊C9C23
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C22⋊C9C2semidirect
3. C2×C3⋊((C22⋊C9)⋊C4)C2semidirect
4. C4×C3⋊((C22⋊C9)⋊C2)C2direct (N × H)
8. C2×C3⋊((C22⋊C9)⋊C2)C4direct (N × H)
11. C3×C4×C22⋊C9C22semidirect
12. C3⋊((C22⋊C9)⋊C4)C22semidirect
23. C2×C4×C22⋊C9D3semidirect
27. C2×C3×C22⋊C9C2×C4semidirect
29. C3⋊((C22⋊C9)⋊C2)C2×C4direct (N × H)
61. C4×C22⋊C9C2×D3semidirect
66. C3×C22⋊C9C22×C4semidirect
69. C23×C3×C4D9semidirect
116. C2×C22⋊C9C4×D3semidirect
136. C22×C3×C4C2×D9semidirect
192. C22⋊C9C2×C4×D3semidirect
197. C23×C4C3⋊D9semidirect
211. C23×C3C4×D9semidirect
235. C2×C3×C4(C22⋊C9)⋊C2semidirect
286. C22×C4C2×C3⋊D9semidirect
306. C22×C3C2×C4×D9semidirect
322. C3×C4C2×(C22⋊C9)⋊C2semidirect
340. C23C4×C3⋊D9semidirect
359. C2×C4C3⋊((C22⋊C9)⋊C2)direct (N × H)
366. C2×C3C4×(C22⋊C9)⋊C2semidirect
384. C22C2×C4×C3⋊D9semidirect
390. C4C2×C3⋊((C22⋊C9)⋊C2)direct (N × H)
391. C3C2×C4×(C22⋊C9)⋊C2semidirect
397. C2C4×C3⋊((C22⋊C9)⋊C2)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊((C22⋊C9)⋊C2)2
C2×C3×C4×C22⋊C92
C2×C3⋊((C22⋊C9)⋊C4)2
C4×C3⋊((C22⋊C9)⋊C2)2
C2×C4×#6383
C2×C4×(C22⋊C9)⋊C23
C2×C4×C3⋊D94