Finite Group Databasebeta

← C22×C3×(C3⋊S4)⋊C3 #72220521

Subgroups — C22×C3×(C3⋊S4)⋊C3 #72220521

Order 2592.

1776 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 176 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⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×(C3⋊S4)⋊C3C216yesmaximal, direct product factor
3C22×C3×C3⋊S4C311yesmaximal, characteristic, complemented
4C22×C32×#67—31nomaximal
5C22×(C3⋊S4)⋊C3C313yesmaximal, direct product factor
6C22×C3×(C3⋊D3)⋊C3—41nomaximal
7C2×C3×C22⋊H(3,𝔽3)C2213yescomplemented
8C3×(C3⋊S4)⋊C3C2214yesdirect product factor
9C24×C33D311yescharacteristic, complemented
10C23×C32×D3—31no—
11C22×C32×A4—21no—
12C22×C32×A4C2×C311yescharacteristic, complemented
13C2×C3×C3⋊S4C2×C316yescomplemented
14C2×C32×#67—36no—
15C2×C32×#67—36no—
16C22×C32×Dic3—31no—
17C22×C22⋊H(3,𝔽3)—23no—
18C22×C22⋊H(3,𝔽3)C2×C313yescomplemented
19C2×(C3⋊S4)⋊C3C2×C3118yesdirect product factor
20C22×C3×H(3,𝔽3)—41no—
21C2×C3×(C3⋊D3)⋊C3—46no—
22C3×C22⋊H(3,𝔽3)C2311yescharacteristic, complemented
23C22×C3⋊S4C3211yescharacteristic, complemented
24C22×C3×S4—91no—
25C22×C32×D4—91no—
26C22×C3×#67—31no—
27C22×C3×#67—33no—
28C23×C33—31no—
29C23×C33—33no—
30C23×C33C2×D313yescomplemented
31C22×C3×C3⋊D3—41no—
32C22×C32×D3—36no—
33C22×C32×D3—63no—
34C22×C32×D3—61no—
35C2×C32×A4—23no—
36C2×C32×A4C22×C313yescomplemented
37C3×C3⋊S4C22×C314yescomplemented
38C32×#67—312no—
39C32×#67—34no—
40C2×C32×Dic3—36no—
41C22×(C3⋊D3)⋊C3—43no—
42C2×C22⋊H(3,𝔽3)—29no—
43C2×C22⋊H(3,𝔽3)C22×C319yescomplemented
44(C3⋊S4)⋊C3C22×C3112yesdirect product factor
45C2×C3×H(3,𝔽3)—43no—
46C3×(C3⋊D3)⋊C3—44no—
47C24×C32—31no—
48C24×C32—61no—
49C24×C32—91no—
50C24×C32C3×D311yescharacteristic
51C24×C32C3×D313yescomplemented
52C23×C3×D3—31no—
53C23×C3×D3—33no—
54C22×C3×A4—23no—
55C22×C3×A4—21no—
56C22×C3×A4—31no—
57C22×C3×A4—61no—
58C22×C3×A4—62no—
59C22×C3×A4—61no—
60C22×C3×A4C2×C3211yescharacteristic, complemented
61C2×C3⋊S4C2×C3216yescomplemented
62C2×C3×S4—96no—
63C2×C32×D4—96no—
64C2×C32×D4—96no—
65C22×C32×C4—91no—
66C2×C3×#67—36no—
67C2×C3×#67—318no—
68C2×C3×#67—36no—
69C2×C3×#67—318no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—33no—
72C22×C33—33no—
73C22×C33—33no—
74C22×C33—33no—
75C22×C33—41no—
76C22×C33—61no—
77C22×C33—81no—
78C22×C33S411yescharacteristic, complemented
79C22×C33C22×D311yescharacteristic, complemented
80C2×C3×C3⋊D3—46no—
81C2×C32×D3—34no—
82C2×C32×D3—66no—
83C2×C32×D3—66no—
84C32×A4—21no—
85C32×A4C23×C311yescharacteristic, complemented
86C32×Dic3—34no—
87C22×H(3,𝔽3)—43no—
88C22×H(3,𝔽3)—83no—
89C2×(C3⋊D3)⋊C3—418no—
90C22⋊H(3,𝔽3)—23no—
91C22⋊H(3,𝔽3)C23×C313yescomplemented
92C22×S4—91no—
93C22×C3×D4—93no—
94C22×C3×D4—91no—
95C22×#67—31no—
96C3×H(3,𝔽3)—41noSylow
97C23×C32—31no—
98C23×C32—33no—
99C23×C32—33no—
100C23×C32—31no—
101C23×C32—33no—
102C23×C32—33no—
103C23×C32—39no—
104C23×C32—63no—
105C23×C32—61no—
106C23×C32—63no—
107C23×C32—63no—
108C23×C32—69no—
109C23×C32—96no—
110C23×C32—183no—
111C23×C32—181no—
112C23×C32C2×C3×D313yes—
113C23×C32C2×C3×D319yescomplemented
114C22×C3⋊D3—41no—
115C22×C3×D3—36no—
116C22×C3×D3—318no—
117C22×C3×D3—63no—
118C22×C3×D3—69no—
119C22×C3×D3—61no—
120C22×C3×D3—63no—
121C22×C3×D3—361no—
122C2×C3×A4—23no—
123C2×C3×A4—29no—
124C2×C3×A4—33no—
125C2×C3×A4—63no—
126C2×C3×A4—66no—
127C2×C3×A4—63no—
128C2×C3×A4C22×C3213yescomplemented
129C3⋊S4C22×C3214yescomplemented
130C3×S4—94no—
131C32×D4—912no—
132C32×D4—94no—
133C2×C32×C4—96no—
134C3×#67—312no—
135C3×#67—336no—
136C3×#67—34no—
137C3×#67—312no—
138C2×C3×Dic3—36no—
139C2×C3×Dic3—318no—
140C2×C33—31no—
141C2×C33—33no—
142C2×C33—43no—
143C2×C33—83no—
144C2×C33C2×S413yescomplemented
145C3×C3⋊D3—44no—
146C32×D3—64no—
147C2×H(3,𝔽3)—49no—
148C2×H(3,𝔽3)—89no—
149(C3⋊D3)⋊C3—412no—
150C24×C3—21no—
151C24×C3—33no—
152C24×C3—93no—
153C24×C3—91no—
154C24×C3(C3⋊D3)⋊C311yescharacteristic, complemented
155C24×C3C32×D311yescharacteristic
156C23×D3—31no—
157C22×A4—31no—
158C22×A4—61no—
159C22×A4—63no—
160C2×S4—96no—
161C2×C3×D4—96no—
162C2×C3×D4—918no—
163C2×C3×D4—918no—
164C2×C3×D4—96no—
165C22×C3×C4—93no—
166C22×C3×C4—91no—
167C2×#67—36no—
168C2×#67—36no—
169C22×Dic3—31no—
170C22×C32—33no—
171C22×C32—33no—
172C22×C32—33no—
173C22×C32—31no—
174C22×C32—31no—
175C22×C32—33no—
176C22×C32—39no—
177C22×C32—33no—
178C22×C32—33no—
179C22×C32—39no—
180C22×C32—39no—
181C22×C32—41no—
182C22×C32—61no—
183C22×C32—61no—
184C22×C32—63no—
185C22×C32—63no—
186C22×C32—63no—
187C22×C32—69no—
188C22×C32—69no—
189C22×C32—69no—
190C22×C32—61no—
191C22×C32—63no—
192C22×C32—63no—
193C22×C32—66no—
194C22×C32—83no—
195C22×C32—81no—
196C22×C32—94no—
197C22×C32—121no—
198C22×C32—186no—
199C22×C32—186no—
200C22×C32—241no—
201C22×C32—242no—
202C22×C32—241no—
203C22×C32C3×S411yescharacteristic
204C22×C32C3×S413yescomplemented
205C22×C32C22×C3×D311yescharacteristic
206C22×C32C22×C3×D313yescomplemented
207C2×C3⋊D3—46no—
208C2×C3×D3—34no—
209C2×C3×D3—312no—
210C2×C3×D3—66no—
211C2×C3×D3—618no—
212C2×C3×D3—66no—
213C2×C3×D3—618no—
214C2×C3×D3—366no—
215C3×A4—21no—
216C3×A4—23no—
217C3×A4—31no—
218C3×A4—61no—
219C3×A4—62no—
220C3×A4—61no—
221C3×A4C23×C3211yescharacteristic, complemented
222C32×C4—94no—
223C3×Dic3—34no—
224C3×Dic3—312no—
225C22×D4—91noSylow
226C33—41no—
227C33—81no—
228C33C22×S411yescharacteristic, complemented
229H(3,𝔽3)—43no—
230H(3,𝔽3)—83no—
231C23×C3—23no—
232C23×C3—33no—
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—33no—
236C23×C3—33no—
237C23×C3—39no—
238C23×C3—39no—
239C23×C3—61no—
240C23×C3—63no—
241C23×C3—69no—
242C23×C3—63no—
243C23×C3—918no—
244C23×C3—96no—
245C23×C3—189no—
246C23×C3—183no—
247C23×C3—183no—
248C23×C3—181no—
249C23×C3C2×(C3⋊D3)⋊C313yescomplemented
250C23×C3C2×C32×D313yes—
251C22×D3—36no—
252C22×D3—63no—
253C22×D3—61no—
254C22×D3—361no—
255C2×A4—33no—
256C2×A4—63no—
257C2×A4—69no—
258S4—94no—
259C3×D4—912no—
260C3×D4—936no—
261C3×D4—912no—
262C3×D4—94no—
263C2×C3×C4—918no—
264C2×C3×C4—96no—
265#67—312no—
266#67—34no—
267C2×Dic3—36no—
268C2×C32—31no—
269C2×C32—33no—
270C2×C32—33no—
271C2×C32—31no—
272C2×C32—33no—
273C2×C32—33no—
274C2×C32—39no—
275C2×C32—43no—
276C2×C32—63no—
277C2×C32—61no—
278C2×C32—63no—
279C2×C32—63no—
280C2×C32—69no—
281C2×C32—89no—
282C2×C32—83no—
283C2×C32—123no—
284C2×C32—184no—
285C2×C32—243no—
286C2×C32—246no—
287C2×C32—243no—
288C2×C32C2×C3×S413yes—
289C2×C32C2×C3×S419yescomplemented
290C3⋊D3—44no—
291C3×D3—64no—
292C3×D3—612no—
293C3×D3—364no—
294C24—91no—
295C24C3×(C3⋊D3)⋊C311yescharacteristic, complemented
296C2×D4—96no—
297C2×D4—96no—
298C22×C4—91no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—33no—
302C22×C3—33no—
303C22×C3—39no—
304C22×C3—39no—
305C22×C3—33no—
306C22×C3—33no—
307C22×C3—33no—
308C22×C3—33no—
309C22×C3—39no—
310C22×C3—33no—
311C22×C3—33no—
312C22×C3—63no—
313C22×C3—63no—
314C22×C3—69no—
315C22×C3—69no—
316C22×C3—69no—
317C22×C3—63no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—69no—
321C22×C3—62no—
322C22×C3—912no—
323C22×C3—94no—
324C22×C3—121no—
325C22×C3—1818no—
326C22×C3—186no—
327C22×C3—1818no—
328C22×C3—186no—
329C22×C3—241no—
330C22×C3—243no—
331C22×C3(C3⋊S4)⋊C311yescharacteristic, direct product factor
332C22×C3C22×(C3⋊D3)⋊C311yescharacteristic, complemented
333C22×C3C32×S411yescharacteristic
334C22×C3C22×C32×D311yescharacteristic
335C2×D3—34no—
336C2×D3—66no—
337C2×D3—66no—
338C2×D3—366no—
339A4—31no—
340A4—61no—
341A4—63no—
342C3×C4—912no—
343C3×C4—94no—
344Dic3—34no—
345C32—31no—
346C32—41no—
347C32—61no—
348C32—83no—
349C32—81no—
350C32—121no—
351C32—241no—
352C32—242no—
353C32—241no—
354C32C22×C3×S411yescharacteristic
355C32C22×C3×S413yescomplemented
356C23—31no—
357C23—33no—
358C23—96no—
359C23—183no—
360C23—181no—
361C23C2×C3×(C3⋊D3)⋊C313yescomplemented
362D4—912no—
363D4—94no—
364C2×C4—96no—
365C2×C3—23no—
366C2×C3—33no—
367C2×C3—39no—
368C2×C3—31no—
369C2×C3—33no—
370C2×C3—31no—
371C2×C3—33no—
372C2×C3—39no—
373C2×C3—61no—
374C2×C3—63no—
375C2×C3—63no—
376C2×C3—69no—
377C2×C3—123no—
378C2×C3—1812no—
379C2×C3—184no—
380C2×C3—243no—
381C2×C3—249no—
382C2×C3C2×(C3⋊S4)⋊C313yesdirect product factor
383C2×C3C2×C32×S413yes—
384D3—64no—
385D3—364no—
386C22—33no—
387C22—33no—
388C22—33no—
389C22—61no—
390C22—94no—
391C22—186no—
392C22—186no—
393C22C3×(C3⋊S4)⋊C311yescharacteristic, direct product factor
394C22C22×C3×(C3⋊D3)⋊C311yescharacteristic, complemented
395C4—94no—
396C3—21no—
397C3—33no—
398C3—121no—
399C3—241no—
400C3—243no—
401C3C22×(C3⋊S4)⋊C311yescharacteristic, direct product factor
402C3C22×C32×S411yescharacteristic
403C2—31no—
404C2—33no—
405C2—184no—
406C2C2×C3×(C3⋊S4)⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C3(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×A4C23×C32
FrattiniC3C22×C32×S4
FittingC24×C33D3
socleC24×C32C3×D3
LCSG ▷ C3×A4
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊H(3,𝔽3)C2semidirect
2. C2×C3×(C3⋊S4)⋊C3C2direct (N × H)
3. C22×C3×C3⋊S4C3semidirect
5. C22×(C3⋊S4)⋊C3C3direct (N × H)
7. C2×C3×C22⋊H(3,𝔽3)C22semidirect
8. C3×(C3⋊S4)⋊C3C22direct (N × H)
9. C24×C33D3semidirect
12. C22×C32×A4C2×C3semidirect
13. C2×C3×C3⋊S4C2×C3semidirect
18. C22×C22⋊H(3,𝔽3)C2×C3semidirect
19. C2×(C3⋊S4)⋊C3C2×C3direct (N × H)
22. C3×C22⋊H(3,𝔽3)C23semidirect
23. C22×C3⋊S4C32semidirect
30. C23×C33C2×D3semidirect
36. C2×C32×A4C22×C3semidirect
37. C3×C3⋊S4C22×C3semidirect
43. C2×C22⋊H(3,𝔽3)C22×C3semidirect
44. (C3⋊S4)⋊C3C22×C3direct (N × H)
51. C24×C32C3×D3semidirect
60. C22×C3×A4C2×C32semidirect
61. C2×C3⋊S4C2×C32semidirect
78. C22×C33S4semidirect
79. C22×C33C22×D3semidirect
85. C32×A4C23×C3semidirect
91. C22⋊H(3,𝔽3)C23×C3semidirect
113. C23×C32C2×C3×D3semidirect
128. C2×C3×A4C22×C32semidirect
129. C3⋊S4C22×C32semidirect
144. C2×C33C2×S4semidirect
154. C24×C3(C3⋊D3)⋊C3semidirect
204. C22×C32C3×S4semidirect
206. C22×C32C22×C3×D3semidirect
221. C3×A4C23×C32semidirect
228. C33C22×S4semidirect
249. C23×C3C2×(C3⋊D3)⋊C3semidirect
289. C2×C32C2×C3×S4semidirect
295. C24C3×(C3⋊D3)⋊C3semidirect
331. C22×C3(C3⋊S4)⋊C3direct (N × H)
332. C22×C3C22×(C3⋊D3)⋊C3semidirect
355. C32C22×C3×S4semidirect
361. C23C2×C3×(C3⋊D3)⋊C3semidirect
382. C2×C3C2×(C3⋊S4)⋊C3direct (N × H)
393. C22C3×(C3⋊S4)⋊C3direct (N × H)
394. C22C22×C3×(C3⋊D3)⋊C3semidirect
401. C3C22×(C3⋊S4)⋊C3direct (N × H)
406. C2C2×C3×(C3⋊S4)⋊C3direct (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⋊H(3,𝔽3)2
C2×C3×(C3⋊S4)⋊C32
C22×C3×C3⋊S43
C22×C32×#673
C22×(C3⋊S4)⋊C33
C22×C3×(C3⋊D3)⋊C34