Finite Group Databasebeta

← C2×C4×(C3⋊S4)⋊C3 #13164395

Subgroups — C2×C4×(C3⋊S4)⋊C3 #13164395

Order 1728.

944 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 102 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⋊S4)⋊C3C211yesmaximal, characteristic
2C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×(C3⋊SL(2,ℤ/4))⋊C3C211yesmaximal, characteristic, complemented
4C4×(C3⋊S4)⋊C3C214yesmaximal, direct product factor
5C2×C4×C3⋊S4C311yesmaximal, characteristic, complemented
6C2×C3×C4×#67—31nomaximal
7C22×C22⋊H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊S4)⋊C3C414yesdirect product factor
9C2×(C3⋊S4)⋊C3C2212yes—
10C2×C4×(C3⋊D3)⋊C3—41nomaximal
11C4×C22⋊H(3,𝔽3)C2212yescomplemented
12(C3⋊SL(2,ℤ/4))⋊C3C2212yescomplemented
13C22×C3⋊S4C2×C311yescharacteristic
14C23×C32×C4D311yescharacteristic, complemented
15C22×C3×#67—31no—
16C22×C3×C4×D3—31no—
17C2×C3×C4×A4—21no—
18C2×C3×C4×A4C2×C311yescharacteristic, complemented
19C2×C3⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
20C4×C3⋊S4C2×C314yescomplemented
21C2×C3×#217—31no—
22C3×C4×#67—34no—
23C3×C4×#67—34no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×Dic3⋊C4—31no—
26C2×C3×C4×Dic3—31no—
27C22×(C3⋊D3)⋊C3—41no—
28C2×C22⋊H(3,𝔽3)C2×C412yescomplemented
29C2×C22⋊H(3,𝔽3)C2311yescharacteristic
30(C3⋊S4)⋊C3C2×C414yesdirect product factor
31C2×C4×H(3,𝔽3)—41no—
32C2×(C3⋊Dic3)⋊C3—41no—
33C4×(C3⋊D3)⋊C3—44no—
34C2×C4×S4—91no—
35C2×C3×C4×D4—91no—
36C2×C4×#67—31no—
37C24×C32C2×D311yescharacteristic
38C23×C3×D3—31no—
39C22×C3×A4—21no—
40C22×C3×A4C22×C311yescharacteristic
41C2×C3⋊S4C3×C414yescomplemented
42C2×C3⋊S4C22×C312yes—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4—32no—
46C22×C32×C4C2×D312yescomplemented
47C2×C4×C3⋊D3—41no—
48C2×C3×#67—32no—
49C2×C3×#67—34no—
50C2×C3×#67—34no—
51C2×C3×#67—32no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—31no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—62no—
58C2×C3×C4×D3—61no—
59C3×C4×A4—22no—
60C3×C4×A4C22×C312yescomplemented
61C3⋊SL(2,ℤ/4)C22×C312yescomplemented
62C3×#217—32no—
63C3×#217—32no—
64C3×(C2×D3)⋊C4—34no—
65C3×Dic3⋊C4—34no—
66C3×C4×Dic3—34no—
67C22×H(3,𝔽3)—41no—
68C2×(C3⋊D3)⋊C3—44no—
69C2×(C3⋊D3)⋊C3—42no—
70C22⋊H(3,𝔽3)C22×C411yescharacteristic, complemented
71C4×H(3,𝔽3)—42no—
72(C3⋊Dic3)⋊C3—42no—
73C22×S4—91no—
74C22×C3×D4—91no—
75C23×C3×C4—31no—
76C23×C3×C4—91no—
77C23×C3×C4C3×D311yescharacteristic
78C22×#67—31no—
79C22×C4×D3—31no—
80C2×C4×A4—31no—
81C2×C4×A4—61no—
82C2×SL(2,ℤ/4)—91no—
83C4×S4—94no—
84C3×C4×D4—94no—
85C3×C4×D4—94no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C22⋊C4—91no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C42—91no—
90C2×#217—31no—
91C4×#67—34no—
92C4×#67—34no—
93C2×(C2×D3)⋊C4—31no—
94C2×Dic3⋊C4—31no—
95C2×C4×Dic3—31no—
96C23×C32—31no—
97C23×C32—31no—
98C23×C32—32no—
99C23×C32C4×D312yescomplemented
100C23×C32C22×D311yescharacteristic
101C22×C3⋊D3—41no—
102C22×C3×D3—32no—
103C22×C3×D3—34no—
104C22×C3×D3—62no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C2×C3×A4—22no—
108C2×C3×A4—21no—
109C2×C3×A4C2×C3×C412yescomplemented
110C2×C3×A4C23×C311yescharacteristic
111C3⋊S4C2×C3×C414yescomplemented
112C2×C32×C4—31no—
113C2×C32×C4—32no—
114C2×C32×C4—32no—
115C2×C32×C4—32no—
116C2×C32×C4—41no—
117C2×C32×C4—61no—
118C2×C32×C4—81no—
119C2×C32×C4S411yescharacteristic, complemented
120C2×C3⋊Dic3—41no—
121C4×C3⋊D3—44no—
122C3×#67—38no—
123C3×#67—34no—
124C3×#67—34no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—61no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C2×C4×D4—91noSylow
135C2×H(3,𝔽3)—42no—
136C2×H(3,𝔽3)—41no—
137(C3⋊D3)⋊C3—44no—
138C24×C3—31no—
139C24×C3—91no—
140C24×C3C2×C3×D311yescharacteristic
141C23×D3—31no—
142C22×A4—31no—
143C22×A4—61no—
144C2×S4—94no—
145C2×S4—92no—
146C2×C3×D4—92no—
147C2×C3×D4—92no—
148C2×C3×D4—94no—
149C2×C3×D4—94no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—32no—
154C22×C3×C4—31no—
155C22×C3×C4—32no—
156C22×C3×C4—32no—
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C22×C3×C4—62no—
160C22×C3×C4—94no—
161C22×C3×C4—91no—
162C22×C3×C4—91no—
163C22×C3×C4—91no—
164C22×C3×C4—181no—
165C22×C3×C4—182no—
166C22×C3×C4—181no—
167C22×C3×C4C2×C3×D312yes—
168C2×#67—32no—
169C2×#67—34no—
170C2×#67—34no—
171C2×#67—32no—
172C22×Dic3—31no—
173C22×Dic3—31no—
174C22×Dic3—31no—
175C2×C4×D3—34no—
176C2×C4×D3—61no—
177C2×C4×D3—62no—
178C2×C4×D3—61no—
179C2×C4×D3—361no—
180C4×A4—32no—
181C4×A4—62no—
182SL(2,ℤ/4)—92no—
183C3×C4⋊C4—94no—
184C3×C22⋊C4—94no—
185C3×C22⋊C4—92no—
186C3×C22⋊C4—92no—
187C3×C42—94no—
188#217—32no—
189#217—32no—
190(C2×D3)⋊C4—34no—
191Dic3⋊C4—34no—
192C4×Dic3—34no—
193C22×C32—32no—
194C22×C32—31no—
195C22×C32—32no—
196C22×C32—32no—
197C22×C32—31no—
198C22×C32—31no—
199C22×C32—41no—
200C22×C32—61no—
201C22×C32—81no—
202C22×C32C2×C4×D311yescharacteristic, complemented
203C22×C32C2×S411yescharacteristic
204C2×C3⋊D3—44no—
205C2×C3⋊D3—42no—
206C2×C3×D3—34no—
207C2×C3×D3—62no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—62no—
211C3×A4—21no—
212C3×A4C22×C3×C411yescharacteristic, complemented
213C32×C4—32no—
214C32×C4—42no—
215C32×C4—82no—
216C32×C4C2×S412yescomplemented
217C3⋊Dic3—42no—
218C3×Dic3—34no—
219C3×Dic3—62no—
220C3×Dic3—62no—
221C22×D4—91no—
222C23×C4—91no—
223C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
224C4×D4—94no—
225C4×D4—94no—
226C2×C4⋊C4—91no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—91no—
229C2×C42—91no—
230H(3,𝔽3)—41noSylow
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—31no—
234C23×C3—32no—
235C23×C3—31no—
236C23×C3—32no—
237C23×C3—32no—
238C23×C3—31no—
239C23×C3—61no—
240C23×C3—61no—
241C23×C3—62no—
242C23×C3—92no—
243C23×C3—94no—
244C23×C3—181no—
245C23×C3—181no—
246C23×C3—182no—
247C23×C3C3×C4×D312yes—
248C23×C3C22×C3×D311yescharacteristic
249C22×D3—32no—
250C22×D3—34no—
251C22×D3—61no—
252C22×D3—62no—
253C22×D3—61no—
254C22×D3—361no—
255C2×A4—32no—
256C2×A4—31no—
257C2×A4—62no—
258C2×A4—61no—
259S4—94no—
260C3×D4—98no—
261C3×D4—94no—
262C3×D4—94no—
263C2×C3×C4—32no—
264C2×C3×C4—31no—
265C2×C3×C4—32no—
266C2×C3×C4—31no—
267C2×C3×C4—32no—
268C2×C3×C4—32no—
269C2×C3×C4—32no—
270C2×C3×C4—32no—
271C2×C3×C4—31no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—62no—
278C2×C3×C4—61no—
279C2×C3×C4—92no—
280C2×C3×C4—94no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—121no—
284C2×C3×C4—184no—
285C2×C3×C4—184no—
286C2×C3×C4—182no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—241no—
290C2×C3×C4C3×S411yescharacteristic
291#67—38no—
292#67—34no—
293#67—34no—
294C2×Dic3—32no—
295C2×Dic3—34no—
296C2×Dic3—32no—
297C2×Dic3—32no—
298C2×Dic3—61no—
299C2×Dic3—62no—
300C2×Dic3—61no—
301C2×Dic3—361no—
302C4×D3—64no—
303C4×D3—64no—
304C4×D3—364no—
305C2×C32—31no—
306C2×C32—31no—
307C2×C32—32no—
308C2×C32—41no—
309C2×C32—42no—
310C2×C32—81no—
311C2×C32—82no—
312C2×C32C4×S412yescomplemented
313C2×C32C22×S411yescharacteristic
314C3⋊D3—44no—
315C3×D3—64no—
316C24—91no—
317C24C2×(C3⋊D3)⋊C311yescharacteristic
318C2×D4—92no—
319C2×D4—94no—
320C2×D4—94no—
321C2×D4—92no—
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—32no—
325C22×C4—94no—
326C22×C4—91no—
327C22×C4—91no—
328C22×C4—91no—
329C22×C4—181no—
330C22×C4—182no—
331C22×C4—181no—
332C22×C4C2×(C3⋊D3)⋊C312yescomplemented
333C4⋊C4—94no—
334C22⋊C4—94no—
335C22⋊C4—92no—
336C22⋊C4—92no—
337C42—94no—
338C22×C3—32no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—32no—
343C22×C3—32no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—32no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—62no—
357C22×C3—62no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3—62no—
361C22×C3—94no—
362C22×C3—121no—
363C22×C3—182no—
364C22×C3—182no—
365C22×C3—184no—
366C22×C3—184no—
367C22×C3—241no—
368C22×C3C2×C3×C4×D311yescharacteristic
369C22×C3C2×C3×S411yescharacteristic
370C2×D3—34no—
371C2×D3—62no—
372C2×D3—64no—
373C2×D3—62no—
374C2×D3—64no—
375C2×D3—362no—
376C2×D3—364no—
377A4—31no—
378A4—61no—
379C3×C4—32no—
380C3×C4—32no—
381C3×C4—32no—
382C3×C4—62no—
383C3×C4—94no—
384C3×C4—122no—
385C3×C4—182no—
386C3×C4—182no—
387C3×C4—242no—
388C3×C4C2×C3×S412yes—
389Dic3—34no—
390Dic3—62no—
391Dic3—62no—
392Dic3—362no—
393C32—41no—
394C32—81no—
395C32C2×C4×S411yescharacteristic, complemented
396C23—31no—
397C23—31no—
398C23—32no—
399C23—92no—
400C23—94no—
401C23—181no—
402C23—182no—
403C23—181no—
404C23C4×(C3⋊D3)⋊C312yescomplemented
405C23C22×(C3⋊D3)⋊C311yescharacteristic
406D4—98no—
407D4—94no—
408D4—94no—
409C2×C4—31no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—61no—
414C2×C4—92no—
415C2×C4—94no—
416C2×C4—92no—
417C2×C4—92no—
418C2×C4—184no—
419C2×C4—181no—
420C2×C4—182no—
421C2×C4—181no—
422C2×C4—184no—
423C2×C4(C3⋊S4)⋊C311yescharacteristic, direct product factor
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—32no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—32no—
430C2×C3—31no—
431C2×C3—32no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—62no—
435C2×C3—121no—
436C2×C3—122no—
437C2×C3—184no—
438C2×C3—241no—
439C2×C3—242no—
440C2×C3C3×C4×S412yes—
441C2×C3C22×C3×S411yescharacteristic
442D3—64no—
443D3—364no—
444C22—32no—
445C22—31no—
446C22—31no—
447C22—31no—
448C22—32no—
449C22—32no—
450C22—61no—
451C22—94no—
452C22—182no—
453C22—184no—
454C22—182no—
455C22—184no—
456C22C2×C4×(C3⋊D3)⋊C311yescharacteristic, complemented
457C22C2×(C3⋊S4)⋊C311yescharacteristic
458C4—32no—
459C4—94no—
460C4—182no—
461C4—182no—
462C4C2×(C3⋊S4)⋊C312yesdirect product factor
463C3—31no—
464C3—121no—
465C3—241no—
466C3C2×C3×C4×S411yescharacteristic
467C2—31no—
468C2—31no—
469C2—32no—
470C2—184no—
471C2C4×(C3⋊S4)⋊C312yesdirect product factor
472C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊S4)⋊C3
commutator, hypocenterC3×A4C22×C3×C4
FrattiniC2×C3C22×C3×S4
FittingC23×C32×C4D3
socleC24×C3C2×C3×D3
prime radicalC22×(C3⋊S4)⋊C3C2
cosocleC2×C3×A4C23×C3
LCSG ▷ C3×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊H(3,𝔽3)C2semidirect
3. C2×(C3⋊SL(2,ℤ/4))⋊C3C2semidirect
4. C4×(C3⋊S4)⋊C3C2direct (N × H)
5. C2×C4×C3⋊S4C3semidirect
8. C2×(C3⋊S4)⋊C3C4direct (N × H)
11. C4×C22⋊H(3,𝔽3)C22semidirect
12. (C3⋊SL(2,ℤ/4))⋊C3C22semidirect
14. C23×C32×C4D3semidirect
18. C2×C3×C4×A4C2×C3semidirect
19. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
20. C4×C3⋊S4C2×C3semidirect
28. C2×C22⋊H(3,𝔽3)C2×C4semidirect
30. (C3⋊S4)⋊C3C2×C4direct (N × H)
41. C2×C3⋊S4C3×C4semidirect
46. C22×C32×C4C2×D3semidirect
60. C3×C4×A4C22×C3semidirect
61. C3⋊SL(2,ℤ/4)C22×C3semidirect
70. C22⋊H(3,𝔽3)C22×C4semidirect
99. C23×C32C4×D3semidirect
109. C2×C3×A4C2×C3×C4semidirect
111. C3⋊S4C2×C3×C4semidirect
119. C2×C32×C4S4semidirect
202. C22×C32C2×C4×D3semidirect
212. C3×A4C22×C3×C4semidirect
216. C32×C4C2×S4semidirect
223. C23×C4(C3⋊D3)⋊C3semidirect
312. C2×C32C4×S4semidirect
332. C22×C4C2×(C3⋊D3)⋊C3semidirect
395. C32C2×C4×S4semidirect
404. C23C4×(C3⋊D3)⋊C3semidirect
423. C2×C4(C3⋊S4)⋊C3direct (N × H)
456. C22C2×C4×(C3⋊D3)⋊C3semidirect
462. C4C2×(C3⋊S4)⋊C3direct (N × H)
471. C2C4×(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⋊S4)⋊C32
C2×C4×C22⋊H(3,𝔽3)2
C2×(C3⋊SL(2,ℤ/4))⋊C32
C4×(C3⋊S4)⋊C32
C2×C4×C3⋊S43
C2×C3×C4×#673
C2×C4×(C3⋊D3)⋊C34