Finite Group Databasebeta

← C3×((C3⋊S4)⋊C3)⋊C4 #72222939

Subgroups — C3×((C3⋊S4)⋊C3)⋊C4 #72222939

Order 2592.

592 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 54 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×C3×(C3⋊S4)⋊C3C211yesmaximal, characteristic
2C3×(C3⋊Dic3)⋊A4C211yesmaximal, characteristic, complemented
3C3×#88613C211yesmaximal, characteristic, complemented
4C3×(C3⋊S4)⋊C4—31nomaximal
5C3×#67⋊Dic3—31nomaximal
6((C3⋊S4)⋊C3)⋊C4C311yesmaximal, characteristic, direct product factor
7C2×C3×C22⋊H(3,𝔽3)C2211yescharacteristic
8C3×(C3⋊S4)⋊C3C412yescomplemented
9C3×((C3⋊D3)⋊C3)⋊C4—41nomaximal
10C2×C3×C3⋊S4D311yescharacteristic
11C22×C3×C3⋊Dic3D311yescharacteristic, complemented
12C2×C32×#67—31no—
13C2×C3×D3×Dic3—31no—
14C3×Dic3×A4—31no—
15C32×SL(2,ℤ/4)—31no—
16C32×SL(2,ℤ/4)—61no—
17C2×(C3⋊S4)⋊C3—21no—
18C2×(C3⋊S4)⋊C3C2×C311yescharacteristic
19C32×#217—31no—
20C3×Dic3⋊Dic3—31no—
21C3×(C2×D3)⋊Dic3—31no—
22C3×Dic32—31no—
23(C3⋊Dic3)⋊A4—21no—
24(C3⋊Dic3)⋊A4C2×C311yescharacteristic, complemented
25#88613C2×C311yescharacteristic, complemented
26C2×C3×(C3⋊D3)⋊C3—41no—
27C3×C22⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
28C3×#1084—41no—
29C3×(C3⋊Dic3)⋊C3—41no—
30C3×C4×S4—91no—
31(C3⋊S4)⋊C4—31no—
32C3×D4×Dic3—91no—
33C3×#67⋊C4—91no—
34#67⋊Dic3—31no—
35C23×C33C2×D311yescharacteristic
36C22×C32×D3—31no—
37C2×C32×A4—21no—
38C2×C32×A4C2×D311yescharacteristic
39C3×C3⋊S4Dic312yescomplemented
40C2×C3×C3⋊Dic3—31no—
41C2×C3×C3⋊Dic3—31no—
42C32×#67—32no—
43C32×#67—32no—
44C2×C32×Dic3—31no—
45C2×C32×Dic3—31no—
46C2×C32×Dic3—31no—
47C3×#624—121no—
48C3×D3×Dic3—61no—
49C3×D3×Dic3—61no—
50C2×C22⋊H(3,𝔽3)—21no—
51C2×C22⋊H(3,𝔽3)—21no—
52C2×C22⋊H(3,𝔽3)—41no—
53C2×C22⋊H(3,𝔽3)C22×C311yescharacteristic
54(C3⋊S4)⋊C3—22no—
55(C3⋊S4)⋊C3C3×C412yescomplemented
56((C3⋊D3)⋊C3)⋊C4—41no—
57C2×C3×H(3,𝔽3)—41no—
58C3×(C3⋊D3)⋊C3—42no—
59C2×C3⋊S4C3×D311yescharacteristic
60C2×C3×S4—91no—
61C2×C32×D4—91no—
62C22×C3⋊Dic3C3×D311yescharacteristic, complemented
63C2×C3×#67—31no—
64C2×C3×#67—31no—
65C2×C3×#67—61no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—91no—
68C22×C3×Dic3—91no—
69C2×C3×C4×D3—91no—
70C3×C4×A4—91no—
71C2×D3×Dic3—31no—
72Dic3×A4—31no—
73Dic3×A4—61no—
74C3×SL(2,ℤ/4)—31no—
75C3×SL(2,ℤ/4)—31no—
76C3×SL(2,ℤ/4)—61no—
77C3×SL(2,ℤ/4)—61no—
78C3×SL(2,ℤ/4)—61no—
79C3×SL(2,ℤ/4)—62no—
80C32×C22⋊C4—91no—
81C3×#217—31no—
82C3×#217—31no—
83C3×#217—61no—
84C3×#217—91no—
85C3×(C2×D3)⋊C4—91no—
86C3×C4⋊Dic3—91no—
87C3×Dic3⋊C4—91no—
88C3×C4×Dic3—91no—
89C3×C4×Dic3—91no—
90Dic3⋊Dic3—31no—
91(C2×D3)⋊Dic3—31no—
92Dic32—31no—
93C22×C33—31no—
94C22×C33—31no—
95C22×C33C4×D311yescharacteristic, complemented
96C2×C3×C3⋊D3—41no—
97C2×C32×D3—32no—
98C2×C32×D3—61no—
99C2×C32×D3—61no—
100C32×A4—21no—
101C32×A4C2×Dic311yescharacteristic, complemented
102C3×C3⋊Dic3—31no—
103C3×C3⋊Dic3S411yescharacteristic, complemented
104C32×Dic3—32no—
105C32×Dic3—61no—
106C32×Dic3—61no—
107C32×Dic3—121no—
108C32×Dic3—121no—
109C32×Dic3—241no—
110C2×(C3⋊D3)⋊C3—41no—
111C2×(C3⋊D3)⋊C3—81no—
112C22⋊H(3,𝔽3)—21no—
113C22⋊H(3,𝔽3)—21no—
114C22⋊H(3,𝔽3)—41no—
115C22⋊H(3,𝔽3)C2×C3×C411yescharacteristic, complemented
116#1084—41no—
117(C3⋊Dic3)⋊C3—41no—
118(C3⋊Dic3)⋊C3—81no—
119C4×S4—91no—
120C3×C4×D4—271no—
121D4×Dic3—91no—
122#67⋊C4—91no—
123C3×H(3,𝔽3)—41noSylow
124C23×C32—21no—
125C23×C32—31no—
126C23×C32—61no—
127C23×C32—91no—
128C23×C32D3211yescharacteristic
129C23×C32C2×C3×D311yescharacteristic
130C22×C3×D3—31no—
131C22×C3×D3—31no—
132C22×C3×D3—61no—
133C2×C3×A4—21no—
134C2×C3×A4—21no—
135C2×C3×A4—31no—
136C2×C3×A4—41no—
137C2×C3×A4—61no—
138C2×C3×A4—62no—
139C2×C3×A4—61no—
140C2×C3×A4C2×C3×D311yescharacteristic
141C3⋊S4C3×Dic312yescomplemented
142C3×S4—92no—
143C32×D4—92no—
144C32×D4—92no—
145C2×C32×C4—91no—
146C2×C32×C4—91no—
147C2×C32×C4—91no—
148C2×C3⋊Dic3—31no—
149C2×C3⋊Dic3—31no—
150C3×#67—32no—
151C3×#67—32no—
152C3×#67—32no—
153C3×#67—32no—
154C3×#67—62no—
155C3×#67—62no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—31no—
158C2×C3×Dic3—31no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—91no—
166C2×C3×Dic3—91no—
167C2×C3×Dic3—91no—
168C2×C3×Dic3—91no—
169C2×C3×Dic3—181no—
170C2×C3×Dic3—181no—
171C2×C3×Dic3—181no—
172C2×C3×Dic3—181no—
173C3×C4×D3—181no—
174C3×C4×D3—181no—
175C3×C4×D3—361no—
176#624—121no—
177D3×Dic3—61no—
178D3×Dic3—61no—
179C2×C33—31no—
180C2×C33—31no—
181C2×C33—41no—
182C2×C33—81no—
183C2×C33C2×S411yescharacteristic
184C3×C3⋊D3—42no—
185C32×D3—62no—
186C2×H(3,𝔽3)—41no—
187C2×H(3,𝔽3)—81no—
188C2×H(3,𝔽3)—81no—
189C2×H(3,𝔽3)—161no—
190(C3⋊D3)⋊C3—42no—
191(C3⋊D3)⋊C3—82no—
192C2×S4—91no—
193C2×C3×D4—91no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C22×C3×C4—91no—
197C22×C3×C4—271no—
198C2×#67—31no—
199C22×Dic3—31no—
200C22×Dic3—91no—
201C22×Dic3—91no—
202C2×C4×D3—91no—
203C4×A4—91no—
204C4×A4—181no—
205SL(2,ℤ/4)—31no—
206SL(2,ℤ/4)—61no—
207C3×C4⋊C4—271no—
208C3×C22⋊C4—91no—
209C3×C22⋊C4—91no—
210C3×C22⋊C4—181no—
211C3×C22⋊C4—271no—
212C3×C42—271no—
213#217—31no—
214#217—91no—
215(C2×D3)⋊C4—91no—
216C4⋊Dic3—91no—
217Dic3⋊C4—91no—
218C4×Dic3—91no—
219C4×Dic3—91no—
220C22×C32—21no—
221C22×C32—31no—
222C22×C32—31no—
223C22×C32—31no—
224C22×C32—31no—
225C22×C32—31no—
226C22×C32—31no—
227C22×C32—31no—
228C22×C32—61no—
229C22×C32—61no—
230C22×C32—61no—
231C22×C32—61no—
232C22×C32—61no—
233C22×C32—61no—
234C22×C32—61no—
235C22×C32—92no—
236C22×C32—121no—
237C22×C32—121no—
238C22×C32—181no—
239C22×C32—181no—
240C22×C32D3×Dic311yescharacteristic
241C22×C32C3×C4×D311yescharacteristic, complemented
242C2×C3⋊D3—41no—
243C2×C3×D3—32no—
244C2×C3×D3—32no—
245C2×C3×D3—62no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C2×C3×D3—361no—
253C3×A4—21no—
254C3×A4—21no—
255C3×A4—31no—
256C3×A4—41no—
257C3×A4—61no—
258C3×A4—62no—
259C3×A4—61no—
260C3×A4C2×C3×Dic311yescharacteristic, complemented
261C32×C4—92no—
262C32×C4—181no—
263C32×C4—181no—
264C32×C4—361no—
265C3⋊Dic3—31no—
266C3⋊Dic3C3×S411yescharacteristic, complemented
267C3×Dic3—32no—
268C3×Dic3—32no—
269C3×Dic3—31no—
270C3×Dic3—62no—
271C3×Dic3—61no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C3×Dic3—91no—
276C3×Dic3—91no—
277C3×Dic3—91no—
278C3×Dic3—121no—
279C3×Dic3—121no—
280C3×Dic3—121no—
281C3×Dic3—121no—
282C3×Dic3—121no—
283C3×Dic3—181no—
284C3×Dic3—241no—
285C3×Dic3—241no—
286C3×Dic3—241no—
287C3×Dic3—241no—
288C3×Dic3—242no—
289C4×D4—271noSylow
290C33—41no—
291C33—81no—
292C33C4×S411yescharacteristic, complemented
293H(3,𝔽3)—41no—
294H(3,𝔽3)—81no—
295H(3,𝔽3)—81no—
296H(3,𝔽3)—161no—
297C23×C3—21no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—91no—
301C23×C3—91no—
302C23×C3—181no—
303C23×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C23×C3C3×D3211yescharacteristic
305C22×D3—31no—
306C2×A4—31no—
307C2×A4—61no—
308C2×A4—61no—
309C2×A4—121no—
310S4—92no—
311C3×D4—92no—
312C3×D4—92no—
313C3×D4—92no—
314C3×D4—92no—
315C3×D4—182no—
316C3×D4—182no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4—271no—
327C2×C3×C4—271no—
328C2×C3×C4—541no—
329C2×C3×C4—541no—
330#67—32no—
331#67—32no—
332C2×Dic3—31no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C4×D3—181no—
344C4×D3—181no—
345C4×D3—361no—
346C2×C32—21no—
347C2×C32—31no—
348C2×C32—31no—
349C2×C32—31no—
350C2×C32—31no—
351C2×C32—31no—
352C2×C32—31no—
353C2×C32—31no—
354C2×C32—41no—
355C2×C32—61no—
356C2×C32—61no—
357C2×C32—61no—
358C2×C32—61no—
359C2×C32—61no—
360C2×C32—61no—
361C2×C32—61no—
362C2×C32—81no—
363C2×C32—81no—
364C2×C32—121no—
365C2×C32—121no—
366C2×C32—121no—
367C2×C32—161no—
368C2×C32—182no—
369C2×C32—241no—
370C2×C32—242no—
371C2×C32—241no—
372C2×C32D3×S411yescharacteristic
373C2×C32C2×C3×S411yescharacteristic
374C3⋊D3—42no—
375C3×D3—62no—
376C3×D3—62no—
377C3×D3—122no—
378C3×D3—362no—
379C2×D4—91no—
380C22×C4—91no—
381C22×C4—271no—
382C4⋊C4—271no—
383C22⋊C4—91no—
384C22⋊C4—271no—
385C42—271no—
386C22×C3—21no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—92no—
402C22×C3—92no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—182no—
406C22×C3—181no—
407C22×C3—181no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
413C22×C3C3×D3×Dic311yescharacteristic
414C2×D3—32no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—361no—
418A4—31no—
419A4—61no—
420A4—61no—
421A4—121no—
422C3×C4—92no—
423C3×C4—92no—
424C3×C4—91no—
425C3×C4—182no—
426C3×C4—181no—
427C3×C4—181no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4—271no—
431C3×C4—361no—
432C3×C4—361no—
433C3×C4—361no—
434C3×C4—721no—
435Dic3—32no—
436Dic3—31no—
437Dic3—61no—
438Dic3—61no—
439Dic3—91no—
440Dic3—91no—
441Dic3—91no—
442Dic3—121no—
443Dic3—181no—
444Dic3—241no—
445C32—21no—
446C32—31no—
447C32—41no—
448C32—61no—
449C32—81no—
450C32—81no—
451C32—121no—
452C32—161no—
453C32—241no—
454C32—242no—
455C32—241no—
456C32Dic3×S411yescharacteristic
457C32C3×C4×S411yescharacteristic, complemented
458C23—91no—
459C23C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460D4—92no—
461D4—92no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—541no—
468C2×C4—541no—
469C2×C3—21no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—61no—
482C2×C3—61no—
483C2×C3—61no—
484C2×C3—121no—
485C2×C3—121no—
486C2×C3—121no—
487C2×C3—182no—
488C2×C3—182no—
489C2×C3—241no—
490C2×C3—241no—
491C2×C3—362no—
492C2×C3—481no—
493C2×C3((C3⋊S4)⋊C3)⋊C211yescharacteristic
494C2×C3C3×D3×S411yescharacteristic
495D3—62no—
496D3—362no—
497C22—31no—
498C22—31no—
499C22—92no—
500C22—181no—
501C22—181no—
502C22C3×((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
503C4—92no—
504C4—91no—
505C4—181no—
506C4—181no—
507C4—271no—
508C3—21no—
509C3—31no—
510C3—61no—
511C3—121no—
512C3—241no—
513C3—241no—
514C3—481no—
515C3((C3⋊S4)⋊C3)⋊C411yescharacteristic, direct product factor
516C3C3×Dic3×S411yescharacteristic
517C2—31no—
518C2—31no—
519C2—182no—
520C2C3×((C3⋊S4)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2×C3((C3⋊S4)⋊C3)⋊C2
commutator, hypocenterC22⋊H(3,𝔽3)C2×C3×C4
FrattiniC2×C3C3×D3×S4
FittingC23×C33C2×D3
socleC23×C32D32
prime radicalC2×C3×(C3⋊S4)⋊C3C2
cosocleC2×C22⋊H(3,𝔽3)C22×C3
LCSG ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×(C3⋊Dic3)⋊A4C2semidirect
3. C3×#88613C2semidirect
6. ((C3⋊S4)⋊C3)⋊C4C3direct (N × H)
8. C3×(C3⋊S4)⋊C3C4semidirect
11. C22×C3×C3⋊Dic3D3semidirect
24. (C3⋊Dic3)⋊A4C2×C3semidirect
25. #88613C2×C3semidirect
27. C3×C22⋊H(3,𝔽3)C2×C4semidirect
39. C3×C3⋊S4Dic3semidirect
55. (C3⋊S4)⋊C3C3×C4semidirect
62. C22×C3⋊Dic3C3×D3semidirect
95. C22×C33C4×D3semidirect
101. C32×A4C2×Dic3semidirect
103. C3×C3⋊Dic3S4semidirect
115. C22⋊H(3,𝔽3)C2×C3×C4semidirect
141. C3⋊S4C3×Dic3semidirect
241. C22×C32C3×C4×D3semidirect
260. C3×A4C2×C3×Dic3semidirect
266. C3⋊Dic3C3×S4semidirect
292. C33C4×S4semidirect
412. C22×C3((C3⋊D3)⋊C3)⋊C4semidirect
457. C32C3×C4×S4semidirect
502. C22C3×((C3⋊D3)⋊C3)⋊C4semidirect
515. C3((C3⋊S4)⋊C3)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×(C3⋊S4)⋊C32
C3×(C3⋊Dic3)⋊A42
C3×#886132
C3×(C3⋊S4)⋊C43
C3×#67⋊Dic33
((C3⋊S4)⋊C3)⋊C43
C3×((C3⋊D3)⋊C3)⋊C44