Finite Group Databasebeta

← C3×#11710729 #72276744

Subgroups — C3×#11710729 #72276744

Order 2304.

3474 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 3474 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C3×#85645C213yesmaximal, complemented
2#11710729C313yesmaximal, direct product factor
3C3×#62817C311yesmaximal, characteristic, complemented
4C3×C22⋊(C22⋊A4)C2211yescharacteristic, complemented
5C3×#5450—42nomaximal
6C2×C3×#120⋊C3—43nomaximal
7C3×#3532—32no—
8C3×#3532—36no—
9C3×#3490—36no—
10C3×#3484—33no—
11C3×#3484—33no—
12C22×C3×#120C2×C313yescomplemented
13#85645C2×C319yescomplemented
14C2×C3×C22⋊A4—43no—
15C3×#120⋊C3—46no—
16C3×#120⋊C3—46no—
17#62817C3211yescharacteristic, complemented, Sylow
18C26×C3C22×C311yescharacteristic, complemented
19C22⋊(C22⋊A4)C22×C313yescomplemented
20C2×C3×H(5,𝔽2)—31no—
21C2×C3×H(5,𝔽2)—31no—
22C23×C3×D4—33no—
23C3×H(3,𝔽4)—32no—
24C3×H(3,𝔽4)A412yescomplemented
25C3×#560—32no—
26C3×#560—36no—
27C3×#551—312no—
28C3×#551—36no—
29C3×#551—36no—
30C3×#546—312no—
31C3×#546—312no—
32C3×#546—312no—
33C3×#546—312no—
34C3×D42—36no—
35C3×D42—312no—
36C3×D42—36no—
37C3×#543—32no—
38C3×#538—312no—
39C3×#538—312no—
40C3×#535—36no—
41C3×#535—36no—
42C3×#534—36no—
43C3×#534—36no—
44C3×#534—33no—
45C3×#534—33no—
46C3×#525—36no—
47C3×#525—36no—
48C2×C3×#120—36no—
49C2×C3×#120—36no—
50C2×C3×#120—36no—
51C2×C3×#120—36no—
52C2×C3×#120—33no—
53C2×C3×#120A413yescomplemented
54C3×#518—33no—
55C3×#518—33no—
56C22×C3×C22⋊C4—33no—
57#5450—46no—
58C2×#120⋊C3—49no—
59C3×C22⋊A4—43no—
60C3×C22⋊A4—42no—
61C3×C22⋊A4—164no—
62C22×C3×A4—161no—
63#3532—32no—
64#3532—36no—
65#3490—36no—
66#3484—33no—
67#3484—33no—
68C22×#120C2×C3213yescomplemented
69C25×C3—33no—
70C25×C3—32no—
71C25×C3—124no—
72C25×C3C2×A413yescomplemented
73C2×C22⋊A4—49no—
74C3×H(5,𝔽2)—124no—
75C3×H(5,𝔽2)—124no—
76C2×C3×#41—36no—
77C2×C3×#41—36no—
78C22×C3×D4—33no—
79C22×C3×D4—33no—
80C22×C3×D4—33no—
81C22×C3×D4—33no—
82C22×C3×D4—312no—
83C22×C3×D4—312no—
84C22×C3×D4—36no—
85C22×C3×D4—33no—
86C22×C3×D4—33no—
87C22×C3×D4—1212no—
88C23×C3×C4—33no—
89C3×#127—32no—
90C3×#127—36no—
91C3×#126—34no—
92C3×#126—312no—
93C3×#124—36no—
94C3×#124—312no—
95C3×#124—36no—
96C3×#123—36no—
97C3×#123—36no—
98C3×#123—312no—
99C3×#123—312no—
100C3×#123—36no—
101C3×#123—36no—
102C3×C22⋊Q8—312no—
103C3×C22⋊Q8—312no—
104C3×#121—312no—
105C3×#121—312no—
106C3×#121—312no—
107C3×#121—312no—
108C3×#121—312no—
109C3×#121—312no—
110C3×#120—36no—
111C3×#120—36no—
112C3×#120—312no—
113C3×#120—36no—
114C3×#120—36no—
115C3×#120—36no—
116C3×#120—312no—
117C3×#120—36no—
118C3×#120—312no—
119C3×#120—36no—
120C3×#120—36no—
121C3×#120—32no—
122C3×#120—36no—
123C3×#120C2×A416yescomplemented
124C3×#120C2×A416yescomplemented
125C3×C4×D4—312no—
126C3×C4×D4—312no—
127C3×C4×D4—312no—
128C3×C4×D4—312no—
129C3×#117—36no—
130C3×#117—36no—
131C2×C3×C22⋊C4—36no—
132C2×C3×C22⋊C4—36no—
133C2×C3×C22⋊C4—312no—
134C2×C3×C22⋊C4—312no—
135C2×C3×C22⋊C4—36no—
136C2×C3×C22⋊C4—33no—
137C2×C3×C22⋊C4—33no—
138#120⋊C3—418no—
139#120⋊C3—418no—
140C2×C3×A4—163no—
141C2×C3×A4—326no—
142C26C22×C3211yescharacteristic, complemented
143C2×H(5,𝔽2)—31no—
144C2×H(5,𝔽2)—31no—
145C23×D4—33no—
146H(3,𝔽4)—32no—
147H(3,𝔽4)C3×A412yescomplemented
148#560—32no—
149#560—36no—
150#551—312no—
151#551—36no—
152#551—36no—
153#546—312no—
154#546—312no—
155#546—312no—
156#546—312no—
157D42—36no—
158D42—36no—
159D42—312no—
160#543—32no—
161#538—312no—
162#538—312no—
163#535—36no—
164#535—36no—
165#534—36no—
166#534—36no—
167#534—33no—
168#534—33no—
169#525—36no—
170#525—36no—
171C2×#120—33no—
172C2×#120—36no—
173C2×#120—36no—
174C2×#120—36no—
175C2×#120—36no—
176C2×#120C3×A413yescomplemented
177#518—33no—
178#518—33no—
179C22×C22⋊C4—33no—
180C24×C3—31no—
181C24×C3—36no—
182C24×C3—33no—
183C24×C3—31no—
184C24×C3—33no—
185C24×C3—33no—
186C24×C3—44no—
187C24×C3—612no—
188C24×C3—126no—
189C24×C3—1212no—
190C24×C3—1212no—
191C24×C3—1212no—
192C24×C3—128no—
193C24×C3C22×A413yes—
194C24×C3C22×A412yes—
195C24×C3C22⋊A411yescharacteristic, complemented
196C22⋊A4—49no—
197C22⋊A4—46no—
198C22⋊A4—1612no—
199C22×A4—163no—
200C3×#41—1212no—
201C3×#41—1212no—
202C2×C3×Q8—32no—
203C2×C3×Q8—32no—
204C2×C3×D4—312no—
205C2×C3×D4—36no—
206C2×C3×D4—36no—
207C2×C3×D4—36no—
208C2×C3×D4—312no—
209C2×C3×D4—36no—
210C2×C3×D4—36no—
211C2×C3×D4—36no—
212C2×C3×D4—612no—
213C2×C3×D4—612no—
214C2×C3×D4—126no—
215C2×C3×D4—126no—
216C2×C3×D4—1224no—
217C2×C3×D4—1224no—
218C2×C3×D4—126no—
219C2×C3×D4—126no—
220C2×C3×D4—1212no—
221C22×C3×C4—33no—
222C22×C3×C4—33no—
223C22×C3×C4—312no—
224C22×C3×C4—36no—
225C22×C3×C4—33no—
226C22×C3×C4—33no—
227C22×C3×C4—126no—
228C3×C4⋊C4—312no—
229C3×C4⋊C4—312no—
230C3×C22⋊C4—36no—
231C3×C22⋊C4—312no—
232C3×C22⋊C4—312no—
233C3×C22⋊C4—36no—
234C3×C22⋊C4—312no—
235C3×C22⋊C4—36no—
236C3×C22⋊C4—36no—
237C3×C22⋊C4—36no—
238C3×C22⋊C4—36no—
239C3×C42—36no—
240C3×C42—32no—
241C22×C32—641no—
242C3×A4—161no—
243C3×A4—326no—
244C3×A4—642no—
245C25—33no—
246C25—32no—
247C25—124no—
248C25C2×C3×A413yescomplemented
249H(5,𝔽2)—124no—
250H(5,𝔽2)—124no—
251C2×#41—36no—
252C2×#41—36no—
253C22×D4—33no—
254C22×D4—33no—
255C22×D4—33no—
256C22×D4—33no—
257C22×D4—312no—
258C22×D4—312no—
259C22×D4—36no—
260C22×D4—33no—
261C22×D4—33no—
262C22×D4—1212no—
263C23×C4—33no—
264#127—32no—
265#127—36no—
266#126—34no—
267#126—312no—
268#124—36no—
269#124—312no—
270#124—36no—
271#123—312no—
272#123—36no—
273#123—36no—
274#123—36no—
275#123—36no—
276#123—312no—
277C22⋊Q8—312no—
278C22⋊Q8—312no—
279#121—312no—
280#121—312no—
281#121—312no—
282#121—312no—
283#121—312no—
284#121—312no—
285#120—36no—
286#120—312no—
287#120—36no—
288#120—36no—
289#120—36no—
290#120—312no—
291#120—36no—
292#120—312no—
293#120—36no—
294#120—36no—
295#120—36no—
296#120—32no—
297#120—36no—
298#120C2×C3×A416yescomplemented
299#120C2×C3×A416yescomplemented
300C4×D4—312no—
301C4×D4—312no—
302C4×D4—312no—
303C4×D4—312no—
304#117—36no—
305#117—36no—
306C2×C22⋊C4—36no—
307C2×C22⋊C4—36no—
308C2×C22⋊C4—312no—
309C2×C22⋊C4—312no—
310C2×C22⋊C4—36no—
311C2×C22⋊C4—33no—
312C2×C22⋊C4—33no—
313C23×C3—34no—
314C23×C3—33no—
315C23×C3—32no—
316C23×C3—33no—
317C23×C3—66no—
318C23×C3—624no—
319C23×C3—66no—
320C23×C3—66no—
321C23×C3—66no—
322C23×C3—86no—
323C23×C3—1212no—
324C23×C3—122no—
325C23×C3—128no—
326C23×C3—122no—
327C23×C3—126no—
328C23×C3—126no—
329C23×C3—121no—
330C23×C3—126no—
331C23×C3—126no—
332C23×C3—121no—
333C23×C3—122no—
334C23×C3—1236no—
335C23×C3—1224no—
336C23×C3—246no—
337C23×C3C2×C22⋊A413yes—
338C2×A4—169no—
339C2×A4—3218no—
340C3×Q8—122no—
341C3×Q8—122no—
342C3×D4—612no—
343C3×D4—624no—
344C3×D4—612no—
345C3×D4—126no—
346C3×D4—126no—
347C3×D4—1212no—
348C3×D4—126no—
349C3×D4—126no—
350C2×C3×C4—36no—
351C2×C3×C4—36no—
352C2×C3×C4—66no—
353C2×C3×C4—66no—
354C2×C3×C4—1212no—
355C2×C3×C4—123no—
356C2×C3×C4—126no—
357C2×C3×C4—123no—
358C2×C32—643no—
359C24—31no—
360C24—36no—
361C24—33no—
362C24—31no—
363C24—33no—
364C24—33no—
365C24—44no—
366C24—612no—
367C24—126no—
368C24—1212no—
369C24—1212no—
370C24—1212no—
371C24—128no—
372C24C22×C3×A413yes—
373C24C22×C3×A412yes—
374C24C3×C22⋊A411yescharacteristic, complemented
375#41—1212no—
376#41—1212no—
377C2×Q8—32no—
378C2×Q8—32no—
379C2×D4—312no—
380C2×D4—36no—
381C2×D4—36no—
382C2×D4—36no—
383C2×D4—312no—
384C2×D4—36no—
385C2×D4—36no—
386C2×D4—36no—
387C2×D4—612no—
388C2×D4—612no—
389C2×D4—126no—
390C2×D4—126no—
391C2×D4—1224no—
392C2×D4—1224no—
393C2×D4—126no—
394C2×D4—126no—
395C2×D4—1212no—
396C22×C4—33no—
397C22×C4—33no—
398C22×C4—312no—
399C22×C4—36no—
400C22×C4—33no—
401C22×C4—33no—
402C22×C4—126no—
403C4⋊C4—312no—
404C4⋊C4—312no—
405C22⋊C4—36no—
406C22⋊C4—312no—
407C22⋊C4—312no—
408C22⋊C4—36no—
409C22⋊C4—312no—
410C22⋊C4—36no—
411C22⋊C4—36no—
412C22⋊C4—36no—
413C22⋊C4—36no—
414C42—36no—
415C42—32no—
416C22×C3—26no—
417C22×C3—36no—
418C22×C3—42no—
419C22×C3—63no—
420C22×C3—66no—
421C22×C3—63no—
422C22×C3—66no—
423C22×C3—66no—
424C22×C3—66no—
425C22×C3—124no—
426C22×C3—1224no—
427C22×C3—1212no—
428C22×C3—122no—
429C22×C3—161no—
430C22×C3—246no—
431C22×C3—246no—
432C22×C3—481no—
433C22×C3—643no—
434C22×C3C22×C22⋊A411yescharacteristic
435A4—163no—
436A4—3218no—
437A4—646no—
438C3×C4—66no—
439C3×C4—66no—
440C32—641noSylow
441C23—34no—
442C23—33no—
443C23—32no—
444C23—33no—
445C23—66no—
446C23—624no—
447C23—66no—
448C23—66no—
449C23—66no—
450C23—86no—
451C23—1212no—
452C23—122no—
453C23—128no—
454C23—122no—
455C23—126no—
456C23—126no—
457C23—121no—
458C23—126no—
459C23—126no—
460C23—121no—
461C23—122no—
462C23—1236no—
463C23—1224no—
464C23—246no—
465C23C2×C3×C22⋊A413yes—
466Q8—122no—
467Q8—122no—
468D4—612no—
469D4—624no—
470D4—612no—
471D4—126no—
472D4—126no—
473D4—1212no—
474D4—126no—
475D4—126no—
476C2×C4—36no—
477C2×C4—36no—
478C2×C4—66no—
479C2×C4—66no—
480C2×C4—1212no—
481C2×C4—123no—
482C2×C4—126no—
483C2×C4—123no—
484C2×C3—31no—
485C2×C3—43no—
486C2×C3—66no—
487C2×C3—122no—
488C2×C3—123no—
489C2×C3—649no—
490C22—26no—
491C22—36no—
492C22—42no—
493C22—63no—
494C22—66no—
495C22—63no—
496C22—66no—
497C22—66no—
498C22—66no—
499C22—124no—
500C22—1224no—
501C22—1212no—
502C22—122no—
503C22—161no—
504C22—246no—
505C22—246no—
506C22—481no—
507C22C22×C3×C22⋊A411yescharacteristic
508C4—66no—
509C4—66no—
510C3—643no—
511C3#1171072911yescharacteristic, direct product factor
512C2—31no—
513C2—43no—
514C2—66no—
515C2—122no—
516C2—123no—
subgroupG/H
center, hypercenterC3#11710729
commutator, cosocle, hypocenterC26C22×C32
FrattiniC22C22×C3×C22⋊A4
FittingC3×#62817C3
socleC22×C3C22×C22⋊A4
LCSG ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#85645C2semidirect
2. #11710729C3direct (N × H)
3. C3×#62817C3semidirect
4. C3×C22⋊(C22⋊A4)C22semidirect
12. C22×C3×#120C2×C3semidirect
13. #85645C2×C3semidirect
17. #62817C32semidirect
18. C26×C3C22×C3semidirect
19. C22⋊(C22⋊A4)C22×C3semidirect
24. C3×H(3,𝔽4)A4semidirect
53. C2×C3×#120A4semidirect
68. C22×#120C2×C32semidirect
72. C25×C3C2×A4semidirect
123. C3×#120C2×A4semidirect
124. C3×#120C2×A4semidirect
142. C26C22×C32semidirect
147. H(3,𝔽4)C3×A4semidirect
176. C2×#120C3×A4semidirect
195. C24×C3C22⋊A4semidirect
248. C25C2×C3×A4semidirect
298. #120C2×C3×A4semidirect
299. #120C2×C3×A4semidirect
374. C24C3×C22⋊A4semidirect
511. C3#11710729direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#856452
#117107293
C3×#628173
C3×#54504
C2×C3×#120⋊C34