Finite Group Databasebeta

← C3×#87024 #11909874

Subgroups — C3×#87024 #11909874

Order 1152.

2084 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C3×#5732C213yesmaximal, complemented
3C3×#5728C213yesmaximal, complemented
4C3×#5698C216yesmaximal, complemented
5C3×#5694C216yesmaximal, complemented
6C3×#5580C213yesmaximal
7C3×Q8×D12C211yesmaximal, characteristic, complemented
8C3×Q8×D12C212yesmaximal, complemented
9C3×#5572C212yesmaximal, complemented
10C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
11C3×#5570C213yesmaximal, complemented
12C3×#3523—31nomaximal
13#87024C311yesmaximal, characteristic, direct product factor
14C2×C3×D3×Q8C2211yescharacteristic
15C2×C3×D3×Q8C2212yes—
16C32×C4⋊Q8C2213yes—
17C32×#125C2216yes—
18C32×C4×Q8C2213yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2212yes—
21C3×#947C2213yes—
22C3×#946C2213yes—
23C3×Q8×Dic3C2216yes—
24C3×Q8×Dic3C2211yescharacteristic
25C3×#898C2216yes—
26C3×#898C22112yes—
27C3×#897C2216yes—
28C3×#897C2216yes—
29C3×#897C2216yes—
30C3×#895C2211yescharacteristic
31C3×#894C2216yes—
32C3×#892C2216yes—
33C3×#892C2216yes—
34C3×D3×C4⋊C4C2213yes—
35C3×#890C2213yes—
36C3×#890C22112yes—
37C3×#890C2216yes—
38C3×#887C2213yes—
39C3×#887C2216yes—
40C3×C4×D12C2216yes—
41C3×C4×D12C2213yes—
42C3×#872C2216yes—
43C3×C42×D3C2213yes—
44C3×#869C2216yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2213yes—
47C3×C4×Dic6C2216yes—
48C3×#557—21no—
49C3×#557—33no—
50C3×#557D311yescharacteristic, complemented
51C3×#557C2×C311yescharacteristic, complemented
52C3×#556—33no—
53C3×#552—36no—
54C3×D4×Q8—32no—
55C3×D4×Q8—31no—
56C3×#541—36no—
57C3×#533—33no—
58C3×#529—33no—
59C3×#519—32no—
60C2×C3×C4×Q8—31no—
61#5732C2×C313yescomplemented
62#5728C2×C313yescomplemented
63#5698C2×C316yescomplemented
64#5694C2×C316yescomplemented
65#5580C2×C313yes—
66Q8×D12C2×C312yescomplemented
67Q8×D12C2×C311yescharacteristic, complemented
68#5572C2×C312yescomplemented
69C4×D3×Q8C2×C311yescharacteristic, complemented
70#5570C2×C313yescomplemented
71C2×C32×Q8C2313yes—
72C2×C32×Q8C2311yescharacteristic
73C3×D3×Q8—24no—
74C3×D3×Q8—28no—
75C2×C3×D12C2312yes—
76C2×C3×D12C2311yescharacteristic
77C2×C3×C4×D3C2313yes—
78C2×C3×C4×D3C2313yes—
79C2×C3×C4×D3C2316yes—
80C2×C3×C4×D3C2311yescharacteristic
81C2×C3×Dic6C2313yes—
82C2×C3×Dic6C2316yes—
83C32×C4⋊C4C2313yes—
84C32×C4⋊C4C2316yes—
85C32×C4⋊C4C2311yescharacteristic
86C32×C4⋊C4C2316yes—
87C32×C4⋊C4C2316yes—
88C32×C42C2316yes—
89C32×C42C2313yes—
90C3×(C2×D3)⋊C4C2316yes—
91C3×(C2×D3)⋊C4C2316yes—
92C3×(C2×D3)⋊C4C2316yes—
93C3×(C2×D3)⋊C4C2312yes—
94C3×(C2×D3)⋊C4C2316yes—
95C3×C4⋊Dic3C2316yes—
96C3×C4⋊Dic3C2316yes—
97C3×C4⋊Dic3C23112yes—
98C3×C4⋊Dic3C2313yes—
99C3×C4⋊Dic3C2313yes—
100C3×Dic3⋊C4C2316yes—
101C3×Dic3⋊C4C2316yes—
102C3×Dic3⋊C4C2316yes—
103C3×C4×Dic3C2313yes—
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C23112yes—
106C3×C4×Dic3C2313yes—
107#3523—31noSylow
108C22×C3×Q8—31no—
109C22×C3×Q8—32no—
110C2×D3×Q8C22×C311yescharacteristic
111C2×D3×Q8C22×C312yes—
112C3×C4⋊Q8—23no—
113C3×C4⋊Q8—33no—
114C3×C4⋊Q8—36no—
115C3×C4⋊Q8C2×D313yes—
116C3×C4⋊Q8C22×C313yes—
117C3×#126—36no—
118C3×#126—312no—
119C3×#125—26no—
120C3×#125—33no—
121C3×#125—36no—
122C3×#125—312no—
123C3×#125C2×D316yes—
124C3×#125C22×C316yes—
125C3×#124—33no—
126C3×#123—36no—
127C3×C22⋊Q8—33no—
128C3×C22⋊Q8—36no—
129C3×C22⋊Q8—36no—
130C3×C22⋊Q8—36no—
131C3×#121—31no—
132C3×C4×Q8—21no—
133C3×C4×Q8—22no—
134C3×C4×Q8—23no—
135C3×C4×Q8—31no—
136C3×C4×Q8—36no—
137C3×C4×Q8—33no—
138C3×C4×Q8—36no—
139C3×C4×Q8—33no—
140C3×C4×Q8—36no—
141C3×C4×Q8C2×D311yescharacteristic
142C3×C4×Q8C2×D312yes—
143C3×C4×Q8C2×D313yes—
144C3×C4×Q8C22×C313yes—
145C3×C4×Q8C22×C311yescharacteristic
146C3×C4×Q8C22×C312yes—
147C3×C4×D4—36no—
148C3×C4×D4—33no—
149C3×#117—36no—
150C3×#117—36no—
151C3×#117—36no—
152C2×C3×C4⋊C4—33no—
153C2×C3×C42—33no—
154#947C22×C313yes—
155#946C22×C313yes—
156Q8×Dic3C22×C316yes—
157Q8×Dic3C22×C311yescharacteristic
158#898C22×C3112yes—
159#898C22×C316yes—
160#897C22×C316yes—
161#897C22×C316yes—
162#897C22×C316yes—
163#895C22×C311yescharacteristic
164#894C22×C316yes—
165#892C22×C316yes—
166#892C22×C316yes—
167D3×C4⋊C4C22×C313yes—
168#890C22×C3112yes—
169#890C22×C316yes—
170#890C22×C313yes—
171#887C22×C313yes—
172#887C22×C316yes—
173C4×D12C22×C316yes—
174C4×D12C22×C313yes—
175#872C22×C316yes—
176C42×D3C22×C313yes—
177#869C22×C316yes—
178#869C22×C313yes—
179C4×Dic6C22×C316yes—
180C4×Dic6C22×C313yes—
181C22×C3×D3C2412yes—
182C22×C3×D3C2411yescharacteristic
183C32×Q8—26no—
184C32×Q8#4114yes—
185C2×C32×C4C2416yes—
186C2×C32×C4C2413yes—
187C2×C32×C4C2412yes—
188C2×C32×C4C2413yes—
189C2×C32×C4C2411yescharacteristic
190C2×C3×Dic3C2411yescharacteristic
191C2×C3×Dic3C2413yes—
192C2×C3×Dic3C2413yes—
193C2×C3×Dic3C2416yes—
194C3×D12—24no—
195C3×D12—22no—
196C3×C4×D3—26no—
197C3×C4×D3—26no—
198C3×C4×D3—212no—
199C3×C4×D3—22no—
200C3×Dic6—26no—
201C3×Dic6—212no—
202#557—33no—
203#557C3×D311yescharacteristic, complemented
204#556—33no—
205#552—36no—
206D4×Q8—32no—
207D4×Q8—31no—
208#541—36no—
209#533—33no—
210#529—33no—
211#519—32no—
212C2×C4×Q8—31no—
213C2×C3×Q8—21no—
214C2×C3×Q8—23no—
215C2×C3×Q8—33no—
216C2×C3×Q8—36no—
217C2×C3×Q8—64no—
218C2×C3×Q8—68no—
219C2×C3×Q8C22×D313yes—
220C2×C3×Q8C22×D311yescharacteristic
221C2×C3×Q8C23×C313yes—
222C2×C3×Q8C23×C311yescharacteristic
223C2×C3×D4—32no—
224C2×C3×D4—31no—
225C22×C3×C4—31no—
226C22×C3×C4—33no—
227C22×C3×C4—33no—
228C22×C3×C4—36no—
229D3×Q8—24no—
230D3×Q8—28no—
231C2×D12C23×C312yes—
232C2×D12C23×C311yescharacteristic
233C2×C4×D3C23×C311yescharacteristic
234C2×C4×D3C23×C313yes—
235C2×C4×D3C23×C313yes—
236C2×C4×D3C23×C316yes—
237C2×Dic6C23×C316yes—
238C2×Dic6C23×C313yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—23no—
241C3×C4⋊C4—26no—
242C3×C4⋊C4—26no—
243C3×C4⋊C4—26no—
244C3×C4⋊C4—36no—
245C3×C4⋊C4—36no—
246C3×C4⋊C4—36no—
247C3×C4⋊C4—33no—
248C3×C4⋊C4—36no—
249C3×C4⋊C4—33no—
250C3×C4⋊C4—36no—
251C3×C4⋊C4—312no—
252C3×C4⋊C4C22×D313yes—
253C3×C4⋊C4C22×D316yes—
254C3×C4⋊C4C22×D316yes—
255C3×C4⋊C4C22×D311yescharacteristic
256C3×C4⋊C4C22×D316yes—
257C3×C4⋊C4C23×C313yes—
258C3×C4⋊C4C23×C316yes—
259C3×C4⋊C4C23×C311yescharacteristic
260C3×C4⋊C4C23×C316yes—
261C3×C4⋊C4C23×C316yes—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—36no—
264C3×C22⋊C4—36no—
265C3×C22⋊C4—36no—
266C3×C22⋊C4—36no—
267C3×C42—26no—
268C3×C42—23no—
269C3×C42—33no—
270C3×C42—33no—
271C3×C42—33no—
272C3×C42—312no—
273C3×C42C22×D313yes—
274C3×C42C22×D316yes—
275C3×C42C23×C313yes—
276C3×C42C23×C316yes—
277(C2×D3)⋊C4C23×C312yes—
278(C2×D3)⋊C4C23×C316yes—
279(C2×D3)⋊C4C23×C316yes—
280(C2×D3)⋊C4C23×C316yes—
281(C2×D3)⋊C4C23×C316yes—
282C4⋊Dic3C23×C313yes—
283C4⋊Dic3C23×C316yes—
284C4⋊Dic3C23×C313yes—
285C4⋊Dic3C23×C316yes—
286C4⋊Dic3C23×C3112yes—
287Dic3⋊C4C23×C316yes—
288Dic3⋊C4C23×C316yes—
289Dic3⋊C4C23×C316yes—
290C4×Dic3C23×C313yes—
291C4×Dic3C23×C3112yes—
292C4×Dic3C23×C313yes—
293C4×Dic3C23×C313yes—
294C22×C32C2511yescharacteristic
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—22no—
298C2×C3×D3—22no—
299C2×C3×D3—22no—
300C2×C3×D32−1+412yes—
301C32×C4—23no—
302C32×C4—22no—
303C32×C4—26no—
304C32×C4C2×#4116yes—
305C32×C42−1+412yes—
306C3×Dic3—23no—
307C3×Dic3—26no—
308C3×Dic3—21no—
309C3×Dic3—23no—
310C22×Q8—31no—
311C22×Q8—32no—
312C4⋊Q8—33no—
313C4⋊Q8—36no—
314C4⋊Q8C2×C3×D313yes—
315#126—36no—
316#126—312no—
317#125—33no—
318#125—36no—
319#125—312no—
320#125C2×C3×D316yes—
321#124—33no—
322#123—36no—
323C22⋊Q8—33no—
324C22⋊Q8—36no—
325C22⋊Q8—36no—
326C22⋊Q8—36no—
327#121—31no—
328C4×Q8—31no—
329C4×Q8—36no—
330C4×Q8—33no—
331C4×Q8—36no—
332C4×Q8—33no—
333C4×Q8—36no—
334C4×Q8C2×C3×D311yescharacteristic
335C4×Q8C2×C3×D312yes—
336C4×Q8C2×C3×D313yes—
337C4×D4—36no—
338C4×D4—33no—
339#117—36no—
340#117—36no—
341#117—36no—
342C2×C4⋊C4—33no—
343C2×C42—33no—
344C23×C3—31no—
345C23×C3—32no—
346C22×D3C24×C312yes—
347C22×D3C24×C311yescharacteristic
348C3×Q8—26no—
349C3×Q8—26no—
350C3×Q8—24no—
351C3×Q8—46no—
352C3×Q8—66no—
353C3×Q8—612no—
354C3×Q8#23914yes—
355C3×Q8C3×#4114yes—
356C3×D4—64no—
357C3×D4—62no—
358C2×C3×C4—23no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—23no—
362C2×C3×C4—26no—
363C2×C3×C4—31no—
364C2×C3×C4—33no—
365C2×C3×C4—33no—
366C2×C3×C4—36no—
367C2×C3×C4—66no—
368C2×C3×C4—66no—
369C2×C3×C4—612no—
370C2×C3×C4—62no—
371C2×C3×C4C23×D316yes—
372C2×C3×C4C23×D313yes—
373C2×C3×C4C23×D312yes—
374C2×C3×C4C23×D313yes—
375C2×C3×C4C23×D311yescharacteristic
376C2×C3×C4C24×C316yes—
377C2×C3×C4C24×C313yes—
378C2×C3×C4C24×C312yes—
379C2×C3×C4C24×C313yes—
380C2×C3×C4C24×C311yescharacteristic
381C2×Dic3C24×C311yescharacteristic
382C2×Dic3C24×C313yes—
383C2×Dic3C24×C313yes—
384C2×Dic3C24×C316yes—
385D12—24no—
386D12—22no—
387C4×D3—26no—
388C4×D3—26no—
389C4×D3—212no—
390C4×D3—22no—
391Dic6—212no—
392Dic6—26no—
393C2×C32C22×#4111yescharacteristic
394C2×C32C2×2−1+411yescharacteristic
395C2×C32C2×2−1+411yescharacteristic
396C3×D3—22no—
397C3×D3—42no—
398C2×Q8—33no—
399C2×Q8—36no—
400C2×Q8—64no—
401C2×Q8—68no—
402C2×Q8C22×C3×D313yes—
403C2×Q8C22×C3×D311yescharacteristic
404C2×D4—32no—
405C2×D4—31no—
406C22×C4—31no—
407C22×C4—33no—
408C22×C4—33no—
409C22×C4—36no—
410C4⋊C4—33no—
411C4⋊C4—36no—
412C4⋊C4—36no—
413C4⋊C4—36no—
414C4⋊C4—36no—
415C4⋊C4—33no—
416C4⋊C4—36no—
417C4⋊C4—312no—
418C4⋊C4C22×C3×D313yes—
419C4⋊C4C22×C3×D316yes—
420C4⋊C4C22×C3×D316yes—
421C4⋊C4C22×C3×D311yescharacteristic
422C4⋊C4C22×C3×D316yes—
423C22⋊C4—32no—
424C22⋊C4—36no—
425C22⋊C4—36no—
426C22⋊C4—36no—
427C22⋊C4—36no—
428C42—33no—
429C42—33no—
430C42—33no—
431C42—312no—
432C42C22×C3×D313yes—
433C42C22×C3×D316yes—
434C22×C3—21no—
435C22×C3—32no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—62no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3C24×D311yescharacteristic
442C22×C3C25×C311yescharacteristic
443C2×D3—21no—
444C2×D3—21no—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—22no—
448C2×D3C3×2−1+412yes—
449C3×C4—26no—
450C3×C4—23no—
451C3×C4—22no—
452C3×C4—23no—
453C3×C4—22no—
454C3×C4—26no—
455C3×C4—26no—
456C3×C4—22no—
457C3×C4—46no—
458C3×C4—43no—
459C3×C4—42no—
460C3×C4—63no—
461C3×C4—66no—
462C3×C4—61no—
463C3×C4—63no—
464C3×C4C2×#23916yes—
465C3×C4#101012yes—
466C3×C4C2×C3×#4116yes—
467C3×C4C3×2−1+412yes—
468Dic3—23no—
469Dic3—26no—
470Dic3—21no—
471Dic3—23no—
472C32#352311yescharacteristic, complemented, Sylow
473C23—31no—
474C23—32no—
475Q8—26no—
476Q8—66no—
477Q8—612no—
478Q8C3×#23914yes—
479D4—64no—
480D4—62no—
481C2×C4—31no—
482C2×C4—33no—
483C2×C4—33no—
484C2×C4—36no—
485C2×C4—66no—
486C2×C4—66no—
487C2×C4—612no—
488C2×C4—62no—
489C2×C4C23×C3×D316yes—
490C2×C4C23×C3×D313yes—
491C2×C4C23×C3×D312yes—
492C2×C4C23×C3×D313yes—
493C2×C4C23×C3×D311yescharacteristic
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—62no—
498C2×C3—122no—
499C2×C3C22×#23911yescharacteristic
500C2×C3C2×#101011yescharacteristic
501C2×C3D3×2−1+411yescharacteristic
502C2×C3C22×C3×#4111yescharacteristic
503C2×C3C2×C3×2−1+411yescharacteristic
504C2×C3C2×C3×2−1+411yescharacteristic
505D3—22no—
506D3—42no—
507C22—32no—
508C22—61no—
509C22—61no—
510C22—62no—
511C22—62no—
512C22—62no—
513C22C24×C3×D311yescharacteristic
514C4—26no—
515C4—23no—
516C4—22no—
517C4—63no—
518C4—66no—
519C4—61no—
520C4—63no—
521C4C2×C3×#23916yes—
522C4C3×#101012yes—
523C3—21no—
524C3#8702411yescharacteristic, direct product factor
525C3C3×#352311yescharacteristic, complemented
526C2—62no—
527C2—122no—
528C2C22×C3×#23911yescharacteristic
529C2C2×C3×#101011yescharacteristic
530C2C3×D3×2−1+411yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#557C2
socleC22×C32C25
prime radicalC3×#895C22
hypercenterC3×#557D3
hypocenterC3C3×#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
2. C3×#5732C2semidirect
3. C3×#5728C2semidirect
4. C3×#5698C2semidirect
5. C3×#5694C2semidirect
7. C3×Q8×D12C2semidirect
8. C3×Q8×D12C2semidirect
9. C3×#5572C2semidirect
10. C3×C4×D3×Q8C2semidirect
11. C3×#5570C2semidirect
13. #87024C3direct (N × H)
50. C3×#557D3semidirect
51. C3×#557C2×C3semidirect
61. #5732C2×C3semidirect
62. #5728C2×C3semidirect
63. #5698C2×C3semidirect
64. #5694C2×C3semidirect
66. Q8×D12C2×C3semidirect
67. Q8×D12C2×C3semidirect
68. #5572C2×C3semidirect
69. C4×D3×Q8C2×C3semidirect
70. #5570C2×C3semidirect
203. #557C3×D3semidirect
472. C32#3523semidirect
524. C3#87024direct (N × H)
525. C3C3×#3523semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
C3×#57322
C3×#57282
C3×#56982
C3×#56942
C3×#55802
C3×Q8×D122
C3×#55722
C3×C4×D3×Q82
C3×#55702
C3×#35233
#870243