Finite Group Databasebeta

← C3×#87046 #11906265

Subgroups — C3×#87046 #11906265

Order 1152.

2034 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q82C211yesmaximal, characteristic, complemented
2C3×#5729C216yesmaximal
3C3×#5728C213yesmaximal
4C3×#5722C213yesmaximal, complemented
5C3×#5721C216yesmaximal, complemented
6C3×#5578C213yesmaximal
7C3×#5572C212yesmaximal
8C3×C4×D3×Q8C211yesmaximal, characteristic
9C3×#5570C211yesmaximal, characteristic, complemented
10C3×#5569C212yesmaximal, complemented
11C3×#5567C213yesmaximal, complemented
12C3×#3502—31nomaximal
13#87046C311yesmaximal, characteristic, direct product factor
14C2×C3×D3×Q8C2211yescharacteristic
15C32×C4⋊Q8C2216yes—
16C32×C4⋊Q8C2213yes—
17C32×C4×Q8C2213yes—
18C32×C4×Q8C2212yes—
19C32×C4×Q8C2211yescharacteristic
20C3×#946C2211yescharacteristic
21C3×#946C2216yes—
22C3×Q8×Dic3C2211yescharacteristic
23C3×Q8×Dic3C2212yes—
24C3×Q8×Dic3C2216yes—
25C3×#944C2213yes—
26C3×#944C22112yes—
27C3×#897C2213yes—
28C3×#896C2216yes—
29C3×#892C2216yes—
30C3×D3×C4⋊C4C2213yes—
31C3×#890C2216yes—
32C3×#890C2216yes—
33C3×#889C22112yes—
34C3×#889C2216yes—
35C3×#888C2216yes—
36C3×#888C2213yes—
37C3×#887C2216yes—
38C3×#887C2213yes—
39C3×#887C2216yes—
40C3×#872C2216yes—
41C3×C42×D3C2213yes—
42C3×#870C2216yes—
43C3×#869C2213yes—
44C3×C4×Dic6C2213yes—
45C3×C4×Dic6C2216yes—
46C3×C4×Dic6C2216yes—
47C3×C4×Dic6C2213yes—
48C3×Q82—21no—
49C3×Q82—33no—
50C3×Q82D311yescharacteristic, complemented
51C3×Q82C2×C311yescharacteristic, complemented
52C3×#557—36no—
53C3×#557—33no—
54C3×#557—32no—
55C3×#557—31no—
56C3×#544—36no—
57C3×#533—33no—
58C3×#533—33no—
59C3×#519—32no—
60C2×C3×C4×Q8—31no—
61#5729C2×C316yes—
62#5728C2×C313yes—
63#5722C2×C313yescomplemented
64#5721C2×C316yescomplemented
65#5578C2×C313yes—
66#5572C2×C312yes—
67C4×D3×Q8C2×C311yescharacteristic
68#5570C2×C311yescharacteristic, complemented
69#5569C2×C312yescomplemented
70#5567C2×C313yescomplemented
71C2×C32×Q8C2311yescharacteristic
72C2×C32×Q8C2316yes—
73C2×C32×Q8C2311yescharacteristic
74C3×D3×Q8Q818yescomplemented
75C2×C3×C4×D3C2311yescharacteristic
76C2×C3×C4×D3C2313yes—
77C2×C3×C4×D3C2313yes—
78C2×C3×Dic6C2313yes—
79C2×C3×Dic6C2316yes—
80C2×C3×Dic6C2313yes—
81C32×C4⋊C4C2313yes—
82C32×C4⋊C4C2316yes—
83C32×C4⋊C4C2313yes—
84C32×C4⋊C4C2316yes—
85C32×C42C2313yes—
86C32×C42C2316yes—
87C3×(C2×D3)⋊C4C2316yes—
88C3×(C2×D3)⋊C4C2312yes—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×C4⋊Dic3C2313yes—
91C3×C4⋊Dic3C2316yes—
92C3×C4⋊Dic3C2316yes—
93C3×C4⋊Dic3C2316yes—
94C3×Dic3⋊C4C2316yes—
95C3×Dic3⋊C4C23112yes—
96C3×Dic3⋊C4C2316yes—
97C3×Dic3⋊C4C2316yes—
98C3×Dic3⋊C4C2312yes—
99C3×Dic3⋊C4C2316yes—
100C3×Dic3⋊C4C2316yes—
101C3×C4×Dic3C2313yes—
102C3×C4×Dic3C2313yes—
103C3×C4×Dic3C23112yes—
104C3×C4×Dic3C2316yes—
105C3×C4×Dic3C2313yes—
106#3502—31noSylow
107C22×C3×Q8—31no—
108C2×D3×Q8C22×C311yescharacteristic
109C3×C4⋊Q8—23no—
110C3×C4⋊Q8—26no—
111C3×C4⋊Q8—312no—
112C3×C4⋊Q8—33no—
113C3×C4⋊Q8—33no—
114C3×C4⋊Q8—36no—
115C3×C4⋊Q8—33no—
116C3×C4⋊Q8C2×D313yes—
117C3×C4⋊Q8C2×D316yes—
118C3×C4⋊Q8C22×C316yes—
119C3×C4⋊Q8C22×C313yes—
120C3×#125—36no—
121C3×#125—312no—
122C3×#125—36no—
123C3×#125—36no—
124C3×#125—36no—
125C3×C22⋊Q8—31no—
126C3×C22⋊Q8—36no—
127C3×C22⋊Q8—36no—
128C3×C22⋊Q8—33no—
129C3×C4×Q8—21no—
130C3×C4×Q8—22no—
131C3×C4×Q8—23no—
132C3×C4×Q8—31no—
133C3×C4×Q8—32no—
134C3×C4×Q8—36no—
135C3×C4×Q8—36no—
136C3×C4×Q8—36no—
137C3×C4×Q8—33no—
138C3×C4×Q8—33no—
139C3×C4×Q8—33no—
140C3×C4×Q8—36no—
141C3×C4×Q8—36no—
142C3×C4×Q8C2×D311yescharacteristic
143C3×C4×Q8C2×D312yes—
144C3×C4×Q8C2×D313yes—
145C3×C4×Q8C22×C313yes—
146C3×C4×Q8C22×C312yes—
147C3×C4×Q8C22×C311yescharacteristic
148C3×#117—36no—
149C3×#117—36no—
150C2×C3×C4⋊C4—33no—
151C2×C3×C42—33no—
152#946C22×C316yes—
153#946C22×C311yescharacteristic
154Q8×Dic3C22×C316yes—
155Q8×Dic3C22×C311yescharacteristic
156Q8×Dic3C22×C312yes—
157#944C22×C3112yes—
158#944C22×C313yes—
159#897C22×C313yes—
160#896C22×C316yes—
161#892C22×C316yes—
162D3×C4⋊C4C22×C313yes—
163#890C22×C316yes—
164#890C22×C316yes—
165#889C22×C316yes—
166#889C22×C3112yes—
167#888C22×C313yes—
168#888C22×C316yes—
169#887C22×C313yes—
170#887C22×C316yes—
171#887C22×C316yes—
172#872C22×C316yes—
173C42×D3C22×C313yes—
174#870C22×C316yes—
175#869C22×C313yes—
176C4×Dic6C22×C316yes—
177C4×Dic6C22×C316yes—
178C4×Dic6C22×C313yes—
179C4×Dic6C22×C313yes—
180C22×C3×D3C2411yescharacteristic
181C32×Q8—22no—
182C32×Q8—46no—
183C32×Q8C2×Q814yes—
184C2×C32×C4C2416yes—
185C2×C32×C4C2413yes—
186C2×C32×C4C2412yes—
187C2×C32×C4C2413yes—
188C2×C32×C4C2411yescharacteristic
189C2×C3×Dic3C2411yescharacteristic
190C2×C3×Dic3C2412yes—
191C2×C3×Dic3C2416yes—
192C2×C3×Dic3C2413yes—
193C2×C3×Dic3C2413yes—
194C3×C4×D3—22no—
195C3×C4×D3—43no—
196C3×C4×D3C2×Q8112yes—
197C3×Dic6—26no—
198C3×Dic6—46no—
199C3×Dic6C2×Q8112yes—
200Q82—33no—
201Q82C3×D311yescharacteristic, complemented
202#557—36no—
203#557—33no—
204#557—32no—
205#557—31no—
206#544—36no—
207#533—33no—
208#533—33no—
209#519—32no—
210C2×C4×Q8—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—26no—
214C2×C3×Q8—38no—
215C2×C3×Q8—33no—
216C2×C3×Q8—33no—
217C2×C3×Q8—36no—
218C2×C3×Q8C22×D311yescharacteristic
219C2×C3×Q8C22×D316yes—
220C2×C3×Q8C22×D311yescharacteristic
221C2×C3×Q8C23×C311yescharacteristic
222C2×C3×Q8C23×C316yes—
223C2×C3×Q8C23×C311yescharacteristic
224C22×C3×C4—33no—
225C22×C3×C4—31no—
226C22×C3×C4—33no—
227D3×Q8C3×Q818yescomplemented
228C2×C4×D3C23×C311yescharacteristic
229C2×C4×D3C23×C313yes—
230C2×C4×D3C23×C313yes—
231C2×Dic6C23×C313yes—
232C2×Dic6C23×C316yes—
233C2×Dic6C23×C313yes—
234C3×C4⋊C4—23no—
235C3×C4⋊C4—26no—
236C3×C4⋊C4—23no—
237C3×C4⋊C4—26no—
238C3×C4⋊C4—36no—
239C3×C4⋊C4—36no—
240C3×C4⋊C4—36no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—36no—
243C3×C4⋊C4—36no—
244C3×C4⋊C4—312no—
245C3×C4⋊C4—33no—
246C3×C4⋊C4—36no—
247C3×C4⋊C4—36no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—36no—
250C3×C4⋊C4C22×D313yes—
251C3×C4⋊C4C22×D316yes—
252C3×C4⋊C4C22×D313yes—
253C3×C4⋊C4C22×D316yes—
254C3×C4⋊C4C23×C313yes—
255C3×C4⋊C4C23×C316yes—
256C3×C4⋊C4C23×C313yes—
257C3×C4⋊C4C23×C316yes—
258C3×C22⋊C4—36no—
259C3×C22⋊C4—32no—
260C3×C42—23no—
261C3×C42—26no—
262C3×C42—33no—
263C3×C42—33no—
264C3×C42—33no—
265C3×C42—312no—
266C3×C42—36no—
267C3×C42C22×D313yes—
268C3×C42C22×D316yes—
269C3×C42C23×C313yes—
270C3×C42C23×C316yes—
271(C2×D3)⋊C4C23×C312yes—
272(C2×D3)⋊C4C23×C316yes—
273C4⋊Dic3C23×C311yescharacteristic
274C4⋊Dic3C23×C316yes—
275C4⋊Dic3C23×C313yes—
276C4⋊Dic3C23×C316yes—
277C4⋊Dic3C23×C316yes—
278Dic3⋊C4C23×C312yes—
279Dic3⋊C4C23×C316yes—
280Dic3⋊C4C23×C316yes—
281Dic3⋊C4C23×C316yes—
282Dic3⋊C4C23×C316yes—
283Dic3⋊C4C23×C3112yes—
284Dic3⋊C4C23×C316yes—
285C4×Dic3C23×C313yes—
286C4×Dic3C23×C313yes—
287C4×Dic3C23×C316yes—
288C4×Dic3C23×C3112yes—
289C4×Dic3C23×C313yes—
290C22×C32C2511yescharacteristic
291C2×C3×D3—21no—
292C2×C3×D3—21no—
293C2×C3×D3C22×Q812yes—
294C32×C4—22no—
295C32×C4—26no—
296C32×C4—23no—
297C32×C4C22×Q816yes—
298C32×C42−1+412yes—
299C3×Dic3—26no—
300C3×Dic3—21no—
301C3×Dic3—22no—
302C3×Dic3—23no—
303C3×Dic3C22×Q816yes—
304C22×Q8—31no—
305C4⋊Q8—33no—
306C4⋊Q8—312no—
307C4⋊Q8—33no—
308C4⋊Q8—36no—
309C4⋊Q8—33no—
310C4⋊Q8C2×C3×D313yes—
311C4⋊Q8C2×C3×D316yes—
312#125—36no—
313#125—36no—
314#125—312no—
315#125—36no—
316#125—36no—
317C22⋊Q8—31no—
318C22⋊Q8—36no—
319C22⋊Q8—33no—
320C22⋊Q8—36no—
321C4×Q8—31no—
322C4×Q8—32no—
323C4×Q8—36no—
324C4×Q8—33no—
325C4×Q8—36no—
326C4×Q8—36no—
327C4×Q8—33no—
328C4×Q8—36no—
329C4×Q8—36no—
330C4×Q8—33no—
331C4×Q8C2×C3×D311yescharacteristic
332C4×Q8C2×C3×D312yes—
333C4×Q8C2×C3×D313yes—
334#117—36no—
335#117—36no—
336C2×C4⋊C4—33no—
337C2×C42—33no—
338C23×C3—31no—
339C22×D3C24×C311yescharacteristic
340C3×Q8—24no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—24no—
344C3×Q8—312no—
345C3×Q8—46no—
346C3×Q8—46no—
347C3×Q8—66no—
348C3×Q8—86no—
349C3×Q8—126no—
350C3×Q8D3×Q814yes—
351C3×Q8C2×C3×Q814yes—
352C2×C3×C4—22no—
353C2×C3×C4—23no—
354C2×C3×C4—21no—
355C2×C3×C4—26no—
356C2×C3×C4—23no—
357C2×C3×C4—33no—
358C2×C3×C4—312no—
359C2×C3×C4—31no—
360C2×C3×C4—36no—
361C2×C3×C4—32no—
362C2×C3×C4—33no—
363C2×C3×C4—62no—
364C2×C3×C4—123no—
365C2×C3×C4C23×D316yes—
366C2×C3×C4C23×D313yes—
367C2×C3×C4C23×D312yes—
368C2×C3×C4C23×D313yes—
369C2×C3×C4C23×D311yescharacteristic
370C2×C3×C4C24×C316yes—
371C2×C3×C4C24×C313yes—
372C2×C3×C4C24×C312yes—
373C2×C3×C4C24×C313yes—
374C2×C3×C4C24×C311yescharacteristic
375C2×Dic3C24×C311yescharacteristic
376C2×Dic3C24×C312yes—
377C2×Dic3C24×C316yes—
378C2×Dic3C24×C313yes—
379C2×Dic3C24×C313yes—
380C4×D3—22no—
381C4×D3—43no—
382C4×D3C2×C3×Q8112yes—
383Dic6—26no—
384Dic6—46no—
385Dic6C2×C3×Q8112yes—
386C2×C32C23×Q811yescharacteristic
387C2×C32C2×2−1+411yescharacteristic
388C2×C32#58511yescharacteristic
389C3×D3—22no—
390C2×Q8—38no—
391C2×Q8—33no—
392C2×Q8—33no—
393C2×Q8—36no—
394C2×Q8C22×C3×D311yescharacteristic
395C2×Q8C22×C3×D316yes—
396C2×Q8C22×C3×D311yescharacteristic
397C22×C4—33no—
398C22×C4—31no—
399C22×C4—33no—
400C4⋊C4—36no—
401C4⋊C4—33no—
402C4⋊C4—36no—
403C4⋊C4—36no—
404C4⋊C4—36no—
405C4⋊C4—32no—
406C4⋊C4—36no—
407C4⋊C4—31no—
408C4⋊C4—36no—
409C4⋊C4—36no—
410C4⋊C4—36no—
411C4⋊C4—312no—
412C4⋊C4C22×C3×D313yes—
413C4⋊C4C22×C3×D316yes—
414C4⋊C4C22×C3×D313yes—
415C4⋊C4C22×C3×D316yes—
416C22⋊C4—36no—
417C22⋊C4—32no—
418C42—33no—
419C42—33no—
420C42—33no—
421C42—312no—
422C42—36no—
423C42C22×C3×D313yes—
424C42C22×C3×D316yes—
425C22×C3—21no—
426C22×C3—32no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3C24×D311yescharacteristic
430C22×C3C25×C311yescharacteristic
431C2×D3—21no—
432C2×D3—21no—
433C2×D3C22×C3×Q812yes—
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—23no—
437C3×C4—26no—
438C3×C4—22no—
439C3×C4—26no—
440C3×C4—23no—
441C3×C4—26no—
442C3×C4—22no—
443C3×C4—36no—
444C3×C4—43no—
445C3×C4—46no—
446C3×C4—61no—
447C3×C4—66no—
448C3×C4—63no—
449C3×C4—62no—
450C3×C4C2×D3×Q816yes—
451C3×C4#100712yes—
452C3×C4C22×C3×Q816yes—
453C3×C4C3×2−1+412yes—
454Dic3—26no—
455Dic3—21no—
456Dic3—22no—
457Dic3—23no—
458Dic3C22×C3×Q816yes—
459C32#350211yescharacteristic, complemented, Sylow
460C23—31no—
461Q8—22no—
462Q8—312no—
463Q8—46no—
464Q8—66no—
465Q8—126no—
466Q8C3×D3×Q814yes—
467C2×C4—33no—
468C2×C4—312no—
469C2×C4—31no—
470C2×C4—36no—
471C2×C4—32no—
472C2×C4—33no—
473C2×C4—62no—
474C2×C4—123no—
475C2×C4C23×C3×D316yes—
476C2×C4C23×C3×D313yes—
477C2×C4C23×C3×D312yes—
478C2×C4C23×C3×D313yes—
479C2×C4C23×C3×D311yescharacteristic
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—62no—
484C2×C3C22×D3×Q811yescharacteristic
485C2×C3C2×#100711yescharacteristic
486C2×C3#596611yescharacteristic
487C2×C3C23×C3×Q811yescharacteristic
488C2×C3C2×C3×2−1+411yescharacteristic
489C2×C3C3×#58511yescharacteristic
490D3—22no—
491C22—32no—
492C22—61no—
493C22—61no—
494C22C24×C3×D311yescharacteristic
495C4—22no—
496C4—23no—
497C4—26no—
498C4—36no—
499C4—61no—
500C4—66no—
501C4—63no—
502C4—62no—
503C4C2×C3×D3×Q816yes—
504C4C3×#100712yes—
505C3—21no—
506C3#8704611yescharacteristic, direct product factor
507C3C3×#350211yescharacteristic, complemented
508C2—62no—
509C2C22×C3×D3×Q811yescharacteristic
510C2C2×C3×#100711yescharacteristic
511C2C3×#596611yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×Q82C2
socleC22×C32C25
prime radicalC22×C3×D3C24
hypercenterC3×Q82D3
hypocenterC3C3×#3502
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×Q82

Complements

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

NH ≅ G/Ntype
1. C32×Q82C2semidirect
4. C3×#5722C2semidirect
5. C3×#5721C2semidirect
9. C3×#5570C2semidirect
10. C3×#5569C2semidirect
11. C3×#5567C2semidirect
13. #87046C3direct (N × H)
50. C3×Q82D3semidirect
51. C3×Q82C2×C3semidirect
63. #5722C2×C3semidirect
64. #5721C2×C3semidirect
68. #5570C2×C3semidirect
69. #5569C2×C3semidirect
70. #5567C2×C3semidirect
74. C3×D3×Q8Q8semidirect
201. Q82C3×D3semidirect
227. D3×Q8C3×Q8semidirect
459. C32#3502semidirect
506. C3#87046direct (N × H)
507. C3C3×#3502semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
C3×#57292
C3×#57282
C3×#57222
C3×#57212
C3×#55782
C3×#55722
C3×C4×D3×Q82
C3×#55702
C3×#55692
C3×#55672
C3×#35023
#870463