Finite Group Databasebeta

← C3×#87041 #11906333

Subgroups — C3×#87041 #11906333

Order 1152.

2226 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q82C211yesmaximal, characteristic, complemented
2C3×#5730C213yesmaximal, complemented
3C3×#5729C216yesmaximal, complemented
4C3×#5727C216yesmaximal, complemented
5C3×D3×C4⋊Q8C213yesmaximal, complemented
6C3×#5579C213yesmaximal, complemented
7C3×#5578C213yesmaximal, complemented
8C3×#5574C212yesmaximal
9C3×C4×#239C211yesmaximal, characteristic
10C3×#5569C212yesmaximal, complemented
11C3×Q8×Dic6C211yesmaximal, characteristic, complemented
12C3×#3503—31nomaximal
13#87041C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2211yescharacteristic
15C2×C3×D3×Q8C2216yes—
16C32×C4⋊Q8C2213yes—
17C32×C4⋊Q8C2216yes—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×Q8C2212yes—
20C32×C4×Q8C2213yes—
21C3×#946C2213yes—
22C3×#946C22112yes—
23C3×Q8×Dic3C2212yes—
24C3×Q8×Dic3C2211yescharacteristic
25C3×#944C2211yescharacteristic
26C3×#944C2216yes—
27C3×#897C2213yes—
28C3×#896C2216yes—
29C3×#896C22112yes—
30C3×#893C2213yes—
31C3×#893C2216yes—
32C3×#892C2213yes—
33C3×D3×C4⋊C4C2216yes—
34C3×D3×C4⋊C4C2216yes—
35C3×#890C2216yes—
36C3×#889C2216yes—
37C3×#888C2216yes—
38C3×#888C2216yes—
39C3×#888C2213yes—
40C3×C4×D12C2213yes—
41C3×C4×D12C2216yes—
42C3×#872C2216yes—
43C3×C42×D3C2213yes—
44C3×#870C2216yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2213yes—
47C3×C4×Dic6C2216yes—
48C3×Q82—21no—
49C3×Q82—31no—
50C3×Q82D311yescharacteristic, complemented
51C3×Q82C2×C311yescharacteristic, complemented
52C3×#557—32no—
53C3×#554—33no—
54C3×#554—33no—
55C3×D4×Q8—36no—
56C3×#544—36no—
57C3×#533—33no—
58C2×C3×C4⋊Q8—33no—
59C3×#520—32no—
60C3×C4×#41—31no—
61#5730C2×C313yescomplemented
62#5729C2×C316yescomplemented
63#5727C2×C316yescomplemented
64D3×C4⋊Q8C2×C313yescomplemented
65#5579C2×C313yescomplemented
66#5578C2×C313yescomplemented
67#5574C2×C312yes—
68C4×#239C2×C311yescharacteristic
69#5569C2×C312yescomplemented
70Q8×Dic6C2×C311yescharacteristic, complemented
71C2×C32×Q8C2311yescharacteristic
72C2×C32×Q8C2316yes—
73C2×C32×Q8C2311yescharacteristic
74C3×#239Q818yescomplemented
75C3×D3×Q8—412no—
76C2×C3×D12C2313yes—
77C2×C3×C4×D3C2313yes—
78C2×C3×C4×D3C2313yes—
79C2×C3×C4×D3C23112yes—
80C2×C3×C4×D3C2313yes—
81C2×C3×Dic6C2316yes—
82C2×C3×Dic6C2311yescharacteristic
83C2×C3×Dic6C2316yes—
84C32×C4⋊C4C2313yes—
85C32×C4⋊C4C2316yes—
86C32×C4⋊C4C2313yes—
87C32×C4⋊C4C2316yes—
88C32×C42C2313yes—
89C32×C42C2316yes—
90C3×(C2×D3)⋊C4C2316yes—
91C3×(C2×D3)⋊C4C2316yes—
92C3×(C2×D3)⋊C4C2316yes—
93C3×C4⋊Dic3C2316yes—
94C3×C4⋊Dic3C2313yes—
95C3×C4⋊Dic3C2316yes—
96C3×C4⋊Dic3C2313yes—
97C3×Dic3⋊C4C2316yes—
98C3×Dic3⋊C4C2316yes—
99C3×Dic3⋊C4C23112yes—
100C3×Dic3⋊C4C2312yes—
101C3×Dic3⋊C4C2316yes—
102C3×Dic3⋊C4C2312yes—
103C3×C4×Dic3C2311yescharacteristic
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C2316yes—
106C3×C4×Dic3C2313yes—
107#3503—31noSylow
108C2×C3×#41—31no—
109C22×C3×Q8—36no—
110C2×#239C22×C311yescharacteristic
111C2×D3×Q8C22×C316yes—
112C3×C4⋊Q8—23no—
113C3×C4⋊Q8—26no—
114C3×C4⋊Q8—36no—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—33no—
118C3×C4⋊Q8—36no—
119C3×C4⋊Q8—33no—
120C3×C4⋊Q8C2×D313yes—
121C3×C4⋊Q8C2×D316yes—
122C3×C4⋊Q8C22×C313yes—
123C3×C4⋊Q8C22×C316yes—
124C3×#125—36no—
125C3×#125—36no—
126C3×#125—36no—
127C3×C22⋊Q8—33no—
128C3×C22⋊Q8—312no—
129C3×C22⋊Q8—36no—
130C3×C22⋊Q8—312no—
131C3×C22⋊Q8—33no—
132C3×C4×Q8—21no—
133C3×C4×Q8—22no—
134C3×C4×Q8—23no—
135C3×C4×Q8—32no—
136C3×C4×Q8—31no—
137C3×C4×Q8—33no—
138C3×C4×Q8—36no—
139C3×C4×Q8C2×D311yescharacteristic
140C3×C4×Q8C2×D312yes—
141C3×C4×Q8C2×D313yes—
142C3×C4×Q8C22×C311yescharacteristic
143C3×C4×Q8C22×C312yes—
144C3×C4×Q8C22×C313yes—
145C3×C4×D4—33no—
146C3×C4×D4—36no—
147C3×C4×D4—33no—
148C3×C4×D4—36no—
149C3×#117—33no—
150C3×#117—36no—
151C2×C3×C4⋊C4—36no—
152C2×C3×C4⋊C4—36no—
153C2×C3×C42—33no—
154#946C22×C3112yes—
155#946C22×C313yes—
156Q8×Dic3C22×C312yes—
157Q8×Dic3C22×C311yescharacteristic
158#944C22×C316yes—
159#944C22×C311yescharacteristic
160#897C22×C313yes—
161#896C22×C316yes—
162#896C22×C3112yes—
163#893C22×C313yes—
164#893C22×C316yes—
165#892C22×C313yes—
166D3×C4⋊C4C22×C316yes—
167D3×C4⋊C4C22×C316yes—
168#890C22×C316yes—
169#889C22×C316yes—
170#888C22×C313yes—
171#888C22×C316yes—
172#888C22×C316yes—
173C4×D12C22×C316yes—
174C4×D12C22×C313yes—
175#872C22×C316yes—
176C42×D3C22×C313yes—
177#870C22×C316yes—
178#869C22×C313yes—
179C4×Dic6C22×C316yes—
180C4×Dic6C22×C313yes—
181C22×C3×D3C2413yes—
182C32×Q8—22no—
183C32×Q8—46no—
184C32×Q8C2×Q814yescomplemented
185C2×C32×C4C2413yes—
186C2×C32×C4C2412yes—
187C2×C32×C4C2416yes—
188C2×C32×C4C2413yes—
189C2×C32×C4C2411yescharacteristic
190C2×C3×Dic3C2412yes—
191C2×C3×Dic3C2413yes—
192C2×C3×Dic3C2416yes—
193C2×C3×Dic3C2411yescharacteristic
194C2×C3×Dic3C2411yescharacteristic
195C3×D12C2×Q8112yescomplemented
196C3×C4×D3—26no—
197C3×C4×D3—26no—
198C3×C4×D3—412no—
199C3×C4×D3C2×Q8112yescomplemented
200C3×Dic6—22no—
201C3×Dic6—212no—
202C3×Dic6—46no—
203Q82—31no—
204Q82C3×D311yescharacteristic, complemented
205#557—32no—
206#554—33no—
207#554—33no—
208D4×Q8—36no—
209#544—36no—
210#533—33no—
211C2×C4⋊Q8—33no—
212#520—32no—
213C4×#41—31no—
214C3×#41—38no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—26no—
218C2×C3×Q8—31no—
219C2×C3×Q8—36no—
220C2×C3×Q8—36no—
221C2×C3×Q8—1212no—
222C2×C3×Q8C22×D311yescharacteristic
223C2×C3×Q8C22×D316yes—
224C2×C3×Q8C22×D311yescharacteristic
225C2×C3×Q8C23×C311yescharacteristic
226C2×C3×Q8C23×C316yes—
227C2×C3×Q8C23×C311yescharacteristic
228C2×C3×D4—33no—
229C22×C3×C4—33no—
230C22×C3×C4—33no—
231C22×C3×C4—33no—
232C22×C3×C4—312no—
233#239C3×Q818yescomplemented
234D3×Q8—412no—
235C2×D12C23×C313yes—
236C2×C4×D3C23×C313yes—
237C2×C4×D3C23×C313yes—
238C2×C4×D3C23×C3112yes—
239C2×C4×D3C23×C313yes—
240C2×Dic6C23×C311yescharacteristic
241C2×Dic6C23×C316yes—
242C2×Dic6C23×C316yes—
243C3×C4⋊C4—23no—
244C3×C4⋊C4—26no—
245C3×C4⋊C4—23no—
246C3×C4⋊C4—26no—
247C3×C4⋊C4—36no—
248C3×C4⋊C4—312no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—32no—
251C3×C4⋊C4—36no—
252C3×C4⋊C4—36no—
253C3×C4⋊C4—33no—
254C3×C4⋊C4—36no—
255C3×C4⋊C4—33no—
256C3×C4⋊C4—36no—
257C3×C4⋊C4C22×D313yes—
258C3×C4⋊C4C22×D316yes—
259C3×C4⋊C4C22×D313yes—
260C3×C4⋊C4C22×D316yes—
261C3×C4⋊C4C23×C313yes—
262C3×C4⋊C4C23×C316yes—
263C3×C4⋊C4C23×C313yes—
264C3×C4⋊C4C23×C316yes—
265C3×C22⋊C4—36no—
266C3×C22⋊C4—36no—
267C3×C22⋊C4—36no—
268C3×C42—23no—
269C3×C42—26no—
270C3×C42—31no—
271C3×C42—33no—
272C3×C42—33no—
273C3×C42—36no—
274C3×C42C22×D313yes—
275C3×C42C22×D316yes—
276C3×C42C23×C313yes—
277C3×C42C23×C316yes—
278(C2×D3)⋊C4C23×C316yes—
279(C2×D3)⋊C4C23×C316yes—
280(C2×D3)⋊C4C23×C316yes—
281C4⋊Dic3C23×C313yes—
282C4⋊Dic3C23×C316yes—
283C4⋊Dic3C23×C316yes—
284C4⋊Dic3C23×C313yes—
285Dic3⋊C4C23×C312yes—
286Dic3⋊C4C23×C316yes—
287Dic3⋊C4C23×C316yes—
288Dic3⋊C4C23×C3112yes—
289Dic3⋊C4C23×C312yes—
290Dic3⋊C4C23×C316yes—
291C4×Dic3C23×C313yes—
292C4×Dic3C23×C316yes—
293C4×Dic3C23×C311yescharacteristic
294C4×Dic3C23×C313yes—
295C22×C32C2511yescharacteristic
296C2×C3×D3—23no—
297C2×C3×D3—23no—
298C2×C3×D3C22×Q816yes—
299C32×C4—22no—
300C32×C4—26no—
301C32×C4—23no—
302C32×C4C22×Q816yes—
303C32×C42−1+412yes—
304C3×Dic3—26no—
305C3×Dic3—22no—
306C3×Dic3—23no—
307C3×Dic3C22×Q812yes—
308C3×Dic3H(5,𝔽2)12yes—
309C2×#41—31no—
310C22×Q8—36no—
311C4⋊Q8—36no—
312C4⋊Q8—31no—
313C4⋊Q8—33no—
314C4⋊Q8—36no—
315C4⋊Q8—36no—
316C4⋊Q8—33no—
317C4⋊Q8C2×C3×D313yes—
318C4⋊Q8C2×C3×D316yes—
319#125—36no—
320#125—36no—
321#125—36no—
322C22⋊Q8—33no—
323C22⋊Q8—312no—
324C22⋊Q8—36no—
325C22⋊Q8—33no—
326C22⋊Q8—312no—
327C4×Q8—31no—
328C4×Q8—32no—
329C4×Q8—36no—
330C4×Q8—33no—
331C4×Q8C2×C3×D311yescharacteristic
332C4×Q8C2×C3×D312yes—
333C4×Q8C2×C3×D313yes—
334C4×D4—33no—
335C4×D4—36no—
336C4×D4—33no—
337C4×D4—36no—
338#117—33no—
339#117—36no—
340C2×C4⋊C4—36no—
341C2×C4⋊C4—36no—
342C2×C42—33no—
343C23×C3—33no—
344C22×D3C24×C313yes—
345C3×Q8—24no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—24no—
349C3×Q8—46no—
350C3×Q8—46no—
351C3×Q8—612no—
352C3×Q8—62no—
353C3×Q8—86no—
354C3×Q8—126no—
355C3×Q8D3×Q814yescomplemented
356C3×Q8C2×C3×Q814yescomplemented
357C3×D4—312no—
358C2×C3×C4—22no—
359C2×C3×C4—23no—
360C2×C3×C4—21no—
361C2×C3×C4—23no—
362C2×C3×C4—26no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—312no—
366C2×C3×C4—32no—
367C2×C3×C4—33no—
368C2×C3×C4—36no—
369C2×C3×C4—66no—
370C2×C3×C4—66no—
371C2×C3×C4—1212no—
372C2×C3×C4C23×D313yes—
373C2×C3×C4C23×D312yes—
374C2×C3×C4C23×D316yes—
375C2×C3×C4C23×D313yes—
376C2×C3×C4C23×D311yescharacteristic
377C2×C3×C4C24×C313yes—
378C2×C3×C4C24×C312yes—
379C2×C3×C4C24×C316yes—
380C2×C3×C4C24×C313yes—
381C2×C3×C4C24×C311yescharacteristic
382C2×Dic3C24×C312yes—
383C2×Dic3C24×C313yes—
384C2×Dic3C24×C316yes—
385C2×Dic3C24×C311yescharacteristic
386C2×Dic3C24×C311yescharacteristic
387D12C2×C3×Q8112yescomplemented
388C4×D3—26no—
389C4×D3—26no—
390C4×D3—412no—
391C4×D3C2×C3×Q8112yescomplemented
392Dic6—22no—
393Dic6—212no—
394Dic6—46no—
395C2×C32C23×Q811yescharacteristic
396C2×C32C2×H(5,𝔽2)11yescharacteristic
397C2×C32C2×2−1+411yescharacteristic
398C3×D3—26no—
399#41—38no—
400C2×Q8—31no—
401C2×Q8—36no—
402C2×Q8—36no—
403C2×Q8—1212no—
404C2×Q8C22×C3×D311yescharacteristic
405C2×Q8C22×C3×D316yes—
406C2×Q8C22×C3×D311yescharacteristic
407C2×D4—33no—
408C22×C4—33no—
409C22×C4—33no—
410C22×C4—33no—
411C22×C4—312no—
412C4⋊C4—36no—
413C4⋊C4—33no—
414C4⋊C4—36no—
415C4⋊C4—312no—
416C4⋊C4—32no—
417C4⋊C4—33no—
418C4⋊C4—32no—
419C4⋊C4—36no—
420C4⋊C4—36no—
421C4⋊C4—36no—
422C4⋊C4C22×C3×D313yes—
423C4⋊C4C22×C3×D316yes—
424C4⋊C4C22×C3×D313yes—
425C4⋊C4C22×C3×D316yes—
426C22⋊C4—36no—
427C22⋊C4—36no—
428C22⋊C4—36no—
429C42—31no—
430C42—33no—
431C42—33no—
432C42—36no—
433C42C22×C3×D313yes—
434C42C22×C3×D316yes—
435C22×C3—21no—
436C22×C3—36no—
437C22×C3—63no—
438C22×C3—63no—
439C22×C3C24×D311yescharacteristic
440C22×C3C25×C311yescharacteristic
441C2×D3—23no—
442C2×D3—23no—
443C2×D3C22×C3×Q816yes—
444C3×C4—24no—
445C3×C4—22no—
446C3×C4—23no—
447C3×C4—26no—
448C3×C4—26no—
449C3×C4—23no—
450C3×C4—22no—
451C3×C4—26no—
452C3×C4—22no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—43no—
456C3×C4—46no—
457C3×C4—63no—
458C3×C4—62no—
459C3×C4—66no—
460C3×C4C2×D3×Q816yes—
461C3×C4#100712yes—
462C3×C4C22×C3×Q816yes—
463C3×C4C3×2−1+412yes—
464Dic3—26no—
465Dic3—22no—
466Dic3—23no—
467Dic3C22×C3×Q812yes—
468Dic3C3×H(5,𝔽2)12yes—
469C32#350311yescharacteristic, complemented, Sylow
470C23—33no—
471Q8—22no—
472Q8—46no—
473Q8—612no—
474Q8—62no—
475Q8—126no—
476Q8C3×D3×Q814yescomplemented
477D4—312no—
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—312no—
481C2×C4—32no—
482C2×C4—33no—
483C2×C4—36no—
484C2×C4—66no—
485C2×C4—66no—
486C2×C4—1212no—
487C2×C4C23×C3×D313yes—
488C2×C4C23×C3×D312yes—
489C2×C4C23×C3×D316yes—
490C2×C4C23×C3×D313yes—
491C2×C4C23×C3×D311yescharacteristic
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—66no—
496C2×C3C22×D3×Q811yescharacteristic
497C2×C3C2×#100711yescharacteristic
498C2×C3D3×H(5,𝔽2)11yescharacteristic
499C2×C3C23×C3×Q811yescharacteristic
500C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
501C2×C3C2×C3×2−1+411yescharacteristic
502D3—26no—
503C22—36no—
504C22—63no—
505C22—63no—
506C22C24×C3×D311yescharacteristic
507C4—22no—
508C4—23no—
509C4—26no—
510C4—32no—
511C4—32no—
512C4—63no—
513C4—62no—
514C4—66no—
515C4C2×C3×D3×Q816yes—
516C4C3×#100712yes—
517C3—21no—
518C3#8704111yescharacteristic, direct product factor
519C3C3×#350311yescharacteristic, complemented
520C2—66no—
521C2C22×C3×D3×Q811yescharacteristic
522C2C2×C3×#100711yescharacteristic
523C2C3×D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×Q82C2
socleC22×C32C25
prime radicalC2×C3×#239C22
hypercenterC3×Q82D3
hypocenterC3C3×#3503
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
2. C3×#5730C2semidirect
3. C3×#5729C2semidirect
4. C3×#5727C2semidirect
5. C3×D3×C4⋊Q8C2semidirect
6. C3×#5579C2semidirect
7. C3×#5578C2semidirect
10. C3×#5569C2semidirect
11. C3×Q8×Dic6C2semidirect
13. #87041C3direct (N × H)
50. C3×Q82D3semidirect
51. C3×Q82C2×C3semidirect
61. #5730C2×C3semidirect
62. #5729C2×C3semidirect
63. #5727C2×C3semidirect
64. D3×C4⋊Q8C2×C3semidirect
65. #5579C2×C3semidirect
66. #5578C2×C3semidirect
69. #5569C2×C3semidirect
70. Q8×Dic6C2×C3semidirect
74. C3×#239Q8semidirect
184. C32×Q8C2×Q8semidirect
195. C3×D12C2×Q8semidirect
199. C3×C4×D3C2×Q8semidirect
204. Q82C3×D3semidirect
233. #239C3×Q8semidirect
355. C3×Q8D3×Q8semidirect
356. C3×Q8C2×C3×Q8semidirect
387. D12C2×C3×Q8semidirect
391. C4×D3C2×C3×Q8semidirect
469. C32#3503semidirect
476. Q8C3×D3×Q8semidirect
518. C3#87041direct (N × H)
519. C3C3×#3503semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
C3×#57302
C3×#57292
C3×#57272
C3×D3×C4⋊Q82
C3×#55792
C3×#55782
C3×#55742
C3×C4×#2392
C3×#55692
C3×Q8×Dic62
C3×#35033
#870413