Finite Group Databasebeta

← C3×#86753 #11909856

Subgroups — C3×#86753 #11909856

Order 1152.

2124 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Q8C211yesmaximal, characteristic, complemented
2C3×Q8×#67C212yesmaximal, complemented
3C3×#5812C212yesmaximal, complemented
4C3×#5728C213yesmaximal, complemented
5C3×#5721C213yesmaximal, complemented
6C3×#5639C216yesmaximal, complemented
7C3×#5625C216yesmaximal, complemented
8C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
9C3×#5570C211yesmaximal, characteristic, complemented
10C3×#5557C213yesmaximal
11C3×#5542C213yesmaximal, complemented
12C3×#3523—31nomaximal
13#86753C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2212yes—
15C2×C3×D3×Q8C2211yescharacteristic
16C32×C4⋊Q8C2213yes—
17C32×C22⋊Q8C2216yes—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×D4C2213yes—
20C3×#946C2216yes—
21C3×Q8×Dic3C2211yescharacteristic
22C3×Q8×Dic3C2212yes—
23C3×Q8×Dic3C2216yes—
24C3×Q8×Dic3C2216yes—
25C3×#944C2216yes—
26C3×#938C2211yescharacteristic
27C3×#936C2213yes—
28C3×C4×#67C2216yes—
29C3×#926C2216yes—
30C3×#926C2216yes—
31C3×#924C2216yes—
32C3×#897C2213yes—
33C3×#892C2216yes—
34C3×D3×C4⋊C4C2213yes—
35C3×#890C2213yes—
36C3×#890C2216yes—
37C3×#890C2216yes—
38C3×#889C22112yes—
39C3×#887C2216yes—
40C3×#887C2213yes—
41C3×#883C2216yes—
42C3×#879C22112yes—
43C3×#879C2216yes—
44C3×C42×D3C2213yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2213yes—
47C3×C4×Dic6C2213yes—
48C3×#557—33no—
49C3×#557—31no—
50C3×#556—33no—
51C3×#552—36no—
52C3×D4×Q8—21no—
53C3×D4×Q8—32no—
54C3×D4×Q8D311yescharacteristic, complemented
55C3×D4×Q8C2×C311yescharacteristic, complemented
56C3×#541—36no—
57C3×#533—33no—
58C3×#529—33no—
59C3×#519—32no—
60C2×C3×C4×Q8—31no—
61Q8×#67C2×C312yescomplemented
62#5812C2×C312yescomplemented
63#5728C2×C313yescomplemented
64#5721C2×C313yescomplemented
65#5639C2×C316yescomplemented
66#5625C2×C316yescomplemented
67C4×D3×Q8C2×C311yescharacteristic, complemented
68#5570C2×C311yescharacteristic, complemented
69#5557C2×C313yes—
70#5542C2×C313yescomplemented
71C2×C32×Q8—28no—
72C2×C32×Q8C2316yes—
73C2×C32×Q8C2311yescharacteristic
74C2×C32×D4C2311yescharacteristic
75C22×C32×C4C2316yes—
76C2×C3×#67C2312yes—
77C3×D3×Q8—24no—
78C2×C3×C4×D3C2313yes—
79C2×C3×C4×D3C2313yes—
80C2×C3×C4×D3C2311yescharacteristic
81C2×C3×Dic6C2313yes—
82C2×C3×Dic6C2313yes—
83C32×C4⋊C4C2313yes—
84C32×C4⋊C4C2313yes—
85C32×C4⋊C4C2316yes—
86C32×C22⋊C4C2316yes—
87C32×C42C2313yes—
88C3×#217C2316yes—
89C3×#217C2312yes—
90C3×#217C2316yes—
91C3×(C2×D3)⋊C4C2316yes—
92C3×C4⋊Dic3C2316yes—
93C3×C4⋊Dic3C2316yes—
94C3×C4⋊Dic3C2316yes—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2316yes—
97C3×C4⋊Dic3C2313yes—
98C3×Dic3⋊C4C2316yes—
99C3×Dic3⋊C4C2316yes—
100C3×Dic3⋊C4C23112yes—
101C3×Dic3⋊C4C2316yes—
102C3×C4×Dic3C2313yes—
103C3×C4×Dic3C23112yes—
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C2313yes—
106C3×C4×Dic3C2316yes—
107#3523—31noSylow
108C22×C3×Q8—22no—
109C22×C3×Q8—31no—
110C22×C3×Q8C2×D312yes—
111C22×C3×Q8C22×C312yes—
112C2×D3×Q8C22×C311yescharacteristic
113C3×C4⋊Q8—23no—
114C3×C4⋊Q8—36no—
115C3×C4⋊Q8—33no—
116C3×C4⋊Q8C2×D313yes—
117C3×C4⋊Q8C22×C313yes—
118C3×#126—36no—
119C3×#126—312no—
120C3×#125—312no—
121C3×#125—33no—
122C3×#125—36no—
123C3×#125—36no—
124C3×#124—33no—
125C3×#123—36no—
126C3×C22⋊Q8—26no—
127C3×C22⋊Q8—36no—
128C3×C22⋊Q8—36no—
129C3×C22⋊Q8—33no—
130C3×C22⋊Q8C2×D316yes—
131C3×C22⋊Q8C22×C316yes—
132C3×#121—31no—
133C3×C4×Q8—21no—
134C3×C4×Q8—31no—
135C3×C4×Q8—36no—
136C3×C4×Q8—32no—
137C3×C4×Q8—36no—
138C3×C4×Q8—33no—
139C3×C4×Q8—36no—
140C3×C4×Q8—33no—
141C3×C4×Q8—33no—
142C3×C4×Q8C2×D311yescharacteristic
143C3×C4×Q8C22×C311yescharacteristic
144C3×C4×D4—23no—
145C3×C4×D4—36no—
146C3×C4×D4C2×D313yes—
147C3×C4×D4C22×C313yes—
148C3×#117—36no—
149C3×#117—36no—
150C3×#117—36no—
151C2×C3×C4⋊C4—33no—
152C2×C3×C42—33no—
153#946C22×C316yes—
154Q8×Dic3C22×C316yes—
155Q8×Dic3C22×C316yes—
156Q8×Dic3C22×C311yescharacteristic
157Q8×Dic3C22×C312yes—
158#944C22×C316yes—
159#938C22×C311yescharacteristic
160#936C22×C313yes—
161C4×#67C22×C316yes—
162#926C22×C316yes—
163#926C22×C316yes—
164#924C22×C316yes—
165#897C22×C313yes—
166#892C22×C316yes—
167D3×C4⋊C4C22×C313yes—
168#890C22×C316yes—
169#890C22×C313yes—
170#890C22×C316yes—
171#889C22×C3112yes—
172#887C22×C316yes—
173#887C22×C313yes—
174#883C22×C316yes—
175#879C22×C3112yes—
176#879C22×C316yes—
177C42×D3C22×C313yes—
178#869C22×C313yes—
179C4×Dic6C22×C313yes—
180C4×Dic6C22×C313yes—
181C23×C32C2412yes—
182C22×C3×D3C2411yescharacteristic
183C32×Q8—212no—
184C32×Q8#4114yes—
185C32×D4—22no—
186C2×C32×C4—212no—
187C2×C32×C4C2416yes—
188C2×C32×C4C2413yes—
189C2×C32×C4C2413yes—
190C2×C32×C4C2411yescharacteristic
191C3×#67—24no—
192C2×C3×Dic3C2413yes—
193C2×C3×Dic3C2411yescharacteristic
194C2×C3×Dic3C2412yes—
195C2×C3×Dic3C2416yes—
196C2×C3×Dic3C2413yes—
197C3×C4×D3—26no—
198C3×C4×D3—26no—
199C3×C4×D3—22no—
200C3×Dic6—26no—
201C3×Dic6—26no—
202#557—33no—
203#557—31no—
204#556—33no—
205#552—36no—
206D4×Q8—32no—
207D4×Q8C3×D311yescharacteristic, complemented
208#541—36no—
209#533—33no—
210#529—33no—
211#519—32no—
212C2×C4×Q8—31no—
213C2×C3×Q8—28no—
214C2×C3×Q8—28no—
215C2×C3×Q8—21no—
216C2×C3×Q8—26no—
217C2×C3×Q8—33no—
218C2×C3×Q8—33no—
219C2×C3×Q8—48no—
220C2×C3×Q8—64no—
221C2×C3×Q8C22×D316yes—
222C2×C3×Q8C22×D311yescharacteristic
223C2×C3×Q8C23×C316yes—
224C2×C3×Q8C23×C311yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4—32no—
227C2×C3×D4C22×D311yescharacteristic
228C2×C3×D4C23×C311yescharacteristic
229C22×C3×C4—26no—
230C22×C3×C4—31no—
231C22×C3×C4—33no—
232C22×C3×C4—33no—
233C22×C3×C4C22×D316yes—
234C22×C3×C4C23×C316yes—
235C2×#67C23×C312yes—
236D3×Q8—24no—
237C2×C4×D3C23×C311yescharacteristic
238C2×C4×D3C23×C313yes—
239C2×C4×D3C23×C313yes—
240C2×Dic6C23×C313yes—
241C2×Dic6C23×C313yes—
242C3×C4⋊C4—23no—
243C3×C4⋊C4—23no—
244C3×C4⋊C4—26no—
245C3×C4⋊C4—36no—
246C3×C4⋊C4—36no—
247C3×C4⋊C4—312no—
248C3×C4⋊C4—36no—
249C3×C4⋊C4—33no—
250C3×C4⋊C4—36no—
251C3×C4⋊C4—36no—
252C3×C4⋊C4—36no—
253C3×C4⋊C4—36no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4C22×D313yes—
256C3×C4⋊C4C22×D316yes—
257C3×C4⋊C4C22×D313yes—
258C3×C4⋊C4C23×C313yes—
259C3×C4⋊C4C23×C313yes—
260C3×C4⋊C4C23×C316yes—
261C3×C22⋊C4—26no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—36no—
264C3×C22⋊C4—36no—
265C3×C22⋊C4—36no—
266C3×C22⋊C4C22×D316yes—
267C3×C22⋊C4C23×C316yes—
268C3×C42—23no—
269C3×C42—33no—
270C3×C42—33no—
271C3×C42—33no—
272C3×C42—312no—
273C3×C42—36no—
274C3×C42C22×D313yes—
275C3×C42C23×C313yes—
276#217C23×C316yes—
277#217C23×C312yes—
278#217C23×C316yes—
279(C2×D3)⋊C4C23×C316yes—
280C4⋊Dic3C23×C311yescharacteristic
281C4⋊Dic3C23×C313yes—
282C4⋊Dic3C23×C316yes—
283C4⋊Dic3C23×C316yes—
284C4⋊Dic3C23×C316yes—
285C4⋊Dic3C23×C316yes—
286Dic3⋊C4C23×C316yes—
287Dic3⋊C4C23×C316yes—
288Dic3⋊C4C23×C3112yes—
289Dic3⋊C4C23×C316yes—
290C4×Dic3C23×C313yes—
291C4×Dic3C23×C316yes—
292C4×Dic3C23×C313yes—
293C4×Dic3C23×C313yes—
294C4×Dic3C23×C3112yes—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32C2511yescharacteristic
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D32−1+412yes—
302C32×C4—26no—
303C32×C4—23no—
304C32×C4C2×#4116yes—
305C32×C42−1+412yes—
306C3×Dic3—23no—
307C3×Dic3—26no—
308C3×Dic3—21no—
309C3×Dic3—23no—
310C3×Dic3—22no—
311C22×Q8—31no—
312C22×Q8C2×C3×D312yes—
313C4⋊Q8—36no—
314C4⋊Q8—33no—
315C4⋊Q8C2×C3×D313yes—
316#126—36no—
317#126—312no—
318#125—33no—
319#125—36no—
320#125—312no—
321#125—36no—
322#124—33no—
323#123—36no—
324C22⋊Q8—36no—
325C22⋊Q8—36no—
326C22⋊Q8—33no—
327C22⋊Q8C2×C3×D316yes—
328#121—31no—
329C4×Q8—31no—
330C4×Q8—36no—
331C4×Q8—32no—
332C4×Q8—36no—
333C4×Q8—33no—
334C4×Q8—36no—
335C4×Q8—33no—
336C4×Q8—33no—
337C4×Q8C2×C3×D311yescharacteristic
338C4×D4—36no—
339C4×D4C2×C3×D313yes—
340#117—36no—
341#117—36no—
342#117—36no—
343C2×C4⋊C4—33no—
344C2×C42—33no—
345C23×C3—22no—
346C23×C3—31no—
347C23×C3C23×D312yes—
348C23×C3C24×C312yes—
349C22×D3C24×C311yescharacteristic
350C3×Q8—212no—
351C3×Q8—212no—
352C3×Q8—24no—
353C3×Q8—412no—
354C3×Q8—66no—
355C3×Q8—66no—
356C3×Q8#23714yes—
357C3×Q8C3×#4114yes—
358C3×D4—24no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—64no—
362C2×C3×C4—212no—
363C2×C3×C4—212no—
364C2×C3×C4—21no—
365C2×C3×C4—23no—
366C2×C3×C4—23no—
367C2×C3×C4—26no—
368C2×C3×C4—31no—
369C2×C3×C4—33no—
370C2×C3×C4—36no—
371C2×C3×C4—32no—
372C2×C3×C4—33no—
373C2×C3×C4—412no—
374C2×C3×C4—66no—
375C2×C3×C4—66no—
376C2×C3×C4—62no—
377C2×C3×C4C23×D316yes—
378C2×C3×C4C23×D313yes—
379C2×C3×C4C23×D313yes—
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C24×C316yes—
382C2×C3×C4C24×C313yes—
383C2×C3×C4C24×C313yes—
384C2×C3×C4C24×C311yescharacteristic
385#67—24no—
386C2×Dic3C24×C313yes—
387C2×Dic3C24×C311yescharacteristic
388C2×Dic3C24×C312yes—
389C2×Dic3C24×C316yes—
390C2×Dic3C24×C313yes—
391C4×D3—26no—
392C4×D3—26no—
393C4×D3—22no—
394Dic6—26no—
395Dic6—26no—
396C2×C32—42no—
397C2×C32C22×#4111yescharacteristic
398C2×C32C2×2−1+411yescharacteristic
399C2×C32C2×2−1+411yescharacteristic
400C3×D3—22no—
401C2×Q8—28no—
402C2×Q8—33no—
403C2×Q8—33no—
404C2×Q8—64no—
405C2×Q8C22×C3×D316yes—
406C2×Q8C22×C3×D311yescharacteristic
407C2×D4—32no—
408C2×D4C22×C3×D311yescharacteristic
409C22×C4—31no—
410C22×C4—33no—
411C22×C4—33no—
412C22×C4C22×C3×D316yes—
413C4⋊C4—33no—
414C4⋊C4—36no—
415C4⋊C4—36no—
416C4⋊C4—36no—
417C4⋊C4—36no—
418C4⋊C4—36no—
419C4⋊C4—36no—
420C4⋊C4—31no—
421C4⋊C4—312no—
422C4⋊C4—36no—
423C4⋊C4C22×C3×D313yes—
424C4⋊C4C22×C3×D316yes—
425C4⋊C4C22×C3×D313yes—
426C22⋊C4—32no—
427C22⋊C4—36no—
428C22⋊C4—36no—
429C22⋊C4—36no—
430C22⋊C4C22×C3×D316yes—
431C42—33no—
432C42—33no—
433C42—33no—
434C42—312no—
435C42—36no—
436C42C22×C3×D313yes—
437C22×C3—24no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—32no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3C24×D311yescharacteristic
451C22×C3C25×C311yescharacteristic
452C2×D3—21no—
453C2×D3—21no—
454C2×D3C3×2−1+412yes—
455C3×C4—26no—
456C3×C4—23no—
457C3×C4—26no—
458C3×C4—23no—
459C3×C4—26no—
460C3×C4—22no—
461C3×C4—46no—
462C3×C4—43no—
463C3×C4—63no—
464C3×C4—61no—
465C3×C4—66no—
466C3×C4—63no—
467C3×C4—62no—
468C3×C4C2×#23716yes—
469C3×C4#100712yes—
470C3×C4C2×C3×#4116yes—
471C3×C4C3×2−1+412yes—
472Dic3—23no—
473Dic3—26no—
474Dic3—21no—
475Dic3—23no—
476Dic3—22no—
477C32#352311yescharacteristic, complemented, Sylow
478C23—31no—
479C23C23×C3×D312yes—
480Q8—212no—
481Q8—66no—
482Q8—66no—
483Q8C3×#23714yes—
484D4—22no—
485D4—64no—
486C2×C4—212no—
487C2×C4—31no—
488C2×C4—33no—
489C2×C4—36no—
490C2×C4—32no—
491C2×C4—33no—
492C2×C4—66no—
493C2×C4—66no—
494C2×C4—62no—
495C2×C4C23×C3×D316yes—
496C2×C4C23×C3×D313yes—
497C2×C4C23×C3×D313yes—
498C2×C4C23×C3×D311yescharacteristic
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—62no—
506C2×C3C22×#23711yescharacteristic
507C2×C3C2×#100711yescharacteristic
508C2×C3D3×2−1+411yescharacteristic
509C2×C3C22×C3×#4111yescharacteristic
510C2×C3C2×C3×2−1+411yescharacteristic
511C2×C3C2×C3×2−1+411yescharacteristic
512D3—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—32no—
517C22—61no—
518C22—61no—
519C22C24×C3×D311yescharacteristic
520C4—26no—
521C4—23no—
522C4—63no—
523C4—61no—
524C4—66no—
525C4—63no—
526C4—62no—
527C4C2×C3×#23716yes—
528C4C3×#100712yes—
529C3—21no—
530C3#8675311yescharacteristic, direct product factor
531C3C3×#352311yescharacteristic, complemented
532C2—42no—
533C2—62no—
534C2C22×C3×#23711yescharacteristic
535C2C2×C3×#100711yescharacteristic
536C2C3×D3×2−1+411yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×D4×Q8C2
socleC22×C32C25
prime radicalC3×#938C22
hypercenterC3×D4×Q8D3
hypocenterC3C3×#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C32×D4×Q8C2semidirect
2. C3×Q8×#67C2semidirect
3. C3×#5812C2semidirect
4. C3×#5728C2semidirect
5. C3×#5721C2semidirect
6. C3×#5639C2semidirect
7. C3×#5625C2semidirect
8. C3×C4×D3×Q8C2semidirect
9. C3×#5570C2semidirect
11. C3×#5542C2semidirect
13. #86753C3direct (N × H)
54. C3×D4×Q8D3semidirect
55. C3×D4×Q8C2×C3semidirect
61. Q8×#67C2×C3semidirect
62. #5812C2×C3semidirect
63. #5728C2×C3semidirect
64. #5721C2×C3semidirect
65. #5639C2×C3semidirect
66. #5625C2×C3semidirect
67. C4×D3×Q8C2×C3semidirect
68. #5570C2×C3semidirect
70. #5542C2×C3semidirect
207. D4×Q8C3×D3semidirect
477. C32#3523semidirect
530. C3#86753direct (N × H)
531. C3C3×#3523semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Q82
C3×Q8×#672
C3×#58122
C3×#57282
C3×#57212
C3×#56392
C3×#56252
C3×C4×D3×Q82
C3×#55702
C3×#55572
C3×#55422
C3×#35233
#867533