Finite Group Databasebeta

← C3×#81614 #11888015

Subgroups — C3×#81614 #11888015

Order 1152.

821 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5732C211yesmaximal, characteristic, complemented
2C3×#5726C211yesmaximal, characteristic
3C32×#487C211yesmaximal, characteristic, complemented
4C3×#5090C211yesmaximal, characteristic, complemented
5C3×#5087C211yesmaximal, characteristic, complemented
6C3×#4815C212yesmaximal, complemented
7C3×#4813C212yesmaximal, complemented
8C3×#4808C212yesmaximal, complemented
9C3×#4804C212yesmaximal, complemented
10C3×C4×#204C211yesmaximal, characteristic, complemented
11C3×C8×D12C211yesmaximal, characteristic, complemented
12C3×#3302—31nomaximal
13#81614C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2212yes—
15C32×C4⋊Q8C2211yescharacteristic
16C32×C4⋊Q8C2211yescharacteristic
17C3×#947—21no—
18C3×#946C2212yes—
19C3×Q8×Dic3—21no—
20C2×C3×#216C2212yes—
21C2×C3×#204C2211yescharacteristic
22C2×C3×D3×C8C2212yes—
23C3×#898—22no—
24C3×#897C2212yes—
25C3×#895—21no—
26C3×#894—22no—
27C3×#892C2212yes—
28C3×D3×C4⋊C4—21no—
29C3×C4×D12C2211yescharacteristic
30C3×C4×D12C2211yescharacteristic
31C3×C4×Dic6C2211yescharacteristic
32C32×#103C2212yes—
33C32×#103C2212yes—
34C32×C4×C8C2211yescharacteristic
35C3×#835C2212yes—
36C3×#821C2211yescharacteristic
37C3×(C2×D3)⋊C8C2212yes—
38C3×#817C2211yescharacteristic
39C3×#816C2211yescharacteristic
40C3×#809C2212yes—
41C3×#807C2212yes—
42C3×C4⋊(C3⋊C8)C2211yescharacteristic
43C3×#556—31no—
44C3×#547—31no—
45#5732C2×C311yescharacteristic, complemented
46#5726C2×C311yescharacteristic
47C3×#487—21no—
48C3×#487D311yescharacteristic, complemented
49C3×#487C2×C311yescharacteristic, complemented
50C3×#485—32no—
51C3×#474—31no—
52C3×#467—32no—
53C3×#462—31no—
54C3×#452—32no—
55C3×C4×SD16—31no—
56C3×C8×D4—31no—
57C3×#416—32no—
58#5090C2×C311yescharacteristic, complemented
59#5087C2×C311yescharacteristic, complemented
60#4815C2×C312yescomplemented
61#4813C2×C312yescomplemented
62#4808C2×C312yescomplemented
63#4804C2×C312yescomplemented
64C4×#204C2×C311yescharacteristic, complemented
65C8×D12C2×C311yescharacteristic, complemented
66C2×C32×Q8—21no—
67C2×C32×Q8C2312yes—
68C3×#239—24no—
69C3×#239—42no—
70C2×C3×D12—22no—
71C2×C3×D12D411yescharacteristic
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3C2312yes—
76C2×C3×Dic6C2311yescharacteristic
77C2×C32×C8C2311yescharacteristic
78C2×C32×C8C2311yescharacteristic
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2312yes—
83C32×C42C2311yescharacteristic
84C3×#216—42no—
85C3×(C2×D3)⋊C4—22no—
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4D412yes—
90C3×C4⋊Dic3—22no—
91C3×C4⋊Dic3D411yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×C4⋊Dic3C2311yescharacteristic
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—22no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3—22no—
98C2×C3×C3⋊C8C2312yes—
99C3×#204—22no—
100C3×D3×C8—42no—
101#3302—31noSylow
102C2×C3×#41—32no—
103C2×#239C22×C312yes—
104C2×C3×Q16—32no—
105C2×C3×SD16—31no—
106C22×C3×C8—32no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8—21no—
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×C4⋊Q8C2×D311yescharacteristic
111C3×C4⋊Q8C22×C311yescharacteristic
112C3×C4⋊Q8C22×C311yescharacteristic
113C3×#126—62no—
114C3×#124—61no—
115C3×#123—62no—
116C3×C22⋊Q8—32no—
117C3×C22⋊Q8—32no—
118C3×#121—61no—
119C3×C4×Q8—31no—
120C3×C4×Q8—61no—
121C3×C4×D4—31no—
122C3×C4×D4—31no—
123C3×#117—32no—
124C2×C3×C4⋊C4—61no—
125#947—21no—
126#946C22×C312yes—
127Q8×Dic3—21no—
128C2×#216C22×C312yes—
129C2×#204C22×C311yescharacteristic
130C2×D3×C8C22×C312yes—
131#898—22no—
132#897C22×C312yes—
133#895—21no—
134#894—22no—
135#892C22×C312yes—
136D3×C4⋊C4—21no—
137C4×D12C22×C311yescharacteristic
138C4×D12C22×C311yescharacteristic
139C4×Dic6C22×C311yescharacteristic
140C3×#107—32no—
141C3×#106—31no—
142C3×C4⋊C8—31no—
143C3×#103—22no—
144C3×#103—22no—
145C3×#103—32no—
146C3×#103—31no—
147C3×#103C2×D312yes—
148C3×#103C2×D312yes—
149C3×#103C22×C312yes—
150C3×#103C22×C312yes—
151C3×#102—31no—
152C3×#102—32no—
153C3×C22⋊C8—32no—
154C3×C4×C8—21no—
155C3×C4×C8C2×D311yescharacteristic
156C3×C4×C8C22×C311yescharacteristic
157#835C22×C312yes—
158#821C22×C311yescharacteristic
159(C2×D3)⋊C8C22×C312yes—
160#817C22×C311yescharacteristic
161#816C22×C311yescharacteristic
162#809C22×C312yes—
163#807C22×C312yes—
164C4⋊(C3⋊C8)C22×C311yescharacteristic
165C22×C3×D3—21no—
166C22×C3×D3C2×D412yes—
167C32×Q8—22no—
168C32×Q8—22no—
169C32×Q8—42no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3C2×D412yes—
179C3×D12—22no—
180C3×D12—21no—
181C3×D12—44no—
182C3×D12#4112yes—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—44no—
186C3×C4×D3—42no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C32×C8—21no—
190C32×C8—21no—
191C3×C3⋊C8—22no—
192#556—31no—
193#547—31no—
194#487C3×D311yescharacteristic, complemented
195#485—32no—
196#474—31no—
197#467—32no—
198#462—31no—
199#452—32no—
200C4×SD16—31no—
201C8×D4—31no—
202#416—32no—
203C3×#41—64no—
204C3×#41—122no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—31no—
209C2×C3×Q8—41no—
210C2×C3×Q8C22×D312yes—
211C2×C3×Q8C23×C312yes—
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—62no—
215C22×C3×C4—32no—
216C22×C3×C4—62no—
217C22×C3×C4—61no—
218#239—24no—
219#239—42no—
220C2×D12—22no—
221C2×D12C3×D411yescharacteristic
222C2×D12C23×C311yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—22no—
225C2×C4×D3C23×C312yes—
226C2×Dic6C23×C311yescharacteristic
227C3×Q16—122no—
228C3×SD16—62no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—32no—
232C2×C3×C8—122no—
233C2×C3×C8C22×D311yescharacteristic
234C2×C3×C8C22×D311yescharacteristic
235C2×C3×C8C23×C311yescharacteristic
236C2×C3×C8C23×C311yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4C22×D312yes—
254C3×C4⋊C4C23×C312yes—
255C3×C22⋊C4—32no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—62no—
258C3×C22⋊C4—62no—
259C3×C22⋊C4—61no—
260C3×C42—21no—
261C3×C42—62no—
262C3×C42—61no—
263C3×C42C22×D311yescharacteristic
264C3×C42C23×C311yescharacteristic
265#216—42no—
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4C3×D412yes—
271C4⋊Dic3—22no—
272C4⋊Dic3C3×D411yescharacteristic
273C4⋊Dic3C23×C311yescharacteristic
274C4⋊Dic3C23×C311yescharacteristic
275Dic3⋊C4—21no—
276Dic3⋊C4—22no—
277C4×Dic3—21no—
278C4×Dic3—22no—
279C2×C3⋊C8C23×C312yes—
280#204—22no—
281D3×C8—42no—
282C22×C32C22×D411yescharacteristic
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—22no—
286C2×C3×D3—22no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C32×C4—21no—
290C32×C4—42no—
291C32×C4—42no—
292C32×C4#13512yes—
293C32×C4C2×#4111yescharacteristic
294C32×C4H(5,𝔽2)11yescharacteristic
295C3×Dic3—22no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C2×#41—32no—
300C2×Q16—32no—
301C2×SD16—31no—
302C22×C8—32no—
303C4⋊Q8C2×C3×D311yescharacteristic
304C4⋊Q8C2×C3×D311yescharacteristic
305#126—62no—
306#124—61no—
307#123—62no—
308C22⋊Q8—32no—
309C22⋊Q8—32no—
310#121—61no—
311C4×Q8—31no—
312C4×Q8—61no—
313C4×D4—31no—
314C4×D4—31no—
315#117—32no—
316C2×C4⋊C4—61no—
317#107—32no—
318#106—31no—
319C4⋊C8—31no—
320#103—32no—
321#103—31no—
322#103C2×C3×D312yes—
323#103C2×C3×D312yes—
324#102—31no—
325#102—32no—
326C22⋊C8—32no—
327C4×C8C2×C3×D311yescharacteristic
328C23×C3—32no—
329C23×C3—61no—
330C22×D3—21no—
331C22×D3C2×C3×D412yes—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—82no—
343C3×D4—32no—
344C3×D4—62no—
345C3×D4—61no—
346C3×D4—124no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—32no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—124no—
363C2×C3×C4—122no—
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4D3×D411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×C3×C4C2×C3×D411yescharacteristic
368C2×C3×C4C23×D311yescharacteristic
369C2×C3×C4C24×C311yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3C2×C3×D412yes—
374D12—22no—
375D12—21no—
376D12—44no—
377D12C3×#4112yes—
378C4×D3—22no—
379C4×D3—22no—
380C4×D3—44no—
381C4×D3—42no—
382Dic6—21no—
383Dic6—21no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—62no—
391C3⋊C8—22no—
392C2×C32#54611yescharacteristic
393C2×C32C2×#13511yescharacteristic
394C2×C32#57811yescharacteristic
395C3×D3—42no—
396C3×D3—42no—
397#41—64no—
398#41—122no—
399C2×Q8—21no—
400C2×Q8—31no—
401C2×Q8C22×C3×D312yes—
402C2×D4—31no—
403C2×D4—31no—
404C2×D4—62no—
405C22×C4—32no—
406C22×C4—62no—
407C22×C4—61no—
408Q16—122no—
409SD16—62no—
410C2×C8—32no—
411C2×C8—122no—
412C2×C8C22×C3×D311yescharacteristic
413C2×C8C22×C3×D311yescharacteristic
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—31no—
418C4⋊C4—31no—
419C4⋊C4—31no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4C22×C3×D312yes—
424C22⋊C4—32no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—61no—
429C42—62no—
430C42—61no—
431C42C22×C3×D311yescharacteristic
432C22×C3—21no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3C2×D3×D411yescharacteristic
440C22×C3C22×C3×D411yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—41no—
446C2×D3—41no—
447C3×C4—22no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—62no—
458C3×C4—82no—
459C3×C4—82no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4#91612yes—
464C3×C4C3×#13512yes—
465C3×C4D3×#4111yescharacteristic
466C3×C4#100911yescharacteristic
467C3×C4C2×C3×#4111yescharacteristic
468C3×C4C3×H(5,𝔽2)11yescharacteristic
469Dic3—22no—
470Dic3—41no—
471Dic3—41no—
472Dic3—41no—
473C32#330211yescharacteristic, complemented, Sylow
474C23—32no—
475C23—61no—
476Q8—22no—
477Q8—22no—
478Q8—42no—
479Q8—61no—
480Q8—61no—
481D4—32no—
482D4—62no—
483D4—61no—
484D4—124no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—32no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—124no—
494C2×C4—122no—
495C2×C4C3×D3×D411yescharacteristic
496C2×C4C3×D3×D411yescharacteristic
497C2×C4C23×C3×D311yescharacteristic
498C8—21no—
499C8—21no—
500C8—62no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—122no—
505C2×C3—122no—
506C2×C3#567611yescharacteristic
507C2×C3C2×#91611yescharacteristic
508C2×C3#576611yescharacteristic
509C2×C3C3×#54611yescharacteristic
510C2×C3C2×C3×#13511yescharacteristic
511C2×C3C3×#57811yescharacteristic
512D3—42no—
513D3—42no—
514C22—62no—
515C22—62no—
516C22—62no—
517C22—62no—
518C22—121no—
519C22—121no—
520C22C2×C3×D3×D411yescharacteristic
521C4—21no—
522C4—42no—
523C4—42no—
524C4—62no—
525C4—121no—
526C4—121no—
527C4—121no—
528C4C3×#91612yes—
529C4C3×D3×#4111yescharacteristic
530C4C3×#100911yescharacteristic
531C3—21no—
532C3#8161411yescharacteristic, direct product factor
533C3C3×#330211yescharacteristic, complemented
534C2—122no—
535C2—122no—
536C2C3×#567611yescharacteristic
537C2C2×C3×#91611yescharacteristic
538C2C3×#576611yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#487C2
socleC22×C32C22×D4
prime radicalC3×#5726C2
hypercenterC3×#487D3
hypocenterC3C3×#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#487

Complements

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

NH ≅ G/Ntype
1. C3×#5732C2semidirect
3. C32×#487C2semidirect
4. C3×#5090C2semidirect
5. C3×#5087C2semidirect
6. C3×#4815C2semidirect
7. C3×#4813C2semidirect
8. C3×#4808C2semidirect
9. C3×#4804C2semidirect
10. C3×C4×#204C2semidirect
11. C3×C8×D12C2semidirect
13. #81614C3direct (N × H)
45. #5732C2×C3semidirect
48. C3×#487D3semidirect
49. C3×#487C2×C3semidirect
58. #5090C2×C3semidirect
59. #5087C2×C3semidirect
60. #4815C2×C3semidirect
61. #4813C2×C3semidirect
62. #4808C2×C3semidirect
63. #4804C2×C3semidirect
64. C4×#204C2×C3semidirect
65. C8×D12C2×C3semidirect
194. #487C3×D3semidirect
473. C32#3302semidirect
532. C3#81614direct (N × H)
533. C3C3×#3302semidirect

Maximal subgroups

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

subgroupindex
C3×#57322
C3×#57262
C32×#4872
C3×#50902
C3×#50872
C3×#48152
C3×#48132
C3×#48082
C3×#48042
C3×C4×#2042
C3×C8×D122
C3×#33023
#816143