Finite Group Databasebeta

← C3×#80302 #11886839

Subgroups — C3×#80302 #11886839

Order 1152.

991 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5837C212yesmaximal, complemented
2C3×#5778C212yesmaximal, complemented
3C32×#456C211yesmaximal, characteristic, complemented
4C3×#5203C212yesmaximal, complemented
5C3×#5095C211yesmaximal, characteristic, complemented
6C3×#5085C211yesmaximal, characteristic, complemented
7C3×#4824C212yesmaximal, complemented
8C3×#4822C212yesmaximal, complemented
9C3×#4754C212yesmaximal, complemented
10C3×#3291—31nomaximal
11#80302C311yesmaximal, characteristic, direct product factor
12C32×2−1+4C2211yescharacteristic
13C3×#1009C2212yescomplemented
14C3×#1007C2211yescharacteristic
15C32×#137C2212yescomplemented
16C32×C4⋊Q8C2211yescharacteristic, complemented
17C3×#950—22no—
18C3×#950C2212yes—
19C3×#949—22no—
20C3×#949C2212yes—
21C3×#948C2212yescomplemented
22C3×#942—22no—
23C3×#942C2212yescomplemented
24C2×C3×#215—22no—
25C2×C3×#215C2212yescomplemented
26C2×C3×#215C2212yescomplemented
27C3×#919—22no—
28C3×#918—22no—
29C3×#914—22no—
30C3×D3×SD16—22no—
31C3×#908C2212yes—
32C3×#907C2212yescomplemented
33C3×#874C2211yescharacteristic
34C32×#104C2212yescomplemented
35C32×#101C2211yescharacteristic, complemented
36C3×#836C2211yescharacteristic, complemented
37C3×#823C2212yescomplemented
38C3×#822C2212yescomplemented
39C3×#805C2212yescomplemented
40C3×C4×C3⋊C8C2211yescharacteristic, complemented
41C3×#577—32no—
42C3×#577—32no—
43#5837C2×C312yescomplemented
44#5778C2×C312yescomplemented
45C3×#492—31no—
46C3×#471—32no—
47C3×#471—32no—
48C3×#456—21no—
49C3×#456—31no—
50C3×#456D311yescharacteristic, complemented
51C3×#456C2×C311yescharacteristic, complemented
52C3×#453—32no—
53C3×#443—32no—
54#5203C2×C312yescomplemented
55#5095C2×C311yescharacteristic, complemented
56#5085C2×C311yescharacteristic, complemented
57#4824C2×C312yescomplemented
58#4822C2×C312yescomplemented
59#4754C2×C312yescomplemented
60C32×#41—22no—
61C32×#41—41no—
62C32×#41C2312yes—
63C2×C32×Q8—22no—
64C2×C32×Q8C2311yescharacteristic
65C2×C32×Q8C2311yescharacteristic
66C3×#239—22no—
67C3×#239—22no—
68C3×D3×Q8—22no—
69C3×D3×D4—22no—
70C3×D3×D4—42no—
71C3×#235—22no—
72C3×#235—41no—
73C3×#235C2312yes—
74C2×C3×D12—22no—
75C2×C3×D12—41no—
76C2×C3×D12C2312yes—
77C32×Q16—22no—
78C32×SD16—22no—
79C32×M16C2312yes—
80C32×C4⋊C4—41no—
81C32×C42C2311yescharacteristic
82C3×#216—22no—
83C3×#216D412yescomplemented
84C3×#215—22no—
85C3×#215—22no—
86C3×#215—22no—
87C3×#215—22no—
88C3×#215—22no—
89C3×#215—22no—
90C3×#215D412yescomplemented
91C3×#214D412yescomplemented
92C3×#213—22no—
93C3×#213D412yescomplemented
94C3×#208—22no—
95C3×#208C2312yes—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2312yes—
98C3×D24—22no—
99C3×#204—22no—
100C3×ΓL(1,𝔽25)—22no—
101C3×D3×C8—22no—
102#3291—31noSylow
103C3×2−1+4—21no—
104C3×2−1+4—31no—
105C3×2−1+4C2×D311yescharacteristic
106C3×2−1+4C22×C311yescharacteristic
107C3×H(5,𝔽2)—32no—
108#1009C22×C312yescomplemented
109#1007C22×C311yescharacteristic
110C3×#137—22no—
111C3×#137—32no—
112C3×#137—62no—
113C3×#137—62no—
114C3×#137C2×D312yescomplemented
115C3×#137C22×C312yescomplemented
116C3×AGL(1,ℤ/8)—32no—
117C3×AGL(1,ℤ/8)—32no—
118C3×AGL(1,ℤ/8)—62no—
119C3×AGL(1,ℤ/8)—62no—
120C3×#135—32no—
121C3×#135—62no—
122C3×#135—62no—
123C2×C3×SD16—32no—
124C2×C3×SD16—32no—
125C2×C3×SD16—62no—
126C2×C3×SD16—62no—
127C3×#131—32no—
128C3×#131—32no—
129C3×C4⋊Q8—21no—
130C3×C4⋊Q8C2×D311yescharacteristic, complemented
131C3×C4⋊Q8C22×C311yescharacteristic, complemented
132C3×#127—31no—
133#950—22no—
134#950C22×C312yes—
135#949—22no—
136#949C22×C312yes—
137#948C22×C312yescomplemented
138#942—22no—
139#942C22×C312yescomplemented
140C2×#215—22no—
141C2×#215C22×C312yescomplemented
142C2×#215C22×C312yescomplemented
143#919—22no—
144#918—22no—
145#914—22no—
146D3×SD16—22no—
147#908C22×C312yes—
148#907C22×C312yescomplemented
149#874C22×C311yescharacteristic
150C3×#108—32no—
151C3×#104—22no—
152C3×#104—32no—
153C3×#104C2×D312yescomplemented
154C3×#104C22×C312yescomplemented
155C3×#101—21no—
156C3×#101—31no—
157C3×#101C2×D311yescharacteristic, complemented
158C3×#101C22×C311yescharacteristic, complemented
159C3×#100—32no—
160C3×C4×C8—31no—
161#836C22×C311yescharacteristic, complemented
162#823C22×C312yescomplemented
163#822C22×C312yescomplemented
164#805C22×C312yescomplemented
165C4×C3⋊C8C22×C311yescharacteristic, complemented
166C22×C3×D3—22no—
167C22×C3×D3—42no—
168C32×Q8—22no—
169C32×Q8—22no—
170C32×Q8—41no—
171C32×Q8C2×D412yes—
172C32×D4—22no—
173C32×D4—41no—
174C32×D4C2×D412yes—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—22no—
178C2×C32×C4—21no—
179C2×C32×C4—42no—
180C2×C32×C4C2411yescharacteristic
181C3×#67—22no—
182C3×#67—42no—
183C3×D12—22no—
184C3×D12—22no—
185C3×D12—42no—
186C3×D12—41no—
187C3×D12—82no—
188C3×D12C2×D412yes—
189C3×C4×D3—22no—
190C3×C4×D3—42no—
191C3×C4×D3—42no—
192C3×Dic6—41no—
193C3×Dic6C2×D412yes—
194C32×C8—22no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8C2×D412yes—
197C3×C3⋊C8C2×D412yes—
198#577—32no—
199#577—32no—
200#492—31no—
201#471—32no—
202#471—32no—
203#456—31no—
204#456C3×D311yescharacteristic, complemented
205#453—32no—
206#443—32no—
207C3×#41—22no—
208C3×#41—22no—
209C3×#41—22no—
210C3×#41—32no—
211C3×#41—42no—
212C3×#41—41no—
213C3×#41—41no—
214C3×#41—42no—
215C3×#41—62no—
216C3×#41—62no—
217C3×#41—62no—
218C3×#41—121no—
219C3×#41C22×D312yes—
220C3×#41C23×C312yes—
221C2×C3×Q8—22no—
222C2×C3×Q8—22no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—42no—
226C2×C3×Q8—62no—
227C2×C3×Q8C22×D311yescharacteristic
228C2×C3×Q8C22×D311yescharacteristic
229C2×C3×Q8C23×C311yescharacteristic
230C2×C3×Q8C23×C311yescharacteristic
231C2×C3×D4—32no—
232C2×C3×D4—62no—
233C2×C3×D4—62no—
234C2×C3×D4—121no—
235C2×C3×D4—122no—
236#239—22no—
237#239—22no—
238D3×Q8—22no—
239D3×D4—22no—
240D3×D4—42no—
241#235—22no—
242#235—41no—
243#235C23×C312yes—
244C2×D12—22no—
245C2×D12—41no—
246C2×D12C23×C312yes—
247C3×Q16—22no—
248C3×Q16—22no—
249C3×Q16—32no—
250C3×Q16—42no—
251C3×Q16—62no—
252C3×SD16—22no—
253C3×SD16—22no—
254C3×SD16—32no—
255C3×SD16—32no—
256C3×SD16—42no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—62no—
260C3×SD16—62no—
261C3×SD16—62no—
262C3×SD16—62no—
263C3×SD16—62no—
264C3×D8—32no—
265C3×D8—62no—
266C3×D8—62no—
267C3×M16—22no—
268C3×M16—32no—
269C3×M16—62no—
270C3×M16—62no—
271C3×M16C22×D312yes—
272C3×M16C23×C312yes—
273C2×C3×C8—32no—
274C2×C3×C8—62no—
275C2×C3×C8—62no—
276C3×C4⋊C4—41no—
277C3×C4⋊C4—41no—
278C3×C4⋊C4—42no—
279C3×C42—21no—
280C3×C42C22×D311yescharacteristic
281C3×C42C23×C311yescharacteristic
282#216—22no—
283#216C3×D412yescomplemented
284#215—22no—
285#215—22no—
286#215—22no—
287#215—22no—
288#215—22no—
289#215—22no—
290#215C3×D412yescomplemented
291#214C3×D412yescomplemented
292#213—22no—
293#213C3×D412yescomplemented
294#208—22no—
295#208C23×C312yes—
296C2×C3⋊C8—22no—
297C2×C3⋊C8C23×C312yes—
298D24—22no—
299#204—22no—
300ΓL(1,𝔽25)—22no—
301D3×C8—22no—
302C22×C32—22no—
303C22×C32H(5,𝔽2)11yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—42no—
306C2×C3×D3—82no—
307C2×C3×D3—162no—
308C32×C4—22no—
309C32×C4—22no—
310C32×C4—41no—
311C32×C4—41no—
312C32×C4C22×D412yes—
313C3×Dic3—22no—
3142−1+4—31no—
3152−1+4C2×C3×D311yescharacteristic
316H(5,𝔽2)—32no—
317#137—32no—
318#137—62no—
319#137—62no—
320#137C2×C3×D312yescomplemented
321AGL(1,ℤ/8)—32no—
322AGL(1,ℤ/8)—32no—
323AGL(1,ℤ/8)—62no—
324AGL(1,ℤ/8)—62no—
325#135—32no—
326#135—62no—
327#135—62no—
328C2×SD16—32no—
329C2×SD16—32no—
330C2×SD16—62no—
331C2×SD16—62no—
332#131—32no—
333#131—32no—
334C4⋊Q8C2×C3×D311yescharacteristic, complemented
335#127—31no—
336#108—32no—
337#104—32no—
338#104C2×C3×D312yescomplemented
339#101—31no—
340#101C2×C3×D311yescharacteristic, complemented
341#100—32no—
342C4×C8—31no—
343C23×C3—62no—
344C23×C3—122no—
345C22×D3—22no—
346C22×D3—42no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—32no—
353C3×Q8—42no—
354C3×Q8—42no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—42no—
358C3×Q8—121no—
359C3×Q8D3×D412yes—
360C3×Q8C2×C3×D412yes—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—32no—
365C3×D4—42no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—42no—
369C3×D4—62no—
370C3×D4—62no—
371C3×D4—62no—
372C3×D4—122no—
373C3×D4—121no—
374C3×D4—122no—
375C3×D4—242no—
376C3×D4D3×D412yes—
377C3×D4C2×C3×D412yes—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—44no—
394C2×C3×C4—62no—
395C2×C3×C4—122no—
396C2×C3×C4—122no—
397C2×C3×C4C23×D311yescharacteristic
398C2×C3×C4C24×C311yescharacteristic
399#67—22no—
400#67—42no—
401D12—22no—
402D12—22no—
403D12—42no—
404D12—41no—
405D12—82no—
406D12C2×C3×D412yes—
407C4×D3—22no—
408C4×D3—42no—
409C4×D3—42no—
410Dic6—41no—
411Dic6C2×C3×D412yes—
412C3×C8—22no—
413C3×C8—22no—
414C3×C8—32no—
415C3×C8—32no—
416C3×C8—42no—
417C3×C8—62no—
418C3⋊C8—22no—
419C3⋊C8C2×C3×D412yes—
420C3⋊C8C2×C3×D412yes—
421C2×C32—21no—
422C2×C32—42no—
423C2×C32D4211yescharacteristic
424C3×D3—42no—
425C3×D3—82no—
426#41—22no—
427#41—32no—
428#41—41no—
429#41—62no—
430#41—62no—
431#41—62no—
432#41—121no—
433#41C22×C3×D312yes—
434C2×Q8—22no—
435C2×Q8—62no—
436C2×Q8C22×C3×D311yescharacteristic
437C2×Q8C22×C3×D311yescharacteristic
438C2×D4—32no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—121no—
442C2×D4—122no—
443Q16—22no—
444Q16—32no—
445Q16—62no—
446SD16—22no—
447SD16—32no—
448SD16—32no—
449SD16—62no—
450SD16—62no—
451SD16—62no—
452SD16—62no—
453SD16—62no—
454SD16—62no—
455SD16—62no—
456D8—32no—
457D8—62no—
458D8—62no—
459M16—32no—
460M16—62no—
461M16—62no—
462M16C22×C3×D312yes—
463C2×C8—32no—
464C2×C8—62no—
465C2×C8—62no—
466C4⋊C4—41no—
467C42C22×C3×D311yescharacteristic
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—42no—
472C22×C3—62no—
473C22×C3—122no—
474C22×C3—242no—
475C22×C3—482no—
476C22×C3#100411yescharacteristic
477C22×C3C3×H(5,𝔽2)11yescharacteristic
478C2×D3—22no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—162no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—62no—
495C3×C4—81no—
496C3×C4C2×D3×D412yes—
497C3×C4C22×C3×D412yes—
498Dic3—22no—
499C32#329111yescharacteristic, complemented, Sylow
500C23—62no—
501C23—122no—
502Q8—22no—
503Q8—22no—
504Q8—32no—
505Q8—41no—
506Q8—121no—
507Q8C3×D3×D412yes—
508D4—22no—
509D4—32no—
510D4—41no—
511D4—62no—
512D4—62no—
513D4—62no—
514D4—122no—
515D4—121no—
516D4—122no—
517D4—242no—
518D4C3×D3×D412yes—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—62no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4C23×C3×D311yescharacteristic
528C8—22no—
529C8—32no—
530C8—32no—
531C8—62no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—41no—
536C2×C3—42no—
537C2×C3—42no—
538C2×C3—82no—
539C2×C3—122no—
540C2×C3—242no—
541C2×C3#559011yescharacteristic
542C2×C3C3×D4211yescharacteristic
543D3—42no—
544D3—82no—
545C22—22no—
546C22—62no—
547C22—122no—
548C22—242no—
549C22—482no—
550C22C3×#100411yescharacteristic
551C4—22no—
552C4—22no—
553C4—41no—
554C4—41no—
555C4—62no—
556C4C2×C3×D3×D412yes—
557C3—21no—
558C3#8030211yescharacteristic, direct product factor
559C3C3×#329111yescharacteristic, complemented
560C2—21no—
561C2—42no—
562C2—122no—
563C2—242no—
564C2C3×#559011yescharacteristic
subgroupG/H
centerC2×C3#5590
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#456C2
socleC2×C32D42
hypercenterC3×#456D3
hypocenterC3C3×#3291
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#456

Complements

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

NH ≅ G/Ntype
1. C3×#5837C2semidirect
2. C3×#5778C2semidirect
3. C32×#456C2semidirect
4. C3×#5203C2semidirect
5. C3×#5095C2semidirect
6. C3×#5085C2semidirect
7. C3×#4824C2semidirect
8. C3×#4822C2semidirect
9. C3×#4754C2semidirect
11. #80302C3direct (N × H)
13. C3×#1009C22semidirect
15. C32×#137C22semidirect
16. C32×C4⋊Q8C22semidirect
21. C3×#948C22semidirect
23. C3×#942C22semidirect
25. C2×C3×#215C22semidirect
26. C2×C3×#215C22semidirect
32. C3×#907C22semidirect
34. C32×#104C22semidirect
35. C32×#101C22semidirect
36. C3×#836C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
39. C3×#805C22semidirect
40. C3×C4×C3⋊C8C22semidirect
43. #5837C2×C3semidirect
44. #5778C2×C3semidirect
50. C3×#456D3semidirect
51. C3×#456C2×C3semidirect
54. #5203C2×C3semidirect
55. #5095C2×C3semidirect
56. #5085C2×C3semidirect
57. #4824C2×C3semidirect
58. #4822C2×C3semidirect
59. #4754C2×C3semidirect
83. C3×#216D4semidirect
90. C3×#215D4semidirect
91. C3×#214D4semidirect
93. C3×#213D4semidirect
108. #1009C22×C3semidirect
114. C3×#137C2×D3semidirect
115. C3×#137C22×C3semidirect
130. C3×C4⋊Q8C2×D3semidirect
131. C3×C4⋊Q8C22×C3semidirect
137. #948C22×C3semidirect
139. #942C22×C3semidirect
141. C2×#215C22×C3semidirect
142. C2×#215C22×C3semidirect
148. #907C22×C3semidirect
153. C3×#104C2×D3semidirect
154. C3×#104C22×C3semidirect
157. C3×#101C2×D3semidirect
158. C3×#101C22×C3semidirect
161. #836C22×C3semidirect
162. #823C22×C3semidirect
163. #822C22×C3semidirect
164. #805C22×C3semidirect
165. C4×C3⋊C8C22×C3semidirect
204. #456C3×D3semidirect
283. #216C3×D4semidirect
290. #215C3×D4semidirect
291. #214C3×D4semidirect
293. #213C3×D4semidirect
320. #137C2×C3×D3semidirect
334. C4⋊Q8C2×C3×D3semidirect
338. #104C2×C3×D3semidirect
340. #101C2×C3×D3semidirect
499. C32#3291semidirect
558. C3#80302direct (N × H)
559. C3C3×#3291semidirect

Maximal subgroups

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

subgroupindex
C3×#58372
C3×#57782
C32×#4562
C3×#52032
C3×#50952
C3×#50852
C3×#48242
C3×#48222
C3×#47542
C3×#32913
#803023