Finite Group Databasebeta

← C3×#80288 #11886667

Subgroups — C3×#80288 #11886667

Order 1152.

969 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5838C212yesmaximal, complemented
2C3×#5775C212yesmaximal, complemented
3C32×#454C211yesmaximal, characteristic, complemented
4C3×#5200C212yesmaximal, complemented
5C3×#5061C211yesmaximal, characteristic, complemented
6C3×#5060C211yesmaximal, characteristic, complemented
7C3×#4826C212yesmaximal, complemented
8C3×#4823C212yesmaximal, complemented
9C3×#4755C212yesmaximal, complemented
10C3×#3289—31nomaximal
11#80288C311yesmaximal, characteristic, direct product factor
12C32×2−1+4C2211yescharacteristic, complemented
13C3×#1010C2212yes—
14C3×#1004C2211yescharacteristic
15C32×AGL(1,ℤ/8)C2212yes—
16C32×#124C2211yescharacteristic
17C3×#951—22no—
18C3×#951C2212yescomplemented
19C3×#950—22no—
20C3×#950C2212yes—
21C3×#948C2212yescomplemented
22C2×C3×#216—22no—
23C2×C3×#216C2212yescomplemented
24C3×#942—22no—
25C3×#942C2212yes—
26C2×C3×#214C2212yes—
27C3×#915—22no—
28C3×D3×SD16—22no—
29C3×#912—22no—
30C3×#911—22no—
31C3×#909C2212yes—
32C3×#907C2212yes—
33C3×#869C2211yescharacteristic, complemented
34C32×#104C2212yescomplemented
35C32×#101C2211yescharacteristic, complemented
36C3×#833C2211yescharacteristic
37C3×#824C2212yescomplemented
38C3×#822C2212yescomplemented
39C3×#805C2212yes—
40C3×#803C2211yescharacteristic, complemented
41C3×#578—32no—
42C3×#577—32no—
43#5838C2×C312yescomplemented
44#5775C2×C312yescomplemented
45C3×#497—31no—
46C3×#473—32no—
47C3×#471—32no—
48C3×#456—32no—
49C3×#455—31no—
50C3×#454—21no—
51C3×#454D311yescharacteristic, complemented
52C3×#454C2×C311yescharacteristic, complemented
53C3×#444—32no—
54#5200C2×C312yescomplemented
55#5061C2×C311yescharacteristic, complemented
56#5060C2×C311yescharacteristic, complemented
57#4826C2×C312yescomplemented
58#4823C2×C312yescomplemented
59#4755C2×C312yescomplemented
60C32×#41—22no—
61C32×#41—41no—
62C32×#41C2312yes—
63C2×C32×Q8—22no—
64C2×C32×Q8C2311yescharacteristic
65C2×C32×D4C2311yescharacteristic
66C2×C3×#67—41no—
67C3×D3×Q8—22no—
68C3×#237—22no—
69C3×#237—22no—
70C3×#237—42no—
71C3×D3×D4—22no—
72C3×#235—22no—
73C3×#235C2312yes—
74C2×C3×Dic6—22no—
75C2×C3×Dic6C2312yes—
76C32×SD16—22no—
77C32×D8—22no—
78C32×M16C2312yes—
79C32×C22⋊C4—41no—
80C32×C42C2311yescharacteristic
81C3×#216—22no—
82C3×#216—22no—
83C3×#216—22no—
84C3×#216—22no—
85C3×#216D412yes—
86C3×#215—22no—
87C3×#215D412yes—
88C3×#214—22no—
89C3×#214—22no—
90C3×#214—22no—
91C3×#214D412yes—
92C3×#213D412yes—
93C3×C4⋊Dic3—41no—
94C3×#208—22no—
95C3×#208C2312yes—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2312yes—
98C3×Dic12—22no—
99C3×#204—22no—
100C3×ΓL(1,𝔽25)—22no—
101C3×D3×C8—22no—
102#3289—31noSylow
103C3×2−1+4—21no—
104C3×2−1+4—32no—
105C3×2−1+4C2×D311yescharacteristic, complemented
106C3×2−1+4C22×C311yescharacteristic, complemented
107C3×H(5,𝔽2)—31no—
108#1010C22×C312yes—
109#1004C22×C311yescharacteristic
110C3×#137—32no—
111C3×#137—32no—
112C3×#137—32no—
113C3×#137—62no—
114C3×#137—62no—
115C3×#137—62no—
116C3×AGL(1,ℤ/8)—22no—
117C3×AGL(1,ℤ/8)—62no—
118C3×AGL(1,ℤ/8)C2×D312yes—
119C3×AGL(1,ℤ/8)C22×C312yes—
120C3×#135—32no—
121C3×#135—62no—
122C3×#135—62no—
123C2×C3×Q16—32no—
124C2×C3×Q16—62no—
125C2×C3×SD16—32no—
126C2×C3×SD16—62no—
127C3×#131—32no—
128C3×#131—32no—
129C3×C4⋊Q8—31no—
130C3×#124—21no—
131C3×#124C2×D311yescharacteristic
132C3×#124C22×C311yescharacteristic
133#951—22no—
134#951C22×C312yescomplemented
135#950—22no—
136#950C22×C312yes—
137#948C22×C312yescomplemented
138C2×#216—22no—
139C2×#216C22×C312yescomplemented
140#942—22no—
141#942C22×C312yes—
142C2×#214C22×C312yes—
143#915—22no—
144D3×SD16—22no—
145#912—22no—
146#911—22no—
147#909C22×C312yes—
148#907C22×C312yes—
149#869C22×C311yescharacteristic, complemented
150C3×#108—32no—
151C3×#104—22no—
152C3×#104—32no—
153C3×#104C2×D312yescomplemented
154C3×#104C22×C312yescomplemented
155C3×#101—21no—
156C3×#101—32no—
157C3×#101C2×D311yescharacteristic, complemented
158C3×#101C22×C311yescharacteristic, complemented
159C3×#100—31no—
160C3×#97—31no—
161#833C22×C311yescharacteristic
162#824C22×C312yescomplemented
163#822C22×C312yescomplemented
164#805C22×C312yes—
165#803C22×C311yescharacteristic, complemented
166C23×C32—21no—
167C22×C3×D3—41no—
168C32×Q8—22no—
169C32×Q8—41no—
170C32×Q8C2×D412yes—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—41no—
174C32×D4C2×D412yes—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—42no—
179C2×C32×C4C2411yescharacteristic
180C3×#67—22no—
181C3×#67—42no—
182C3×#67—42no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—42no—
186C3×D12C2×D412yes—
187C3×C4×D3—22no—
188C3×C4×D3—42no—
189C3×Dic6—22no—
190C3×Dic6—22no—
191C3×Dic6—42no—
192C3×Dic6C2×D412yes—
193C32×C8—22no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8C2×D412yes—
196C3×C3⋊C8C2×D412yes—
197#578—32no—
198#577—32no—
199#497—31no—
200#473—32no—
201#471—32no—
202#456—32no—
203#455—31no—
204#454C3×D311yescharacteristic, complemented
205#444—32no—
206C3×#41—22no—
207C3×#41—22no—
208C3×#41—22no—
209C3×#41—32no—
210C3×#41—42no—
211C3×#41—41no—
212C3×#41—41no—
213C3×#41—42no—
214C3×#41—62no—
215C3×#41—62no—
216C3×#41—62no—
217C3×#41—122no—
218C3×#41C22×D312yes—
219C3×#41C23×C312yes—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—21no—
223C2×C3×Q8—32no—
224C2×C3×Q8—42no—
225C2×C3×Q8—62no—
226C2×C3×Q8—62no—
227C2×C3×Q8C22×D311yescharacteristic
228C2×C3×Q8C23×C311yescharacteristic
229C2×C3×D4—21no—
230C2×C3×D4—62no—
231C2×C3×D4—121no—
232C2×C3×D4C22×D311yescharacteristic
233C2×C3×D4C23×C311yescharacteristic
234C2×#67—41no—
235D3×Q8—22no—
236#237—22no—
237#237—22no—
238#237—42no—
239D3×D4—22no—
240#235—22no—
241#235C23×C312yes—
242C2×Dic6—22no—
243C2×Dic6C23×C312yes—
244C3×Q16—32no—
245C3×Q16—62no—
246C3×Q16—62no—
247C3×Q16—62no—
248C3×Q16—62no—
249C3×Q16—62no—
250C3×SD16—22no—
251C3×SD16—22no—
252C3×SD16—32no—
253C3×SD16—32no—
254C3×SD16—42no—
255C3×SD16—62no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—62no—
260C3×D8—22no—
261C3×D8—22no—
262C3×D8—32no—
263C3×D8—42no—
264C3×M16—22no—
265C3×M16—32no—
266C3×M16—62no—
267C3×M16—62no—
268C3×M16C22×D312yes—
269C3×M16C23×C312yes—
270C2×C3×C8—32no—
271C2×C3×C8—62no—
272C2×C3×C8—62no—
273C3×C4⋊C4—121no—
274C3×C22⋊C4—41no—
275C3×C22⋊C4—41no—
276C3×C22⋊C4—42no—
277C3×C42—21no—
278C3×C42C22×D311yescharacteristic
279C3×C42C23×C311yescharacteristic
280#216—22no—
281#216—22no—
282#216—22no—
283#216—22no—
284#216C3×D412yes—
285#215—22no—
286#215C3×D412yes—
287#214—22no—
288#214—22no—
289#214—22no—
290#214C3×D412yes—
291#213C3×D412yes—
292C4⋊Dic3—41no—
293#208—22no—
294#208C23×C312yes—
295C2×C3⋊C8—22no—
296C2×C3⋊C8C23×C312yes—
297Dic12—22no—
298#204—22no—
299ΓL(1,𝔽25)—22no—
300D3×C8—22no—
301C22×C32—22no—
302C22×C32—41no—
303C22×C32—81no—
304C22×C32H(5,𝔽2)11yescharacteristic
305C2×C3×D3—22no—
306C2×C3×D3—161no—
307C32×C4—22no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4C22×D412yes—
311C3×Dic3—22no—
312C3×Dic3—42no—
3132−1+4—32no—
3142−1+4C2×C3×D311yescharacteristic, complemented
315H(5,𝔽2)—31no—
316#137—32no—
317#137—32no—
318#137—32no—
319#137—62no—
320#137—62no—
321#137—62no—
322AGL(1,ℤ/8)—62no—
323AGL(1,ℤ/8)C2×C3×D312yes—
324#135—32no—
325#135—62no—
326#135—62no—
327C2×Q16—32no—
328C2×Q16—62no—
329C2×SD16—32no—
330C2×SD16—62no—
331#131—32no—
332#131—32no—
333C4⋊Q8—31no—
334#124C2×C3×D311yescharacteristic
335#108—32no—
336#104—32no—
337#104C2×C3×D312yescomplemented
338#101—32no—
339#101C2×C3×D311yescharacteristic, complemented
340#100—31no—
341#97—31no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—41no—
345C23×C3—121no—
346C22×D3—41no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—32no—
351C3×Q8—42no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—42no—
355C3×Q8—62no—
356C3×Q8—62no—
357C3×Q8—122no—
358C3×Q8D3×D412yes—
359C3×Q8C2×C3×D412yes—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—32no—
366C3×D4—42no—
367C3×D4—42no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—42no—
371C3×D4—62no—
372C3×D4—122no—
373C3×D4—122no—
374C3×D4D3×D412yes—
375C3×D4C2×C3×D412yes—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—42no—
385C2×C3×C4—41no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—44no—
389C2×C3×C4—62no—
390C2×C3×C4—62no—
391C2×C3×C4—122no—
392C2×C3×C4—121no—
393C2×C3×C4—122no—
394C2×C3×C4C23×D311yescharacteristic
395C2×C3×C4C24×C311yescharacteristic
396#67—22no—
397#67—42no—
398#67—42no—
399C2×Dic3—22no—
400C2×Dic3—41no—
401C2×Dic3—42no—
402D12C2×C3×D412yes—
403C4×D3—22no—
404C4×D3—42no—
405Dic6—22no—
406Dic6—22no—
407Dic6—42no—
408Dic6C2×C3×D412yes—
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—32no—
412C3×C8—32no—
413C3×C8—42no—
414C3×C8—62no—
415C3⋊C8—22no—
416C3⋊C8C2×C3×D412yes—
417C3⋊C8C2×C3×D412yes—
418C2×C32—21no—
419C2×C32—42no—
420C2×C32—81no—
421C2×C32D4211yescharacteristic
422C3×D3—42no—
423#41—22no—
424#41—32no—
425#41—41no—
426#41—62no—
427#41—62no—
428#41—62no—
429#41—122no—
430#41C22×C3×D312yes—
431C2×Q8—22no—
432C2×Q8—32no—
433C2×Q8—62no—
434C2×Q8—62no—
435C2×Q8C22×C3×D311yescharacteristic
436C2×D4—62no—
437C2×D4—121no—
438C2×D4C22×C3×D311yescharacteristic
439Q16—32no—
440Q16—62no—
441Q16—62no—
442Q16—62no—
443Q16—62no—
444Q16—62no—
445SD16—22no—
446SD16—32no—
447SD16—32no—
448SD16—62no—
449SD16—62no—
450SD16—62no—
451SD16—62no—
452SD16—62no—
453D8—22no—
454D8—32no—
455M16—32no—
456M16—62no—
457M16—62no—
458M16C22×C3×D312yes—
459C2×C8—32no—
460C2×C8—62no—
461C2×C8—62no—
462C4⋊C4—121no—
463C22⋊C4—41no—
464C42C22×C3×D311yescharacteristic
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—42no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—62no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—82no—
476C22×C3—481no—
477C22×C3#100411yescharacteristic
478C22×C3C3×H(5,𝔽2)11yescharacteristic
479C2×D3—22no—
480C2×D3—161no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—62no—
491C3×C4—81no—
492C3×C4—122no—
493C3×C4C2×D3×D412yes—
494C3×C4C22×C3×D412yes—
495Dic3—22no—
496Dic3—42no—
497C32#328911yescharacteristic, complemented, Sylow
498C23—21no—
499C23—121no—
500Q8—22no—
501Q8—32no—
502Q8—41no—
503Q8—62no—
504Q8—62no—
505Q8—122no—
506Q8C3×D3×D412yes—
507D4—22no—
508D4—22no—
509D4—32no—
510D4—41no—
511D4—62no—
512D4—122no—
513D4—122no—
514D4C3×D3×D412yes—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4C23×C3×D311yescharacteristic
525C8—22no—
526C8—32no—
527C8—32no—
528C8—62no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—41no—
533C2×C3—42no—
534C2×C3—42no—
535C2×C3—82no—
536C2×C3—81no—
537C2×C3—81no—
538C2×C3—122no—
539C2×C3—161no—
540C2×C3#559011yescharacteristic
541C2×C3C3×D4211yescharacteristic
542D3—42no—
543C22—22no—
544C22—41no—
545C22—62no—
546C22—81no—
547C22—481no—
548C22C3×#100411yescharacteristic
549C4—22no—
550C4—41no—
551C4—41no—
552C4—62no—
553C4—122no—
554C4C2×C3×D3×D412yes—
555C3—21no—
556C3#8028811yescharacteristic, direct product factor
557C3C3×#328911yescharacteristic, complemented
558C2—21no—
559C2—42no—
560C2—81no—
561C2—122no—
562C2C3×#559011yescharacteristic
subgroupG/H
centerC2×C3#5590
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#454C2
socleC2×C32D42
hypercenterC3×#454D3
hypocenterC3C3×#3289
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#454

Complements

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

NH ≅ G/Ntype
1. C3×#5838C2semidirect
2. C3×#5775C2semidirect
3. C32×#454C2semidirect
4. C3×#5200C2semidirect
5. C3×#5061C2semidirect
6. C3×#5060C2semidirect
7. C3×#4826C2semidirect
8. C3×#4823C2semidirect
9. C3×#4755C2semidirect
11. #80288C3direct (N × H)
12. C32×2−1+4C22semidirect
18. C3×#951C22semidirect
21. C3×#948C22semidirect
23. C2×C3×#216C22semidirect
33. C3×#869C22semidirect
34. C32×#104C22semidirect
35. C32×#101C22semidirect
37. C3×#824C22semidirect
38. C3×#822C22semidirect
40. C3×#803C22semidirect
43. #5838C2×C3semidirect
44. #5775C2×C3semidirect
51. C3×#454D3semidirect
52. C3×#454C2×C3semidirect
54. #5200C2×C3semidirect
55. #5061C2×C3semidirect
56. #5060C2×C3semidirect
57. #4826C2×C3semidirect
58. #4823C2×C3semidirect
59. #4755C2×C3semidirect
105. C3×2−1+4C2×D3semidirect
106. C3×2−1+4C22×C3semidirect
134. #951C22×C3semidirect
137. #948C22×C3semidirect
139. C2×#216C22×C3semidirect
149. #869C22×C3semidirect
153. C3×#104C2×D3semidirect
154. C3×#104C22×C3semidirect
157. C3×#101C2×D3semidirect
158. C3×#101C22×C3semidirect
162. #824C22×C3semidirect
163. #822C22×C3semidirect
165. #803C22×C3semidirect
204. #454C3×D3semidirect
314. 2−1+4C2×C3×D3semidirect
337. #104C2×C3×D3semidirect
339. #101C2×C3×D3semidirect
497. C32#3289semidirect
556. C3#80288direct (N × H)
557. C3C3×#3289semidirect

Maximal subgroups

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

subgroupindex
C3×#58382
C3×#57752
C32×#4542
C3×#52002
C3×#50612
C3×#50602
C3×#48262
C3×#48232
C3×#47552
C3×#32893
#802883