Finite Group Databasebeta

← C3×#78718 #11865204

Subgroups — C3×#78718 #11865204

Order 1152.

1483 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#909C212yesmaximal, complemented
4C32×#413C211yesmaximal, characteristic, complemented
5C3×#5139C212yesmaximal, complemented
6C3×#4752C214yesmaximal, complemented
7C3×#4746C214yesmaximal, complemented
8C3×#3022—31nomaximal
9#78718C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007C2212yes—
15C3×#1004—22no—
16C3×#1004C2212yes—
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19C22×C3×Dic6C2211yescharacteristic, complemented
20C2×C32×M16C2212yescomplemented
21C3×#944C2212yescomplemented
22C3×#936C2212yescomplemented
23C3×#935—22no—
24C3×#926C2211yescharacteristic, complemented
25C3×#924—22no—
26C3×#909—24no—
27C3×#909C2218yescomplemented
28C2×C3×Dic12—22no—
29C2×C3×#204—22no—
30C32×#101C2212yescomplemented
31C32×#100C2212yescomplemented
32C3×#825C2218yescomplemented
33C3×#585—31no—
34#5964C2×C311yescharacteristic
35C2×C3×#137—32no—
36C3×#536—31no—
37#5825C2×C311yescharacteristic, complemented
38C2×#909C2×C312yescomplemented
39C3×#456—34no—
40C3×#455—34no—
41C3×#421—32no—
42C3×#413—21no—
43C3×#413D311yescharacteristic, complemented
44C3×#413C2×C311yescharacteristic, complemented
45#5139C2×C312yescomplemented
46#4752C2×C314yescomplemented
47#4746C2×C314yescomplemented
48C32×#41—22no—
49C32×#41—22no—
50C2×C32×Q8C2311yescharacteristic
51C2×C32×D4D412yes—
52C2×C32×D4C2311yescharacteristic
53C22×C32×C4D412yes—
54C22×C32×C4C2311yescharacteristic
55C2×C3×#67—24no—
56C2×C3×#67—22no—
57C22×C3×Dic3—21no—
58C3×#239—24no—
59C3×D3×Q8—24no—
60C3×#237—28no—
61C3×#237—24no—
62C3×D3×D4—28no—
63C3×D3×D4—24no—
64C3×#235—24no—
65C3×#235—24no—
66C3×#235—22no—
67C3×#235—24no—
68C3×#235—22no—
69C3×#235D414yes—
70C3×#235C2314yes—
71C2×C3×D12—22no—
72C2×C3×D12D412yes—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3—22no—
75C2×C3×Dic6—22no—
76C2×C3×Dic6—22no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6D412yes—
80C2×C3×Dic6C2312yes—
81C32×M16—22no—
82C32×M16C2314yes—
83C2×C32×C8—22no—
84C3×#217—24no—
85C3×#217—21no—
86C3×C4⋊Dic3—21no—
87C3×Dic3⋊C4—24no—
88C3×C4×Dic3C2312yes—
89C3×Dic12—28no—
90C3×#204—28no—
91#3022—31noSylow
92C3×2−1+4—32no—
93C3×2−1+4—62no—
94C3×H(5,𝔽2)—32no—
95C3×H(5,𝔽2)—62no—
96C3×H(5,𝔽2)—62no—
97C2×C3×#41—21no—
98C2×C3×#41—32no—
99C2×C3×#41—64no—
100C2×C3×#41—62no—
101C2×C3×#41C2×D311yescharacteristic
102C2×C3×#41C22×C311yescharacteristic
103C22×C3×Q8—31no—
104#1010—22no—
105#1009—22no—
106D3×#41—24no—
107#1007C22×C312yes—
108#1004—22no—
109#1004C22×C312yes—
110C2×#235—22no—
111C2×#235C22×C312yes—
112C22×Dic6C22×C311yescharacteristic, complemented
113C3×#137—38no—
114C3×#137—64no—
115C2×C3×Q16—62no—
116C2×C3×SD16—62no—
117C2×C3×M16—22no—
118C2×C3×M16C2×D312yescomplemented
119C2×C3×M16C22×C312yescomplemented
120C3×C4⋊Q8—32no—
121C3×#124—32no—
122C3×#123—62no—
123C3×C22⋊Q8—62no—
124C3×#117—31no—
125#944C22×C312yescomplemented
126#936C22×C312yescomplemented
127#935—22no—
128#926C22×C311yescharacteristic, complemented
129#924—22no—
130#909—24no—
131#909C22×C318yescomplemented
132C2×Dic12—22no—
133C2×#204—22no—
134C3×#104—38no—
135C3×#101—22no—
136C3×#101C2×D312yescomplemented
137C3×#101C22×C312yescomplemented
138C3×#100—22no—
139C3×#100C2×D312yescomplemented
140C3×#100C22×C312yescomplemented
141#825C22×C318yescomplemented
142C23×C32C2×D412yes—
143C23×C32C2×D411yescharacteristic
144C22×C3×D3—24no—
145C22×C3×D3—22no—
146C32×Q8—22no—
147C32×D4—24no—
148C32×D4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2×D412yes—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2411yescharacteristic
156C3×#67—28no—
157C3×#67—24no—
158C3×#67—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—42no—
164C3×D12—24no—
165C3×D12—22no—
166C3×D12C2×D414yes—
167C3×C4×D3—28no—
168C3×C4×D3—24no—
169C3×C4×D3—24no—
170C3×Dic6—24no—
171C3×Dic6—22no—
172C3×Dic6—24no—
173C3×Dic6—41no—
174C3×Dic6—41no—
175C3×Dic6C2×D414yes—
176C32×C8—24no—
177#585—31no—
178C2×#137—32no—
179#536—31no—
180#456—34no—
181#455—34no—
182#421—32no—
183#413C3×D311yescharacteristic, complemented
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—34no—
189C3×#41—34no—
190C3×#41—42no—
191C3×#41—42no—
192C3×#41—64no—
193C3×#41—64no—
194C3×#41—64no—
195C3×#41—68no—
196C3×#41—64no—
197C3×#41—62no—
198C3×#41—64no—
199C3×#41—62no—
200C2×C3×Q8—21no—
201C2×C3×Q8—32no—
202C2×C3×Q8—32no—
203C2×C3×Q8—62no—
204C2×C3×Q8—64no—
205C2×C3×Q8—61no—
206C2×C3×Q8—62no—
207C2×C3×Q8—62no—
208C2×C3×Q8C22×D311yescharacteristic
209C2×C3×Q8C23×C311yescharacteristic
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—32no—
213C2×C3×D4—62no—
214C2×C3×D4—64no—
215C2×C3×D4—68no—
216C2×C3×D4—62no—
217C2×C3×D4—64no—
218C2×C3×D4D1212yes—
219C2×C3×D4C3×D412yes—
220C2×C3×D4C22×D311yescharacteristic
221C2×C3×D4C23×C311yescharacteristic
222C22×C3×C4—22no—
223C22×C3×C4—21no—
224C22×C3×C4—64no—
225C22×C3×C4—62no—
226C22×C3×C4—61no—
227C22×C3×C4D1212yes—
228C22×C3×C4C3×D412yes—
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C23×C311yescharacteristic
231C2×#67—24no—
232C2×#67—22no—
233C22×Dic3—21no—
234#239—24no—
235D3×Q8—24no—
236#237—28no—
237#237—24no—
238D3×D4—28no—
239D3×D4—24no—
240#235—24no—
241#235—24no—
242#235—22no—
243#235—24no—
244#235—22no—
245#235C3×D414yes—
246#235C23×C314yes—
247C2×D12—22no—
248C2×D12C3×D412yes—
249C2×C4×D3—24no—
250C2×C4×D3—22no—
251C2×Dic6—22no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—21no—
255C2×Dic6C3×D412yes—
256C2×Dic6C23×C312yes—
257C3×Q16—68no—
258C3×SD16—68no—
259C3×M16—22no—
260C3×M16—22no—
261C3×M16—24no—
262C3×M16—42no—
263C3×M16C22×D314yes—
264C3×M16C23×C314yes—
265C2×C3×C8—22no—
266C2×C3×C8—22no—
267C2×C3×C8—42no—
268C3×C4⋊C4—64no—
269C3×C4⋊C4—61no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—64no—
272C3×C42—32no—
273#217—24no—
274#217—21no—
275C4⋊Dic3—21no—
276Dic3⋊C4—24no—
277C4×Dic3C23×C312yes—
278Dic12—28no—
279#204—28no—
280C22×C32—22no—
281C22×C32—41no—
282C22×C32—42no—
283C22×C32#12012yes—
284C22×C32C22×D411yescharacteristic
285C2×C3×D3—24no—
286C2×C3×D3—82no—
287C2×C3×D3—84no—
288C32×C4—22no—
289C32×C4#12011yescharacteristic
290C32×C4#12011yescharacteristic
291C32×C4C22×D412yes—
292C3×Dic3—24no—
293C3×Dic3—42no—
294C3×Dic3—42no—
2952−1+4—32no—
2962−1+4—62no—
297H(5,𝔽2)—32no—
298H(5,𝔽2)—62no—
299H(5,𝔽2)—62no—
300C2×#41—32no—
301C2×#41—64no—
302C2×#41—62no—
303C2×#41C2×C3×D311yescharacteristic
304C22×Q8—31no—
305#137—38no—
306#137—64no—
307C2×Q16—62no—
308C2×SD16—62no—
309C2×M16C2×C3×D312yescomplemented
310C4⋊Q8—32no—
311#124—32no—
312#123—62no—
313C22⋊Q8—62no—
314#117—31no—
315#104—38no—
316#101C2×C3×D312yescomplemented
317#100C2×C3×D312yescomplemented
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—64no—
321C23×C3—62no—
322C23×C3C2×D1212yes—
323C23×C3C2×D1211yescharacteristic
324C23×C3C2×C3×D412yes—
325C23×C3C2×C3×D411yescharacteristic
326C22×D3—24no—
327C22×D3—22no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—34no—
331C3×Q8—42no—
332C3×Q8—64no—
333C3×Q8—62no—
334C3×Q8—64no—
335C3×Q8—121no—
336C3×Q8—121no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—34no—
342C3×D4—42no—
343C3×D4—44no—
344C3×D4—64no—
345C3×D4—68no—
346C3×D4—64no—
347C3×D4—64no—
348C3×D4—62no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—68no—
361C2×C3×C4—64no—
362C2×C3×C4—62no—
363C2×C3×C4—64no—
364C2×C3×C4—64no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—122no—
368C2×C3×C4C2×D1212yes—
369C2×C3×C4C2×D1211yescharacteristic
370C2×C3×C4D3×D412yes—
371C2×C3×C4D3×D412yes—
372C2×C3×C4C2×C3×D412yes—
373C2×C3×C4C2×C3×D411yescharacteristic
374C2×C3×C4C2×C3×D412yes—
375C2×C3×C4C2×C3×D412yes—
376C2×C3×C4C23×D311yescharacteristic
377C2×C3×C4C24×C311yescharacteristic
378#67—28no—
379#67—24no—
380#67—24no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—42no—
386D12—24no—
387D12—22no—
388D12C2×C3×D414yes—
389C4×D3—28no—
390C4×D3—24no—
391C4×D3—24no—
392Dic6—24no—
393Dic6—22no—
394Dic6—24no—
395Dic6—41no—
396Dic6—41no—
397Dic6C2×C3×D414yes—
398C3×C8—24no—
399C3×C8—24no—
400C3×C8—44no—
401C2×C32—21no—
402C2×C32—22no—
403C2×C32—42no—
404C2×C32C2×#12011yescharacteristic
405C3×D3—44no—
406#41—22no—
407#41—22no—
408#41—34no—
409#41—34no—
410#41—64no—
411#41—64no—
412#41—64no—
413#41—68no—
414#41—64no—
415#41—62no—
416#41—64no—
417#41—62no—
418C2×Q8—32no—
419C2×Q8—32no—
420C2×Q8—62no—
421C2×Q8—64no—
422C2×Q8—61no—
423C2×Q8—62no—
424C2×Q8—62no—
425C2×Q8C22×C3×D311yescharacteristic
426C2×D4—32no—
427C2×D4—62no—
428C2×D4—64no—
429C2×D4—68no—
430C2×D4—62no—
431C2×D4—64no—
432C2×D4C3×D1212yes—
433C2×D4C22×C3×D311yescharacteristic
434C22×C4—64no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4C3×D1212yes—
438C22×C4C22×C3×D311yescharacteristic
439Q16—68no—
440SD16—68no—
441M16—22no—
442M16C22×C3×D314yes—
443C2×C8—22no—
444C4⋊C4—61no—
445C4⋊C4—64no—
446C22⋊C4—61no—
447C22⋊C4—64no—
448C42—32no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—64no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—242no—
462C22×C3—244no—
463C22×C3#88212yes—
464C22×C3C3×#12012yes—
465C22×C3C22×D1211yescharacteristic
466C22×C3C22×C3×D411yescharacteristic
467C2×D3—24no—
468C2×D3—82no—
469C2×D3—84no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—64no—
477C3×C4—122no—
478C3×C4—122no—
479C3×C4#88211yescharacteristic
480C3×C4#88211yescharacteristic
481C3×C4C3×#12011yescharacteristic
482C3×C4C3×#12011yescharacteristic
483C3×C4C2×D3×D412yes—
484C3×C4C22×C3×D412yes—
485Dic3—24no—
486Dic3—42no—
487Dic3—42no—
488C32#302211yescharacteristic, complemented, Sylow
489C23—64no—
490C23—62no—
491C23C2×C3×D1212yes—
492C23C2×C3×D1211yescharacteristic
493Q8—22no—
494Q8—34no—
495Q8—64no—
496Q8—62no—
497Q8—64no—
498Q8—121no—
499Q8—121no—
500D4—24no—
501D4—22no—
502D4—34no—
503D4—64no—
504D4—68no—
505D4—64no—
506D4—64no—
507D4—62no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—68no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—122no—
518C2×C4C2×C3×D1212yes—
519C2×C4C2×C3×D1211yescharacteristic
520C2×C4C3×D3×D412yes—
521C2×C4C3×D3×D412yes—
522C2×C4C23×C3×D311yescharacteristic
523C8—24no—
524C2×C3—21no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—42no—
531C2×C3—42no—
532C2×C3—42no—
533C2×C3—82no—
534C2×C3—124no—
535C2×C3C2×#88211yescharacteristic
536C2×C3C2×C3×#12011yescharacteristic
537D3—44no—
538C22—22no—
539C22—41no—
540C22—42no—
541C22—64no—
542C22—242no—
543C22—244no—
544C22C3×#88212yes—
545C22C22×C3×D1211yescharacteristic
546C4—22no—
547C4—64no—
548C4—122no—
549C4—122no—
550C4C3×#88211yescharacteristic
551C4C3×#88211yescharacteristic
552C4C2×C3×D3×D412yes—
553C3—21no—
554C3#7871811yescharacteristic, direct product factor
555C3C3×#302211yescharacteristic, complemented
556C2—21no—
557C2—22no—
558C2—42no—
559C2—124no—
560C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC2×C3C2×#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#413C2
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC3×#413D3
hypocenterC3C3×#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×C3×#909C2semidirect
4. C32×#413C2semidirect
5. C3×#5139C2semidirect
6. C3×#4752C2semidirect
7. C3×#4746C2semidirect
9. #78718C3direct (N × H)
19. C22×C3×Dic6C22semidirect
20. C2×C32×M16C22semidirect
21. C3×#944C22semidirect
22. C3×#936C22semidirect
24. C3×#926C22semidirect
27. C3×#909C22semidirect
30. C32×#101C22semidirect
31. C32×#100C22semidirect
32. C3×#825C22semidirect
37. #5825C2×C3semidirect
38. C2×#909C2×C3semidirect
43. C3×#413D3semidirect
44. C3×#413C2×C3semidirect
45. #5139C2×C3semidirect
46. #4752C2×C3semidirect
47. #4746C2×C3semidirect
112. C22×Dic6C22×C3semidirect
118. C2×C3×M16C2×D3semidirect
119. C2×C3×M16C22×C3semidirect
125. #944C22×C3semidirect
126. #936C22×C3semidirect
128. #926C22×C3semidirect
131. #909C22×C3semidirect
136. C3×#101C2×D3semidirect
137. C3×#101C22×C3semidirect
139. C3×#100C2×D3semidirect
140. C3×#100C22×C3semidirect
141. #825C22×C3semidirect
183. #413C3×D3semidirect
309. C2×M16C2×C3×D3semidirect
316. #101C2×C3×D3semidirect
317. #100C2×C3×D3semidirect
488. C32#3022semidirect
554. C3#78718direct (N × H)
555. C3C3×#3022semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C3×#58252
C2×C3×#9092
C32×#4132
C3×#51392
C3×#47522
C3×#47462
C3×#30223
#787183