Finite Group Databasebeta

← C3×#78709 #11865091

Subgroups — C3×#78709 #11865091

Order 1152.

1469 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C211yesmaximal, characteristic, complemented
2C3×#5816C211yesmaximal, characteristic, complemented
3C3×#5748C212yesmaximal, complemented
4C2×C32×#101C211yesmaximal, characteristic, complemented
5C3×#5139C212yesmaximal, complemented
6C3×#4753C214yesmaximal, complemented
7C3×#4752C214yesmaximal, complemented
8C3×#3020—31nomaximal
9#78709C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C3×#1007—22no—
12C3×#1007—24no—
13C3×#1007C2214yes—
14C2×C3×#239—22no—
15C2×C3×D3×Q8—22no—
16C2×C3×#235—22no—
17C2×C3×#235C2212yes—
18C2×C3×#235C2211yescharacteristic
19C2×C32×M16C2212yes—
20C3×#947C2212yescomplemented
21C3×#946—22no—
22C3×#944C2212yescomplemented
23C3×#929—22no—
24C3×#926C2211yescharacteristic
25C3×#909—22no—
26C3×#909C2214yescomplemented
27C3×#908—22no—
28C3×#908C2214yescomplemented
29C3×#904—24no—
30C32×#101C2214yescomplemented
31C3×#825C2218yescomplemented
32C2×C3×2−1+4—31no—
33C2×#1007C2×C311yescharacteristic, complemented
34C3×#575—32no—
35C3×#536—31no—
36#5816C2×C311yescharacteristic, complemented
37#5748C2×C312yescomplemented
38C3×#456—34no—
39C3×#454—34no—
40C3×#421—32no—
41C2×C3×#101—21no—
42C2×C3×#101D311yescharacteristic, complemented
43C2×C3×#101C2×C311yescharacteristic, complemented
44#5139C2×C312yescomplemented
45#4753C2×C314yescomplemented
46#4752C2×C314yescomplemented
47C2×C32×Q8—22no—
48C2×C32×Q8—22no—
49C2×C32×Q8D412yes—
50C2×C32×Q8C2312yes—
51C22×C32×C4D412yes—
52C22×C32×C4C2311yescharacteristic
53C2×C3×#67—22no—
54C2×C3×#67—21no—
55C3×#239—28no—
56C3×#239—28no—
57C3×D3×Q8—28no—
58C3×D3×Q8—28no—
59C3×#235—28no—
60C3×#235—28no—
61C3×#235—22no—
62C3×#235—22no—
63C3×#235—22no—
64C3×#235—22no—
65C3×#235D414yes—
66C3×#235C2314yes—
67C2×C3×D12—22no—
68C2×C3×D12D412yes—
69C2×C3×D12C2311yescharacteristic
70C2×C3×C4×D3—24no—
71C2×C3×C4×D3—22no—
72C2×C3×C4×D3—21no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6D412yes—
75C2×C3×Dic6C2311yescharacteristic
76C32×M16—22no—
77C32×M16C2314yes—
78C2×C32×C8—22no—
79C3×#217—21no—
80C3×(C2×D3)⋊C4—24no—
81C3×C4⋊Dic3—21no—
82C3×Dic3⋊C4—24no—
83C3×C4×Dic3C2312yes—
84C3×Dic12—24no—
85C3×D24—24no—
86C3×#204—24no—
87C3×#204—24no—
88#3020—31noSylow
89C3×2−1+4—34no—
90C3×2−1+4—64no—
91C3×2−1+4—62no—
92C2×C3×#41—32no—
93C2×C3×#41—31no—
94C2×C3×#41—62no—
95C2×C3×#41—62no—
96C22×C3×Q8—21no—
97C22×C3×Q8—62no—
98C22×C3×Q8C2×D311yescharacteristic
99C22×C3×Q8C22×C311yescharacteristic
100#1007—22no—
101#1007—24no—
102#1007C22×C314yes—
103C2×#239—22no—
104C2×D3×Q8—22no—
105C2×#235—22no—
106C2×#235C22×C312yes—
107C2×#235C22×C311yescharacteristic
108C3×#137—34no—
109C3×#137—62no—
110C3×AGL(1,ℤ/8)—34no—
111C3×AGL(1,ℤ/8)—62no—
112C3×#135—64no—
113C2×C3×M16—22no—
114C2×C3×M16C2×D312yes—
115C2×C3×M16C22×C312yes—
116C3×C4⋊Q8—32no—
117C3×#124—32no—
118C3×#123—62no—
119C3×C22⋊Q8—62no—
120C3×#117—31no—
121#947C22×C312yescomplemented
122#946—22no—
123#944C22×C312yescomplemented
124#929—22no—
125#926C22×C311yescharacteristic
126#909—22no—
127#909C22×C314yescomplemented
128#908—22no—
129#908C22×C314yescomplemented
130#904—24no—
131C3×#104—38no—
132C3×#101—24no—
133C3×#101C2×D314yescomplemented
134C3×#101C22×C314yescomplemented
135#825C22×C318yescomplemented
136C23×C32C2×D411yescharacteristic
137C22×C3×D3—22no—
138C22×C3×D3—21no—
139C32×Q8—24no—
140C32×Q8—24no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4C2×D414yes—
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D412yes—
146C2×C32×C4C2×D412yes—
147C2×C32×C4C2411yescharacteristic
148C3×#67—24no—
149C3×#67—24no—
150C3×#67—41no—
151C3×#67—41no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—21no—
155C3×D12—28no—
156C3×D12—22no—
157C3×D12—22no—
158C3×D12C2×D414yes—
159C3×C4×D3—216no—
160C3×C4×D3—24no—
161C3×C4×D3—24no—
162C3×C4×D3—42no—
163C3×Dic6—28no—
164C3×Dic6—22no—
165C3×Dic6—22no—
166C3×Dic6C2×D414yes—
167C32×C8—24no—
168C2×2−1+4—31no—
169#575—32no—
170#536—31no—
171#456—34no—
172#454—34no—
173#421—32no—
174C2×#101C3×D311yescharacteristic, complemented
175C3×#41—34no—
176C3×#41—34no—
177C3×#41—68no—
178C3×#41—68no—
179C3×#41—68no—
180C3×#41—62no—
181C3×#41—68no—
182C3×#41—62no—
183C3×#41—62no—
184C3×#41—62no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—32no—
192C2×C3×Q8—31no—
193C2×C3×Q8—42no—
194C2×C3×Q8—42no—
195C2×C3×Q8—62no—
196C2×C3×Q8—68no—
197C2×C3×Q8—68no—
198C2×C3×Q8D1212yes—
199C2×C3×Q8C3×D412yes—
200C2×C3×Q8C22×D312yes—
201C2×C3×Q8C23×C312yes—
202C2×C3×D4—32no—
203C2×C3×D4—31no—
204C2×C3×D4—62no—
205C2×C3×D4—62no—
206C2×C3×D4—61no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—64no—
210C22×C3×C4—62no—
211C22×C3×C4—61no—
212C22×C3×C4D1212yes—
213C22×C3×C4C3×D412yes—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C2×#67—22no—
217C2×#67—21no—
218#239—28no—
219#239—28no—
220D3×Q8—28no—
221D3×Q8—28no—
222#235—28no—
223#235—28no—
224#235—22no—
225#235—22no—
226#235—22no—
227#235—22no—
228#235C3×D414yes—
229#235C23×C314yes—
230C2×D12—22no—
231C2×D12C3×D412yes—
232C2×D12C23×C311yescharacteristic
233C2×C4×D3—24no—
234C2×C4×D3—22no—
235C2×C4×D3—21no—
236C2×Dic6—22no—
237C2×Dic6C3×D412yes—
238C2×Dic6C23×C311yescharacteristic
239C3×Q16—64no—
240C3×SD16—64no—
241C3×SD16—64no—
242C3×D8—64no—
243C3×M16—22no—
244C3×M16—22no—
245C3×M16—24no—
246C3×M16—42no—
247C3×M16C22×D314yes—
248C3×M16C23×C314yes—
249C2×C3×C8—22no—
250C2×C3×C8—22no—
251C2×C3×C8—42no—
252C3×C4⋊C4—64no—
253C3×C4⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—64no—
256C3×C42—32no—
257#217—21no—
258(C2×D3)⋊C4—24no—
259C4⋊Dic3—21no—
260Dic3⋊C4—24no—
261C4×Dic3C23×C312yes—
262Dic12—24no—
263D24—24no—
264#204—24no—
265#204—24no—
266C22×C32—41no—
267C22×C32#12011yescharacteristic
268C22×C32#12011yescharacteristic
269C22×C32C22×D411yescharacteristic
270C2×C3×D3—24no—
271C2×C3×D3—44no—
272C2×C3×D3—42no—
273C2×C3×D3—41no—
274C32×C4—24no—
275C32×C4#12012yes—
276C32×C4C22×D412yes—
277C3×Dic3—24no—
278C3×Dic3—42no—
279C3×Dic3—41no—
2802−1+4—34no—
2812−1+4—64no—
2822−1+4—62no—
283C2×#41—32no—
284C2×#41—31no—
285C2×#41—62no—
286C2×#41—62no—
287C22×Q8—62no—
288C22×Q8C2×C3×D311yescharacteristic
289#137—34no—
290#137—62no—
291AGL(1,ℤ/8)—34no—
292AGL(1,ℤ/8)—62no—
293#135—64no—
294C2×M16C2×C3×D312yes—
295C4⋊Q8—32no—
296#124—32no—
297#123—62no—
298C22⋊Q8—62no—
299#117—31no—
300#104—38no—
301#101C2×C3×D314yescomplemented
302C23×C3—21no—
303C23×C3—62no—
304C23×C3—61no—
305C23×C3C2×D1211yescharacteristic
306C23×C3C2×C3×D411yescharacteristic
307C22×D3—22no—
308C22×D3—21no—
309C3×Q8—24no—
310C3×Q8—24no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—34no—
314C3×Q8—44no—
315C3×Q8—44no—
316C3×Q8—68no—
317C3×Q8—62no—
318C3×Q8—62no—
319C3×D4—34no—
320C3×D4—68no—
321C3×D4—64no—
322C3×D4—64no—
323C3×D4—62no—
324C3×D4—62no—
325C3×D4—121no—
326C3×D4—121no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—616no—
339C2×C3×C4—62no—
340C2×C3×C4—64no—
341C2×C3×C4—64no—
342C2×C3×C4—62no—
343C2×C3×C4—61no—
344C2×C3×C4—122no—
345C2×C3×C4C2×D1214yes—
346C2×C3×C4C2×D1211yescharacteristic
347C2×C3×C4D3×D412yes—
348C2×C3×C4D3×D412yes—
349C2×C3×C4C2×C3×D414yes—
350C2×C3×C4C2×C3×D411yescharacteristic
351C2×C3×C4C2×C3×D412yes—
352C2×C3×C4C2×C3×D412yes—
353C2×C3×C4C23×D311yescharacteristic
354C2×C3×C4C24×C311yescharacteristic
355#67—24no—
356#67—24no—
357#67—41no—
358#67—41no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362D12—28no—
363D12—22no—
364D12—22no—
365D12C2×C3×D414yes—
366C4×D3—216no—
367C4×D3—24no—
368C4×D3—24no—
369C4×D3—42no—
370Dic6—28no—
371Dic6—22no—
372Dic6—22no—
373Dic6C2×C3×D414yes—
374C3×C8—24no—
375C3×C8—24no—
376C3×C8—44no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32C2×#12011yescharacteristic
381C3×D3—44no—
382C3×D3—81no—
383#41—34no—
384#41—34no—
385#41—68no—
386#41—68no—
387#41—68no—
388#41—62no—
389#41—68no—
390#41—62no—
391#41—62no—
392#41—62no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—31no—
396C2×Q8—32no—
397C2×Q8—62no—
398C2×Q8—68no—
399C2×Q8—68no—
400C2×Q8C3×D1212yes—
401C2×Q8C22×C3×D312yes—
402C2×D4—31no—
403C2×D4—32no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—61no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4C3×D1212yes—
411C22×C4C22×C3×D311yescharacteristic
412Q16—64no—
413SD16—64no—
414SD16—64no—
415D8—64no—
416M16—22no—
417M16C22×C3×D314yes—
418C2×C8—22no—
419C4⋊C4—61no—
420C4⋊C4—64no—
421C22⋊C4—61no—
422C22⋊C4—64no—
423C42—32no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3—64no—
431C22×C3—124no—
432C22×C3—122no—
433C22×C3—121no—
434C22×C3#88211yescharacteristic
435C22×C3#88211yescharacteristic
436C22×C3C3×#12011yescharacteristic
437C22×C3C3×#12011yescharacteristic
438C22×C3C22×D1211yescharacteristic
439C22×C3C22×C3×D411yescharacteristic
440C2×D3—24no—
441C2×D3—44no—
442C2×D3—42no—
443C2×D3—41no—
444C3×C4—24no—
445C3×C4—24no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—44no—
449C3×C4—64no—
450C3×C4—122no—
451C3×C4—121no—
452C3×C4#88212yes—
453C3×C4C3×#12012yes—
454C3×C4C2×D3×D412yes—
455C3×C4C22×C3×D412yes—
456Dic3—24no—
457Dic3—42no—
458Dic3—41no—
459C32#302011yescharacteristic, complemented, Sylow
460C23—62no—
461C23—61no—
462C23C2×C3×D1211yescharacteristic
463Q8—24no—
464Q8—24no—
465Q8—34no—
466Q8—68no—
467Q8—62no—
468Q8—62no—
469D4—34no—
470D4—68no—
471D4—64no—
472D4—64no—
473D4—62no—
474D4—62no—
475D4—121no—
476D4—121no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—616no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—122no—
486C2×C4C2×C3×D1214yes—
487C2×C4C2×C3×D1211yescharacteristic
488C2×C4C3×D3×D412yes—
489C2×C4C3×D3×D412yes—
490C2×C4C23×C3×D311yescharacteristic
491C8—24no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—124no—
503C2×C3—241no—
504C2×C3C2×#88211yescharacteristic
505C2×C3C2×C3×#12011yescharacteristic
506D3—44no—
507D3—81no—
508C22—41no—
509C22—64no—
510C22—124no—
511C22—122no—
512C22—121no—
513C22C3×#88211yescharacteristic
514C22C3×#88211yescharacteristic
515C22C22×C3×D1211yescharacteristic
516C4—24no—
517C4—64no—
518C4—122no—
519C4—121no—
520C4C3×#88212yes—
521C4C2×C3×D3×D412yes—
522C3—21no—
523C3#7870911yescharacteristic, direct product factor
524C3C3×#302011yescharacteristic, complemented
525C2—21no—
526C2—21no—
527C2—21no—
528C2—124no—
529C2—241no—
530C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC2×C3C2×#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#101C2
socleC2×C32C2×#120
hypercenterC2×C3×#101D3
hypocenterC3C3×#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C3×#5816C2semidirect
3. C3×#5748C2semidirect
4. C2×C32×#101C2semidirect
5. C3×#5139C2semidirect
6. C3×#4753C2semidirect
7. C3×#4752C2semidirect
9. #78709C3direct (N × H)
20. C3×#947C22semidirect
22. C3×#944C22semidirect
26. C3×#909C22semidirect
28. C3×#908C22semidirect
30. C32×#101C22semidirect
31. C3×#825C22semidirect
33. C2×#1007C2×C3semidirect
36. #5816C2×C3semidirect
37. #5748C2×C3semidirect
42. C2×C3×#101D3semidirect
43. C2×C3×#101C2×C3semidirect
44. #5139C2×C3semidirect
45. #4753C2×C3semidirect
46. #4752C2×C3semidirect
121. #947C22×C3semidirect
123. #944C22×C3semidirect
127. #909C22×C3semidirect
129. #908C22×C3semidirect
133. C3×#101C2×D3semidirect
134. C3×#101C22×C3semidirect
135. #825C22×C3semidirect
174. C2×#101C3×D3semidirect
301. #101C2×C3×D3semidirect
459. C32#3020semidirect
523. C3#78709direct (N × H)
524. C3C3×#3020semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C3×#58162
C3×#57482
C2×C32×#1012
C3×#51392
C3×#47532
C3×#47522
C3×#30203
#787093