Finite Group Databasebeta

← C3×#78689 #11857563

Subgroups — C3×#78689 #11857563

Order 1152.

1511 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic
2C3×#5516C212yesmaximal, complemented
3C32×#410C211yesmaximal, characteristic, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5C3×#5006C212yesmaximal, complemented
6C3×#4740C218yesmaximal, complemented
7C3×#2884—31nomaximal
8#78689C311yesmaximal, characteristic, direct product factor
9C2×C32×#41C2211yescharacteristic
10C3×#1010—24no—
11C3×#1010—24no—
12C2×C3×D3×Q8C412yescomplemented
13C2×C3×#237C412yescomplemented
14C2×C3×#237C2214yes—
15C2×C3×#235C412yescomplemented
16C2×C3×#235C2211yescharacteristic
17C22×C3×Dic6C412yescomplemented
18C22×C3×Dic6C2211yescharacteristic
19C32×#117C2212yes—
20C3×#926C2212yes—
21C3×#887—24no—
22C3×#877C2218yes—
23C3×C4×Dic6—24no—
24C32×#99C2214yes—
25C3×#834C2214yes—
26C3×#806C2218yes—
27C2×C3×2−1+4—31no—
28C2×#1010C2×C311yescharacteristic
29C3×#519—32no—
30#5516C2×C312yescomplemented
31C3×#410—21no—
32C3×#410—31no—
33C3×#410—32no—
34C3×#410—38no—
35C3×#410D311yescharacteristic, complemented
36C3×#410C2×C311yescharacteristic, complemented
37#5240C2×C311yescharacteristic, complemented
38#5006C2×C312yescomplemented
39#4740C2×C318yescomplemented
40C32×#41—22no—
41C32×#41—22no—
42C2×C32×Q8C2×C411yescharacteristic
43C2×C32×D4C2×C411yescharacteristic
44C2×C32×D4C2312yes—
45C22×C32×C4C2×C412yes—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67C2×C414yes—
48C2×C3×#67C2312yes—
49C22×C3×Dic3C2×C414yes—
50C22×C3×Dic3C2312yes—
51C3×D3×Q8—28no—
52C3×#237—216no—
53C3×#237—28no—
54C3×#235—22no—
55C3×#235—22no—
56C3×#235—24no—
57C3×#235—24no—
58C2×C3×D12C2×C411yescharacteristic
59C2×C3×C4×D3C2×C414yes—
60C2×C3×C4×D3C2×C412yes—
61C2×C3×Dic6—24no—
62C2×C3×Dic6—24no—
63C2×C3×Dic6C2×C411yescharacteristic
64C2×C3×Dic6C2×C414yes—
65C2×C3×Dic6C2×C412yes—
66C2×C3×Dic6C2×C412yes—
67C2×C3×Dic6D414yes—
68C2×C3×Dic6D414yes—
69C32×C4⋊C4—22no—
70C32×C22⋊C4C2314yes—
71C32×C42—22no—
72C3×#217C2314yes—
73C3×C4⋊Dic3—22no—
74C3×Dic3⋊C4—28no—
75C3×C4×Dic3—22no—
76C3×C4×Dic3—28no—
77#2884—31noSylow
78C3×2−1+4—64no—
79C3×2−1+4—64no—
80C2×C3×#41—21no—
81C2×C3×#41—32no—
82C2×C3×#41—34no—
83C2×C3×#41—32no—
84C2×C3×#41—31no—
85C2×C3×#41C2×D311yescharacteristic
86C2×C3×#41C22×C311yescharacteristic
87C22×C3×Q8—32no—
88C22×C3×Q8—32no—
89C22×C3×Q8—31no—
90#1010—24no—
91#1010—24no—
92C2×D3×Q8C3×C412yescomplemented
93C2×#237C3×C412yescomplemented
94C2×#237C22×C314yes—
95C2×#235C3×C412yescomplemented
96C2×#235C22×C311yescharacteristic
97C22×Dic6C3×C412yescomplemented
98C22×Dic6C22×C311yescharacteristic
99C3×C4×Q8—64no—
100C3×C4×Q8—64no—
101C3×#117—22no—
102C3×#117—32no—
103C3×#117—38no—
104C3×#117C2×D312yes—
105C3×#117C22×C312yes—
106#926C22×C312yes—
107#887—24no—
108#877C22×C318yes—
109C4×Dic6—24no—
110C3×#99—24no—
111C3×#99—34no—
112C3×#99—38no—
113C3×#99C2×D314yes—
114C3×#99C22×C314yes—
115#834C22×C314yes—
116#806C22×C318yes—
117C23×C32C22×C412yes—
118C23×C32C2411yescharacteristic
119C22×C3×D3C22×C412yes—
120C32×Q8—22no—
121C32×D4—24no—
122C32×D4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—24no—
126C2×C32×C4C22×C412yes—
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D412yes—
130C2×C32×C4C2×D412yes—
131C3×#67—24no—
132C3×#67—28no—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3C22×C412yes—
136C2×C3×Dic3C22×C414yes—
137C2×C3×Dic3C2×D418yes—
138C3×D12—21no—
139C3×D12—21no—
140C3×C4×D3—24no—
141C3×C4×D3—28no—
142C3×Dic6—21no—
143C3×Dic6—21no—
144C3×Dic6—28no—
145C3×Dic6C22⋊C414yes—
146C3×Dic6C22⋊C414yes—
147C3×Dic6C22⋊C418yes—
148C2×2−1+4—31no—
149#519—32no—
150#410—31no—
151#410—32no—
152#410—38no—
153#410C3×D311yescharacteristic, complemented
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—42no—
159C3×#41—42no—
160C3×#41—64no—
161C3×#41—64no—
162C3×#41—616no—
163C3×#41—68no—
164C3×#41—62no—
165C3×#41—62no—
166C2×C3×Q8—21no—
167C2×C3×Q8—32no—
168C2×C3×Q8—32no—
169C2×C3×Q8—34no—
170C2×C3×Q8—31no—
171C2×C3×Q8—34no—
172C2×C3×Q8—34no—
173C2×C3×Q8—64no—
174C2×C3×Q8—64no—
175C2×C3×Q8—68no—
176C2×C3×Q8C4×D311yescharacteristic
177C2×C3×Q8C2×C3×C411yescharacteristic
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—34no—
181C2×C3×D4—32no—
182C2×C3×D4—31no—
183C2×C3×D4C4×D311yescharacteristic
184C2×C3×D4C2×C3×C411yescharacteristic
185C2×C3×D4C22×D312yes—
186C2×C3×D4C23×C312yes—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—32no—
190C22×C3×C4—34no—
191C22×C3×C4—34no—
192C22×C3×C4—32no—
193C22×C3×C4C4×D312yes—
194C22×C3×C4C2×C3×C412yes—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C23×C311yescharacteristic
197C2×#67C2×C3×C414yes—
198C2×#67C23×C312yes—
199C22×Dic3C2×C3×C414yes—
200C22×Dic3C23×C312yes—
201D3×Q8—28no—
202#237—216no—
203#237—28no—
204#235—22no—
205#235—22no—
206#235—24no—
207#235—24no—
208C2×D12C2×C3×C411yescharacteristic
209C2×C4×D3C2×C3×C414yes—
210C2×C4×D3C2×C3×C412yes—
211C2×Dic6—24no—
212C2×Dic6—24no—
213C2×Dic6C2×C3×C411yescharacteristic
214C2×Dic6C2×C3×C414yes—
215C2×Dic6C2×C3×C412yes—
216C2×Dic6C2×C3×C412yes—
217C2×Dic6C3×D414yes—
218C2×Dic6C3×D414yes—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—68no—
223C3×C4⋊C4—62no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—34no—
226C3×C22⋊C4C22×D314yes—
227C3×C22⋊C4C23×C314yes—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—42no—
231C3×C42—68no—
232C3×C42—62no—
233#217C23×C314yes—
234C4⋊Dic3—22no—
235Dic3⋊C4—28no—
236C4×Dic3—22no—
237C4×Dic3—28no—
238C22×C32—22no—
239C22×C32—41no—
240C22×C32—42no—
241C22×C32C23×C411yescharacteristic
242C22×C32C22×D412yes—
243C2×C3×D3—22no—
244C2×C3×D3—42no—
245C32×C4—22no—
246C32×C4—44no—
247C32×C4C2×C22⋊C411yescharacteristic
248C32×C4C2×C22⋊C411yescharacteristic
249C32×C4C2×C22⋊C412yes—
250C3×Dic3—22no—
251C3×Dic3—44no—
252C3×Dic3C2×C22⋊C418yes—
2532−1+4—64no—
2542−1+4—64no—
255C2×#41—32no—
256C2×#41—34no—
257C2×#41—32no—
258C2×#41—31no—
259C2×#41C2×C3×D311yescharacteristic
260C22×Q8—32no—
261C22×Q8—32no—
262C22×Q8—31no—
263C4×Q8—64no—
264C4×Q8—64no—
265#117—32no—
266#117—38no—
267#117C2×C3×D312yes—
268#99—34no—
269#99—38no—
270#99C2×C3×D314yes—
271C23×C3—21no—
272C23×C3—22no—
273C23×C3—32no—
274C23×C3C2×C4×D312yes—
275C23×C3C22×C3×C412yes—
276C23×C3C23×D311yescharacteristic
277C23×C3C24×C311yescharacteristic
278C22×D3C22×C3×C412yes—
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—34no—
282C3×Q8—34no—
283C3×Q8—38no—
284C3×Q8—42no—
285C3×Q8—68no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—42no—
293C3×D4—44no—
294C3×D4—68no—
295C3×D4—64no—
296C3×D4—61no—
297C3×D4—61no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—34no—
310C2×C3×C4—38no—
311C2×C3×C4—32no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—44no—
315C2×C3×C4—68no—
316C2×C3×C4—68no—
317C2×C3×C4—64no—
318C2×C3×C4—64no—
319C2×C3×C4C2×C4×D312yes—
320C2×C3×C4C2×C4×D311yescharacteristic
321C2×C3×C4C2×C4×D311yescharacteristic
322C2×C3×C4D3×D412yes—
323C2×C3×C4D3×D412yes—
324C2×C3×C4C22×C3×C412yes—
325C2×C3×C4C22×C3×C411yescharacteristic
326C2×C3×C4C22×C3×C411yescharacteristic
327C2×C3×C4C2×C3×D412yes—
328C2×C3×C4C2×C3×D412yes—
329#67—24no—
330#67—28no—
331C2×Dic3—28no—
332C2×Dic3—24no—
333C2×Dic3C22×C3×C412yes—
334C2×Dic3C22×C3×C414yes—
335C2×Dic3C2×C3×D418yes—
336D12—21no—
337D12—21no—
338C4×D3—24no—
339C4×D3—28no—
340Dic6—21no—
341Dic6—21no—
342Dic6—28no—
343Dic6C3×C22⋊C414yes—
344Dic6C3×C22⋊C414yes—
345Dic6C3×C22⋊C418yes—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—42no—
349C2×C32C22×C22⋊C411yescharacteristic
350C3×D3—42no—
351#41—22no—
352#41—22no—
353#41—64no—
354#41—64no—
355#41—616no—
356#41—68no—
357#41—62no—
358#41—62no—
359C2×Q8—32no—
360C2×Q8—32no—
361C2×Q8—34no—
362C2×Q8—31no—
363C2×Q8—34no—
364C2×Q8—34no—
365C2×Q8—64no—
366C2×Q8—64no—
367C2×Q8—68no—
368C2×Q8C3×C4×D311yescharacteristic
369C2×D4—32no—
370C2×D4—34no—
371C2×D4—31no—
372C2×D4C3×C4×D311yescharacteristic
373C2×D4C22×C3×D312yes—
374C22×C4—32no—
375C22×C4—34no—
376C22×C4—34no—
377C22×C4—32no—
378C22×C4C3×C4×D312yes—
379C22×C4C22×C3×D311yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—68no—
382C4⋊C4—62no—
383C22⋊C4—34no—
384C22⋊C4C22×C3×D314yes—
385C42—22no—
386C42—68no—
387C42—62no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—42no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—62no—
398C22×C3—81no—
399C22×C3—82no—
400C22×C3—122no—
401C22×C3C22×C4×D311yescharacteristic
402C22×C3C2×D3×D412yes—
403C22×C3C23×C3×C411yescharacteristic
404C22×C3C22×C3×D412yes—
405C2×D3—22no—
406C2×D3—42no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—38no—
413C3×C4—42no—
414C3×C4—44no—
415C3×C4—44no—
416C3×C4—62no—
417C3×C4—84no—
418C3×C4—124no—
419C3×C4D3×C22⋊C411yescharacteristic
420C3×C4D3×C22⋊C411yescharacteristic
421C3×C4D3×C22⋊C412yes—
422C3×C4C2×C3×C22⋊C411yescharacteristic
423C3×C4C2×C3×C22⋊C411yescharacteristic
424C3×C4C2×C3×C22⋊C412yes—
425Dic3—22no—
426Dic3—44no—
427Dic3C2×C3×C22⋊C418yes—
428C32#288411yescharacteristic, complemented, Sylow
429C23—32no—
430C23C2×C3×C4×D312yes—
431C23C23×C3×D311yescharacteristic
432Q8—22no—
433Q8—34no—
434Q8—34no—
435Q8—38no—
436Q8—68no—
437Q8—61no—
438Q8—61no—
439D4—24no—
440D4—22no—
441D4—68no—
442D4—64no—
443D4—61no—
444D4—61no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—34no—
449C2×C4—38no—
450C2×C4—32no—
451C2×C4—68no—
452C2×C4—68no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4C2×C3×C4×D312yes—
456C2×C4C2×C3×C4×D311yescharacteristic
457C2×C4C2×C3×C4×D311yescharacteristic
458C2×C4C3×D3×D412yes—
459C2×C4C3×D3×D412yes—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—42no—
466C2×C3—41no—
467C2×C3—42no—
468C2×C3—42no—
469C2×C3—82no—
470C2×C3—122no—
471C2×C3C2×D3×C22⋊C411yescharacteristic
472C2×C3C22×C3×C22⋊C411yescharacteristic
473D3—42no—
474C22—22no—
475C22—41no—
476C22—42no—
477C22—62no—
478C22—122no—
479C22C22×C3×C4×D311yescharacteristic
480C22C2×C3×D3×D412yes—
481C4—22no—
482C4—38no—
483C4—44no—
484C4—62no—
485C4—124no—
486C4C3×D3×C22⋊C411yescharacteristic
487C4C3×D3×C22⋊C411yescharacteristic
488C4C3×D3×C22⋊C412yes—
489C3—21no—
490C3#7868911yescharacteristic, direct product factor
491C3C3×#288411yescharacteristic, complemented
492C2—22no—
493C2—21no—
494C2—42no—
495C2—122no—
496C2C2×C3×D3×C22⋊C411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×C22⋊C4
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#410C2
socleC2×C32C22×C22⋊C4
prime radicalC2×C3×#1010C2
cosocleC23×C3C24×C3
hypercenterC3×#410D3
hypocenterC3C3×#2884
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#5516C2semidirect
3. C32×#410C2semidirect
4. C3×#5240C2semidirect
5. C3×#5006C2semidirect
6. C3×#4740C2semidirect
8. #78689C3direct (N × H)
12. C2×C3×D3×Q8C4semidirect
13. C2×C3×#237C4semidirect
15. C2×C3×#235C4semidirect
17. C22×C3×Dic6C4semidirect
30. #5516C2×C3semidirect
35. C3×#410D3semidirect
36. C3×#410C2×C3semidirect
37. #5240C2×C3semidirect
38. #5006C2×C3semidirect
39. #4740C2×C3semidirect
92. C2×D3×Q8C3×C4semidirect
93. C2×#237C3×C4semidirect
95. C2×#235C3×C4semidirect
97. C22×Dic6C3×C4semidirect
153. #410C3×D3semidirect
428. C32#2884semidirect
490. C3#78689direct (N × H)
491. C3C3×#2884semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C3×#55162
C32×#4102
C3×#52402
C3×#50062
C3×#47402
C3×#28843
#786893