Finite Group Databasebeta

← C3×#70363 #11786681

Subgroups — C3×#70363 #11786681

Order 1152.

689 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5827C211yesmaximal, characteristic
2C2×C32×#104C211yesmaximal, characteristic
3C3×#5232C211yesmaximal, characteristic
4C3×#5121C211yesmaximal, characteristic, complemented
5C3×#4935C211yesmaximal, characteristic, complemented
6C3×#4554C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1859—31nomaximal
9#70363C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13C3×Q8×Dic3—21no—
14C3×D4×Dic3—21no—
15C3×D4×Dic3—22no—
16C3×#926—21no—
17C3×#926C2211yescharacteristic
18C2×C3×C4⋊Dic3—21no—
19C2×C3×C4⋊Dic3—22no—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C32×#104C414yescomplemented
24C3×#835—21no—
25C3×#832—21no—
26C3×#818—21no—
27C3×#817—21no—
28C3×#520—31no—
29#5827C2×C311yescharacteristic
30C3×#428—31no—
31C2×C3×#104—21no—
32C2×C3×#104D311yescharacteristic
33C2×C3×#104C2×C311yescharacteristic
34C3×#417—31no—
35C3×#384—31no—
36#5232C2×C311yescharacteristic
37#5121C2×C311yescharacteristic, complemented
38#4935C2×C311yescharacteristic, complemented
39C3×#343—31no—
40C3×#339—31no—
41#4554C2×C311yescharacteristic, complemented
42#4523C2×C311yescharacteristic, complemented
43C32×#41—21no—
44C32×#41—21no—
45C32×#41C2×C412yes—
46C32×#41Q812yes—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4—21no—
49C2×C32×D4D411yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—22no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3D412yes—
56C32×M16—21no—
57C32×M16C2×C412yescomplemented
58C2×C32×C8—21no—
59C32×C42—21no—
60C32×C42C2×C412yescomplemented
61C3×#217—22no—
62C3×#217—21no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—44no—
69C3×C4⋊Dic3D411yescharacteristic
70C3×C4⋊Dic3D411yescharacteristic
71C3×C4×Dic3—22no—
72C3×C4×Dic3—21no—
73C3×#208—21no—
74C3×#208—21no—
75C2×C3×C3⋊C8—21no—
76#1859—31noSylow
77C2×C3×#41—21no—
78C2×C3×#41C2×D311yescharacteristic
79C2×C3×#41C22×C311yescharacteristic
80C2×C3×M16—21no—
81C2×C3×M16—31no—
82C2×C3×M16C2×D311yescharacteristic
83C2×C3×M16C22×C311yescharacteristic
84C3×C4×Q8—61no—
85C3×C4×D4—62no—
86C3×C4×D4—61no—
87C3×#117—31no—
88C3×#117—61no—
89C2×C3×C4⋊C4—31no—
90C2×C3×C4⋊C4—31no—
91C2×C3×C4⋊C4—61no—
92C2×C3×C4⋊C4—62no—
93C2×C3×C42—21no—
94C2×C3×C42C2×D311yescharacteristic
95C2×C3×C42C22×C311yescharacteristic
96Q8×Dic3—21no—
97D4×Dic3—21no—
98D4×Dic3—22no—
99#926—21no—
100#926C22×C311yescharacteristic
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3C22×C311yescharacteristic
104C2×C4⋊Dic3C22×C311yescharacteristic
105C2×#208C22×C311yescharacteristic
106C3×#107—61no—
107C3×#106—61no—
108C3×#104—24no—
109C3×#104Dic314yescomplemented
110C3×#104C3×C414yescomplemented
111C3×#103—61no—
112C3×#102—61no—
113#835—21no—
114#832—21no—
115#818—21no—
116#817—21no—
117C23×C32—21no—
118C23×C32C2×D411yescharacteristic
119C32×Q8—21no—
120C32×Q8C4⋊C412yescomplemented
121C32×D4—22no—
122C32×D4—22no—
123C32×D4—21no—
124C32×D4C4⋊C412yescomplemented
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4C22×C411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×Q811yescharacteristic
135C2×C32×C4#4111yescharacteristic
136C2×C32×C4#4111yescharacteristic
137C2×C3×Dic3—21no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—82no—
144C32×C8—22no—
145C3×C3⋊C8—41no—
146#520—31no—
147#428—31no—
148C2×#104C3×D311yescharacteristic
149#417—31no—
150#384—31no—
151#343—31no—
152#339—31no—
153C3×#41—21no—
154C3×#41—21no—
155C3×#41—21no—
156C3×#41—21no—
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—41no—
160C3×#41—41no—
161C3×#41Dic612yes—
162C3×#41C2×Dic312yes—
163C3×#41C2×C3×C412yes—
164C3×#41C3×Q812yes—
165C2×C3×Q8—21no—
166C2×C3×Q8D1211yescharacteristic
167C2×C3×Q8C3×D411yescharacteristic
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—41no—
172C2×C3×D4D1211yescharacteristic
173C2×C3×D4C3×D411yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—32no—
180C22×C3×C4—41no—
181C22×C3×C4—41no—
182C22×C3×C4—62no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3C3×D412yes—
189C3×M16—21no—
190C3×M16—21no—
191C3×M16—22no—
192C3×M16—41no—
193C3×M16—61no—
194C3×M16—61no—
195C3×M16C2×Dic312yescomplemented
196C3×M16C2×C3×C412yescomplemented
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—41no—
200C2×C3×C8—61no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—62no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—124no—
209C3×C22⋊C4—62no—
210C3×C22⋊C4—61no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—22no—
214C3×C42—41no—
215C3×C42—62no—
216C3×C42—61no—
217C3×C42C2×Dic312yescomplemented
218C3×C42C2×C3×C412yescomplemented
219#217—22no—
220#217—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—44no—
227C4⋊Dic3C3×D411yescharacteristic
228C4⋊Dic3C3×D411yescharacteristic
229C4×Dic3—22no—
230C4×Dic3—21no—
231#208—21no—
232#208—21no—
233C2×C3⋊C8—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—22no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32C4×D411yescharacteristic
240C22×C32#12011yescharacteristic
241C22×C32C22⋊Q811yescharacteristic
242C32×C4—24no—
243C32×C4—22no—
244C32×C4C2×C4⋊C411yescharacteristic
245C32×C4C4×D411yescharacteristic
246C32×C4C22⋊Q811yescharacteristic
247C32×C4#12311yescharacteristic
248C3×Dic3—42no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—81no—
252C2×#41C2×C3×D311yescharacteristic
253C2×M16—31no—
254C2×M16C2×C3×D311yescharacteristic
255C4×Q8—61no—
256C4×D4—62no—
257C4×D4—61no—
258#117—31no—
259#117—61no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C42C2×C3×D311yescharacteristic
265#107—61no—
266#106—61no—
267#104C3×Dic314yescomplemented
268#103—61no—
269#102—61no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—41no—
274C23×C3D3×D411yescharacteristic
275C23×C3C2×C3×D411yescharacteristic
276C3×Q8—21no—
277C3×Q8—21no—
278C3×Q8—22no—
279C3×Q8—41no—
280C3×Q8C4⋊Dic312yescomplemented
281C3×Q8C3×C4⋊C412yescomplemented
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—21no—
287C3×D4—21no—
288C3×D4—22no—
289C3×D4—41no—
290C3×D4—42no—
291C3×D4—42no—
292C3×D4C4⋊Dic312yescomplemented
293C3×D4C3×C4⋊C412yescomplemented
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—62no—
321C2×C3×C4—122no—
322C2×C3×C4—122no—
323C2×C3×C4—121no—
324C2×C3×C4—242no—
325C2×C3×C4C2×Dic611yescharacteristic
326C2×C3×C4C2×D1211yescharacteristic
327C2×C3×C4D3×D411yescharacteristic
328C2×C3×C4#23711yescharacteristic
329C2×C3×C4#23711yescharacteristic
330C2×C3×C4C22×Dic311yescharacteristic
331C2×C3×C4C22×C3×C411yescharacteristic
332C2×C3×C4C2×C3×D411yescharacteristic
333C2×C3×C4C2×C3×D411yescharacteristic
334C2×C3×C4C2×C3×Q811yescharacteristic
335C2×C3×C4C3×#4111yescharacteristic
336C2×C3×C4C3×#4111yescharacteristic
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—41no—
343C2×Dic3—82no—
344C3×C8—22no—
345C3×C8—22no—
346C3×C8—42no—
347C3×C8—121no—
348C3⋊C8—41no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—42no—
352C2×C32#38511yescharacteristic
353C2×C32#45311yescharacteristic
354C2×C32#45611yescharacteristic
355#41—21no—
356#41—21no—
357#41C3×Dic612yes—
358#41C2×C3×Dic312yes—
359C2×Q8C3×D1211yescharacteristic
360C2×D4—21no—
361C2×D4C3×D1211yescharacteristic
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—32no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4C22×C3×D311yescharacteristic
368M16—21no—
369M16—61no—
370M16—61no—
371M16C2×C3×Dic312yescomplemented
372C2×C8—21no—
373C2×C8—61no—
374C4⋊C4—31no—
375C4⋊C4—31no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—124no—
382C22⋊C4—62no—
383C22⋊C4—61no—
384C42—21no—
385C42—62no—
386C42—61no—
387C42C2×C3×Dic312yescomplemented
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—42no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3#87811yescharacteristic
407C22×C3#88211yescharacteristic
408C22×C3D4×Dic311yescharacteristic
409C22×C3C3×C4×D411yescharacteristic
410C22×C3C3×#12011yescharacteristic
411C22×C3C3×C22⋊Q811yescharacteristic
412C3×C4—24no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—42no—
421C3×C4—44no—
422C3×C4—122no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—241no—
426C3×C4#87811yescharacteristic
427C3×C4#88611yescharacteristic
428C3×C4C2×C4⋊Dic311yescharacteristic
429C3×C4D4×Dic311yescharacteristic
430C3×C4C2×C3×C4⋊C411yescharacteristic
431C3×C4C3×C4×D411yescharacteristic
432C3×C4C3×C22⋊Q811yescharacteristic
433C3×C4C3×#12311yescharacteristic
434Dic3—42no—
435Dic3—41no—
436Dic3—41no—
437Dic3—81no—
438C32#185911yescharacteristic, complemented, Sylow
439C23—21no—
440C23C3×D3×D411yescharacteristic
441Q8—21no—
442Q8C3×C4⋊Dic312yescomplemented
443D4—22no—
444D4—22no—
445D4—21no—
446D4C3×C4⋊Dic312yescomplemented
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—62no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—121no—
459C2×C4—242no—
460C2×C4C2×C3×Dic611yescharacteristic
461C2×C4C2×C3×D1211yescharacteristic
462C2×C4C3×D3×D411yescharacteristic
463C2×C4C3×#23711yescharacteristic
464C2×C4C3×#23711yescharacteristic
465C2×C4C22×C3×Dic311yescharacteristic
466C8—22no—
467C8—121no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—82no—
480C2×C3#482211yescharacteristic
481C2×C3#482611yescharacteristic
482C2×C3#495011yescharacteristic
483C2×C3C3×#38511yescharacteristic
484C2×C3C3×#45311yescharacteristic
485C2×C3C3×#45611yescharacteristic
486C22—21no—
487C22—21no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22C3×#87811yescharacteristic
492C22C3×#88211yescharacteristic
493C22C3×D4×Dic311yescharacteristic
494C4—24no—
495C4—22no—
496C4—122no—
497C4—121no—
498C4—121no—
499C4—241no—
500C4C3×#87811yescharacteristic
501C4C3×#88611yescharacteristic
502C4C2×C3×C4⋊Dic311yescharacteristic
503C4C3×D4×Dic311yescharacteristic
504C3—21no—
505C3#7036311yescharacteristic, direct product factor
506C3C3×#185911yescharacteristic, complemented
507C2—21no—
508C2—21no—
509C2—42no—
510C2C3×#482211yescharacteristic
511C2C3×#482611yescharacteristic
512C2C3×#495011yescharacteristic
subgroupG/H
centerC22×C3#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#104C2
socleC22×C32#120
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#104D3
hypocenterC3C3×#1859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#104

Complements

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

NH ≅ G/Ntype
4. C3×#5121C2semidirect
5. C3×#4935C2semidirect
6. C3×#4554C2semidirect
7. C3×#4523C2semidirect
9. #70363C3direct (N × H)
23. C32×#104C4semidirect
37. #5121C2×C3semidirect
38. #4935C2×C3semidirect
41. #4554C2×C3semidirect
42. #4523C2×C3semidirect
57. C32×M16C2×C4semidirect
60. C32×C42C2×C4semidirect
109. C3×#104Dic3semidirect
110. C3×#104C3×C4semidirect
120. C32×Q8C4⋊C4semidirect
124. C32×D4C4⋊C4semidirect
195. C3×M16C2×Dic3semidirect
196. C3×M16C2×C3×C4semidirect
217. C3×C42C2×Dic3semidirect
218. C3×C42C2×C3×C4semidirect
267. #104C3×Dic3semidirect
280. C3×Q8C4⋊Dic3semidirect
281. C3×Q8C3×C4⋊C4semidirect
292. C3×D4C4⋊Dic3semidirect
293. C3×D4C3×C4⋊C4semidirect
371. M16C2×C3×Dic3semidirect
387. C42C2×C3×Dic3semidirect
438. C32#1859semidirect
442. Q8C3×C4⋊Dic3semidirect
446. D4C3×C4⋊Dic3semidirect
505. C3#70363direct (N × H)
506. C3C3×#1859semidirect

Maximal subgroups

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

subgroupindex
C3×#58272
C2×C32×#1042
C3×#52322
C3×#51212
C3×#49352
C3×#45542
C3×#45232
C3×#18593
#703633