Finite Group Databasebeta

← C3×#70364 #11786730

Subgroups — C3×#70364 #11786730

Order 1152.

657 conjugacy classes of subgroups.

The subgroup lattice has 510 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×#4936C211yesmaximal, characteristic, complemented
6C3×#4554C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1860—31nomaximal
9#70364C311yesmaximal, 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⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C32×#104C414yescomplemented
23C3×#835—21no—
24C3×#832—21no—
25C3×C3⋊#95—22no—
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×#382—31no—
36#5232C2×C311yescharacteristic
37#5121C2×C311yescharacteristic, complemented
38#4936C2×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⋊Dic3D411yescharacteristic
69C3×C4⋊Dic3D411yescharacteristic
70C3×C4×Dic3—22no—
71C3×C4×Dic3—21no—
72C3×#208—21no—
73C3×#208—21no—
74C2×C3×C3⋊C8—21no—
75#1860—31noSylow
76C2×C3×#41—21no—
77C2×C3×#41C2×D311yescharacteristic
78C2×C3×#41C22×C311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—31no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C22×C311yescharacteristic
83C3×C4×Q8—61no—
84C3×C4×D4—62no—
85C3×C4×D4—61no—
86C3×#117—31no—
87C3×#117—61no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—31no—
90C2×C3×C4⋊C4—61no—
91C2×C3×C42—21no—
92C2×C3×C42C2×D311yescharacteristic
93C2×C3×C42C22×C311yescharacteristic
94Q8×Dic3—21no—
95D4×Dic3—21no—
96D4×Dic3—22no—
97#926—21no—
98#926C22×C311yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C22×C311yescharacteristic
101C2×C4⋊Dic3C22×C311yescharacteristic
102C2×#208C22×C311yescharacteristic
103C3×#107—61no—
104C3×#106—61no—
105C3×#104—24no—
106C3×#104Dic314yescomplemented
107C3×#104C3×C414yescomplemented
108C3×#103—61no—
109C3×#102—61no—
110C3×#95—62no—
111#835—21no—
112#832—21no—
113C3⋊#95—22no—
114#818—21no—
115#817—21no—
116C23×C32—21no—
117C23×C32C2×D411yescharacteristic
118C32×Q8—21no—
119C32×Q8C4⋊C412yes—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—21no—
123C32×D4C4⋊C412yes—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—41no—
129C2×C32×C4—41no—
130C2×C32×C4C22×C411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×Q811yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C2×C32×C4#4111yescharacteristic
136C2×C3×Dic3—21no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—82no—
143C32×C8—22no—
144C3×C3⋊C8—41no—
145#520—31no—
146#428—31no—
147C2×#104C3×D311yescharacteristic
148#417—31no—
149#382—31no—
150#343—31no—
151#339—31no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41—21no—
155C3×#41—21no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—41no—
159C3×#41—41no—
160C3×#41Dic612yes—
161C3×#41C2×Dic312yes—
162C3×#41C2×C3×C412yes—
163C3×#41C3×Q812yes—
164C2×C3×Q8—21no—
165C2×C3×Q8D1211yescharacteristic
166C2×C3×Q8C3×D411yescharacteristic
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—41no—
171C2×C3×D4D1211yescharacteristic
172C2×C3×D4C3×D411yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—32no—
179C22×C3×C4—41no—
180C22×C3×C4—41no—
181C22×C3×C4—62no—
182C22×C3×C4—61no—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C23×C311yescharacteristic
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3C3×D412yes—
188C3×M16—21no—
189C3×M16—21no—
190C3×M16—22no—
191C3×M16—41no—
192C3×M16—61no—
193C3×M16—61no—
194C3×M16C2×Dic312yescomplemented
195C3×M16C2×C3×C412yescomplemented
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—41no—
199C2×C3×C8—61no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C22⋊C4—62no—
208C3×C22⋊C4—61no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—22no—
212C3×C42—41no—
213C3×C42—62no—
214C3×C42—61no—
215C3×C42C2×Dic312yescomplemented
216C3×C42C2×C3×C412yescomplemented
217#217—22no—
218#217—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3C3×D411yescharacteristic
225C4⋊Dic3C3×D411yescharacteristic
226C4×Dic3—22no—
227C4×Dic3—21no—
228#208—21no—
229#208—21no—
230C2×C3⋊C8—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—22no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32C4×D411yescharacteristic
237C22×C32#12011yescharacteristic
238C22×C32C22⋊Q811yescharacteristic
239C32×C4—22no—
240C32×C4—42no—
241C32×C4C2×C4⋊C411yescharacteristic
242C32×C4C4×D411yescharacteristic
243C32×C4C22⋊Q811yescharacteristic
244C32×C4#12311yescharacteristic
245C3×Dic3—41no—
246C3×Dic3—42no—
247C3×Dic3—41no—
248C3×Dic3—81no—
249C2×#41C2×C3×D311yescharacteristic
250C2×M16—31no—
251C2×M16C2×C3×D311yescharacteristic
252C4×Q8—61no—
253C4×D4—62no—
254C4×D4—61no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—61no—
260C2×C42C2×C3×D311yescharacteristic
261#107—61no—
262#106—61no—
263#104C3×Dic314yescomplemented
264#103—61no—
265#102—61no—
266#95—62no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3D3×D411yescharacteristic
272C23×C3C2×C3×D411yescharacteristic
273C3×Q8—21no—
274C3×Q8—21no—
275C3×Q8—22no—
276C3×Q8—41no—
277C3×Q8C4⋊Dic312yes—
278C3×Q8C3×C4⋊C412yes—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—21no—
284C3×D4—21no—
285C3×D4—22no—
286C3×D4—41no—
287C3×D4—42no—
288C3×D4—42no—
289C3×D4C4⋊Dic312yes—
290C3×D4C3×C4⋊C412yes—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—62no—
318C2×C3×C4—122no—
319C2×C3×C4—122no—
320C2×C3×C4—121no—
321C2×C3×C4—242no—
322C2×C3×C4C2×Dic611yescharacteristic
323C2×C3×C4C2×D1211yescharacteristic
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4#23711yescharacteristic
326C2×C3×C4#23711yescharacteristic
327C2×C3×C4C22×Dic311yescharacteristic
328C2×C3×C4C22×C3×C411yescharacteristic
329C2×C3×C4C2×C3×D411yescharacteristic
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×Q811yescharacteristic
332C2×C3×C4C3×#4111yescharacteristic
333C2×C3×C4C3×#4111yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—21no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—82no—
341C3×C8—22no—
342C3×C8—22no—
343C3×C8—42no—
344C3×C8—121no—
345C3⋊C8—41no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—42no—
349C2×C32#38511yescharacteristic
350C2×C32#45411yescharacteristic
351C2×C32#45511yescharacteristic
352#41—21no—
353#41—21no—
354#41C3×Dic612yes—
355#41C2×C3×Dic312yes—
356C2×Q8C3×D1211yescharacteristic
357C2×D4—21no—
358C2×D4C3×D1211yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—32no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4C22×C3×D311yescharacteristic
365M16—21no—
366M16—61no—
367M16—61no—
368M16C2×C3×Dic312yescomplemented
369C2×C8—21no—
370C2×C8—61no—
371C4⋊C4—31no—
372C4⋊C4—31no—
373C4⋊C4—62no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C22⋊C4—62no—
379C22⋊C4—61no—
380C42—21no—
381C42—62no—
382C42—61no—
383C42C2×C3×Dic312yescomplemented
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3#87811yescharacteristic
403C22×C3#88211yescharacteristic
404C22×C3D4×Dic311yescharacteristic
405C22×C3C3×C4×D411yescharacteristic
406C22×C3C3×#12011yescharacteristic
407C22×C3C3×C22⋊Q811yescharacteristic
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—122no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4—241no—
422C3×C4#87811yescharacteristic
423C3×C4#88611yescharacteristic
424C3×C4C2×C4⋊Dic311yescharacteristic
425C3×C4D4×Dic311yescharacteristic
426C3×C4C2×C3×C4⋊C411yescharacteristic
427C3×C4C3×C4×D411yescharacteristic
428C3×C4C3×C22⋊Q811yescharacteristic
429C3×C4C3×#12311yescharacteristic
430Dic3—41no—
431Dic3—42no—
432Dic3—41no—
433Dic3—81no—
434C32#186011yescharacteristic, complemented, Sylow
435C23—21no—
436C23C3×D3×D411yescharacteristic
437Q8—21no—
438Q8C3×C4⋊Dic312yes—
439D4—22no—
440D4—22no—
441D4—21no—
442D4C3×C4⋊Dic312yes—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—62no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—242no—
456C2×C4C2×C3×Dic611yescharacteristic
457C2×C4C2×C3×D1211yescharacteristic
458C2×C4C3×D3×D411yescharacteristic
459C2×C4C3×#23711yescharacteristic
460C2×C4C3×#23711yescharacteristic
461C2×C4C22×C3×Dic311yescharacteristic
462C8—22no—
463C8—121no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—82no—
476C2×C3#482511yescharacteristic
477C2×C3#482711yescharacteristic
478C2×C3#495011yescharacteristic
479C2×C3C3×#38511yescharacteristic
480C2×C3C3×#45411yescharacteristic
481C2×C3C3×#45511yescharacteristic
482C22—21no—
483C22—21no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22C3×#87811yescharacteristic
488C22C3×#88211yescharacteristic
489C22C3×D4×Dic311yescharacteristic
490C4—22no—
491C4—42no—
492C4—122no—
493C4—121no—
494C4—121no—
495C4—241no—
496C4C3×#87811yescharacteristic
497C4C3×#88611yescharacteristic
498C4C2×C3×C4⋊Dic311yescharacteristic
499C4C3×D4×Dic311yescharacteristic
500C3—21no—
501C3#7036411yescharacteristic, direct product factor
502C3C3×#186011yescharacteristic, complemented
503C2—21no—
504C2—21no—
505C2—42no—
506C2C3×#482511yescharacteristic
507C2C3×#482711yescharacteristic
508C2C3×#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×#1860
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×#4936C2semidirect
6. C3×#4554C2semidirect
7. C3×#4523C2semidirect
9. #70364C3direct (N × H)
22. C32×#104C4semidirect
37. #5121C2×C3semidirect
38. #4936C2×C3semidirect
41. #4554C2×C3semidirect
42. #4523C2×C3semidirect
57. C32×M16C2×C4semidirect
60. C32×C42C2×C4semidirect
106. C3×#104Dic3semidirect
107. C3×#104C3×C4semidirect
194. C3×M16C2×Dic3semidirect
195. C3×M16C2×C3×C4semidirect
215. C3×C42C2×Dic3semidirect
216. C3×C42C2×C3×C4semidirect
263. #104C3×Dic3semidirect
368. M16C2×C3×Dic3semidirect
383. C42C2×C3×Dic3semidirect
434. C32#1860semidirect
501. C3#70364direct (N × H)
502. C3C3×#1860semidirect

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×#49362
C3×#45542
C3×#45232
C3×#18603
#703643