Finite Group Databasebeta

← C3×#70055 #11788554

Subgroups — C3×#70055 #11788554

Order 1152.

743 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5822C211yesmaximal, characteristic
2C3×#5522C211yesmaximal, characteristic, complemented
3C32×#410C211yesmaximal, characteristic, complemented
4C2×C3×#837C211yesmaximal, characteristic, complemented
5C3×#5005C211yesmaximal, characteristic, complemented
6C3×#4533C211yesmaximal, characteristic, complemented
7C3×#4532C211yesmaximal, characteristic, complemented
8C3×#1906—31nomaximal
9#70055C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#951—21no—
15C3×#950—22no—
16C3×#949—21no—
17C3×#942C412yescomplemented
18C3×#932C412yescomplemented
19C2×C3×C4×Dic3C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C3×#893—21no—
23C3×#887—21no—
24C3×#67⋊C4—22no—
25C3×#872—21no—
26C3×C42×D3—21no—
27C32×#99—22no—
28C3×#837—21no—
29C3×#837—21no—
30C3×#805—22no—
31C3×#575—31no—
32C3×C4×#41—31no—
33#5822C2×C311yescharacteristic
34#5522C2×C311yescharacteristic, complemented
35C3×#421—31no—
36C2×C3×#104—31no—
37C3×#410—21no—
38C3×#410D311yescharacteristic, complemented
39C3×#410C2×C311yescharacteristic, complemented
40C2×#837C2×C311yescharacteristic, complemented
41#5005C2×C311yescharacteristic, complemented
42C3×#344—31no—
43C3×#339—31no—
44#4533C2×C311yescharacteristic, complemented
45#4532C2×C311yescharacteristic, complemented
46C32×#41—21no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C2×C32×Q8C2×C411yescharacteristic, complemented
51C2×C32×D4—21no—
52C2×C32×D4C2×C411yescharacteristic, complemented
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C2×C3×#67—21no—
56C22×C3×Dic3—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235C2×C412yes—
60C3×#235D412yes—
61C2×C3×D12D411yescharacteristic
62C2×C3×C4×D3—21no—
63C2×C3×C4×D3—22no—
64C2×C3×Dic6D411yescharacteristic
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4D412yes—
69C32×C42—21no—
70C32×C42—21no—
71C3×#216—22no—
72C3×#215—22no—
73C3×#214—22no—
74C3×#213—22no—
75C3×(C2×D3)⋊C4—22no—
76C3×Dic3⋊C4—22no—
77C3×C4×Dic3—22no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3D411yescharacteristic
80C3×C4×Dic3D411yescharacteristic
81C3×#208—21no—
82C3×#208—21no—
83C3×#208—21no—
84C3×#208C2×C412yescomplemented
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—21no—
87#1906—31noSylow
88C2×C3×#41—21no—
89C2×C3×#41—31no—
90C2×C3×#41C2×D311yescharacteristic
91C2×C3×#41C22×C311yescharacteristic
92C2×#235C22×C311yescharacteristic
93C3×#137—32no—
94C3×#137—61no—
95C3×AGL(1,ℤ/8)—32no—
96C3×AGL(1,ℤ/8)—61no—
97C3×#135—62no—
98C2×C3×M16—31no—
99C2×C3×M16—31no—
100C3×C4×Q8—61no—
101C3×C4×D4—61no—
102C3×C4×D4—62no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—61no—
106C3×#117C2×D311yescharacteristic
107C3×#117C2×D311yescharacteristic
108C3×#117C22×C311yescharacteristic
109C3×#117C22×C311yescharacteristic
110C2×C3×C42—31no—
111C2×C3×C42—61no—
112#951—21no—
113#950—22no—
114#949—21no—
115#942C3×C412yescomplemented
116#932C3×C412yescomplemented
117C2×C4×Dic3C22×C311yescharacteristic
118C2×#208C22×C311yescharacteristic
119C2×#208C22×C311yescharacteristic
120#893—21no—
121#887—21no—
122#67⋊C4—22no—
123#872—21no—
124C42×D3—21no—
125C3×#104—62no—
126C3×#104—61no—
127C3×#104—61no—
128C3×#99—24no—
129C3×#99—22no—
130C3×#99—22no—
131#837—21no—
132#837—21no—
133#805—22no—
134C23×C32—21no—
135C23×C32C2×D411yescharacteristic
136C22×C3×D3—21no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×D4—21no—
140C32×D4—21no—
141C32×D4—41no—
142C32×D4—41no—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4C22×C411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4#4111yescharacteristic
153C2×C32×C4#4111yescharacteristic
154C3×#67—22no—
155C3×#67—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—42no—
160C3×D12—21no—
161C3×D12C22⋊C412yes—
162C3×C4×D3—22no—
163C3×C4×D3—22no—
164C3×C4×D3—44no—
165C3×Dic6—21no—
166C3×Dic6C22⋊C412yes—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—41no—
169#575—31no—
170C4×#41—31no—
171#421—31no—
172C2×#104—31no—
173#410C3×D311yescharacteristic, complemented
174#344—31no—
175#339—31no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—32no—
185C3×#41—32no—
186C3×#41—41no—
187C3×#41—41no—
188C3×#41—41no—
189C3×#41—41no—
190C3×#41—61no—
191C3×#41—61no—
192C2×C3×Q8—21no—
193C2×C3×Q8—31no—
194C2×C3×Q8C4×D311yescharacteristic, complemented
195C2×C3×Q8C2×C3×C411yescharacteristic, complemented
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C2×C3×D4C4×D311yescharacteristic, complemented
203C2×C3×D4C2×C3×C411yescharacteristic, complemented
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—62no—
209C22×C3×C4—61no—
210C22×C3×C4—61no—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C311yescharacteristic
213C2×#67—21no—
214C22×Dic3—21no—
215#235—21no—
216#235—21no—
217#235C2×C3×C412yes—
218#235C3×D412yes—
219C2×D12C3×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—22no—
222C2×Dic6C3×D411yescharacteristic
223C3×Q16—62no—
224C3×SD16—62no—
225C3×SD16—62no—
226C3×D8—62no—
227C3×M16—32no—
228C3×M16—61no—
229C3×M16—61no—
230C3×M16—61no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—62no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4#6712yes—
246C3×C22⋊C4C3×D412yes—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—31no—
252C3×C42—31no—
253C3×C42—41no—
254C3×C42—41no—
255C3×C42—62no—
256C3×C42—61no—
257#216—22no—
258#215—22no—
259#214—22no—
260#213—22no—
261(C2×D3)⋊C4—22no—
262Dic3⋊C4—22no—
263C4×Dic3—22no—
264C4×Dic3—21no—
265C4×Dic3C3×D411yescharacteristic
266C4×Dic3C3×D411yescharacteristic
267#208—21no—
268#208—21no—
269#208—21no—
270#208C2×C3×C412yescomplemented
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32—81no—
276C22×C32C4×D411yescharacteristic
277C22×C32#12011yescharacteristic
278C22×C32#12111yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—42no—
281C32×C4—41no—
282C32×C4—42no—
283C32×C4—81no—
284C32×C4C2×C22⋊C411yescharacteristic
285C32×C4C4×D411yescharacteristic
286C32×C4#12111yescharacteristic
287C32×C4#12311yescharacteristic
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—41no—
291C2×#41—31no—
292C2×#41C2×C3×D311yescharacteristic
293#137—32no—
294#137—61no—
295AGL(1,ℤ/8)—32no—
296AGL(1,ℤ/8)—61no—
297#135—62no—
298C2×M16—31no—
299C2×M16—31no—
300C4×Q8—61no—
301C4×D4—61no—
302C4×D4—62no—
303#117—61no—
304#117C2×C3×D311yescharacteristic
305#117C2×C3×D311yescharacteristic
306C2×C42—31no—
307C2×C42—61no—
308#104—62no—
309#104—61no—
310#104—61no—
311#99—22no—
312C23×C3—22no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—61no—
317C23×C3C2×#6711yescharacteristic
318C23×C3C2×C3×D411yescharacteristic
319C22×D3—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—32no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—61no—
328C3×D4—21no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—32no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—61no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—124no—
371C2×C3×C4—122no—
372C2×C3×C4C2×C4×D311yescharacteristic
373C2×C3×C4#23511yescharacteristic
374C2×C3×C4D3×D411yescharacteristic
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4#23711yescharacteristic
378C2×C3×C4C22×C3×C411yescharacteristic
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C2×C3×D411yescharacteristic
381C2×C3×C4C2×C3×D411yescharacteristic
382C2×C3×C4C3×#4111yescharacteristic
383C2×C3×C4C3×#4111yescharacteristic
384#67—22no—
385#67—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—42no—
390D12—21no—
391D12C3×C22⋊C412yes—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—44no—
395Dic6—21no—
396Dic6C3×C22⋊C412yes—
397C3×C8—62no—
398C3×C8—121no—
399C3⋊C8—22no—
400C3⋊C8—41no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—81no—
405C2×C32#38611yescharacteristic
406C3×D3—42no—
407#41—21no—
408#41—21no—
409#41—21no—
410#41—21no—
411#41—32no—
412#41—32no—
413#41—61no—
414#41—61no—
415C2×Q8—31no—
416C2×Q8C3×C4×D311yescharacteristic, complemented
417C2×D4—21no—
418C2×D4—31no—
419C2×D4—61no—
420C2×D4C3×C4×D311yescharacteristic, complemented
421C22×C4—21no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4C22×C3×D311yescharacteristic
426Q16—62no—
427SD16—62no—
428SD16—62no—
429D8—62no—
430M16—32no—
431M16—61no—
432M16—61no—
433M16—61no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—62no—
439C22⋊C4—21no—
440C22⋊C4—62no—
441C22⋊C4C3×#6712yes—
442C42—21no—
443C42—21no—
444C42—31no—
445C42—31no—
446C42—62no—
447C42—61no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—62no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—122no—
462C22×C3C4×#6711yescharacteristic
463C22×C3#93711yescharacteristic
464C22×C3#93811yescharacteristic
465C22×C3C3×C4×D411yescharacteristic
466C22×C3C3×#12011yescharacteristic
467C22×C3C3×#12111yescharacteristic
468C2×D3—22no—
469C2×D3—42no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—42no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—62no—
480C3×C4—62no—
481C3×C4—82no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—82no—
485C3×C4—121no—
486C3×C4D3×C22⋊C411yescharacteristic
487C3×C4#67⋊C411yescharacteristic
488C3×C4#88311yescharacteristic
489C3×C4#88411yescharacteristic
490C3×C4C2×C3×C22⋊C411yescharacteristic
491C3×C4C3×C4×D411yescharacteristic
492C3×C4C3×#12111yescharacteristic
493C3×C4C3×#12311yescharacteristic
494Dic3—22no—
495Dic3—22no—
496Dic3—41no—
497C32#190611yescharacteristic, complemented, Sylow
498C23—21no—
499C23—61no—
500C23C2×C3×#6711yescharacteristic
501Q8—21no—
502Q8—21no—
503Q8—32no—
504Q8—61no—
505D4—21no—
506D4—21no—
507D4—32no—
508D4—41no—
509D4—41no—
510D4—62no—
511D4—62no—
512D4—61no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—124no—
524C2×C4—122no—
525C2×C4C2×C3×C4×D311yescharacteristic
526C2×C4C3×#23511yescharacteristic
527C2×C4C3×D3×D411yescharacteristic
528C2×C4C3×D3×D411yescharacteristic
529C2×C4C3×D3×D411yescharacteristic
530C2×C4C3×#23711yescharacteristic
531C8—62no—
532C8—121no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3—82no—
544C2×C3—81no—
545C2×C3—81no—
546C2×C3—122no—
547C2×C3#495611yescharacteristic
548C2×C3C3×#38611yescharacteristic
549D3—42no—
550C22—41no—
551C22—41no—
552C22—62no—
553C22—81no—
554C22—122no—
555C22C3×C4×#6711yescharacteristic
556C22C3×#93711yescharacteristic
557C22C3×#93811yescharacteristic
558C4—42no—
559C4—41no—
560C4—62no—
561C4—62no—
562C4—81no—
563C4—121no—
564C4C3×D3×C22⋊C411yescharacteristic
565C4C3×#67⋊C411yescharacteristic
566C4C3×#88311yescharacteristic
567C4C3×#88411yescharacteristic
568C3—21no—
569C3#7005511yescharacteristic, direct product factor
570C3C3×#190611yescharacteristic, complemented
571C2—21no—
572C2—21no—
573C2—21no—
574C2—81no—
575C2—122no—
576C2C3×#495611yescharacteristic
subgroupG/H
centerC3×C4#884
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#410C2
socleC2×C32#386
prime radicalC3×#5822C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#410D3
hypocenterC3C3×#1906
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#5522C2semidirect
3. C32×#410C2semidirect
4. C2×C3×#837C2semidirect
5. C3×#5005C2semidirect
6. C3×#4533C2semidirect
7. C3×#4532C2semidirect
9. #70055C3direct (N × H)
17. C3×#942C4semidirect
18. C3×#932C4semidirect
34. #5522C2×C3semidirect
38. C3×#410D3semidirect
39. C3×#410C2×C3semidirect
40. C2×#837C2×C3semidirect
41. #5005C2×C3semidirect
44. #4533C2×C3semidirect
45. #4532C2×C3semidirect
50. C2×C32×Q8C2×C4semidirect
52. C2×C32×D4C2×C4semidirect
84. C3×#208C2×C4semidirect
115. #942C3×C4semidirect
116. #932C3×C4semidirect
173. #410C3×D3semidirect
194. C2×C3×Q8C4×D3semidirect
195. C2×C3×Q8C2×C3×C4semidirect
202. C2×C3×D4C4×D3semidirect
203. C2×C3×D4C2×C3×C4semidirect
270. #208C2×C3×C4semidirect
416. C2×Q8C3×C4×D3semidirect
420. C2×D4C3×C4×D3semidirect
497. C32#1906semidirect
569. C3#70055direct (N × H)
570. C3C3×#1906semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C3×#55222
C32×#4102
C2×C3×#8372
C3×#50052
C3×#45332
C3×#45322
C3×#19063
#700553