Finite Group Databasebeta

← C3×#70187 #11789894

Subgroups — C3×#70187 #11789894

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#102C211yesmaximal, characteristic
2C3×#5221C211yesmaximal, characteristic
3C2×C3×#832C211yesmaximal, characteristic
4C2×C3×#815C211yesmaximal, characteristic, complemented
5C3×#4984C211yesmaximal, characteristic, complemented
6C3×#4551C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1936—31nomaximal
9#70187C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C3×#217—21no—
14C2×C3×#217—21no—
15C2×C3×C4⋊Dic3—21no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4×Dic3C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C32×#102C414yescomplemented
21C3×#832—22no—
22C3×C3⋊#95—21no—
23C3×#815—22no—
24C2×C3×#102—21no—
25C2×C3×#102—31no—
26C2×C3×#102D311yescharacteristic
27C2×C3×#102C2×C311yescharacteristic
28C2×C3×#97—31no—
29C3×#390—31no—
30C3×#389—31no—
31#5221C2×C311yescharacteristic
32C2×#832C2×C311yescharacteristic
33C2×#815C2×C311yescharacteristic, complemented
34#4984C2×C311yescharacteristic, complemented
35C3×#340—31no—
36C3×#340—31no—
37#4551C2×C311yescharacteristic, complemented
38#4529C2×C311yescharacteristic, complemented
39C24×C32—21no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4C2×C412yes—
46C22×C32×C4—21no—
47C22×C32×C4C2311yescharacteristic
48C22×C3×Dic3—21no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3D411yescharacteristic
51C22×C3×Dic3D411yescharacteristic
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8C2×C412yes—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2×C412yes—
57C3×#217—42no—
58C3×#217—42no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—42no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8D412yes—
69#1936—31noSylow
70C22×C3×D4—21no—
71C22×C3×D4C2×D311yescharacteristic
72C22×C3×D4C22×C311yescharacteristic
73C22×C3×C8—21no—
74C22×C3×C8—31no—
75C22×C3×C8C2×D311yescharacteristic
76C22×C3×C8C22×C311yescharacteristic
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—61no—
81C2×C3×C4⋊C4C2×D311yescharacteristic
82C2×C3×C4⋊C4C22×C311yescharacteristic
83C2×C3×C22⋊C4—61no—
84C2×C3×C22⋊C4—61no—
85C2×C3×C42—31no—
86C2×#217—21no—
87C2×#217—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C22×C311yescharacteristic
90C2×C4⋊Dic3C22×C311yescharacteristic
91C2×C4×Dic3C22×C311yescharacteristic
92C22×C3⋊C8C22×C311yescharacteristic
93C3×#102—24no—
94C3×#102—62no—
95C3×#102Dic314yescomplemented
96C3×#102C3×C414yescomplemented
97C3×#97—62no—
98C3×#95—61no—
99#832—22no—
100C3⋊#95—21no—
101#815—22no—
102C23×C32—22no—
103C23×C32—41no—
104C23×C32—41no—
105C23×C32—41no—
106C23×C32—41no—
107C23×C32—41no—
108C23×C32—41no—
109C23×C32C2×D411yescharacteristic
110C32×D4—22no—
111C32×D4—22no—
112C32×D4—41no—
113C32×D4—41no—
114C2×C32×C4—22no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C32×C8—22no—
136C3×C3⋊C8—22no—
137C2×#102—31no—
138C2×#102C3×D311yescharacteristic
139C2×#97—31no—
140#390—31no—
141#389—31no—
142#340—31no—
143#340—31no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3—41no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4C2×Dic312yes—
164C2×C3×D4C2×C3×C412yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—41no—
171C22×C3×C4—61no—
172C22×C3×C4—61no—
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C23×C311yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3C3×D411yescharacteristic
178C22×Dic3C3×D411yescharacteristic
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8—32no—
185C2×C3×C8—41no—
186C2×C3×C8—41no—
187C2×C3×C8—61no—
188C2×C3×C8—61no—
189C2×C3×C8C2×Dic312yes—
190C2×C3×C8C2×C3×C412yes—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—122no—
200C3×C4⋊C4C2×Dic312yes—
201C3×C4⋊C4C2×C3×C412yes—
202C3×C22⋊C4—122no—
203C3×C22⋊C4—122no—
204C3×C42—61no—
205C3×C42—61no—
206#217—42no—
207#217—42no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—42no—
213C4×Dic3—21no—
214C4×Dic3—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C3×D412yes—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—81no—
222C22×C32—81no—
223C22×C32—81no—
224C22×C32—81no—
225C22×C32C4×D411yescharacteristic
226C22×C32#12111yescharacteristic
227C22×C32#12411yescharacteristic
228C22×C32AGL(1,ℤ/8)11yescharacteristic
229C22×C32AGL(1,ℤ/8)11yescharacteristic
230C22×C32AGL(1,ℤ/8)11yescharacteristic
231C22×C32#13711yescharacteristic
232C32×C4—42no—
233C32×C4#11711yescharacteristic
234C32×C4C4×D411yescharacteristic
235C32×C4#12311yescharacteristic
236C32×C4#12611yescharacteristic
237C3×Dic3—41no—
238C3×Dic3—41no—
239C3×Dic3—81no—
240C3×Dic3—81no—
241C22×D4C2×C3×D311yescharacteristic
242C22×C8—31no—
243C22×C8C2×C3×D311yescharacteristic
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4C2×C3×D311yescharacteristic
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C42—31no—
251#102—62no—
252#102C3×Dic314yescomplemented
253#97—62no—
254#95—61no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—81no—
274C23×C3—81no—
275C23×C3—81no—
276C23×C3—81no—
277C23×C3D3×D411yescharacteristic
278C23×C3C2×C3×D411yescharacteristic
279C3×D4—24no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—41no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—81no—
311C2×C3×C4—81no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4#23511yescharacteristic
319C2×C3×C4#23511yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4#23711yescharacteristic
322C2×C3×C4#23711yescharacteristic
323C2×C3×C4C22×Dic311yescharacteristic
324C2×C3×C4C22×C3×C411yescharacteristic
325C2×C3×C4C2×C3×D411yescharacteristic
326C2×C3×C4C3×#4111yescharacteristic
327C2×C3×C4C3×#4111yescharacteristic
328C2×C3×C4C3×#4111yescharacteristic
329C2×C3×C4C3×#4111yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8—42no—
345C3×C8—62no—
346C3⋊C8—22no—
347C2×C32—82no—
348C2×C32#38811yescharacteristic
349C2×C32#44011yescharacteristic
350C2×C32#44211yescharacteristic
351C2×C32#46811yescharacteristic
352C2×C32#46911yescharacteristic
353C2×C32#48911yescharacteristic
354C2×C32#49011yescharacteristic
355C24—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4C2×C3×Dic312yes—
362C22×C4—21no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4C22×C3×D311yescharacteristic
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—32no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8C2×C3×Dic312yes—
374C4⋊C4—21no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—122no—
380C4⋊C4C2×C3×Dic312yes—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C42—61no—
384C42—61no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—44no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3#88411yescharacteristic
414C22×C3#88511yescharacteristic
415C22×C3#91111yescharacteristic
416C22×C3#91111yescharacteristic
417C22×C3#91411yescharacteristic
418C22×C3#91511yescharacteristic
419C22×C3D4×Dic311yescharacteristic
420C22×C3C3×C4×D411yescharacteristic
421C22×C3C3×#12111yescharacteristic
422C22×C3C3×#12411yescharacteristic
423C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
424C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
425C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
426C22×C3C3×#13711yescharacteristic
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—82no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—241no—
437C3×C4—241no—
438C3×C4#87911yescharacteristic
439C3×C4#88311yescharacteristic
440C3×C4#92611yescharacteristic
441C3×C4D4×Dic311yescharacteristic
442C3×C4C3×#11711yescharacteristic
443C3×C4C3×C4×D411yescharacteristic
444C3×C4C3×#12311yescharacteristic
445C3×C4C3×#12611yescharacteristic
446Dic3—41no—
447Dic3—41no—
448Dic3—81no—
449Dic3—81no—
450C32#193611yescharacteristic, complemented, Sylow
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C3×D3×D411yescharacteristic
459D4—22no—
460D4—22no—
461D4—41no—
462D4—41no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4C3×#23511yescharacteristic
479C2×C4C3×#23511yescharacteristic
480C2×C4C3×D3×D411yescharacteristic
481C2×C4C3×#23711yescharacteristic
482C2×C4C3×#23711yescharacteristic
483C2×C4C22×C3×Dic311yescharacteristic
484C8—22no—
485C8—62no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—84no—
494C2×C3—82no—
495C2×C3—82no—
496C2×C3#476411yescharacteristic
497C2×C3#476511yescharacteristic
498C2×C3#478011yescharacteristic
499C2×C3#478211yescharacteristic
500C2×C3#495111yescharacteristic
501C2×C3#515211yescharacteristic
502C2×C3#516411yescharacteristic
503C2×C3C3×#38811yescharacteristic
504C2×C3C3×#44011yescharacteristic
505C2×C3C3×#44211yescharacteristic
506C2×C3C3×#46811yescharacteristic
507C2×C3C3×#46911yescharacteristic
508C2×C3C3×#48911yescharacteristic
509C2×C3C3×#49011yescharacteristic
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C3×#88411yescharacteristic
518C22C3×#88511yescharacteristic
519C22C3×#91111yescharacteristic
520C22C3×#91111yescharacteristic
521C22C3×#91411yescharacteristic
522C22C3×#91511yescharacteristic
523C22C3×D4×Dic311yescharacteristic
524C4—42no—
525C4—121no—
526C4—121no—
527C4—241no—
528C4—241no—
529C4C3×#87911yescharacteristic
530C4C3×#88311yescharacteristic
531C4C3×#92611yescharacteristic
532C4C3×D4×Dic311yescharacteristic
533C3—21no—
534C3#7018711yescharacteristic, direct product factor
535C3C3×#193611yescharacteristic, complemented
536C2—82no—
537C2C3×#476411yescharacteristic
538C2C3×#476511yescharacteristic
539C2C3×#478011yescharacteristic
540C2C3×#478211yescharacteristic
541C2C3×#495111yescharacteristic
542C2C3×#515211yescharacteristic
543C2C3×#516411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#102D3
hypocenterC3C3×#1936
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
4. C2×C3×#815C2semidirect
5. C3×#4984C2semidirect
6. C3×#4551C2semidirect
7. C3×#4529C2semidirect
9. #70187C3direct (N × H)
20. C32×#102C4semidirect
33. C2×#815C2×C3semidirect
34. #4984C2×C3semidirect
37. #4551C2×C3semidirect
38. #4529C2×C3semidirect
95. C3×#102Dic3semidirect
96. C3×#102C3×C4semidirect
252. #102C3×Dic3semidirect
450. C32#1936semidirect
534. C3#70187direct (N × H)
535. C3C3×#1936semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1022
C3×#52212
C2×C3×#8322
C2×C3×#8152
C3×#49842
C3×#45512
C3×#45292
C3×#19363
#701873