Finite Group Databasebeta

← C3×#70168 #11786828

Subgroups — C3×#70168 #11786828

Order 1152.

825 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C2×C32×#102C211yesmaximal, characteristic
3C2×C3×#832C211yesmaximal, characteristic
4C2×C3×#818C211yesmaximal, characteristic, complemented
5C3×#4984C211yesmaximal, characteristic, complemented
6C3×#4550C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1862—31nomaximal
9#70168C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C23×C3×Dic3—21no—
12C22×C32×C8C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C3×#217—21no—
15C3×D4×Dic3—22no—
16C3×D4×Dic3—22no—
17C2×C3×C4⋊Dic3—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4×Dic3C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C32×#102C414yescomplemented
23C3×#832—22no—
24C3×C3⋊#95—21no—
25C3×#818—22no—
26C2×C3×C4×D4—31no—
27C2×D4×Dic3C2×C311yescharacteristic
28C2×C3×#107—31no—
29C2×C3×#102—21no—
30C2×C3×#102—31no—
31C2×C3×#102D311yescharacteristic
32C2×C3×#102C2×C311yescharacteristic
33C3×#389—31no—
34C2×#832C2×C311yescharacteristic
35C2×#818C2×C311yescharacteristic, complemented
36#4984C2×C311yescharacteristic, complemented
37C3×#340—31no—
38C3×#336—31no—
39#4550C2×C311yescharacteristic, complemented
40#4529C2×C311yescharacteristic, complemented
41C24×C32—21no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4C2×C412yes—
46C2×C32×D4D412yes—
47C2×C32×D4Q812yes—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C22×C3×Dic3—24no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3—42no—
54C22×C3×Dic3—42no—
55C22×C3×Dic3D411yescharacteristic
56C22×C3×Dic3D411yescharacteristic
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8C2×C412yescomplemented
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×C412yescomplemented
62C3×#217—24no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—42no—
67C3×C4⋊Dic3D412yes—
68C3×C4×Dic3—21no—
69C3×C4×Dic3—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73#1862—31noSylow
74C22×C3×D4—21no—
75C22×C3×D4C2×D311yescharacteristic
76C22×C3×D4C22×C311yescharacteristic
77C23×C3×C4—61no—
78C23×Dic3—21no—
79C22×C3×C8—21no—
80C22×C3×C8—31no—
81C22×C3×C8C2×D311yescharacteristic
82C22×C3×C8C22×C311yescharacteristic
83C3×C4×D4—62no—
84C3×C4×D4—62no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C4⋊C4—61no—
89C2×C3×C4⋊C4C2×D311yescharacteristic
90C2×C3×C4⋊C4C22×C311yescharacteristic
91C2×C3×C22⋊C4—61no—
92C2×C3×C42—31no—
93C2×#217—21no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C22×C311yescharacteristic
98C2×C4⋊Dic3C22×C311yescharacteristic
99C2×C4×Dic3C22×C311yescharacteristic
100C22×C3⋊C8C22×C311yescharacteristic
101C3×#107—62no—
102C3×#102—24no—
103C3×#102—62no—
104C3×#102Dic314yescomplemented
105C3×#102C3×C414yescomplemented
106C3×#95—61no—
107#832—22no—
108C3⋊#95—21no—
109#818—22no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—41no—
114C23×C32—41no—
115C23×C32—41no—
116C23×C32—41no—
117C23×C32C2×D411yescharacteristic
118C32×D4—22no—
119C32×D4—22no—
120C32×D4—22no—
121C32×D4C4⋊C414yescomplemented
122C2×C32×C4—22no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4C22×C411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×Q811yescharacteristic
129C2×C32×C4#4111yescharacteristic
130C2×C32×C4#4111yescharacteristic
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—48no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3D812yes—
144C32×C8—22no—
145C3×C3⋊C8—41no—
146C2×C4×D4—31no—
147C2×#107—31no—
148C2×#102—31no—
149C2×#102C3×D311yescharacteristic
150#389—31no—
151#340—31no—
152#336—31no—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3—41no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C2×C3×D4Dic612yes—
169C2×C3×D4D1212yes—
170C2×C3×D4C2×Dic312yes—
171C2×C3×D4C2×C3×C412yes—
172C2×C3×D4C3×D412yes—
173C2×C3×D4C3×Q812yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—41no—
180C22×C3×C4—64no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—122no—
184C22×C3×C4—122no—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C23×C311yescharacteristic
187C22×Dic3—24no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3C3×D411yescharacteristic
193C22×Dic3C3×D411yescharacteristic
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—22no—
199C2×C3×C8—41no—
200C2×C3×C8—41no—
201C2×C3×C8—61no—
202C2×C3×C8—61no—
203C2×C3×C8—61no—
204C2×C3×C8C2×Dic312yescomplemented
205C2×C3×C8C2×C3×C412yescomplemented
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—122no—
215C3×C4⋊C4C2×Dic312yescomplemented
216C3×C4⋊C4C2×C3×C412yescomplemented
217C3×C22⋊C4—64no—
218C3×C42—61no—
219C3×C42—61no—
220#217—24no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—42no—
225C4⋊Dic3C3×D412yes—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C22×C32—24no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32C4×D411yescharacteristic
239C22×C32#12011yescharacteristic
240C22×C32C22⋊Q811yescharacteristic
241C22×C32C2×D811yescharacteristic
242C22×C32#13511yescharacteristic
243C22×C32AGL(1,ℤ/8)11yescharacteristic
244C22×C32#13711yescharacteristic
245C32×C4—42no—
246C32×C4C2×C4⋊C411yescharacteristic
247C32×C4C4×D411yescharacteristic
248C32×C4C22⋊Q811yescharacteristic
249C32×C4#12311yescharacteristic
250C3×Dic3—22no—
251C3×Dic3—42no—
252C3×Dic3—41no—
253C3×Dic3—81no—
254C22×D4C2×C3×D311yescharacteristic
255C23×C4—61no—
256C22×C8—31no—
257C22×C8C2×C3×D311yescharacteristic
258C4×D4—62no—
259C4×D4—62no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C2×C3×D311yescharacteristic
264C2×C22⋊C4—61no—
265C2×C42—31no—
266#107—62no—
267#102—62no—
268#102C3×Dic314yescomplemented
269#95—61no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—21no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—81no—
289C23×C3—81no—
290C23×C3—81no—
291C23×C3—81no—
292C23×C3D3×D411yescharacteristic
293C23×C3C2×C3×D411yescharacteristic
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—22no—
300C3×D4—24no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4C4⋊Dic314yescomplemented
305C3×D4C3×C4⋊C414yescomplemented
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—32no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—62no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—81no—
327C2×C3×C4—81no—
328C2×C3×C4—128no—
329C2×C3×C4—121no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4—121no—
333C2×C3×C4—121no—
334C2×C3×C4C2×Dic611yescharacteristic
335C2×C3×C4C2×D1211yescharacteristic
336C2×C3×C4D3×D411yescharacteristic
337C2×C3×C4#23711yescharacteristic
338C2×C3×C4#23711yescharacteristic
339C2×C3×C4C22×Dic311yescharacteristic
340C2×C3×C4C22×C3×C411yescharacteristic
341C2×C3×C4C2×C3×D411yescharacteristic
342C2×C3×C4C2×C3×D411yescharacteristic
343C2×C3×C4C2×C3×Q811yescharacteristic
344C2×C3×C4C3×#4111yescharacteristic
345C2×C3×C4C3×#4111yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—22no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—48no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3C3×D812yes—
359C3×C8—22no—
360C3×C8—22no—
361C3×C8—42no—
362C3×C8—121no—
363C3⋊C8—41no—
364C2×C32—44no—
365C2×C32#38511yescharacteristic
366C2×C32C4×D811yescharacteristic
367C2×C32#44011yescharacteristic
368C2×C32#44711yescharacteristic
369C2×C32#45211yescharacteristic
370C2×C32#47411yescharacteristic
371C2×C32#47811yescharacteristic
372C24—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C3×Dic612yes—
377C2×D4C3×D1212yes—
378C2×D4C2×C3×Dic312yes—
379C22×C4—21no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—64no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4C22×C3×D311yescharacteristic
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8C2×C3×Dic312yescomplemented
394C4⋊C4—21no—
395C4⋊C4—32no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—122no—
400C4⋊C4C2×C3×Dic312yescomplemented
401C22⋊C4—64no—
402C42—61no—
403C42—61no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3#87811yescharacteristic
433C22×C3#88211yescharacteristic
434C22×C3D3×D811yescharacteristic
435C22×C3#91211yescharacteristic
436C22×C3#91411yescharacteristic
437C22×C3#91511yescharacteristic
438C22×C3D4×Dic311yescharacteristic
439C22×C3C3×C4×D411yescharacteristic
440C22×C3C3×#12011yescharacteristic
441C22×C3C3×C22⋊Q811yescharacteristic
442C22×C3C2×C3×D811yescharacteristic
443C22×C3C3×#13511yescharacteristic
444C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
445C22×C3C3×#13711yescharacteristic
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—62no—
453C3×C4—82no—
454C3×C4—122no—
455C3×C4—121no—
456C3×C4—241no—
457C3×C4#87811yescharacteristic
458C3×C4#88611yescharacteristic
459C3×C4C2×C4⋊Dic311yescharacteristic
460C3×C4D4×Dic311yescharacteristic
461C3×C4C2×C3×C4⋊C411yescharacteristic
462C3×C4C3×C4×D411yescharacteristic
463C3×C4C3×C22⋊Q811yescharacteristic
464C3×C4C3×#12311yescharacteristic
465Dic3—22no—
466Dic3—42no—
467Dic3—41no—
468Dic3—81no—
469C32#186211yescharacteristic, complemented, Sylow
470C23—22no—
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C3×D3×D411yescharacteristic
478D4—22no—
479D4—22no—
480D4—22no—
481D4C3×C4⋊Dic314yescomplemented
482C2×C4—22no—
483C2×C4—32no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—128no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4C2×C3×Dic611yescharacteristic
499C2×C4C2×C3×D1211yescharacteristic
500C2×C4C3×D3×D411yescharacteristic
501C2×C4C3×#23711yescharacteristic
502C2×C4C3×#23711yescharacteristic
503C2×C4C22×C3×Dic311yescharacteristic
504C8—22no—
505C8—121no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—44no—
514C2×C3—44no—
515C2×C3—84no—
516C2×C3#475911yescharacteristic
517C2×C3#476611yescharacteristic
518C2×C3#477311yescharacteristic
519C2×C3#478311yescharacteristic
520C2×C3#495011yescharacteristic
521C2×C3Dic3×D811yescharacteristic
522C2×C3#516411yescharacteristic
523C2×C3C3×#38511yescharacteristic
524C2×C3C3×C4×D811yescharacteristic
525C2×C3C3×#44011yescharacteristic
526C2×C3C3×#44711yescharacteristic
527C2×C3C3×#45211yescharacteristic
528C2×C3C3×#47411yescharacteristic
529C2×C3C3×#47811yescharacteristic
530C22—24no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22C3×#87811yescharacteristic
538C22C3×#88211yescharacteristic
539C22C3×D3×D811yescharacteristic
540C22C3×#91211yescharacteristic
541C22C3×#91411yescharacteristic
542C22C3×#91511yescharacteristic
543C22C3×D4×Dic311yescharacteristic
544C4—42no—
545C4—62no—
546C4—122no—
547C4—121no—
548C4—241no—
549C4C3×#87811yescharacteristic
550C4C3×#88611yescharacteristic
551C4C2×C3×C4⋊Dic311yescharacteristic
552C4C3×D4×Dic311yescharacteristic
553C3—21no—
554C3#7016811yescharacteristic, direct product factor
555C3C3×#186211yescharacteristic, complemented
556C2—44no—
557C2C3×#475911yescharacteristic
558C2C3×#476611yescharacteristic
559C2C3×#477311yescharacteristic
560C2C3×#478311yescharacteristic
561C2C3×#495011yescharacteristic
562C2C3×Dic3×D811yescharacteristic
563C2C3×#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×#1862
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×#818C2semidirect
5. C3×#4984C2semidirect
6. C3×#4550C2semidirect
7. C3×#4529C2semidirect
9. #70168C3direct (N × H)
22. C32×#102C4semidirect
35. C2×#818C2×C3semidirect
36. #4984C2×C3semidirect
39. #4550C2×C3semidirect
40. #4529C2×C3semidirect
59. C2×C32×C8C2×C4semidirect
61. C32×C4⋊C4C2×C4semidirect
104. C3×#102Dic3semidirect
105. C3×#102C3×C4semidirect
121. C32×D4C4⋊C4semidirect
204. C2×C3×C8C2×Dic3semidirect
205. C2×C3×C8C2×C3×C4semidirect
215. C3×C4⋊C4C2×Dic3semidirect
216. C3×C4⋊C4C2×C3×C4semidirect
268. #102C3×Dic3semidirect
304. C3×D4C4⋊Dic3semidirect
305. C3×D4C3×C4⋊C4semidirect
393. C2×C8C2×C3×Dic3semidirect
400. C4⋊C4C2×C3×Dic3semidirect
469. C32#1862semidirect
481. D4C3×C4⋊Dic3semidirect
554. C3#70168direct (N × H)
555. C3C3×#1862semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C2×C32×#1022
C2×C3×#8322
C2×C3×#8182
C3×#49842
C3×#45502
C3×#45292
C3×#18623
#701683