Finite Group Databasebeta

← C3×#78870 #11868786

Subgroups — C3×#78870 #11868786

Order 1152.

913 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Dic3×#41C211yesmaximal, characteristic
2C2×C3×#888C211yesmaximal, characteristic, complemented
3C32×#417C211yesmaximal, characteristic
4C3×#5232C211yesmaximal, characteristic
5C3×#5121C211yesmaximal, characteristic, complemented
6C3×#5118C211yesmaximal, characteristic, complemented
7C3×#4964C211yesmaximal, characteristic, complemented
8C3×#4793C212yesmaximal, complemented
9C3×#4791C212yesmaximal, complemented
10C3×#4761C212yesmaximal, complemented
11C3×#4759C212yesmaximal, complemented
12C3×#3079—31nomaximal
13#78870C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C22×C3×Dic6—21no—
16C2×C32×M16C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C3×Q8×Dic3C2211yescharacteristic
19C3×Q8×Dic3C2211yescharacteristic
20C3×D4×Dic3—21no—
21C3×D4×Dic3—21no—
22C3×D4×Dic3C2211yescharacteristic
23C3×D4×Dic3C2211yescharacteristic
24C3×#926—21no—
25C3×#926C2211yescharacteristic
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×Dic3⋊C4—21no—
28C2×C3×C4×Dic3—21no—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×#208C2211yescharacteristic
31C3×#888—21no—
32C3×#888—21no—
33C3×#888C2212yes—
34C3×#888C2212yes—
35C32×#103C2212yes—
36C32×#102C2212yes—
37C3×#835C2212yes—
38C3×#832C2212yes—
39C3×#818C2212yes—
40C3×#817C2212yes—
41C3×Dic3⋊C8C2212yes—
42C3×Dic3⋊C8C2212yes—
43C3×#808C2212yes—
44C3×#807C2212yes—
45C2×C3×C4⋊Q8—31no—
46C3×C4×#41—31no—
47Dic3×#41C2×C311yescharacteristic
48C2×#888C2×C311yescharacteristic, complemented
49C3×#477—32no—
50C3×#476—32no—
51C3×#475—32no—
52C3×#474—32no—
53C3×#428—31no—
54C3×#428—31no—
55C3×#423—31no—
56C3×#417—21no—
57C3×#417—31no—
58C3×#417D311yescharacteristic
59C3×#417C2×C311yescharacteristic
60#5232C2×C311yescharacteristic
61#5121C2×C311yescharacteristic, complemented
62#5118C2×C311yescharacteristic, complemented
63#4964C2×C311yescharacteristic, complemented
64#4793C2×C312yescomplemented
65#4791C2×C312yescomplemented
66#4761C2×C312yescomplemented
67#4759C2×C312yescomplemented
68C32×#41—22no—
69C32×#41Q814yescomplemented
70C2×C32×Q8C2311yescharacteristic
71C2×C32×D4—21no—
72C2×C32×D4C2311yescharacteristic
73C22×C32×C4—21no—
74C22×C32×C4—21no—
75C22×C32×C4C2311yescharacteristic
76C22×C3×Dic3—21no—
77C22×C3×Dic3—21no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3D411yescharacteristic
80C22×C3×Dic3D411yescharacteristic
81C2×C3×Dic6—22no—
82C2×C3×Dic6—22no—
83C2×C3×Dic6—44no—
84C32×M16—22no—
85C2×C32×C8C2312yes—
86C32×C4⋊C4—21no—
87C32×C4⋊C4C2312yes—
88C3×#217—21no—
89C3×#217—21no—
90C3×#217—21no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2312yes—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2311yescharacteristic
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C3×#208—22no—
107C2×C3×C3⋊C8C2312yes—
108#3079—31noSylow
109C2×C3×#41—21no—
110C2×C3×#41C2×D311yescharacteristic
111C2×C3×#41C22×C311yescharacteristic
112C22×C3×Q8—61no—
113C22×Dic6—21no—
114C2×C3×M16—21no—
115C2×C3×M16—31no—
116C2×C3×M16C2×D311yescharacteristic
117C2×C3×M16C22×C311yescharacteristic
118C3×C4⋊Q8—32no—
119C3×C4⋊Q8—32no—
120C3×C4⋊Q8—61no—
121C3×C4⋊Q8—61no—
122C3×C4×Q8—31no—
123C3×C4×Q8—31no—
124C3×C4×D4—31no—
125C3×C4×D4—31no—
126C3×C4×D4—61no—
127C3×C4×D4—61no—
128C3×#117—31no—
129C3×#117—61no—
130C2×C3×C4⋊C4—21no—
131C2×C3×C4⋊C4—31no—
132C2×C3×C4⋊C4—61no—
133C2×C3×C4⋊C4C2×D311yescharacteristic
134C2×C3×C4⋊C4C22×C311yescharacteristic
135C2×C3×C42—31no—
136C2×C3×C42—61no—
137Q8×Dic3C22×C311yescharacteristic
138Q8×Dic3C22×C311yescharacteristic
139D4×Dic3—21no—
140D4×Dic3—21no—
141D4×Dic3C22×C311yescharacteristic
142D4×Dic3C22×C311yescharacteristic
143#926—21no—
144#926C22×C311yescharacteristic
145C2×C4⋊Dic3C22×C311yescharacteristic
146C2×Dic3⋊C4—21no—
147C2×C4×Dic3—21no—
148C2×C4×Dic3C22×C311yescharacteristic
149C2×#208C22×C311yescharacteristic
150#888—21no—
151#888—21no—
152#888C22×C312yes—
153#888C22×C312yes—
154C3×#107—32no—
155C3×#107—32no—
156C3×#106—32no—
157C3×#106—32no—
158C3×C4⋊C8—32no—
159C3×C4⋊C8—32no—
160C3×#103—22no—
161C3×#103—32no—
162C3×#103C2×D312yes—
163C3×#103C22×C312yes—
164C3×#102—22no—
165C3×#102—32no—
166C3×#102C2×D312yes—
167C3×#102C22×C312yes—
168#835C22×C312yes—
169#832C22×C312yes—
170#818C22×C312yes—
171#817C22×C312yes—
172Dic3⋊C8C22×C312yes—
173Dic3⋊C8C22×C312yes—
174#808C22×C312yes—
175#807C22×C312yes—
176C23×C32—21no—
177C23×C32C2×D411yescharacteristic
178C32×Q8—21no—
179C32×Q8C2×Q812yescomplemented
180C32×D4—24no—
181C32×D4—21no—
182C32×D4C2×Q812yescomplemented
183C2×C32×C4—21no—
184C2×C32×C4—24no—
185C2×C32×C4—22no—
186C2×C32×C4—42no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2×Q812yes—
189C2×C32×C4#4112yes—
190C2×C32×C4C2411yescharacteristic
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—42no—
197C2×C3×Dic3—42no—
198C2×C3×Dic3—42no—
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C2×C3×Dic3C2×D411yescharacteristic
202C2×C3×Dic3C2×D411yescharacteristic
203C3×Dic6—44no—
204C3×Dic6—44no—
205C32×C8—22no—
206C3×C3⋊C8—22no—
207C2×C4⋊Q8—31no—
208C4×#41—31no—
209#477—32no—
210#476—32no—
211#475—32no—
212#474—32no—
213#428—31no—
214#428—31no—
215#423—31no—
216#417—31no—
217#417C3×D311yescharacteristic
218C3×#41—22no—
219C3×#41—22no—
220C3×#41—24no—
221C3×#41—42no—
222C3×#41Dic614yescomplemented
223C3×#41C3×Q814yescomplemented
224C2×C3×Q8—21no—
225C2×C3×Q8—62no—
226C2×C3×Q8—62no—
227C2×C3×Q8—124no—
228C2×C3×Q8C22×D311yescharacteristic
229C2×C3×Q8C23×C311yescharacteristic
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—41no—
234C2×C3×D4C22×D311yescharacteristic
235C2×C3×D4C23×C311yescharacteristic
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—41no—
244C22×C3×C4—41no—
245C22×C3×C4—61no—
246C22×C3×C4—61no—
247C22×C3×C4—61no—
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C23×C311yescharacteristic
250C22×Dic3—21no—
251C22×Dic3—21no—
252C22×Dic3—21no—
253C22×Dic3C3×D411yescharacteristic
254C22×Dic3C3×D411yescharacteristic
255C2×Dic6—22no—
256C2×Dic6—22no—
257C2×Dic6—44no—
258C3×M16—22no—
259C3×M16—22no—
260C3×M16—42no—
261C3×M16—62no—
262C2×C3×C8—22no—
263C2×C3×C8—32no—
264C2×C3×C8C22×D312yes—
265C2×C3×C8C23×C312yes—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4—31no—
271C3×C4⋊C4—31no—
272C3×C4⋊C4—41no—
273C3×C4⋊C4—62no—
274C3×C4⋊C4—62no—
275C3×C4⋊C4—61no—
276C3×C4⋊C4—61no—
277C3×C4⋊C4—61no—
278C3×C4⋊C4C22×D312yes—
279C3×C4⋊C4C23×C312yes—
280C3×C22⋊C4—61no—
281C3×C22⋊C4—61no—
282C3×C22⋊C4—61no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—61no—
288C3×C42—61no—
289C3×C42—61no—
290#217—21no—
291#217—21no—
292#217—21no—
293C4⋊Dic3—21no—
294C4⋊Dic3—21no—
295C4⋊Dic3—21no—
296C4⋊Dic3C23×C311yescharacteristic
297C4⋊Dic3C23×C311yescharacteristic
298C4⋊Dic3C23×C312yes—
299Dic3⋊C4—22no—
300Dic3⋊C4—22no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—21no—
304C4×Dic3C3×D411yescharacteristic
305C4×Dic3C3×D411yescharacteristic
306C4×Dic3C23×C311yescharacteristic
307C4×Dic3C23×C311yescharacteristic
308#208—22no—
309C2×C3⋊C8C23×C312yes—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—22no—
313C22×C32—41no—
314C22×C32—41no—
315C22×C32C22⋊Q812yes—
316C22×C32C22×D411yescharacteristic
317C32×C4—22no—
318C32×C4—42no—
319C32×C4C22⋊Q812yes—
320C32×C4C22×Q811yescharacteristic
321C32×C4C2×#4111yescharacteristic
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—42no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3AGL(1,ℤ/8)12yes—
328C3×Dic3#13712yes—
329C2×#41C2×C3×D311yescharacteristic
330C22×Q8—61no—
331C2×M16—31no—
332C2×M16C2×C3×D311yescharacteristic
333C4⋊Q8—32no—
334C4⋊Q8—32no—
335C4⋊Q8—61no—
336C4⋊Q8—61no—
337C4×Q8—31no—
338C4×Q8—31no—
339C4×D4—31no—
340C4×D4—31no—
341C4×D4—61no—
342C4×D4—61no—
343#117—31no—
344#117—61no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4C2×C3×D311yescharacteristic
348C2×C42—31no—
349C2×C42—61no—
350#107—32no—
351#107—32no—
352#106—32no—
353#106—32no—
354C4⋊C8—32no—
355C4⋊C8—32no—
356#103—32no—
357#103C2×C3×D312yes—
358#102—32no—
359#102C2×C3×D312yes—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—41no—
364C23×C3D3×D411yescharacteristic
365C23×C3C2×C3×D411yescharacteristic
366C3×Q8—21no—
367C3×Q8—21no—
368C3×Q8—22no—
369C3×Q8—41no—
370C3×Q8—124no—
371C3×Q8—124no—
372C3×Q8C2×Dic612yescomplemented
373C3×Q8C2×C3×Q812yescomplemented
374C3×D4—24no—
375C3×D4—24no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—22no—
379C3×D4—41no—
380C3×D4—44no—
381C3×D4C2×Dic612yescomplemented
382C3×D4C2×C3×Q812yescomplemented
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—44no—
398C2×C3×C4—41no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—42no—
402C2×C3×C4—62no—
403C2×C3×C4—62no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—62no—
407C2×C3×C4—82no—
408C2×C3×C4—122no—
409C2×C3×C4—122no—
410C2×C3×C4—122no—
411C2×C3×C4C2×Dic612yes—
412C2×C3×C4D3×D411yescharacteristic
413C2×C3×C4#23712yes—
414C2×C3×C4C2×C3×D411yescharacteristic
415C2×C3×C4C2×C3×Q812yes—
416C2×C3×C4C3×#4112yes—
417C2×C3×C4C23×D311yescharacteristic
418C2×C3×C4C24×C311yescharacteristic
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—42no—
425C2×Dic3—42no—
426C2×Dic3—42no—
427C2×Dic3C2×C3×D411yescharacteristic
428C2×Dic3C2×C3×D411yescharacteristic
429C2×Dic3C2×C3×D411yescharacteristic
430C2×Dic3C2×C3×D411yescharacteristic
431Dic6—44no—
432Dic6—44no—
433C3×C8—22no—
434C3×C8—22no—
435C3×C8—42no—
436C3×C8—62no—
437C3⋊C8—22no—
438C2×C32—22no—
439C2×C32—42no—
440C2×C32C2×C22⋊Q811yescharacteristic
441C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C32C2×#13711yescharacteristic
443#41—22no—
444#41C3×Dic614yescomplemented
445C2×Q8—62no—
446C2×Q8—62no—
447C2×Q8—124no—
448C2×Q8C22×C3×D311yescharacteristic
449C2×D4—21no—
450C2×D4C22×C3×D311yescharacteristic
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4C22×C3×D311yescharacteristic
459M16—22no—
460M16—62no—
461C2×C8—32no—
462C2×C8C22×C3×D312yes—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—31no—
466C4⋊C4—31no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4C22×C3×D312yes—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C42—31no—
477C42—31no—
478C42—31no—
479C42—31no—
480C42—61no—
481C42—61no—
482C42—61no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3#87812yes—
501C22×C3C3×C22⋊Q812yes—
502C22×C3C2×D3×D411yescharacteristic
503C22×C3C22×C3×D411yescharacteristic
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—32no—
510C3×C4—32no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—82no—
517C3×C4—122no—
518C3×C4—121no—
519C3×C4—121no—
520C3×C4#87812yes—
521C3×C4C3×C22⋊Q812yes—
522C3×C4C22×Dic611yescharacteristic
523C3×C4C2×#23711yescharacteristic
524C3×C4C22×C3×Q811yescharacteristic
525C3×C4C2×C3×#4111yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—42no—
529Dic3—41no—
530Dic3—41no—
531Dic3C3×AGL(1,ℤ/8)12yes—
532Dic3C3×#13712yes—
533C32#307911yescharacteristic, complemented, Sylow
534C23—21no—
535C23C3×D3×D411yescharacteristic
536Q8—21no—
537Q8—124no—
538Q8—124no—
539Q8C2×C3×Dic612yescomplemented
540D4—24no—
541D4—21no—
542D4C2×C3×Dic612yescomplemented
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—42no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4C2×C3×Dic612yes—
560C2×C4C3×D3×D411yescharacteristic
561C2×C4C3×#23712yes—
562C2×C4C23×C3×D311yescharacteristic
563C8—22no—
564C8—62no—
565C2×C3—22no—
566C2×C3—22no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—42no—
571C2×C3—42no—
572C2×C3—42no—
573C2×C3—82no—
574C2×C3C2×#87811yescharacteristic
575C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
576C2×C3D3×#13711yescharacteristic
577C2×C3C2×C3×C22⋊Q811yescharacteristic
578C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
579C2×C3C2×C3×#13711yescharacteristic
580C22—21no—
581C22—21no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22C3×#87812yes—
586C22C2×C3×D3×D411yescharacteristic
587C4—22no—
588C4—32no—
589C4—32no—
590C4—42no—
591C4—61no—
592C4—61no—
593C4—122no—
594C4—121no—
595C4—121no—
596C4C3×#87812yes—
597C4C22×C3×Dic611yescharacteristic
598C4C2×C3×#23711yescharacteristic
599C3—21no—
600C3#7887011yescharacteristic, direct product factor
601C3C3×#307911yescharacteristic, complemented
602C2—22no—
603C2—42no—
604C2C2×C3×#87811yescharacteristic
605C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
606C2C3×D3×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#417C2
socleC22×C32C22×D4
prime radicalC2×C32×#41C22
hypercenterC3×#417D3
hypocenterC3C3×#3079
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
2. C2×C3×#888C2semidirect
5. C3×#5121C2semidirect
6. C3×#5118C2semidirect
7. C3×#4964C2semidirect
8. C3×#4793C2semidirect
9. C3×#4791C2semidirect
10. C3×#4761C2semidirect
11. C3×#4759C2semidirect
13. #78870C3direct (N × H)
48. C2×#888C2×C3semidirect
61. #5121C2×C3semidirect
62. #5118C2×C3semidirect
63. #4964C2×C3semidirect
64. #4793C2×C3semidirect
65. #4791C2×C3semidirect
66. #4761C2×C3semidirect
67. #4759C2×C3semidirect
69. C32×#41Q8semidirect
179. C32×Q8C2×Q8semidirect
182. C32×D4C2×Q8semidirect
222. C3×#41Dic6semidirect
223. C3×#41C3×Q8semidirect
372. C3×Q8C2×Dic6semidirect
373. C3×Q8C2×C3×Q8semidirect
381. C3×D4C2×Dic6semidirect
382. C3×D4C2×C3×Q8semidirect
444. #41C3×Dic6semidirect
533. C32#3079semidirect
539. Q8C2×C3×Dic6semidirect
542. D4C2×C3×Dic6semidirect
600. C3#78870direct (N × H)
601. C3C3×#3079semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×#412
C2×C3×#8882
C32×#4172
C3×#52322
C3×#51212
C3×#51182
C3×#49642
C3×#47932
C3×#47912
C3×#47612
C3×#47592
C3×#30793
#788703