Finite Group Databasebeta

← C3×#32958 #10651565

Subgroups — C3×#32958 #10651565

Order 768.

1364 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#1865C212yesmaximal, complemented
6C3×#1862C212yesmaximal, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1803C211yesmaximal, characteristic, complemented
9C3×#1802C211yesmaximal, characteristic, complemented
10C3×#1797C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal
12#32958C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C3×C4×#41—24no—
17C3×C4×#41—24no—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C42—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#477—24no—
29C3×#476—24no—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#391—21no—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#380—21no—
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×C4×#41C2×C311yescharacteristic
49C2×#477C2×C311yescharacteristic, complemented
50C2×#476C2×C311yescharacteristic, complemented
51#2857C2×C311yescharacteristic, complemented
52#1865C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1854C2×C312yescomplemented
55#1803C2×C311yescharacteristic, complemented
56#1802C2×C311yescharacteristic, complemented
57#1797C2×C311yescharacteristic, complemented
58#1788C2×C312yes—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41Q814yescomplemented
65C22×C3×Q8—21no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4D411yescharacteristic
73C23×C3×C4D411yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×C22⋊Q8—44no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×D4—28no—
85C3×C4×D4—28no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4D414yes—
90C3×#117—28no—
91C3×#117—22no—
92C3×#117—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C42—28no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—22no—
112C2×C3×C42—22no—
113C2×C3×C42D411yescharacteristic
114C2×C3×C42D411yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—24no—
118C3×#106—24no—
119C3×C4⋊C8—22no—
120C3×C4⋊C8—22no—
121C3×#103—24no—
122C3×#103—24no—
123C3×#102—24no—
124C3×#102—24no—
125C3×C22⋊C8—24no—
126C3×#95—22no—
127C3×#95—22no—
128C22×#41C22×C311yescharacteristic
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×C22⋊Q8—21no—
131C4×#41—24no—
132C4×#41—24no—
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139C2×#117—21no—
140C2×#117C22×C311yescharacteristic
141C22×C42—21no—
142C22×C42C22×C311yescharacteristic
143#477—24no—
144#476—24no—
145C2×#107C22×C312yes—
146C2×#106C22×C312yes—
147C2×C4⋊C8C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×C22⋊C8C22×C312yes—
154#391—21no—
155#389—21no—
156#389C22×C312yes—
157#384C22×C311yescharacteristic
158#380—21no—
159#380C22×C312yes—
160#340C22×C312yes—
161#340C22×C312yes—
162#336C22×C312yes—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C3×#41—28no—
166C3×#41—28no—
167C3×#41—28no—
168C3×#41—28no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×Q812yes—
177C2×C3×Q8#4112yes—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4C2×D412yes—
185C2×C3×D4C2×Q812yes—
186C2×C3×D4#4112yes—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—41no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×Q812yes—
217C22×C3×C4#4112yes—
218C22×C3×C4C2411yescharacteristic
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—44no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—44no—
243C3×C42—24no—
244C3×C42—24no—
245C3×C42—22no—
246C3×C42—24no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42C2×D412yes—
251C3×C42C2×D412yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41C3×Q814yescomplemented
258C22×Q8—21no—
259C22×Q8C23×C311yescharacteristic
260C22×D4—21no—
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C3×D411yescharacteristic
266C23×C4C3×D411yescharacteristic
267C23×C4C23×C311yescharacteristic
268C22×C8C23×C312yes—
269C22×C8C23×C312yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C22⋊Q8—44no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8C3×D414yes—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4C3×D414yes—
283#117—28no—
284#117—22no—
285#117—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C23×C312yes—
293C2×C4⋊C4C23×C312yes—
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C42—28no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C3×D411yescharacteristic
307C2×C42C3×D411yescharacteristic
308C2×C42C23×C311yescharacteristic
309C2×C42C23×C311yescharacteristic
310#107—24no—
311#106—24no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314#103—24no—
315#103—24no—
316#102—24no—
317#102—24no—
318C22⋊C8—24no—
319#95—22no—
320#95—22no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3C22⋊Q812yes—
335C23×C3C22×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—84no—
340C3×Q8C22⋊Q814yescomplemented
341C3×D4—216no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4C22⋊Q814yescomplemented
346C2×C3×C4—216no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—44no—
366C2×C3×C4—48no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—44no—
372C2×C3×C4—48no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4#12012yes—
380C2×C3×C4#12012yes—
381C2×C3×C4C22⋊Q812yes—
382C2×C3×C4#13512yes—
383C2×C3×C4AGL(1,ℤ/8)12yes—
384C2×C3×C4#13712yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×Q811yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C3×C8—42no—
392C3×C8—42no—
393C24—21no—
394C24C2×C3×D411yescharacteristic
395#41—28no—
396#41—28no—
397#41—28no—
398#41—28no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8C2×C3×D412yes—
406C2×Q8C2×C3×Q812yes—
407C2×Q8C3×#4112yes—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C3×D412yes—
415C2×D4C2×C3×Q812yes—
416C2×D4C3×#4112yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×Q812yes—
447C22×C4C3×#4112yes—
448C22×C4C24×C311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4C2×C3×D412yes—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—44no—
473C42—24no—
474C42—24no—
475C42—22no—
476C42—24no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42C2×C3×D412yes—
481C42C2×C3×D412yes—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—24no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—41no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3C2×#12011yescharacteristic
497C22×C3C2×C22⋊Q811yescharacteristic
498C22×C3#52511yescharacteristic
499C22×C3C2×#13511yescharacteristic
500C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C3×C4—22no—
504C3×C4—24no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—82no—
511C3×C4—82no—
512C3×C4#45214yes—
513C3×C4C2×C22⋊Q811yescharacteristic
514C3×C4C2×C22⋊Q811yescharacteristic
515C3×C4#52511yescharacteristic
516C3×C4#52511yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—21no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C3×C22⋊Q812yes—
531C23C22×C3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—84no—
536Q8C3×C22⋊Q814yescomplemented
537D4—216no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4C3×C22⋊Q814yescomplemented
542C2×C4—216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4C3×#12012yes—
576C2×C4C3×#12012yes—
577C2×C4C3×C22⋊Q812yes—
578C2×C4C3×#13512yes—
579C2×C4C3×AGL(1,ℤ/8)12yes—
580C2×C4C3×#13712yes—
581C2×C4C22×C3×D411yescharacteristic
582C2×C4C22×C3×D411yescharacteristic
583C2×C4C22×C3×Q811yescharacteristic
584C2×C4C2×C3×#4111yescharacteristic
585C2×C4C2×C3×#4111yescharacteristic
586C2×C4C2×C3×#4111yescharacteristic
587C8—42no—
588C8—42no—
589C2×C3—42no—
590C2×C3—44no—
591C2×C3#284811yescharacteristic
592C2×C3C2×#45211yescharacteristic
593C2×C3#300611yescharacteristic
594C2×C3#307611yescharacteristic
595C2×C3#307911yescharacteristic
596C2×C3#310611yescharacteristic
597C2×C3#311611yescharacteristic
598C22—22no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—41no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22C2×C3×#12011yescharacteristic
613C22C2×C3×C22⋊Q811yescharacteristic
614C22C3×#52511yescharacteristic
615C22C2×C3×#13511yescharacteristic
616C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
617C22C2×C3×#13711yescharacteristic
618C22C3×#57511yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4C3×#45214yes—
629C4C2×C3×C22⋊Q811yescharacteristic
630C4C2×C3×C22⋊Q811yescharacteristic
631C4C3×#52511yescharacteristic
632C4C3×#52511yescharacteristic
633C3#3295811yescharacteristic, direct product factor, Sylow
634C2—42no—
635C2—44no—
636C2C3×#284811yescharacteristic
637C2C2×C3×#45211yescharacteristic
638C2C3×#300611yescharacteristic
639C2C3×#307611yescharacteristic
640C2C3×#307911yescharacteristic
641C2C3×#310611yescharacteristic
642C2C3×#311611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. C2×C3×#476C2semidirect
4. C3×#2857C2semidirect
5. C3×#1865C2semidirect
6. C3×#1862C2semidirect
7. C3×#1854C2semidirect
8. C3×#1803C2semidirect
9. C3×#1802C2semidirect
10. C3×#1797C2semidirect
12. #32958C3direct (N × H)
49. C2×#477C2×C3semidirect
50. C2×#476C2×C3semidirect
51. #2857C2×C3semidirect
52. #1865C2×C3semidirect
53. #1862C2×C3semidirect
54. #1854C2×C3semidirect
55. #1803C2×C3semidirect
56. #1802C2×C3semidirect
57. #1797C2×C3semidirect
64. C2×C3×#41Q8semidirect
257. C2×#41C3×Q8semidirect
340. C3×Q8C22⋊Q8semidirect
345. C3×D4C22⋊Q8semidirect
536. Q8C3×C22⋊Q8semidirect
541. D4C3×C22⋊Q8semidirect
633. C3#32958direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×C3×#4772
C2×C3×#4762
C3×#28572
C3×#18652
C3×#18622
C3×#18542
C3×#18032
C3×#18022
C3×#17972
C3×#17882
#329583