Finite Group Databasebeta

← C3×#11063 #10629670

Subgroups — C3×#11063 #10629670

Order 768.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#418C211yesmaximal, characteristic
2C2×C3×#418C211yesmaximal, characteristic
3C3×#2317C211yesmaximal, characteristic, complemented
4C2×C3×#390C211yesmaximal, characteristic
5C2×C3×#343C211yesmaximal, characteristic, complemented
6C2×C3×#339C211yesmaximal, characteristic, complemented
7C2×C3×#339C211yesmaximal, characteristic, complemented
8#11063C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4—22no—
16C22×C3×C42C2211yescharacteristic
17C3×#418—22no—
18C3×#418—22no—
19C3×#418—22no—
20C3×#418C414yescomplemented
21C2×C3×#102—21no—
22C2×C3×#102—21no—
23C2×C3×#102—22no—
24C3×#390—22no—
25C3×#390—22no—
26C3×#389—22no—
27C3×#385—22no—
28C3×#384—21no—
29C3×#382—21no—
30C3×#343—22no—
31C3×#339—22no—
32C3×#339—22no—
33C2×#418C2×C311yescharacteristic
34C2×#418C2×C311yescharacteristic
35#2317C2×C311yescharacteristic, complemented
36C2×#390C2×C311yescharacteristic
37C2×#343C2×C311yescharacteristic, complemented
38C2×#339C2×C311yescharacteristic, complemented
39C2×#339C2×C311yescharacteristic, complemented
40C25×C3—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2×C412yes—
53C22×C3×D4D412yes—
54C22×C3×D4Q812yes—
55C23×C3×C4—21no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16C2×C412yes—
69C2×C3×M16D412yes—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C3×#117—21no—
73C3×#117—21no—
74C3×#117—21no—
75C3×#117—21no—
76C3×#117—21no—
77C3×#117—21no—
78C3×#117C2×C412yes—
79C3×#117D412yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D412yes—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—28no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—48no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42D412yes—
102C3×#102—24no—
103C3×#102—24no—
104C3×#102—28no—
105C3×#95—24no—
106C23×D4C22×C311yescharacteristic
107C22×M16C22×C311yescharacteristic
108C22×M16C22×C311yescharacteristic
109C2×#117C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C42C22×C311yescharacteristic
114#418—22no—
115#418—22no—
116#418—22no—
117#418C3×C414yescomplemented
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—22no—
121#390—22no—
122#390—22no—
123#389—22no—
124#385—22no—
125#384—21no—
126#382—21no—
127#343—22no—
128#339—22no—
129#339—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—44no—
134C24×C3—44no—
135C24×C3—41no—
136C24×C3—41no—
137C24×C3—41no—
138C24×C3—41no—
139C24×C3C2×D411yescharacteristic
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4—41no—
164C2×C3×D4—41no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4C4⋊C414yescomplemented
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—42no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4—44no—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×Q811yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4#4111yescharacteristic
191C22×C3×C4#4111yescharacteristic
192C22×C3×C4#4111yescharacteristic
193C22×C3×C4#4111yescharacteristic
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—41no—
199C3×M16—41no—
200C3×M16—41no—
201C3×M16—41no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C2×C3×C8—41no—
207C2×C3×C8—41no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—84no—
218C3×C4⋊C4C22⋊C414yescomplemented
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—416no—
222C3×C22⋊C4—41no—
223C3×C22⋊C4—41no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—41no—
229C3×C42—41no—
230C3×C42—41no—
231C3×C42—41no—
232C25—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×C3×C412yes—
245C22×D4C3×D412yes—
246C22×D4C3×Q812yes—
247C23×C4—21no—
248C23×C4C3×D412yes—
249C23×C4C23×C311yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16C2×C3×C412yes—
261C2×M16C3×D412yes—
262C22×C8—21no—
263C22×C8—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117C2×C3×C412yes—
271#117C3×D412yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C3×D411yescharacteristic
281C2×C4⋊C4C3×D412yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—48no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C3×D412yes—
294#102—24no—
295#102—24no—
296#102—28no—
297#95—24no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—44no—
302C23×C3—48no—
303C23×C3—44no—
304C23×C3—48no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—84no—
312C23×C3—82no—
313C23×C3—82no—
314C23×C3—84no—
315C23×C3—82no—
316C23×C3—82no—
317C23×C3C4×D411yescharacteristic
318C23×C3#12011yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3#12111yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3C22⋊Q811yescharacteristic
323C23×C3#12411yescharacteristic
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—82no—
348C2×C3×C4C2×C22⋊C411yescharacteristic
349C2×C3×C4C2×C4⋊C411yescharacteristic
350C2×C3×C4#11711yescharacteristic
351C2×C3×C4C4×D411yescharacteristic
352C2×C3×C4C4×D411yescharacteristic
353C2×C3×C4C4×D411yescharacteristic
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12111yescharacteristic
356C2×C3×C4#12111yescharacteristic
357C2×C3×C4#12111yescharacteristic
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4#12111yescharacteristic
361C2×C3×C4C22⋊Q811yescharacteristic
362C2×C3×C4C22⋊Q811yescharacteristic
363C2×C3×C4C22⋊Q811yescharacteristic
364C2×C3×C4C22⋊Q811yescharacteristic
365C2×C3×C4C22⋊Q811yescharacteristic
366C2×C3×C4#12311yescharacteristic
367C2×C3×C4#12311yescharacteristic
368C2×C3×C4#12311yescharacteristic
369C2×C3×C4#12311yescharacteristic
370C2×C3×C4#12311yescharacteristic
371C2×C3×C4#12411yescharacteristic
372C2×C3×C4#12411yescharacteristic
373C2×C3×C4#12611yescharacteristic
374C2×C3×C4#12611yescharacteristic
375C2×C3×C4#12711yescharacteristic
376C3×C8—42no—
377C3×C8—42no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—44no—
382C24—44no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24C2×C3×D411yescharacteristic
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C3×C4⋊C414yescomplemented
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4C22×C3×C411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×Q811yescharacteristic
436C22×C4C3×#4111yescharacteristic
437C22×C4C3×#4111yescharacteristic
438C22×C4C3×#4111yescharacteristic
439C22×C4C3×#4111yescharacteristic
440C22×C4C3×#4111yescharacteristic
441C22×C4C3×#4111yescharacteristic
442M16—22no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—84no—
466C4⋊C4C3×C22⋊C414yescomplemented
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—416no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—48no—
489C22×C3—44no—
490C22×C3—84no—
491C22×C3—84no—
492C22×C3—84no—
493C22×C3—84no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—84no—
497C22×C3#38511yescharacteristic
498C22×C3#38611yescharacteristic
499C22×C3#38811yescharacteristic
500C22×C3#39211yescharacteristic
501C22×C3#39311yescharacteristic
502C22×C3#39411yescharacteristic
503C22×C3#39611yescharacteristic
504C22×C3#45312yes—
505C22×C3#45412yes—
506C3×C4—44no—
507C3×C4—44no—
508C3×C4—81no—
509C3×C4—82no—
510C3×C4—81no—
511C3×C4#38011yescharacteristic
512C3×C4#38511yescharacteristic
513C3×C4#38811yescharacteristic
514C3×C4#39011yescharacteristic
515C3×C4#39411yescharacteristic
516C3×C4#39611yescharacteristic
517C3×C4#39711yescharacteristic
518C3×C4#39911yescharacteristic
519C23—22no—
520C23—22no—
521C23—24no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—82no—
533C23—82no—
534C23—84no—
535C23—84no—
536C23—82no—
537C23—82no—
538C23C3×C4×D411yescharacteristic
539C23C3×#12011yescharacteristic
540C23C3×#12011yescharacteristic
541C23C3×#12111yescharacteristic
542C23C3×#12111yescharacteristic
543C23C3×C22⋊Q811yescharacteristic
544C23C3×#12411yescharacteristic
545D4—24no—
546D4—24no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4C2×C3×C22⋊C411yescharacteristic
570C2×C4C2×C3×C4⋊C411yescharacteristic
571C2×C4C3×#11711yescharacteristic
572C2×C4C3×C4×D411yescharacteristic
573C2×C4C3×C4×D411yescharacteristic
574C2×C4C3×C4×D411yescharacteristic
575C2×C4C3×#12011yescharacteristic
576C2×C4C3×#12111yescharacteristic
577C2×C4C3×#12111yescharacteristic
578C2×C4C3×#12111yescharacteristic
579C2×C4C3×#12111yescharacteristic
580C2×C4C3×#12111yescharacteristic
581C2×C4C3×#12111yescharacteristic
582C2×C4C3×C22⋊Q811yescharacteristic
583C2×C4C3×C22⋊Q811yescharacteristic
584C2×C4C3×C22⋊Q811yescharacteristic
585C2×C4C3×C22⋊Q811yescharacteristic
586C2×C4C3×C22⋊Q811yescharacteristic
587C2×C4C3×#12311yescharacteristic
588C2×C4C3×#12311yescharacteristic
589C2×C4C3×#12311yescharacteristic
590C2×C4C3×#12311yescharacteristic
591C2×C4C3×#12311yescharacteristic
592C2×C4C3×#12411yescharacteristic
593C2×C4C3×#12411yescharacteristic
594C2×C4C3×#12611yescharacteristic
595C2×C4C3×#12611yescharacteristic
596C2×C4C3×#12711yescharacteristic
597C8—42no—
598C8—42no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—84no—
604C2×C3#143911yescharacteristic
605C2×C3#188312yes—
606C2×C3#200211yescharacteristic
607C2×C3#200311yescharacteristic
608C2×C3#202712yes—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—48no—
618C22—44no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—82no—
623C22—82no—
624C22—84no—
625C22—84no—
626C22C3×#38511yescharacteristic
627C22C3×#38611yescharacteristic
628C22C3×#38811yescharacteristic
629C22C3×#39211yescharacteristic
630C22C3×#39311yescharacteristic
631C22C3×#39411yescharacteristic
632C22C3×#39611yescharacteristic
633C22C3×#45312yes—
634C22C3×#45412yes—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4C3×#38011yescharacteristic
641C4C3×#38511yescharacteristic
642C4C3×#38811yescharacteristic
643C4C3×#39011yescharacteristic
644C4C3×#39411yescharacteristic
645C4C3×#39611yescharacteristic
646C4C3×#39711yescharacteristic
647C4C3×#39911yescharacteristic
648C3#1106311yescharacteristic, direct product factor, Sylow
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—84no—
654C2C3×#143911yescharacteristic
655C2C3×#188312yes—
656C2C3×#200211yescharacteristic
657C2C3×#200311yescharacteristic
658C2C3×#202712yes—
subgroupG/H
center, socleC23×C3#120
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2317C2semidirect
5. C2×C3×#343C2semidirect
6. C2×C3×#339C2semidirect
7. C2×C3×#339C2semidirect
8. #11063C3direct (N × H)
20. C3×#418C4semidirect
35. #2317C2×C3semidirect
37. C2×#343C2×C3semidirect
38. C2×#339C2×C3semidirect
39. C2×#339C2×C3semidirect
117. #418C3×C4semidirect
167. C2×C3×D4C4⋊C4semidirect
218. C3×C4⋊C4C22⋊C4semidirect
415. C2×D4C3×C4⋊C4semidirect
466. C4⋊C4C3×C22⋊C4semidirect
648. C3#11063direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4182
C3×#23172
C2×C3×#3902
C2×C3×#3432
C2×C3×#3392
#110633