Finite Group Databasebeta

← C3×#24323 #10642930

Subgroups — C3×#24323 #10642930

Order 768.

1004 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×#3503C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2060C212yesmaximal, complemented
6C3×#2028C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1986C211yesmaximal, characteristic, complemented
9C3×#1882C212yesmaximal
10C3×#1859C212yesmaximal
11C3×#1806C211yesmaximal, characteristic
12#24323C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C3×Q82—21no—
15C3×#557—21no—
16C3×#554—21no—
17C3×#554—22no—
18C3×D4×Q8—22no—
19C3×D4×Q8—21no—
20C3×#544—22no—
21C3×#544C2212yes—
22C3×#533—21no—
23C3×#533C2211yescharacteristic
24C2×C3×C4⋊Q8—21no—
25C2×C3×C4⋊Q8C2211yescharacteristic
26C3×#520C2212yes—
27C3×C4×#41C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C3×#479—22no—
30C3×#477—21no—
31C3×#475—21no—
32C3×#441—21no—
33C3×#440—21no—
34C3×C4×Q16—21no—
35C3×C4×SD16—21no—
36C3×#428C2211yescharacteristic
37C3×#428C2212yes—
38C3×#423C2211yescharacteristic
39C2×C3×#104C2212yes—
40C3×#419C2211yescharacteristic
41C3×#419C2212yes—
42C3×#417C2211yescharacteristic
43C3×C4×M16C2211yescharacteristic
44C3×#395—22no—
45C3×#391C2212yes—
46C3×#389—22no—
47C3×#384C2212yes—
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#343C2212yes—
50C3×#343C2212yes—
51C3×#339C2212yes—
52C3×#339C2212yes—
53C3×#339C2212yes—
54#3503C2×C311yescharacteristic
55#3078C2×C311yescharacteristic, complemented
56C4×#137C2×C311yescharacteristic
57#2567C2×C311yescharacteristic, complemented
58#2060C2×C312yescomplemented
59#2028C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1986C2×C311yescharacteristic, complemented
62#1882C2×C312yes—
63#1859C2×C312yes—
64#1806C2×C311yescharacteristic
65C2×C3×#41C2311yescharacteristic
66C22×C3×Q8—22no—
67C22×C3×Q8D412yes—
68C22×C3×Q8C2311yescharacteristic
69C3×#137—22no—
70C3×#137Q814yescomplemented
71C2×C3×Q16—21no—
72C2×C3×SD16—21no—
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2312yes—
75C2×C3×M16C2311yescharacteristic
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—24no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8D411yescharacteristic
86C3×C4⋊Q8D411yescharacteristic
87C3×#125—24no—
88C3×#125—22no—
89C3×#125—22no—
90C3×#125D412yes—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—24no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8—22no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8D411yescharacteristic
108C3×C4×Q8D411yescharacteristic
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—24no—
113C3×C4×D4D411yescharacteristic
114C3×C4×D4D411yescharacteristic
115C3×#117—22no—
116C3×#117—21no—
117C3×#117C2311yescharacteristic
118C3×#117C2312yes—
119C2×C3×C4⋊C4—22no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4—22no—
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4—24no—
124C2×C3×C4⋊C4D412yes—
125C2×C3×C4⋊C4C2311yescharacteristic
126C2×C3×C4⋊C4C2312yes—
127C2×C3×C42—21no—
128C2×C3×C42—21no—
129C2×C3×C42C2312yes—
130C2×C3×C42C2311yescharacteristic
131C3×#107—21no—
132C3×#107—22no—
133C3×#106—21no—
134C3×#106—22no—
135C3×C4⋊C8—21no—
136C3×C4⋊C8—21no—
137C3×#104—24no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#103—21no—
141C3×#103—22no—
142C3×#103—22no—
143C3×#102—21no—
144C3×#97—21no—
145C3×C4×C8—21no—
146C3×#95—21no—
147C3×#95—24no—
148C2×#137C22×C311yescharacteristic
149Q82—21no—
150#557—21no—
151#554—21no—
152#554—22no—
153D4×Q8—22no—
154D4×Q8—21no—
155#544—22no—
156#544C22×C312yes—
157#533—21no—
158#533C22×C311yescharacteristic
159C2×C4⋊Q8—21no—
160C2×C4⋊Q8C22×C311yescharacteristic
161#520C22×C312yes—
162C4×#41C22×C311yescharacteristic
163C2×C4×Q8C22×C311yescharacteristic
164#479—22no—
165#477—21no—
166#475—21no—
167#441—21no—
168#440—21no—
169C4×Q16—21no—
170C4×SD16—21no—
171#428C22×C312yes—
172#428C22×C311yescharacteristic
173#423C22×C311yescharacteristic
174C2×#104C22×C312yes—
175#419C22×C312yes—
176#419C22×C311yescharacteristic
177#417C22×C311yescharacteristic
178C4×M16C22×C311yescharacteristic
179#395—22no—
180#391C22×C312yes—
181#389—22no—
182#384C22×C312yes—
183C4×C4⋊C4C22×C311yescharacteristic
184#343C22×C312yes—
185#343C22×C312yes—
186#339C22×C312yes—
187#339C22×C312yes—
188#339C22×C312yes—
189C3×#41—21no—
190C3×#41—21no—
191C3×#41C2×Q812yes—
192C3×#41#4112yes—
193C2×C3×Q8—24no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—21no—
197C2×C3×Q8—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—82no—
204C2×C3×Q8—84no—
205C2×C3×Q8C2×D411yescharacteristic
206C2×C3×Q8C2×Q812yescomplemented
207C2×C3×D4—21no—
208C2×C3×D4C2×D411yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—24no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D414yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C3×Q16—24no—
221C3×SD16—24no—
222C3×M16—21no—
223C3×M16—22no—
224C3×M16—21no—
225C3×M16—21no—
226C3×M16—22no—
227C3×M16C2×Q812yescomplemented
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—84no—
251C3×C4⋊C4C2×D412yes—
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C4⋊C4C2×D412yes—
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C22⋊C4—24no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—21no—
258C3×C22⋊C4—21no—
259C3×C22⋊C4—21no—
260C3×C42—24no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—22no—
265C3×C42—22no—
266C3×C42—22no—
267C3×C42—41no—
268C3×C42—41no—
269C3×C42C2×D411yescharacteristic
270C3×C42C2×D411yescharacteristic
271C3×C42C2×D411yescharacteristic
272C3×C42C2×D411yescharacteristic
273C2×#41C23×C311yescharacteristic
274C22×Q8—22no—
275C22×Q8C3×D412yes—
276C22×Q8C23×C311yescharacteristic
277#137—22no—
278#137C3×Q814yescomplemented
279C2×Q16—21no—
280C2×SD16—21no—
281C2×M16C23×C311yescharacteristic
282C2×M16C23×C312yes—
283C2×M16C23×C311yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8C3×D411yescharacteristic
294C4⋊Q8C3×D411yescharacteristic
295#125—24no—
296#125—22no—
297#125—22no—
298#125C3×D412yes—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—24no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C3×D411yescharacteristic
316C4×Q8C3×D411yescharacteristic
317C4×D4—22no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—24no—
321C4×D4C3×D411yescharacteristic
322C4×D4C3×D411yescharacteristic
323#117—22no—
324#117—21no—
325#117C23×C311yescharacteristic
326#117C23×C312yes—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4C3×D412yes—
333C2×C4⋊C4C23×C311yescharacteristic
334C2×C4⋊C4C23×C312yes—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C23×C312yes—
338C2×C42C23×C311yescharacteristic
339#107—21no—
340#107—22no—
341#106—21no—
342#106—22no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345#104—24no—
346#103—21no—
347#103—21no—
348#103—21no—
349#103—22no—
350#103—22no—
351#102—21no—
352#97—21no—
353C4×C8—21no—
354#95—21no—
355#95—24no—
356C23×C3—21no—
357C23×C3C22×D411yescharacteristic
358C3×Q8—21no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—44no—
362C3×Q8—42no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—42no—
366C3×Q8—82no—
367C3×Q8—84no—
368C3×Q8C22⋊Q812yes—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—21no—
372C3×D4C22⋊Q812yes—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—42no—
391C2×C3×C4—44no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—84no—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×Q811yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C3×C8—22no—
406C3×C8—41no—
407C3×C8—42no—
408#41—21no—
409#41—21no—
410#41C2×C3×Q812yes—
411#41C3×#4112yes—
412C2×Q8—24no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—82no—
423C2×Q8—84no—
424C2×Q8C2×C3×D411yescharacteristic
425C2×Q8C2×C3×Q812yescomplemented
426C2×D4—21no—
427C2×D4C2×C3×D411yescharacteristic
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D414yes—
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C24×C311yescharacteristic
439Q16—24no—
440SD16—24no—
441M16—21no—
442M16—22no—
443M16—21no—
444M16—21no—
445M16—22no—
446M16C2×C3×Q812yescomplemented
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—84no—
470C4⋊C4C2×C3×D412yes—
471C4⋊C4C2×C3×D411yescharacteristic
472C4⋊C4C2×C3×D412yes—
473C4⋊C4C2×C3×D411yescharacteristic
474C22⋊C4—24no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—24no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42C2×C3×D411yescharacteristic
489C42C2×C3×D411yescharacteristic
490C42C2×C3×D411yescharacteristic
491C42C2×C3×D411yescharacteristic
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—22no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3C2×#12011yescharacteristic
498C22×C3#54811yescharacteristic
499C22×C3D4×Q811yescharacteristic
500C3×C4—22no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—44no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—44no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4#45612yes—
513C3×C4C2×C22⋊Q811yescharacteristic
514C3×C4#53611yescharacteristic
515C3×C4#54811yescharacteristic
516C3×C4D4×Q811yescharacteristic
517C23—21no—
518C23C22×C3×D411yescharacteristic
519Q8—21no—
520Q8—22no—
521Q8—22no—
522Q8—44no—
523Q8—42no—
524Q8—42no—
525Q8—41no—
526Q8—41no—
527Q8—82no—
528Q8—84no—
529Q8C3×C22⋊Q812yes—
530D4—22no—
531D4—22no—
532D4—21no—
533D4C3×C22⋊Q812yes—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4C3×#12011yescharacteristic
557C2×C4C3×#12011yescharacteristic
558C2×C4C3×#12011yescharacteristic
559C2×C4C3×#12011yescharacteristic
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C22×C3×D411yescharacteristic
562C2×C4C22×C3×Q811yescharacteristic
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C3×2−1+411yescharacteristic
565C2×C4C3×2−1+411yescharacteristic
566C8—22no—
567C8—41no—
568C8—42no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—42no—
572C2×C3#240911yescharacteristic
573C2×C3C2×#45611yescharacteristic
574C2×C3#301911yescharacteristic
575C22—21no—
576C22—21no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22C2×C3×#12011yescharacteristic
581C22C3×#54811yescharacteristic
582C22C3×D4×Q811yescharacteristic
583C4—22no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—44no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—42no—
594C4—81no—
595C4C3×#45612yes—
596C4C2×C3×C22⋊Q811yescharacteristic
597C4C3×#53611yescharacteristic
598C4C3×#54811yescharacteristic
599C4C3×D4×Q811yescharacteristic
600C3#2432311yescharacteristic, direct product factor, Sylow
601C2—21no—
602C2—21no—
603C2—42no—
604C2C3×#240911yescharacteristic
605C2C2×C3×#45611yescharacteristic
606C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3078C2semidirect
4. C3×#2567C2semidirect
5. C3×#2060C2semidirect
6. C3×#2028C2semidirect
7. C3×#1997C2semidirect
8. C3×#1986C2semidirect
12. #24323C3direct (N × H)
55. #3078C2×C3semidirect
57. #2567C2×C3semidirect
58. #2060C2×C3semidirect
59. #2028C2×C3semidirect
60. #1997C2×C3semidirect
61. #1986C2×C3semidirect
70. C3×#137Q8semidirect
206. C2×C3×Q8C2×Q8semidirect
227. C3×M16C2×Q8semidirect
278. #137C3×Q8semidirect
425. C2×Q8C2×C3×Q8semidirect
446. M16C2×C3×Q8semidirect
600. C3#24323direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#30782
C3×C4×#1372
C3×#25672
C3×#20602
C3×#20282
C3×#19972
C3×#19862
C3×#18822
C3×#18592
C3×#18062
#243233