Finite Group Databasebeta

← C3×#61824 #10680431

Subgroups — C3×#61824 #10680431

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3523C211yesmaximal, characteristic, complemented
2C3×Q8×#41C211yesmaximal, characteristic, complemented
3C3×#3383C213yesmaximal, complemented
4C3×#3381C213yesmaximal, complemented
5C3×Q8×SD16C211yesmaximal, characteristic, complemented
6C3×Q8×D8C211yesmaximal, characteristic, complemented
7C3×#3364C213yesmaximal, complemented
8C3×#3363C211yesmaximal, characteristic, complemented
9C3×#3360C213yesmaximal, complemented
10C3×#3359C211yesmaximal, characteristic, complemented
11C3×#3233C213yesmaximal, complemented
12C3×#3231C213yesmaximal, complemented
13C3×#3107C213yesmaximal, complemented
14C3×#3102C213yesmaximal
15C3×#2982C211yesmaximal, characteristic, complemented
16#61824C311yesmaximal, characteristic, direct product factor, Sylow
17C3×Q82C2211yescharacteristic
18C3×#557C2211yescharacteristic
19C3×#557C2213yes—
20C3×#557C2213yes—
21C3×#556C2213yes—
22C3×#554—23no—
23C3×#554C2213yes—
24C3×#552—23no—
25C3×D4×Q8—21no—
26C3×D4×Q8—21no—
27C3×D4×Q8C2211yescharacteristic
28C3×D4×Q8C2211yescharacteristic
29C3×#541—23no—
30C3×#533—23no—
31C3×#533C2213yes—
32C3×#533C2213yes—
33C3×#529C2213yes—
34C3×#519—21no—
35C3×C4×#41C2213yes—
36C2×C3×C4×Q8—21no—
37C2×C3×C4×Q8C2211yescharacteristic
38C3×#500C2213yes—
39C3×#499C2216yes—
40C3×#498C2213yes—
41C3×#488C2216yes—
42C3×#487C2213yes—
43C3×#487C2213yes—
44C3×#485C2213yes—
45C3×#485C2216yes—
46C3×#482C2213yes—
47C3×#479C2216yes—
48C3×#479C2216yes—
49C3×#479C2213yes—
50C3×#478C2213yes—
51C3×#477C2213yes—
52C3×#476C2213yes—
53C3×#475C2213yes—
54C3×#475C2213yes—
55C3×#474C2213yes—
56C3×#474C2213yes—
57C3×#464C2213yes—
58C3×#463C2213yes—
59C3×#462C2213yes—
60C3×#461C2213yes—
61C3×#452C2213yes—
62C3×#449C2211yescharacteristic
63C3×C8×Q8C2211yescharacteristic
64C3×C8×Q8C2213yes—
65C3×C8×Q8C2211yescharacteristic
66C3×C8×Q8C2213yes—
67C3×C4×Q16C2213yes—
68C3×C4×Q16C2216yes—
69C3×C4×SD16C2213yes—
70C3×C4×SD16C2213yes—
71C3×C4×SD16C2216yes—
72C3×C4×D8C2213yes—
73C3×#431C2213yes—
74C3×#431C2213yes—
75#3523C2×C311yescharacteristic, complemented
76Q8×#41C2×C311yescharacteristic, complemented
77#3383C2×C313yescomplemented
78#3381C2×C313yescomplemented
79Q8×SD16C2×C311yescharacteristic, complemented
80Q8×D8C2×C311yescharacteristic, complemented
81#3364C2×C313yescomplemented
82#3363C2×C311yescharacteristic, complemented
83#3360C2×C313yescomplemented
84#3359C2×C311yescharacteristic, complemented
85#3233C2×C313yescomplemented
86#3231C2×C313yescomplemented
87#3107C2×C313yescomplemented
88#3102C2×C313yes—
89#2982C2×C311yescharacteristic, complemented
90C2×C3×#41C2311yescharacteristic
91C22×C3×Q8—21no—
92C22×C3×Q8—21no—
93C22×C3×Q8D411yescharacteristic
94C2×C3×Q16C2313yes—
95C2×C3×SD16C2311yescharacteristic
96C2×C3×SD16C2313yes—
97C2×C3×D8C2311yescharacteristic
98C3×C4⋊Q8—23no—
99C3×C4⋊Q8—23no—
100C3×C4⋊Q8—23no—
101C3×C4⋊Q8C2313yes—
102C3×C4⋊Q8C2313yes—
103C3×C4⋊Q8C2313yes—
104C3×C4⋊Q8C2313yes—
105C3×#126—26no—
106C3×#126—23no—
107C3×#125—26no—
108C3×#125—26no—
109C3×#125—23no—
110C3×#125—23no—
111C3×#125C2313yes—
112C3×#125C2316yes—
113C3×#124C2313yes—
114C3×#123—23no—
115C3×C22⋊Q8—23no—
116C3×C22⋊Q8—23no—
117C3×C22⋊Q8—23no—
118C3×C22⋊Q8—23no—
119C3×C22⋊Q8—23no—
120C3×C22⋊Q8—23no—
121C3×C22⋊Q8—26no—
122C3×C22⋊Q8—23no—
123C3×C22⋊Q8C2313yes—
124C3×#121C2311yescharacteristic
125C3×C4×Q8—21no—
126C3×C4×Q8—21no—
127C3×C4×Q8—23no—
128C3×C4×Q8—23no—
129C3×C4×Q8—23no—
130C3×C4×Q8—23no—
131C3×C4×Q8—23no—
132C3×C4×Q8D411yescharacteristic
133C3×C4×Q8D413yes—
134C3×C4×Q8C2313yes—
135C3×C4×Q8C2313yes—
136C3×C4×Q8C2316yes—
137C3×C4×Q8C2313yes—
138C3×C4×Q8C2311yescharacteristic
139C3×C4×Q8C2313yes—
140C3×C4×D4—23no—
141C3×C4×D4—23no—
142C3×C4×D4—23no—
143C3×C4×D4C2313yes—
144C3×C4×D4C2313yes—
145C3×C4×D4C2313yes—
146C3×#117—23no—
147C3×#117—23no—
148C3×#117—23no—
149C3×#117—23no—
150C3×#117C2313yes—
151C2×C3×C4⋊C4—23no—
152C2×C3×C4⋊C4D413yes—
153C2×C3×C42—23no—
154C2×C3×C42C2313yes—
155C3×#107C2313yes—
156C3×#107C2316yes—
157C3×#107C2313yes—
158C3×#106C2313yes—
159C3×#106C2316yes—
160C3×#106C2313yes—
161C3×C4⋊C8C2313yes—
162C3×C4⋊C8C2313yes—
163C3×C4⋊C8C2313yes—
164C3×C4⋊C8C2313yes—
165C3×#103C2313yes—
166C3×#103C2316yes—
167C3×#103C2313yes—
168C3×#103C2311yescharacteristic
169C3×#103C2316yes—
170C3×#103C2313yes—
171C3×#102C2313yes—
172C3×#102C2313yes—
173C3×#102C2311yescharacteristic
174C3×#102C2313yes—
175C3×C22⋊C8C2311yescharacteristic
176C3×C22⋊C8C2313yes—
177C3×C4×C8C2313yes—
178C3×C4×C8C2313yes—
179C3×C4×C8C2316yes—
180Q82C22×C311yescharacteristic
181#557C22×C311yescharacteristic
182#557C22×C313yes—
183#557C22×C313yes—
184#556C22×C313yes—
185#554—23no—
186#554C22×C313yes—
187#552—23no—
188D4×Q8—21no—
189D4×Q8—21no—
190D4×Q8C22×C311yescharacteristic
191D4×Q8C22×C311yescharacteristic
192#541—23no—
193#533—23no—
194#533C22×C313yes—
195#533C22×C313yes—
196#529C22×C313yes—
197#519—21no—
198C4×#41C22×C313yes—
199C2×C4×Q8—21no—
200C2×C4×Q8C22×C311yescharacteristic
201#500C22×C313yes—
202#499C22×C316yes—
203#498C22×C313yes—
204#488C22×C316yes—
205#487C22×C313yes—
206#487C22×C313yes—
207#485C22×C313yes—
208#485C22×C316yes—
209#482C22×C313yes—
210#479C22×C313yes—
211#479C22×C316yes—
212#479C22×C316yes—
213#478C22×C313yes—
214#477C22×C313yes—
215#476C22×C313yes—
216#475C22×C313yes—
217#475C22×C313yes—
218#474C22×C313yes—
219#474C22×C313yes—
220#464C22×C313yes—
221#463C22×C313yes—
222#462C22×C313yes—
223#461C22×C313yes—
224#452C22×C313yes—
225#449C22×C311yescharacteristic
226C8×Q8C22×C311yescharacteristic
227C8×Q8C22×C313yes—
228C8×Q8C22×C311yescharacteristic
229C8×Q8C22×C313yes—
230C4×Q16C22×C313yes—
231C4×Q16C22×C316yes—
232C4×SD16C22×C313yes—
233C4×SD16C22×C313yes—
234C4×SD16C22×C316yes—
235C4×D8C22×C313yes—
236#431C22×C313yes—
237#431C22×C313yes—
238C3×#41—22no—
239C3×#41—22no—
240C2×C3×Q8—24no—
241C2×C3×Q8—23no—
242C2×C3×Q8—23no—
243C2×C3×Q8—23no—
244C2×C3×Q8—44no—
245C2×C3×Q8—44no—
246C2×C3×Q8C2×D413yes—
247C2×C3×Q8C2×D411yescharacteristic
248C2×C3×Q8C2413yes—
249C2×C3×Q8C2411yescharacteristic
250C2×C3×D4—21no—
251C2×C3×D4—21no—
252C2×C3×D4C2411yescharacteristic
253C2×C3×D4C2411yescharacteristic
254C22×C3×C4—21no—
255C22×C3×C4—23no—
256C22×C3×C4—23no—
257C22×C3×C4—23no—
258C22×C3×C4C2×D413yes—
259C22×C3×C4C2×D413yes—
260C22×C3×C4C2411yescharacteristic
261C3×Q16—26no—
262C3×SD16—22no—
263C3×SD16—26no—
264C3×D8—22no—
265C2×C3×C8C2413yes—
266C2×C3×C8C2411yescharacteristic
267C2×C3×C8C2413yes—
268C2×C3×C8C2411yescharacteristic
269C3×C4⋊C4—26no—
270C3×C4⋊C4—23no—
271C3×C4⋊C4—23no—
272C3×C4⋊C4—23no—
273C3×C4⋊C4—23no—
274C3×C4⋊C4—23no—
275C3×C4⋊C4—23no—
276C3×C4⋊C4—23no—
277C3×C4⋊C4—23no—
278C3×C4⋊C4—23no—
279C3×C4⋊C4—23no—
280C3×C4⋊C4—26no—
281C3×C4⋊C4—26no—
282C3×C4⋊C4—23no—
283C3×C4⋊C4C2×D413yes—
284C3×C4⋊C4C2×D413yes—
285C3×C4⋊C4C2×D416yes—
286C3×C4⋊C4C2413yes—
287C3×C4⋊C4C2413yes—
288C3×C4⋊C4C2413yes—
289C3×C4⋊C4C2411yescharacteristic
290C3×C22⋊C4—23no—
291C3×C22⋊C4—23no—
292C3×C22⋊C4—21no—
293C3×C22⋊C4—23no—
294C3×C22⋊C4—23no—
295C3×C22⋊C4—23no—
296C3×C22⋊C4—23no—
297C3×C22⋊C4—23no—
298C3×C42—23no—
299C3×C42—23no—
300C3×C42—23no—
301C3×C42—26no—
302C3×C42—23no—
303C3×C42C2×D413yes—
304C3×C42C2×D413yes—
305C3×C42C2413yes—
306C3×C42C2413yes—
307C2×#41C23×C311yescharacteristic
308C22×Q8—21no—
309C22×Q8—21no—
310C22×Q8C3×D411yescharacteristic
311C2×Q16C23×C313yes—
312C2×SD16C23×C311yescharacteristic
313C2×SD16C23×C313yes—
314C2×D8C23×C311yescharacteristic
315C4⋊Q8—23no—
316C4⋊Q8—23no—
317C4⋊Q8—23no—
318C4⋊Q8C23×C313yes—
319C4⋊Q8C23×C313yes—
320C4⋊Q8C23×C313yes—
321C4⋊Q8C23×C313yes—
322#126—26no—
323#126—23no—
324#125—26no—
325#125—26no—
326#125—23no—
327#125—23no—
328#125C23×C313yes—
329#125C23×C316yes—
330#124C23×C313yes—
331#123—23no—
332C22⋊Q8—23no—
333C22⋊Q8—23no—
334C22⋊Q8—23no—
335C22⋊Q8—23no—
336C22⋊Q8—23no—
337C22⋊Q8—23no—
338C22⋊Q8—26no—
339C22⋊Q8—23no—
340C22⋊Q8C23×C313yes—
341#121C23×C311yescharacteristic
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—23no—
345C4×Q8—23no—
346C4×Q8—23no—
347C4×Q8—23no—
348C4×Q8—23no—
349C4×Q8C3×D411yescharacteristic
350C4×Q8C3×D413yes—
351C4×Q8C23×C313yes—
352C4×Q8C23×C313yes—
353C4×Q8C23×C313yes—
354C4×Q8C23×C316yes—
355C4×Q8C23×C311yescharacteristic
356C4×Q8C23×C313yes—
357C4×D4—23no—
358C4×D4—23no—
359C4×D4—23no—
360C4×D4C23×C313yes—
361C4×D4C23×C313yes—
362C4×D4C23×C313yes—
363#117—23no—
364#117—23no—
365#117—23no—
366#117—23no—
367#117C23×C313yes—
368C2×C4⋊C4—23no—
369C2×C4⋊C4C3×D413yes—
370C2×C42—23no—
371C2×C42C23×C313yes—
372#107C23×C313yes—
373#107C23×C316yes—
374#107C23×C313yes—
375#106C23×C313yes—
376#106C23×C316yes—
377#106C23×C313yes—
378C4⋊C8C23×C313yes—
379C4⋊C8C23×C313yes—
380C4⋊C8C23×C313yes—
381C4⋊C8C23×C313yes—
382#103C23×C313yes—
383#103C23×C316yes—
384#103C23×C313yes—
385#103C23×C311yescharacteristic
386#103C23×C316yes—
387#103C23×C313yes—
388#102C23×C313yes—
389#102C23×C313yes—
390#102C23×C311yescharacteristic
391#102C23×C313yes—
392C22⋊C8C23×C311yescharacteristic
393C22⋊C8C23×C313yes—
394C4×C8C23×C313yes—
395C4×C8C23×C313yes—
396C4×C8C23×C316yes—
397C23×C3—21no—
398C23×C3—21no—
399C23×C3C22×D411yescharacteristic
400C3×Q8—26no—
401C3×Q8—21no—
402C3×Q8—23no—
403C3×Q8—23no—
404C3×Q8—46no—
405C3×Q8—46no—
406C3×Q8—83no—
407C3×Q8#13514yes—
408C3×Q82−1+412yes—
409C3×D4—24no—
410C3×D4—21no—
411C3×D4—21no—
412C3×D4—21no—
413C3×D4—42no—
414C3×D42−1+412yes—
415C2×C3×C4—24no—
416C2×C3×C4—21no—
417C2×C3×C4—26no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—23no—
421C2×C3×C4—21no—
422C2×C3×C4—23no—
423C2×C3×C4—23no—
424C2×C3×C4—23no—
425C2×C3×C4—43no—
426C2×C3×C4—46no—
427C2×C3×C4—46no—
428C2×C3×C4—83no—
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D413yes—
431C2×C3×C4C22×D413yes—
432C2×C3×C4C22×D413yes—
433C2×C3×C4C22×D413yes—
434C2×C3×C4C2511yescharacteristic
435C3×C8—21no—
436C3×C8—21no—
437C3×C8—23no—
438C3×C8—23no—
439#41—22no—
440#41—22no—
441C2×Q8—24no—
442C2×Q8—23no—
443C2×Q8—23no—
444C2×Q8—23no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8C2×C3×D413yes—
448C2×Q8C2×C3×D411yescharacteristic
449C2×Q8C24×C313yes—
450C2×Q8C24×C311yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4C24×C311yescharacteristic
454C2×D4C24×C311yescharacteristic
455C22×C4—21no—
456C22×C4—23no—
457C22×C4—23no—
458C22×C4—23no—
459C22×C4C2×C3×D413yes—
460C22×C4C2×C3×D413yes—
461C22×C4C24×C311yescharacteristic
462Q16—26no—
463SD16—22no—
464SD16—26no—
465D8—22no—
466C2×C8C24×C313yes—
467C2×C8C24×C311yescharacteristic
468C2×C8C24×C313yes—
469C2×C8C24×C311yescharacteristic
470C4⋊C4—26no—
471C4⋊C4—23no—
472C4⋊C4—23no—
473C4⋊C4—23no—
474C4⋊C4—23no—
475C4⋊C4—23no—
476C4⋊C4—23no—
477C4⋊C4—23no—
478C4⋊C4—23no—
479C4⋊C4—23no—
480C4⋊C4—23no—
481C4⋊C4—26no—
482C4⋊C4—26no—
483C4⋊C4—23no—
484C4⋊C4C2×C3×D413yes—
485C4⋊C4C2×C3×D413yes—
486C4⋊C4C2×C3×D416yes—
487C4⋊C4C24×C313yes—
488C4⋊C4C24×C313yes—
489C4⋊C4C24×C313yes—
490C4⋊C4C24×C311yescharacteristic
491C22⋊C4—23no—
492C22⋊C4—23no—
493C22⋊C4—21no—
494C22⋊C4—23no—
495C22⋊C4—23no—
496C22⋊C4—23no—
497C22⋊C4—23no—
498C22⋊C4—23no—
499C42—23no—
500C42—23no—
501C42—23no—
502C42—26no—
503C42—23no—
504C42C2×C3×D413yes—
505C42C2×C3×D413yes—
506C42C24×C313yes—
507C42C24×C313yes—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—21no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3C23×D411yescharacteristic
518C3×C4—23no—
519C3×C4—23no—
520C3×C4—21no—
521C3×C4—22no—
522C3×C4—23no—
523C3×C4—43no—
524C3×C4—41no—
525C3×C4—43no—
526C3×C4—43no—
527C3×C4—43no—
528C3×C4C2×#13516yes—
529C3×C4C2×2−1+411yescharacteristic
530C3×C4C2×2−1+411yescharacteristic
531C23—21no—
532C23—21no—
533C23C22×C3×D411yescharacteristic
534Q8—26no—
535Q8—21no—
536Q8—23no—
537Q8—23no—
538Q8—46no—
539Q8—46no—
540Q8—83no—
541Q8C3×#13514yes—
542Q8C3×2−1+412yes—
543D4—24no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—42no—
548D4C3×2−1+412yes—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—26no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—23no—
555C2×C4—21no—
556C2×C4—23no—
557C2×C4—23no—
558C2×C4—23no—
559C2×C4—43no—
560C2×C4—46no—
561C2×C4—46no—
562C2×C4—83no—
563C2×C4C22×C3×D411yescharacteristic
564C2×C4C22×C3×D413yes—
565C2×C4C22×C3×D413yes—
566C2×C4C22×C3×D413yes—
567C2×C4C22×C3×D413yes—
568C2×C4C25×C311yescharacteristic
569C8—21no—
570C8—21no—
571C8—23no—
572C8—23no—
573C2×C3—41no—
574C2×C3—42no—
575C2×C3—81no—
576C2×C3#348611yescharacteristic
577C2×C3C22×#13511yescharacteristic
578C2×C3#358611yescharacteristic
579C22—21no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C23×C3×D411yescharacteristic
589C4—21no—
590C4—22no—
591C4—23no—
592C4—23no—
593C4—23no—
594C4—43no—
595C4—43no—
596C4—43no—
597C4—43no—
598C4—41no—
599C4C2×C3×#13516yes—
600C4C2×C3×2−1+411yescharacteristic
601C4C2×C3×2−1+411yescharacteristic
602C3#6182411yescharacteristic, direct product factor, Sylow
603C2—41no—
604C2—42no—
605C2—81no—
606C2C3×#348611yescharacteristic
607C2C22×C3×#13511yescharacteristic
608C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#449C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3523C2semidirect
2. C3×Q8×#41C2semidirect
3. C3×#3383C2semidirect
4. C3×#3381C2semidirect
5. C3×Q8×SD16C2semidirect
6. C3×Q8×D8C2semidirect
7. C3×#3364C2semidirect
8. C3×#3363C2semidirect
9. C3×#3360C2semidirect
10. C3×#3359C2semidirect
11. C3×#3233C2semidirect
12. C3×#3231C2semidirect
13. C3×#3107C2semidirect
15. C3×#2982C2semidirect
16. #61824C3direct (N × H)
75. #3523C2×C3semidirect
76. Q8×#41C2×C3semidirect
77. #3383C2×C3semidirect
78. #3381C2×C3semidirect
79. Q8×SD16C2×C3semidirect
80. Q8×D8C2×C3semidirect
81. #3364C2×C3semidirect
82. #3363C2×C3semidirect
83. #3360C2×C3semidirect
84. #3359C2×C3semidirect
85. #3233C2×C3semidirect
86. #3231C2×C3semidirect
87. #3107C2×C3semidirect
89. #2982C2×C3semidirect
602. C3#61824direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35232
C3×Q8×#412
C3×#33832
C3×#33812
C3×Q8×SD162
C3×Q8×D82
C3×#33642
C3×#33632
C3×#33602
C3×#33592
C3×#32332
C3×#32312
C3×#31072
C3×#31022
C3×#29822
#618243