Finite Group Databasebeta

← C3×#61823 #10680430

Subgroups — C3×#61823 #10680430

Order 768.

1750 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3522C211yesmaximal, characteristic, complemented
2C3×Q8×#41C211yesmaximal, characteristic, complemented
3C3×Q8×Q16C211yesmaximal, characteristic, complemented
4C3×#3381C213yesmaximal, complemented
5C3×#3380C213yesmaximal, complemented
6C3×Q8×SD16C211yesmaximal, characteristic, complemented
7C3×#3365C213yesmaximal, complemented
8C3×#3362C211yesmaximal, characteristic, complemented
9C3×#3361C213yesmaximal, complemented
10C3×#3358C211yesmaximal, characteristic, complemented
11C3×#3233C213yesmaximal, complemented
12C3×#3228C213yesmaximal, complemented
13C3×#3106C213yesmaximal, complemented
14C3×#3102C213yesmaximal, complemented
15C3×#2982C211yesmaximal, characteristic, complemented
16#61823C311yesmaximal, characteristic, direct product factor, Sylow
17C3×#559C2213yes—
18C3×Q82C2211yescharacteristic
19C3×Q82C2211yescharacteristic
20C3×#557C2213yes—
21C3×#554—23no—
22C3×#554C2213yes—
23C3×#554C2213yes—
24C3×#550C2211yescharacteristic
25C3×D4×Q8—21no—
26C3×D4×Q8C2211yescharacteristic
27C3×#548—21no—
28C3×#547—23no—
29C3×#542—23no—
30C3×#533—23no—
31C3×#533C2213yes—
32C3×#532C2213yes—
33C3×#529C2213yes—
34C3×#520—21no—
35C3×C4×#41C2213yes—
36C2×C3×C4×Q8—21no—
37C2×C3×C4×Q8C2211yescharacteristic
38C3×#500C2213yes—
39C3×#499C2216yes—
40C3×#498C2213yes—
41C3×#488C2216yes—
42C3×#486C2213yes—
43C3×#486C2213yes—
44C3×#485C2213yes—
45C3×#482C2213yes—
46C3×#482C2216yes—
47C3×#479C2213yes—
48C3×#478C2216yes—
49C3×#478C2216yes—
50C3×#478C2213yes—
51C3×#477C2213yes—
52C3×#477C2213yes—
53C3×#476C2213yes—
54C3×#476C2213yes—
55C3×#475C2213yes—
56C3×#474C2213yes—
57C3×#464C2213yes—
58C3×#463C2213yes—
59C3×#460C2213yes—
60C3×#459C2213yes—
61C3×#452C2211yescharacteristic
62C3×#449C2213yes—
63C3×C8×Q8C2211yescharacteristic
64C3×C8×Q8C2213yes—
65C3×C8×Q8C2211yescharacteristic
66C3×C8×Q8C2213yes—
67C3×C4×Q16C2213yes—
68C3×C4×SD16C2213yes—
69C3×C4×SD16C2213yes—
70C3×C4×SD16C2216yes—
71C3×C4×D8C2213yes—
72C3×C4×D8C2216yes—
73C3×#431C2213yes—
74C3×#431C2213yes—
75#3522C2×C311yescharacteristic, complemented
76Q8×#41C2×C311yescharacteristic, complemented
77Q8×Q16C2×C311yescharacteristic, complemented
78#3381C2×C313yescomplemented
79#3380C2×C313yescomplemented
80Q8×SD16C2×C311yescharacteristic, complemented
81#3365C2×C313yescomplemented
82#3362C2×C311yescharacteristic, complemented
83#3361C2×C313yescomplemented
84#3358C2×C311yescharacteristic, complemented
85#3233C2×C313yescomplemented
86#3228C2×C313yescomplemented
87#3106C2×C313yescomplemented
88#3102C2×C313yescomplemented
89#2982C2×C311yescharacteristic, complemented
90C2×C3×#41—21no—
91C2×C3×#41—23no—
92C2×C3×#41C2311yescharacteristic
93C22×C3×Q8—21no—
94C22×C3×Q8D411yescharacteristic
95C2×C3×Q16C2311yescharacteristic
96C2×C3×SD16C2313yes—
97C2×C3×SD16C2311yescharacteristic
98C2×C3×D8C2313yes—
99C3×C4⋊Q8—23no—
100C3×C4⋊Q8—23no—
101C3×C4⋊Q8—23no—
102C3×C4⋊Q8C2313yes—
103C3×C4⋊Q8C2313yes—
104C3×C4⋊Q8C2313yes—
105C3×#127C2313yes—
106C3×#126—23no—
107C3×#125—26no—
108C3×#125C2313yes—
109C3×#125C2316yes—
110C3×#124—23no—
111C3×#124—23no—
112C3×#124C2313yes—
113C3×#123—23no—
114C3×#123—26no—
115C3×C22⋊Q8—23no—
116C3×C22⋊Q8—23no—
117C3×C22⋊Q8—23no—
118C3×C22⋊Q8—23no—
119C3×C22⋊Q8—23no—
120C3×C22⋊Q8—26no—
121C3×C22⋊Q8—26no—
122C3×C22⋊Q8C2311yescharacteristic
123C3×#121—23no—
124C3×#121—23no—
125C3×#121C2313yes—
126C3×C4×Q8—21no—
127C3×C4×Q8—21no—
128C3×C4×Q8—23no—
129C3×C4×Q8—23no—
130C3×C4×Q8—23no—
131C3×C4×Q8D411yescharacteristic
132C3×C4×Q8D413yes—
133C3×C4×Q8C2313yes—
134C3×C4×Q8C2313yes—
135C3×C4×Q8C2313yes—
136C3×C4×Q8C2311yescharacteristic
137C3×C4×Q8C2313yes—
138C3×C4×D4—23no—
139C3×C4×D4—23no—
140C3×C4×D4—23no—
141C3×C4×D4—23no—
142C3×C4×D4—23no—
143C3×C4×D4C2316yes—
144C3×C4×D4C2313yes—
145C3×C4×D4C2313yes—
146C3×C4×D4C2313yes—
147C3×#117—23no—
148C3×#117—23no—
149C3×#117C2313yes—
150C2×C3×C4⋊C4—23no—
151C2×C3×C4⋊C4—23no—
152C2×C3×C4⋊C4D413yes—
153C2×C3×C42—23no—
154C2×C3×C42C2313yes—
155C3×#107C2313yes—
156C3×#107C2316yes—
157C3×#107C2313yes—
158C3×#106C2316yes—
159C3×#106C2313yes—
160C3×#106C2313yes—
161C3×C4⋊C8C2313yes—
162C3×C4⋊C8C2313yes—
163C3×C4⋊C8C2313yes—
164C3×C4⋊C8C2313yes—
165C3×#103C2311yescharacteristic
166C3×#103C2313yes—
167C3×#103C2313yes—
168C3×#103C2313yes—
169C3×#102C2311yescharacteristic
170C3×#102C2313yes—
171C3×#102C2316yes—
172C3×#102C2313yes—
173C3×#102C2316yes—
174C3×#102C2313yes—
175C3×C22⋊C8C2313yes—
176C3×C22⋊C8C2311yescharacteristic
177C3×C4×C8C2316yes—
178C3×C4×C8C2313yes—
179C3×C4×C8C2313yes—
180#559C22×C313yes—
181Q82C22×C311yescharacteristic
182Q82C22×C311yescharacteristic
183#557C22×C313yes—
184#554—23no—
185#554C22×C313yes—
186#554C22×C313yes—
187#550C22×C311yescharacteristic
188D4×Q8—21no—
189D4×Q8C22×C311yescharacteristic
190#548—21no—
191#547—23no—
192#542—23no—
193#533—23no—
194#533C22×C313yes—
195#532C22×C313yes—
196#529C22×C313yes—
197#520—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#486C22×C313yes—
206#486C22×C313yes—
207#485C22×C313yes—
208#482C22×C313yes—
209#482C22×C316yes—
210#479C22×C313yes—
211#478C22×C313yes—
212#478C22×C316yes—
213#478C22×C316yes—
214#477C22×C313yes—
215#477C22×C313yes—
216#476C22×C313yes—
217#476C22×C313yes—
218#475C22×C313yes—
219#474C22×C313yes—
220#464C22×C313yes—
221#463C22×C313yes—
222#460C22×C313yes—
223#459C22×C313yes—
224#452C22×C311yescharacteristic
225#449C22×C313yes—
226C8×Q8C22×C311yescharacteristic
227C8×Q8C22×C313yes—
228C8×Q8C22×C311yescharacteristic
229C8×Q8C22×C313yes—
230C4×Q16C22×C313yes—
231C4×SD16C22×C313yes—
232C4×SD16C22×C313yes—
233C4×SD16C22×C316yes—
234C4×D8C22×C313yes—
235C4×D8C22×C316yes—
236#431C22×C313yes—
237#431C22×C313yes—
238C3×#41—22no—
239C3×#41—22no—
240C3×#41—44no—
241C3×#41—86no—
242C2×C3×Q8—24no—
243C2×C3×Q8—23no—
244C2×C3×Q8—23no—
245C2×C3×Q8—23no—
246C2×C3×Q8—44no—
247C2×C3×Q8C2×D413yes—
248C2×C3×Q8C2×D411yescharacteristic
249C2×C3×Q8C2411yescharacteristic
250C2×C3×Q8C2411yescharacteristic
251C2×C3×D4—21no—
252C2×C3×D4—23no—
253C2×C3×D4—23no—
254C2×C3×D4—23no—
255C2×C3×D4C2413yes—
256C2×C3×D4C2411yescharacteristic
257C22×C3×C4—21no—
258C22×C3×C4—23no—
259C22×C3×C4—23no—
260C22×C3×C4—23no—
261C22×C3×C4—26no—
262C22×C3×C4C2×D413yes—
263C22×C3×C4C2×D413yes—
264C22×C3×C4C2411yescharacteristic
265C3×Q16—22no—
266C3×SD16—26no—
267C3×SD16—22no—
268C3×D8—26no—
269C2×C3×C8C2411yescharacteristic
270C2×C3×C8C2413yes—
271C2×C3×C8C2411yescharacteristic
272C2×C3×C8C2413yes—
273C3×C4⋊C4—23no—
274C3×C4⋊C4—23no—
275C3×C4⋊C4—26no—
276C3×C4⋊C4—23no—
277C3×C4⋊C4—21no—
278C3×C4⋊C4—23no—
279C3×C4⋊C4—23no—
280C3×C4⋊C4—23no—
281C3×C4⋊C4—23no—
282C3×C4⋊C4—26no—
283C3×C4⋊C4—23no—
284C3×C4⋊C4C2×D413yes—
285C3×C4⋊C4C2×D413yes—
286C3×C4⋊C4C2×D416yes—
287C3×C4⋊C4C2413yes—
288C3×C4⋊C4C2413yes—
289C3×C4⋊C4C2411yescharacteristic
290C3×C4⋊C4C2413yes—
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×C22⋊C4—23no—
299C3×C22⋊C4—26no—
300C3×C42—23no—
301C3×C42—23no—
302C3×C42—23no—
303C3×C42—23no—
304C3×C42C2×D413yes—
305C3×C42C2×D413yes—
306C3×C42C2413yes—
307C3×C42C2413yes—
308C2×#41—21no—
309C2×#41—23no—
310C2×#41C23×C311yescharacteristic
311C22×Q8—21no—
312C22×Q8C3×D411yescharacteristic
313C2×Q16C23×C311yescharacteristic
314C2×SD16C23×C313yes—
315C2×SD16C23×C311yescharacteristic
316C2×D8C23×C313yes—
317C4⋊Q8—23no—
318C4⋊Q8—23no—
319C4⋊Q8—23no—
320C4⋊Q8C23×C313yes—
321C4⋊Q8C23×C313yes—
322C4⋊Q8C23×C313yes—
323#127C23×C313yes—
324#126—23no—
325#125—26no—
326#125C23×C313yes—
327#125C23×C316yes—
328#124—23no—
329#124—23no—
330#124C23×C313yes—
331#123—23no—
332#123—26no—
333C22⋊Q8—23no—
334C22⋊Q8—23no—
335C22⋊Q8—23no—
336C22⋊Q8—23no—
337C22⋊Q8—23no—
338C22⋊Q8—26no—
339C22⋊Q8—26no—
340C22⋊Q8C23×C311yescharacteristic
341#121—23no—
342#121—23no—
343#121C23×C313yes—
344C4×Q8—21no—
345C4×Q8—21no—
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×C311yescharacteristic
355C4×Q8C23×C313yes—
356C4×D4—23no—
357C4×D4—23no—
358C4×D4—23no—
359C4×D4—23no—
360C4×D4—23no—
361C4×D4C23×C313yes—
362C4×D4C23×C313yes—
363C4×D4C23×C313yes—
364C4×D4C23×C316yes—
365#117—23no—
366#117—23no—
367#117C23×C313yes—
368C2×C4⋊C4—23no—
369C2×C4⋊C4—23no—
370C2×C4⋊C4C3×D413yes—
371C2×C42—23no—
372C2×C42C23×C313yes—
373#107C23×C313yes—
374#107C23×C316yes—
375#107C23×C313yes—
376#106C23×C316yes—
377#106C23×C313yes—
378#106C23×C313yes—
379C4⋊C8C23×C313yes—
380C4⋊C8C23×C313yes—
381C4⋊C8C23×C313yes—
382C4⋊C8C23×C313yes—
383#103C23×C311yescharacteristic
384#103C23×C313yes—
385#103C23×C313yes—
386#103C23×C313yes—
387#102C23×C311yescharacteristic
388#102C23×C313yes—
389#102C23×C316yes—
390#102C23×C313yes—
391#102C23×C316yes—
392#102C23×C313yes—
393C22⋊C8C23×C313yes—
394C22⋊C8C23×C311yescharacteristic
395C4×C8C23×C316yes—
396C4×C8C23×C313yes—
397C4×C8C23×C313yes—
398C23×C3—21no—
399C23×C3—23no—
400C23×C3C22×D411yescharacteristic
401C3×Q8—26no—
402C3×Q8—21no—
403C3×Q8—21no—
404C3×Q8—21no—
405C3×Q8—46no—
406C3×Q8—83no—
407C3×Q8—83no—
408C3×Q8#13514yes—
409C3×Q82−1+412yes—
410C3×D4—24no—
411C3×D4—21no—
412C3×D4—23no—
413C3×D4—23no—
414C3×D4—46no—
415C3×D4—46no—
416C3×D4—83no—
417C3×D42−1+412yes—
418C2×C3×C4—24no—
419C2×C3×C4—21no—
420C2×C3×C4—26no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—23no—
425C2×C3×C4—23no—
426C2×C3×C4—23no—
427C2×C3×C4—21no—
428C2×C3×C4—43no—
429C2×C3×C4—46no—
430C2×C3×C4—46no—
431C2×C3×C4—83no—
432C2×C3×C4—86no—
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D413yes—
435C2×C3×C4C22×D413yes—
436C2×C3×C4C22×D413yes—
437C2×C3×C4C22×D413yes—
438C2×C3×C4C2511yescharacteristic
439C3×C8—23no—
440C3×C8—23no—
441C3×C8—21no—
442C3×C8—21no—
443#41—22no—
444#41—22no—
445#41—44no—
446#41—86no—
447C2×Q8—24no—
448C2×Q8—23no—
449C2×Q8—23no—
450C2×Q8—23no—
451C2×Q8—44no—
452C2×Q8C2×C3×D413yes—
453C2×Q8C2×C3×D411yescharacteristic
454C2×Q8C24×C311yescharacteristic
455C2×Q8C24×C311yescharacteristic
456C2×D4—21no—
457C2×D4—23no—
458C2×D4—23no—
459C2×D4—23no—
460C2×D4C24×C313yes—
461C2×D4C24×C311yescharacteristic
462C22×C4—21no—
463C22×C4—23no—
464C22×C4—23no—
465C22×C4—23no—
466C22×C4—26no—
467C22×C4C2×C3×D413yes—
468C22×C4C2×C3×D413yes—
469C22×C4C24×C311yescharacteristic
470Q16—22no—
471SD16—26no—
472SD16—22no—
473D8—26no—
474C2×C8C24×C311yescharacteristic
475C2×C8C24×C313yes—
476C2×C8C24×C311yescharacteristic
477C2×C8C24×C313yes—
478C4⋊C4—23no—
479C4⋊C4—23no—
480C4⋊C4—26no—
481C4⋊C4—23no—
482C4⋊C4—21no—
483C4⋊C4—23no—
484C4⋊C4—23no—
485C4⋊C4—23no—
486C4⋊C4—23no—
487C4⋊C4—26no—
488C4⋊C4—23no—
489C4⋊C4C2×C3×D413yes—
490C4⋊C4C2×C3×D413yes—
491C4⋊C4C2×C3×D416yes—
492C4⋊C4C24×C313yes—
493C4⋊C4C24×C313yes—
494C4⋊C4C24×C311yescharacteristic
495C4⋊C4C24×C313yes—
496C22⋊C4—23no—
497C22⋊C4—21no—
498C22⋊C4—23no—
499C22⋊C4—23no—
500C22⋊C4—23no—
501C22⋊C4—23no—
502C22⋊C4—23no—
503C22⋊C4—23no—
504C22⋊C4—26no—
505C42—23no—
506C42—23no—
507C42—23no—
508C42—23no—
509C42C2×C3×D413yes—
510C42C2×C3×D413yes—
511C42C24×C313yes—
512C42C24×C313yes—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—22no—
516C22×C3—21no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—43no—
520C22×C3—43no—
521C22×C3—43no—
522C22×C3C23×D411yescharacteristic
523C3×C4—23no—
524C3×C4—23no—
525C3×C4—21no—
526C3×C4—22no—
527C3×C4—23no—
528C3×C4—43no—
529C3×C4—43no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—43no—
533C3×C4C2×#13516yes—
534C3×C4C2×2−1+411yescharacteristic
535C3×C4C2×2−1+411yescharacteristic
536C23—21no—
537C23—23no—
538C23C22×C3×D411yescharacteristic
539Q8—26no—
540Q8—21no—
541Q8—21no—
542Q8—21no—
543Q8—46no—
544Q8—83no—
545Q8—83no—
546Q8C3×#13514yes—
547Q8C3×2−1+412yes—
548D4—24no—
549D4—21no—
550D4—23no—
551D4—23no—
552D4—46no—
553D4—46no—
554D4—83no—
555D4C3×2−1+412yes—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—26no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—43no—
567C2×C4—46no—
568C2×C4—46no—
569C2×C4—86no—
570C2×C4—83no—
571C2×C4C22×C3×D411yescharacteristic
572C2×C4C22×C3×D413yes—
573C2×C4C22×C3×D413yes—
574C2×C4C22×C3×D413yes—
575C2×C4C22×C3×D413yes—
576C2×C4C25×C311yescharacteristic
577C8—23no—
578C8—23no—
579C8—21no—
580C8—21no—
581C2×C3—41no—
582C2×C3—42no—
583C2×C3—83no—
584C2×C3#348611yescharacteristic
585C2×C3C22×#13511yescharacteristic
586C2×C3#358511yescharacteristic
587C22—21no—
588C22—22no—
589C22—21no—
590C22—21no—
591C22—41no—
592C22—41no—
593C22—43no—
594C22—43no—
595C22—43no—
596C22C23×C3×D411yescharacteristic
597C4—21no—
598C4—22no—
599C4—23no—
600C4—23no—
601C4—23no—
602C4—41no—
603C4—41no—
604C4—43no—
605C4—43no—
606C4—43no—
607C4C2×C3×#13516yes—
608C4C2×C3×2−1+411yescharacteristic
609C4C2×C3×2−1+411yescharacteristic
610C3#6182311yescharacteristic, direct product factor, Sylow
611C2—41no—
612C2—42no—
613C2—83no—
614C2C3×#348611yescharacteristic
615C2C22×C3×#13511yescharacteristic
616C2C3×#358511yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
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×#3522C2semidirect
2. C3×Q8×#41C2semidirect
3. C3×Q8×Q16C2semidirect
4. C3×#3381C2semidirect
5. C3×#3380C2semidirect
6. C3×Q8×SD16C2semidirect
7. C3×#3365C2semidirect
8. C3×#3362C2semidirect
9. C3×#3361C2semidirect
10. C3×#3358C2semidirect
11. C3×#3233C2semidirect
12. C3×#3228C2semidirect
13. C3×#3106C2semidirect
14. C3×#3102C2semidirect
15. C3×#2982C2semidirect
16. #61823C3direct (N × H)
75. #3522C2×C3semidirect
76. Q8×#41C2×C3semidirect
77. Q8×Q16C2×C3semidirect
78. #3381C2×C3semidirect
79. #3380C2×C3semidirect
80. Q8×SD16C2×C3semidirect
81. #3365C2×C3semidirect
82. #3362C2×C3semidirect
83. #3361C2×C3semidirect
84. #3358C2×C3semidirect
85. #3233C2×C3semidirect
86. #3228C2×C3semidirect
87. #3106C2×C3semidirect
88. #3102C2×C3semidirect
89. #2982C2×C3semidirect
610. C3#61823direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35222
C3×Q8×#412
C3×Q8×Q162
C3×#33812
C3×#33802
C3×Q8×SD162
C3×#33652
C3×#33622
C3×#33612
C3×#33582
C3×#32332
C3×#32282
C3×#31062
C3×#31022
C3×#29822
#618233