Finite Group Databasebeta

← C3×#61840 #10680447

Subgroups — C3×#61840 #10680447

Order 768.

1928 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
1C3×#3497C211yesmaximal, characteristic, complemented
2C3×Q8×#41C211yesmaximal, characteristic, complemented
3C3×#3386C213yesmaximal, complemented
4C3×#3384C213yesmaximal, complemented
5C3×Q8×SD16C211yesmaximal, characteristic, complemented
6C3×Q8×D8C211yesmaximal, characteristic, complemented
7C3×#3368C213yesmaximal, complemented
8C3×#3366C213yesmaximal, complemented
9C3×#3362C211yesmaximal, characteristic, complemented
10C3×#3358C211yesmaximal, characteristic, complemented
11C3×#3235C213yesmaximal, complemented
12C3×#3228C213yesmaximal, complemented
13C3×#3115C213yesmaximal, complemented
14C3×#3106C213yesmaximal, complemented
15C3×#2984C211yesmaximal, characteristic, complemented
16#61840C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×2−1+4—21no—
18C3×Q82C2211yescharacteristic
19C3×#557C2213yes—
20C3×#554—23no—
21C3×#554C2213yes—
22C3×#550C2211yescharacteristic
23C3×#550C2213yes—
24C3×D4×Q8—21no—
25C3×D4×Q8C2213yes—
26C3×D4×Q8C2211yescharacteristic
27C3×D4×Q8C2211yescharacteristic
28C3×#548—21no—
29C3×#548—23no—
30C3×#537—23no—
31C3×#533—23no—
32C3×#533C2213yes—
33C3×#532C2213yes—
34C3×#532C2213yes—
35C3×C4×#41C2213yes—
36C2×C3×C4×Q8—21no—
37C2×C3×C4×Q8C2211yescharacteristic
38C3×#501C2216yes—
39C3×#500C2213yes—
40C3×#498C2213yes—
41C3×#490C2216yes—
42C3×#486C2213yes—
43C3×#486C2213yes—
44C3×#482C2213yes—
45C3×#482C2216yes—
46C3×#482C2213yes—
47C3×#478C2216yes—
48C3×#478C2216yes—
49C3×#477C2213yes—
50C3×#476C2213yes—
51C3×#476C2213yes—
52C3×#476C2213yes—
53C3×#475C2213yes—
54C3×#474C2213yes—
55C3×#474C2213yes—
56C3×#474C2213yes—
57C3×#460C2213yes—
58C3×#460C2213yes—
59C3×#459C2213yes—
60C3×#459C2213yes—
61C3×#449C2211yescharacteristic
62C3×#449C2213yes—
63C3×#446C2213yes—
64C3×#446C2213yes—
65C3×C8×Q8C2211yescharacteristic
66C3×C8×Q8C2211yescharacteristic
67C3×#442C2216yes—
68C3×#440C2216yes—
69C3×C4×SD16C2213yes—
70C3×C4×SD16C2213yes—
71C3×C4×D8C2213yes—
72C3×C4×D8C2213yes—
73C3×#432C2213yes—
74C3×#431C2213yes—
75#3497C2×C311yescharacteristic, complemented
76Q8×#41C2×C311yescharacteristic, complemented
77#3386C2×C313yescomplemented
78#3384C2×C313yescomplemented
79Q8×SD16C2×C311yescharacteristic, complemented
80Q8×D8C2×C311yescharacteristic, complemented
81#3368C2×C313yescomplemented
82#3366C2×C313yescomplemented
83#3362C2×C311yescharacteristic, complemented
84#3358C2×C311yescharacteristic, complemented
85#3235C2×C313yescomplemented
86#3228C2×C313yescomplemented
87#3115C2×C313yescomplemented
88#3106C2×C313yescomplemented
89#2984C2×C311yescharacteristic, complemented
90C3×2−1+4—48no—
91C2×C3×#41—23no—
92C2×C3×#41—21no—
93C2×C3×#41—23no—
94C2×C3×#41—23no—
95C2×C3×#41C2311yescharacteristic
96C22×C3×Q8—23no—
97C22×C3×Q8—21no—
98C22×C3×Q8—21no—
99C22×C3×Q8D411yescharacteristic
100C2×C3×SD16C2311yescharacteristic
101C2×C3×SD16C2313yes—
102C2×C3×D8C2311yescharacteristic
103C2×C3×D8C2313yes—
104C3×C4⋊Q8—23no—
105C3×C4⋊Q8—23no—
106C3×C4⋊Q8C2313yes—
107C3×C4⋊Q8C2313yes—
108C3×C4⋊Q8C2313yes—
109C3×C4⋊Q8C2313yes—
110C3×#127C2313yes—
111C3×#127C2313yes—
112C3×#125—26no—
113C3×#125C2316yes—
114C3×#124—23no—
115C3×#124—23no—
116C3×C22⋊Q8—23no—
117C3×C22⋊Q8—23no—
118C3×C22⋊Q8—23no—
119C3×C22⋊Q8—23no—
120C3×C22⋊Q8—26no—
121C3×C22⋊Q8—23no—
122C3×C22⋊Q8—26no—
123C3×C22⋊Q8—23no—
124C3×#121—23no—
125C3×#121—23no—
126C3×#121—23no—
127C3×#121—26no—
128C3×#121—23no—
129C3×#121C2311yescharacteristic
130C3×#121C2313yes—
131C3×C4×Q8—21no—
132C3×C4×Q8—23no—
133C3×C4×Q8—23no—
134C3×C4×Q8—23no—
135C3×C4×Q8D411yescharacteristic
136C3×C4×Q8D413yes—
137C3×C4×Q8C2313yes—
138C3×C4×Q8C2313yes—
139C3×C4×Q8C2311yescharacteristic
140C3×C4×Q8C2313yes—
141C3×C4×D4—23no—
142C3×C4×D4—23no—
143C3×C4×D4—23no—
144C3×C4×D4—23no—
145C3×C4×D4C2313yes—
146C3×C4×D4C2313yes—
147C3×C4×D4C2313yes—
148C3×C4×D4C2316yes—
149C3×C4×D4C2313yes—
150C3×#117—23no—
151C3×#117C2313yes—
152C2×C3×C4⋊C4—23no—
153C2×C3×C4⋊C4D413yes—
154C2×C3×C42—23no—
155C2×C3×C42C2313yes—
156C3×#107C2313yes—
157C3×#107C2313yes—
158C3×#107C2316yes—
159C3×#106C2313yes—
160C3×#106C2313yes—
161C3×#106C2316yes—
162C3×C4⋊C8C2313yes—
163C3×C4⋊C8C2313yes—
164C3×C4⋊C8C2313yes—
165C3×C4⋊C8C2313yes—
166C3×#103C2311yescharacteristic
167C3×#103C2313yes—
168C3×#102C2311yescharacteristic
169C3×#102C2313yes—
170C3×#102C2313yes—
171C3×#102C2313yes—
172C3×#102C2313yes—
173C3×#102C2313yes—
174C3×#102C2316yes—
175C3×#102C2316yes—
176C3×C22⋊C8C2311yescharacteristic
177C3×C22⋊C8C2313yes—
178C3×#97C2316yes—
179C3×C4×C8C2313yes—
180C3×C4×C8C2313yes—
181C2×2−1+4—21no—
182Q82C22×C311yescharacteristic
183#557C22×C313yes—
184#554—23no—
185#554C22×C313yes—
186#550C22×C311yescharacteristic
187#550C22×C313yes—
188D4×Q8—21no—
189D4×Q8C22×C311yescharacteristic
190D4×Q8C22×C311yescharacteristic
191D4×Q8C22×C313yes—
192#548—21no—
193#548—23no—
194#537—23no—
195#533—23no—
196#533C22×C313yes—
197#532C22×C313yes—
198#532C22×C313yes—
199C4×#41C22×C313yes—
200C2×C4×Q8—21no—
201C2×C4×Q8C22×C311yescharacteristic
202#501C22×C316yes—
203#500C22×C313yes—
204#498C22×C313yes—
205#490C22×C316yes—
206#486C22×C313yes—
207#486C22×C313yes—
208#482C22×C313yes—
209#482C22×C313yes—
210#482C22×C316yes—
211#478C22×C316yes—
212#478C22×C316yes—
213#477C22×C313yes—
214#476C22×C313yes—
215#476C22×C313yes—
216#476C22×C313yes—
217#475C22×C313yes—
218#474C22×C313yes—
219#474C22×C313yes—
220#474C22×C313yes—
221#460C22×C313yes—
222#460C22×C313yes—
223#459C22×C313yes—
224#459C22×C313yes—
225#449C22×C311yescharacteristic
226#449C22×C313yes—
227#446C22×C313yes—
228#446C22×C313yes—
229C8×Q8C22×C311yescharacteristic
230C8×Q8C22×C311yescharacteristic
231#442C22×C316yes—
232#440C22×C316yes—
233C4×SD16C22×C313yes—
234C4×SD16C22×C313yes—
235C4×D8C22×C313yes—
236C4×D8C22×C313yes—
237#432C22×C313yes—
238#431C22×C313yes—
239C3×#41—22no—
240C3×#41—22no—
241C3×#41—412no—
242C3×#41—412no—
243C3×#41—412no—
244C3×#41—44no—
245C2×C3×Q8—24no—
246C2×C3×Q8—23no—
247C2×C3×Q8—23no—
248C2×C3×Q8—23no—
249C2×C3×Q8—23no—
250C2×C3×Q8—412no—
251C2×C3×Q8—44no—
252C2×C3×Q8—44no—
253C2×C3×Q8C2×D413yes—
254C2×C3×Q8C2×D411yescharacteristic
255C2×C3×Q8C2411yescharacteristic
256C2×C3×D4—21no—
257C2×C3×D4—23no—
258C2×C3×D4—21no—
259C2×C3×D4—23no—
260C2×C3×D4—23no—
261C2×C3×D4—23no—
262C2×C3×D4—23no—
263C2×C3×D4C2411yescharacteristic
264C2×C3×D4C2413yes—
265C2×C3×D4C2411yescharacteristic
266C22×C3×C4—21no—
267C22×C3×C4—23no—
268C22×C3×C4—26no—
269C22×C3×C4—23no—
270C22×C3×C4—23no—
271C22×C3×C4—23no—
272C22×C3×C4C2×D413yes—
273C22×C3×C4C2×D413yes—
274C22×C3×C4C2411yescharacteristic
275C3×SD16—22no—
276C3×SD16—26no—
277C3×D8—22no—
278C3×D8—26no—
279C2×C3×C8C2411yescharacteristic
280C2×C3×C8C2411yescharacteristic
281C2×C3×C8C2413yes—
282C2×C3×C8C2413yes—
283C3×C4⋊C4—26no—
284C3×C4⋊C4—23no—
285C3×C4⋊C4—23no—
286C3×C4⋊C4—23no—
287C3×C4⋊C4—23no—
288C3×C4⋊C4—23no—
289C3×C4⋊C4—23no—
290C3×C4⋊C4—26no—
291C3×C4⋊C4C2×D413yes—
292C3×C4⋊C4C2×D416yes—
293C3×C4⋊C4C2×D413yes—
294C3×C4⋊C4C2413yes—
295C3×C4⋊C4C2413yes—
296C3×C4⋊C4C2411yescharacteristic
297C3×C4⋊C4C2413yes—
298C3×C22⋊C4—23no—
299C3×C22⋊C4—23no—
300C3×C22⋊C4—23no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—23no—
303C3×C22⋊C4—26no—
304C3×C22⋊C4—23no—
305C3×C22⋊C4—23no—
306C3×C42—23no—
307C3×C42—23no—
308C3×C42—23no—
309C3×C42C2×D413yes—
310C3×C42C2×D413yes—
311C3×C42C2413yes—
312C3×C42C2413yes—
3132−1+4—48no—
314C2×#41—23no—
315C2×#41—21no—
316C2×#41—23no—
317C2×#41—23no—
318C2×#41C23×C311yescharacteristic
319C22×Q8—21no—
320C22×Q8—23no—
321C22×Q8—21no—
322C22×Q8C3×D411yescharacteristic
323C2×SD16C23×C311yescharacteristic
324C2×SD16C23×C313yes—
325C2×D8C23×C311yescharacteristic
326C2×D8C23×C313yes—
327C4⋊Q8—23no—
328C4⋊Q8—23no—
329C4⋊Q8C23×C313yes—
330C4⋊Q8C23×C313yes—
331C4⋊Q8C23×C313yes—
332C4⋊Q8C23×C313yes—
333#127C23×C313yes—
334#127C23×C313yes—
335#125—26no—
336#125C23×C316yes—
337#124—23no—
338#124—23no—
339C22⋊Q8—23no—
340C22⋊Q8—23no—
341C22⋊Q8—23no—
342C22⋊Q8—23no—
343C22⋊Q8—26no—
344C22⋊Q8—23no—
345C22⋊Q8—26no—
346C22⋊Q8—23no—
347#121—23no—
348#121—23no—
349#121—23no—
350#121—26no—
351#121—23no—
352#121C23×C311yescharacteristic
353#121C23×C313yes—
354C4×Q8—21no—
355C4×Q8—23no—
356C4×Q8—23no—
357C4×Q8—23no—
358C4×Q8C3×D411yescharacteristic
359C4×Q8C3×D413yes—
360C4×Q8C23×C313yes—
361C4×Q8C23×C313yes—
362C4×Q8C23×C311yescharacteristic
363C4×Q8C23×C313yes—
364C4×D4—23no—
365C4×D4—23no—
366C4×D4—23no—
367C4×D4—23no—
368C4×D4C23×C313yes—
369C4×D4C23×C313yes—
370C4×D4C23×C313yes—
371C4×D4C23×C313yes—
372C4×D4C23×C316yes—
373#117—23no—
374#117C23×C313yes—
375C2×C4⋊C4—23no—
376C2×C4⋊C4C3×D413yes—
377C2×C42—23no—
378C2×C42C23×C313yes—
379#107C23×C313yes—
380#107C23×C313yes—
381#107C23×C316yes—
382#106C23×C313yes—
383#106C23×C313yes—
384#106C23×C316yes—
385C4⋊C8C23×C313yes—
386C4⋊C8C23×C313yes—
387C4⋊C8C23×C313yes—
388C4⋊C8C23×C313yes—
389#103C23×C311yescharacteristic
390#103C23×C313yes—
391#102C23×C311yescharacteristic
392#102C23×C313yes—
393#102C23×C313yes—
394#102C23×C313yes—
395#102C23×C313yes—
396#102C23×C313yes—
397#102C23×C316yes—
398#102C23×C316yes—
399C22⋊C8C23×C311yescharacteristic
400C22⋊C8C23×C313yes—
401#97C23×C316yes—
402C4×C8C23×C313yes—
403C4×C8C23×C313yes—
404C23×C3—21no—
405C23×C3—23no—
406C23×C3—21no—
407C23×C3C22×D411yescharacteristic
408C3×Q8—26no—
409C3×Q8—21no—
410C3×Q8—46no—
411C3×Q8—46no—
412C3×Q8—46no—
413C3×Q8—83no—
414C3×Q8AGL(1,ℤ/8)14yes—
415C3×Q82−1+412yescomplemented
416C3×D4—24no—
417C3×D4—21no—
418C3×D4—21no—
419C3×D4—23no—
420C3×D4—21no—
421C3×D4—23no—
422C3×D4—46no—
423C3×D4—46no—
424C3×D4—42no—
425C3×D4—46no—
426C3×D4—83no—
427C3×D4—83no—
428C3×D42−1+412yescomplemented
429C2×C3×C4—24no—
430C2×C3×C4—21no—
431C2×C3×C4—26no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—23no—
435C2×C3×C4—23no—
436C2×C3×C4—23no—
437C2×C3×C4—21no—
438C2×C3×C4—43no—
439C2×C3×C4—412no—
440C2×C3×C4—46no—
441C2×C3×C4—46no—
442C2×C3×C4—46no—
443C2×C3×C4—83no—
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D413yes—
446C2×C3×C4C22×D413yes—
447C2×C3×C4C22×D413yes—
448C2×C3×C4C22×D413yes—
449C2×C3×C4C2511yescharacteristic
450C3×C8—21no—
451C3×C8—23no—
452C3×C8—21no—
453C3×C8—23no—
454#41—22no—
455#41—22no—
456#41—412no—
457#41—412no—
458#41—412no—
459#41—44no—
460C2×Q8—24no—
461C2×Q8—23no—
462C2×Q8—23no—
463C2×Q8—23no—
464C2×Q8—23no—
465C2×Q8—412no—
466C2×Q8—44no—
467C2×Q8—44no—
468C2×Q8C2×C3×D413yes—
469C2×Q8C2×C3×D411yescharacteristic
470C2×Q8C24×C311yescharacteristic
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—23no—
474C2×D4—23no—
475C2×D4—23no—
476C2×D4—23no—
477C2×D4—23no—
478C2×D4C24×C311yescharacteristic
479C2×D4C24×C313yes—
480C2×D4C24×C311yescharacteristic
481C22×C4—21no—
482C22×C4—23no—
483C22×C4—26no—
484C22×C4—23no—
485C22×C4—23no—
486C22×C4—23no—
487C22×C4C2×C3×D413yes—
488C22×C4C2×C3×D413yes—
489C22×C4C24×C311yescharacteristic
490SD16—22no—
491SD16—26no—
492D8—22no—
493D8—26no—
494C2×C8C24×C311yescharacteristic
495C2×C8C24×C311yescharacteristic
496C2×C8C24×C313yes—
497C2×C8C24×C313yes—
498C4⋊C4—26no—
499C4⋊C4—23no—
500C4⋊C4—23no—
501C4⋊C4—23no—
502C4⋊C4—23no—
503C4⋊C4—23no—
504C4⋊C4—23no—
505C4⋊C4—26no—
506C4⋊C4C2×C3×D413yes—
507C4⋊C4C2×C3×D416yes—
508C4⋊C4C2×C3×D413yes—
509C4⋊C4C24×C313yes—
510C4⋊C4C24×C313yes—
511C4⋊C4C24×C311yescharacteristic
512C4⋊C4C24×C313yes—
513C22⋊C4—23no—
514C22⋊C4—23no—
515C22⋊C4—23no—
516C22⋊C4—21no—
517C22⋊C4—23no—
518C22⋊C4—26no—
519C22⋊C4—23no—
520C22⋊C4—23no—
521C42—23no—
522C42—23no—
523C42—23no—
524C42C2×C3×D413yes—
525C42C2×C3×D413yes—
526C42C24×C313yes—
527C42C24×C313yes—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—22no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—43no—
536C22×C3—43no—
537C22×C3—41no—
538C22×C3—43no—
539C22×C3—41no—
540C22×C3C23×D411yescharacteristic
541C3×C4—23no—
542C3×C4—23no—
543C3×C4—21no—
544C3×C4—22no—
545C3×C4—23no—
546C3×C4—43no—
547C3×C4—43no—
548C3×C4—41no—
549C3×C4—43no—
550C3×C4C2×AGL(1,ℤ/8)16yes—
551C3×C4C2×2−1+411yescharacteristic
552C3×C4C2×2−1+411yescharacteristic
553C23—21no—
554C23—23no—
555C23—21no—
556C23C22×C3×D411yescharacteristic
557Q8—26no—
558Q8—21no—
559Q8—46no—
560Q8—46no—
561Q8—46no—
562Q8—83no—
563Q8C3×AGL(1,ℤ/8)14yes—
564Q8C3×2−1+412yescomplemented
565D4—24no—
566D4—21no—
567D4—21no—
568D4—23no—
569D4—21no—
570D4—23no—
571D4—46no—
572D4—46no—
573D4—42no—
574D4—46no—
575D4—83no—
576D4—83no—
577D4C3×2−1+412yescomplemented
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—26no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—23no—
584C2×C4—23no—
585C2×C4—23no—
586C2×C4—21no—
587C2×C4—43no—
588C2×C4—412no—
589C2×C4—46no—
590C2×C4—46no—
591C2×C4—46no—
592C2×C4—83no—
593C2×C4C22×C3×D411yescharacteristic
594C2×C4C22×C3×D413yes—
595C2×C4C22×C3×D413yes—
596C2×C4C22×C3×D413yes—
597C2×C4C22×C3×D413yes—
598C2×C4C25×C311yescharacteristic
599C8—21no—
600C8—23no—
601C8—21no—
602C8—23no—
603C2×C3—41no—
604C2×C3—42no—
605C2×C3—83no—
606C2×C3—81no—
607C2×C3#348611yescharacteristic
608C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
609C2×C3#358411yescharacteristic
610C22—21no—
611C22—22no—
612C22—21no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—43no—
617C22—41no—
618C22—43no—
619C22—41no—
620C22—41no—
621C22—43no—
622C22C23×C3×D411yescharacteristic
623C4—21no—
624C4—22no—
625C4—23no—
626C4—23no—
627C4—23no—
628C4—43no—
629C4—41no—
630C4—43no—
631C4—43no—
632C4C2×C3×AGL(1,ℤ/8)16yes—
633C4C2×C3×2−1+411yescharacteristic
634C4C2×C3×2−1+411yescharacteristic
635C3#6184011yescharacteristic, direct product factor, Sylow
636C2—41no—
637C2—42no—
638C2—83no—
639C2—81no—
640C2C3×#348611yescharacteristic
641C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
642C2C3×#358411yescharacteristic
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×#3497C2semidirect
2. C3×Q8×#41C2semidirect
3. C3×#3386C2semidirect
4. C3×#3384C2semidirect
5. C3×Q8×SD16C2semidirect
6. C3×Q8×D8C2semidirect
7. C3×#3368C2semidirect
8. C3×#3366C2semidirect
9. C3×#3362C2semidirect
10. C3×#3358C2semidirect
11. C3×#3235C2semidirect
12. C3×#3228C2semidirect
13. C3×#3115C2semidirect
14. C3×#3106C2semidirect
15. C3×#2984C2semidirect
16. #61840C3direct (N × H)
75. #3497C2×C3semidirect
76. Q8×#41C2×C3semidirect
77. #3386C2×C3semidirect
78. #3384C2×C3semidirect
79. Q8×SD16C2×C3semidirect
80. Q8×D8C2×C3semidirect
81. #3368C2×C3semidirect
82. #3366C2×C3semidirect
83. #3362C2×C3semidirect
84. #3358C2×C3semidirect
85. #3235C2×C3semidirect
86. #3228C2×C3semidirect
87. #3115C2×C3semidirect
88. #3106C2×C3semidirect
89. #2984C2×C3semidirect
415. C3×Q82−1+4semidirect
428. C3×D42−1+4semidirect
564. Q8C3×2−1+4semidirect
577. D4C3×2−1+4semidirect
635. C3#61840direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C3×Q8×#412
C3×#33862
C3×#33842
C3×Q8×SD162
C3×Q8×D82
C3×#33682
C3×#33662
C3×#33622
C3×#33582
C3×#32352
C3×#32282
C3×#31152
C3×#31062
C3×#29842
#618403