Finite Group Databasebeta

← C3×#60499 #10679106

Subgroups — C3×#60499 #10679106

Order 768.

1970 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3504C211yesmaximal, characteristic, complemented
2C3×C4×2−1+4C211yesmaximal, characteristic
3C3×#3084C213yesmaximal, complemented
4C3×#3083C213yesmaximal, complemented
5C3×#3082C213yesmaximal, complemented
6C3×#3082C213yesmaximal, complemented
7C3×#3080C213yesmaximal, complemented
8C3×#3078C213yesmaximal, complemented
9C3×#3076C213yesmaximal, complemented
10C2×C3×#479C213yesmaximal, complemented
11C3×#2913C211yesmaximal, characteristic, complemented
12C3×#2912C211yesmaximal, characteristic, complemented
13C3×#2906C211yesmaximal, characteristic, complemented
14C3×#2898C211yesmaximal, characteristic
15C3×#2898C211yesmaximal, characteristic
16#60499C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×2−1+4C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#557—21no—
21C3×#557—21no—
22C3×#554—23no—
23C3×#554—23no—
24C3×#544C2213yes—
25C3×#544C2213yes—
26C3×#533C2213yes—
27C2×C3×#125C2213yes—
28C3×#520—23no—
29C3×#520C2213yes—
30C3×#520C2211yescharacteristic
31C3×#520C2211yescharacteristic
32C3×#520C2211yescharacteristic
33C3×#519—21no—
34C3×#519C2213yes—
35C3×C4×#41—23no—
36C3×C4×#41C2213yes—
37C3×C4×#41C2211yescharacteristic
38C2×C3×C4×Q8—21no—
39C2×C3×C4×Q8C2213yes—
40C3×#479C2213yes—
41C3×#479C2216yes—
42C3×#479C2216yes—
43C3×#479C2216yes—
44C3×#479C2213yes—
45C3×#479C2216yes—
46C3×#478C2211yescharacteristic
47C3×#478C2213yes—
48C3×#478C2216yes—
49C3×#477C2213yes—
50C3×#477C2216yes—
51C3×#476C2213yes—
52C3×#475C2213yes—
53C3×#475C2216yes—
54C3×#474C2213yes—
55C3×#428C2213yes—
56C3×#428C2213yes—
57C3×#428C2213yes—
58C3×#428C2213yes—
59C3×#427C2213yes—
60C3×#427C2213yes—
61C2×C3×#107C2213yes—
62C2×C3×#106C2213yes—
63C3×#424C2213yes—
64C3×#424C2213yes—
65C3×#423C2213yes—
66C2×C3×C4⋊C8C2213yes—
67C3×#419C2213yes—
68C3×#419C2213yes—
69C3×#417C2213yes—
70C3×#417C2213yes—
71C3×#416C2213yes—
72C3×#416C2213yes—
73C2×C3×#103C2213yes—
74C2×C3×#103C2213yes—
75#3504C2×C311yescharacteristic, complemented
76C4×2−1+4C2×C311yescharacteristic
77#3084C2×C313yescomplemented
78#3083C2×C313yescomplemented
79#3082C2×C313yescomplemented
80#3082C2×C313yescomplemented
81#3080C2×C313yescomplemented
82#3078C2×C313yescomplemented
83#3076C2×C313yescomplemented
84C2×#479C2×C313yescomplemented
85#2913C2×C311yescharacteristic, complemented
86#2912C2×C311yescharacteristic, complemented
87#2906C2×C311yescharacteristic, complemented
88#2898C2×C311yescharacteristic
89#2898C2×C311yescharacteristic
90C3×2−1+4—24no—
91C3×2−1+4Q818yescomplemented
92C2×C3×#41—23no—
93C2×C3×#41C2313yes—
94C2×C3×#41C2311yescharacteristic
95C22×C3×Q8—21no—
96C22×C3×Q8C2313yes—
97C3×#131—24no—
98C3×#131—24no—
99C2×C3×M16C2313yes—
100C2×C3×M16C2313yes—
101C22×C3×C8C2313yes—
102C22×C3×C8C2313yes—
103C3×C4⋊Q8—23no—
104C3×C4⋊Q8—23no—
105C3×C4⋊Q8C2313yes—
106C3×C4⋊Q8C2313yes—
107C3×#125—23no—
108C3×#125—23no—
109C3×#125—23no—
110C3×#125—23no—
111C3×#125C2313yes—
112C3×#125C2316yes—
113C3×#125C2311yescharacteristic
114C3×C22⋊Q8—23no—
115C3×C22⋊Q8—23no—
116C3×C22⋊Q8—26no—
117C3×C22⋊Q8—26no—
118C3×C4×Q8—21no—
119C3×C4×Q8—21no—
120C3×C4×Q8—21no—
121C3×C4×Q8—23no—
122C3×C4×Q8—23no—
123C3×C4×Q8—23no—
124C3×C4×Q8—23no—
125C3×C4×Q8—23no—
126C3×C4×Q8—23no—
127C3×C4×Q8D411yescharacteristic
128C3×C4×Q8D411yescharacteristic
129C3×C4×Q8C2313yes—
130C3×C4×Q8C2316yes—
131C3×C4×Q8C2313yes—
132C3×C4×D4—23no—
133C3×C4×D4—23no—
134C3×C4×D4—23no—
135C3×C4×D4—23no—
136C3×C4×D4—23no—
137C3×C4×D4—26no—
138C3×C4×D4D413yes—
139C3×C4×D4D413yes—
140C3×C4×D4C2311yescharacteristic
141C3×C4×D4C2313yes—
142C3×#117—23no—
143C3×#117—26no—
144C3×#117—23no—
145C3×#117C2313yes—
146C3×#117C2313yes—
147C3×#117C2313yes—
148C3×#117C2313yes—
149C2×C3×C4⋊C4—23no—
150C2×C3×C4⋊C4—23no—
151C2×C3×C4⋊C4—23no—
152C2×C3×C4⋊C4—23no—
153C2×C3×C4⋊C4C2313yes—
154C2×C3×C4⋊C4C2313yes—
155C2×C3×C4⋊C4C2313yes—
156C2×C3×C42—23no—
157C2×C3×C42—23no—
158C2×C3×C42C2313yes—
159C3×#107C2313yes—
160C3×#107C2316yes—
161C3×#107C2311yescharacteristic
162C3×#107C2313yes—
163C3×#107C2313yes—
164C3×#106C2313yes—
165C3×#106C2316yes—
166C3×#106C2311yescharacteristic
167C3×#106C2313yes—
168C3×#106C2313yes—
169C3×C4⋊C8C2311yescharacteristic
170C3×C4⋊C8C2313yes—
171C3×C4⋊C8C2313yes—
172C3×C4⋊C8C2313yes—
173C3×C4⋊C8C2316yes—
174C3×#103C2313yes—
175C3×#103C2313yes—
176C3×#103C2316yes—
177C3×#103C2316yes—
178C3×#103C2313yes—
179C3×#103C2313yes—
180C3×#102C2311yescharacteristic
181C3×#102C2311yescharacteristic
182C3×#102C2313yes—
183C3×#102C2313yes—
184C2×2−1+4C22×C311yescharacteristic
185C2×#131C22×C311yescharacteristic
186C2×#131C22×C311yescharacteristic
187#557—21no—
188#557—21no—
189#554—23no—
190#554—23no—
191#544C22×C313yes—
192#544C22×C313yes—
193#533C22×C313yes—
194C2×#125C22×C313yes—
195#520—23no—
196#520C22×C313yes—
197#520C22×C311yescharacteristic
198#520C22×C311yescharacteristic
199#520C22×C311yescharacteristic
200#519—21no—
201#519C22×C313yes—
202C4×#41—23no—
203C4×#41C22×C313yes—
204C4×#41C22×C311yescharacteristic
205C2×C4×Q8—21no—
206C2×C4×Q8C22×C313yes—
207#479C22×C313yes—
208#479C22×C316yes—
209#479C22×C316yes—
210#479C22×C316yes—
211#479C22×C313yes—
212#479C22×C316yes—
213#478C22×C311yescharacteristic
214#478C22×C313yes—
215#478C22×C316yes—
216#477C22×C313yes—
217#477C22×C316yes—
218#476C22×C313yes—
219#475C22×C313yes—
220#475C22×C316yes—
221#474C22×C313yes—
222#428C22×C313yes—
223#428C22×C313yes—
224#428C22×C313yes—
225#428C22×C313yes—
226#427C22×C313yes—
227#427C22×C313yes—
228C2×#107C22×C313yes—
229C2×#106C22×C313yes—
230#424C22×C313yes—
231#424C22×C313yes—
232#423C22×C313yes—
233C2×C4⋊C8C22×C313yes—
234#419C22×C313yes—
235#419C22×C313yes—
236#417C22×C313yes—
237#417C22×C313yes—
238#416C22×C313yes—
239#416C22×C313yes—
240C2×#103C22×C313yes—
241C2×#103C22×C313yes—
242C3×#41—224no—
243C3×#41—26no—
244C3×#41C2×Q8112yes—
245C3×#41C2×Q814yes—
246C3×#41#4114yes—
247C2×C3×Q8—28no—
248C2×C3×Q8—23no—
249C2×C3×Q8—23no—
250C2×C3×Q8—23no—
251C2×C3×Q8—26no—
252C2×C3×Q8C2×D411yescharacteristic
253C2×C3×Q8C2×Q8112yes—
254C2×C3×Q8C2413yes—
255C2×C3×D4—23no—
256C2×C3×D4C2×D413yes—
257C2×C3×D4C2411yescharacteristic
258C22×C3×C4—23no—
259C22×C3×C4—23no—
260C22×C3×C4—23no—
261C22×C3×C4—23no—
262C22×C3×C4—23no—
263C22×C3×C4—21no—
264C22×C3×C4—23no—
265C22×C3×C4C2×D413yes—
266C22×C3×C4C2×D413yes—
267C22×C3×C4C2413yes—
268C3×M16—26no—
269C3×M16—26no—
270C2×C3×C8—26no—
271C2×C3×C8—26no—
272C2×C3×C8C2411yescharacteristic
273C2×C3×C8C2413yes—
274C2×C3×C8C2413yes—
275C2×C3×C8C2411yescharacteristic
276C3×C4⋊C4—26no—
277C3×C4⋊C4—23no—
278C3×C4⋊C4—23no—
279C3×C4⋊C4—23no—
280C3×C4⋊C4—23no—
281C3×C4⋊C4—26no—
282C3×C4⋊C4—26no—
283C3×C4⋊C4—23no—
284C3×C4⋊C4—23no—
285C3×C4⋊C4—23no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—23no—
289C3×C4⋊C4—23no—
290C3×C4⋊C4C2×D413yes—
291C3×C4⋊C4C2×D413yes—
292C3×C4⋊C4C2413yes—
293C3×C4⋊C4C2411yescharacteristic
294C3×C4⋊C4C2413yes—
295C3×C4⋊C4C2411yescharacteristic
296C3×C4⋊C4C2413yes—
297C3×C4⋊C4C2411yescharacteristic
298C3×C22⋊C4—23no—
299C3×C22⋊C4—23no—
300C3×C22⋊C4—23no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—23no—
303C3×C22⋊C4C2×D416yes—
304C3×C42—23no—
305C3×C42—23no—
306C3×C42—26no—
307C3×C42—23no—
308C3×C42—23no—
309C3×C42—23no—
310C3×C42C2×D413yes—
311C3×C42C2×D413yes—
312C3×C42C2411yescharacteristic
313C3×C42C2413yes—
3142−1+4—24no—
3152−1+4C3×Q818yescomplemented
316C2×#41—23no—
317C2×#41C23×C313yes—
318C2×#41C23×C311yescharacteristic
319C22×Q8—21no—
320C22×Q8C23×C313yes—
321#131—24no—
322#131—24no—
323C2×M16C23×C313yes—
324C2×M16C23×C313yes—
325C22×C8C23×C313yes—
326C22×C8C23×C313yes—
327C4⋊Q8—23no—
328C4⋊Q8—23no—
329C4⋊Q8C23×C313yes—
330C4⋊Q8C23×C313yes—
331#125—23no—
332#125—23no—
333#125—23no—
334#125—23no—
335#125C23×C313yes—
336#125C23×C316yes—
337#125C23×C311yescharacteristic
338C22⋊Q8—23no—
339C22⋊Q8—23no—
340C22⋊Q8—26no—
341C22⋊Q8—26no—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—21no—
345C4×Q8—23no—
346C4×Q8—23no—
347C4×Q8—23no—
348C4×Q8—23no—
349C4×Q8—23no—
350C4×Q8—23no—
351C4×Q8C3×D411yescharacteristic
352C4×Q8C3×D411yescharacteristic
353C4×Q8C23×C313yes—
354C4×Q8C23×C316yes—
355C4×Q8C23×C313yes—
356C4×D4—23no—
357C4×D4—23no—
358C4×D4—23no—
359C4×D4—23no—
360C4×D4—23no—
361C4×D4—26no—
362C4×D4C3×D413yes—
363C4×D4C3×D413yes—
364C4×D4C23×C311yescharacteristic
365C4×D4C23×C313yes—
366#117—23no—
367#117—26no—
368#117—23no—
369#117C23×C313yes—
370#117C23×C313yes—
371#117C23×C313yes—
372#117C23×C313yes—
373C2×C4⋊C4—23no—
374C2×C4⋊C4—23no—
375C2×C4⋊C4—23no—
376C2×C4⋊C4—23no—
377C2×C4⋊C4C23×C313yes—
378C2×C4⋊C4C23×C313yes—
379C2×C4⋊C4C23×C313yes—
380C2×C42—23no—
381C2×C42—23no—
382C2×C42C23×C313yes—
383#107C23×C313yes—
384#107C23×C316yes—
385#107C23×C311yescharacteristic
386#107C23×C313yes—
387#107C23×C313yes—
388#106C23×C313yes—
389#106C23×C316yes—
390#106C23×C311yescharacteristic
391#106C23×C313yes—
392#106C23×C313yes—
393C4⋊C8C23×C311yescharacteristic
394C4⋊C8C23×C313yes—
395C4⋊C8C23×C313yes—
396C4⋊C8C23×C313yes—
397C4⋊C8C23×C316yes—
398#103C23×C313yes—
399#103C23×C313yes—
400#103C23×C316yes—
401#103C23×C316yes—
402#103C23×C313yes—
403#103C23×C313yes—
404#102C23×C311yescharacteristic
405#102C23×C311yescharacteristic
406#102C23×C313yes—
407#102C23×C313yes—
408C23×C3—21no—
409C23×C3C22×D413yes—
410C3×Q8—212no—
411C3×Q8—23no—
412C3×Q8—83no—
413C3×Q8—83no—
414C3×Q8C22⋊Q814yes—
415C3×Q8C22×Q816yes—
416C3×D4—212no—
417C3×D4—21no—
418C3×D4C22⋊Q8112yes—
419C3×D4C22×Q812yes—
420C2×C3×C4—212no—
421C2×C3×C4—23no—
422C2×C3×C4—212no—
423C2×C3×C4—21no—
424C2×C3×C4—23no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—23no—
428C2×C3×C4—23no—
429C2×C3×C4—43no—
430C2×C3×C4—43no—
431C2×C3×C4—42no—
432C2×C3×C4—83no—
433C2×C3×C4—83no—
434C2×C3×C4—83no—
435C2×C3×C4—83no—
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D413yes—
438C2×C3×C4C22×D413yes—
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D413yes—
441C2×C3×C4C22×Q816yes—
442C2×C3×C4C2×#4116yes—
443C2×C3×C4C2511yescharacteristic
444C3×C8—21no—
445C3×C8—21no—
446C3×C8—23no—
447C3×C8—23no—
448#41—224no—
449#41—26no—
450#41C2×C3×Q8112yes—
451#41C2×C3×Q814yes—
452#41C3×#4114yes—
453C2×Q8—28no—
454C2×Q8—23no—
455C2×Q8—23no—
456C2×Q8—23no—
457C2×Q8—26no—
458C2×Q8C2×C3×D411yescharacteristic
459C2×Q8C2×C3×Q8112yes—
460C2×Q8C24×C313yes—
461C2×D4—23no—
462C2×D4C2×C3×D413yes—
463C2×D4C24×C311yescharacteristic
464C22×C4—23no—
465C22×C4—23no—
466C22×C4—23no—
467C22×C4—23no—
468C22×C4—23no—
469C22×C4—21no—
470C22×C4—23no—
471C22×C4C2×C3×D413yes—
472C22×C4C2×C3×D413yes—
473C22×C4C24×C313yes—
474M16—26no—
475M16—26no—
476C2×C8—26no—
477C2×C8—26no—
478C2×C8C24×C311yescharacteristic
479C2×C8C24×C313yes—
480C2×C8C24×C313yes—
481C2×C8C24×C311yescharacteristic
482C4⋊C4—26no—
483C4⋊C4—23no—
484C4⋊C4—23no—
485C4⋊C4—23no—
486C4⋊C4—23no—
487C4⋊C4—26no—
488C4⋊C4—26no—
489C4⋊C4—23no—
490C4⋊C4—23no—
491C4⋊C4—23no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—23no—
495C4⋊C4—23no—
496C4⋊C4C2×C3×D413yes—
497C4⋊C4C2×C3×D413yes—
498C4⋊C4C24×C313yes—
499C4⋊C4C24×C311yescharacteristic
500C4⋊C4C24×C313yes—
501C4⋊C4C24×C311yescharacteristic
502C4⋊C4C24×C313yes—
503C4⋊C4C24×C311yescharacteristic
504C22⋊C4—23no—
505C22⋊C4—23no—
506C22⋊C4—23no—
507C22⋊C4—21no—
508C22⋊C4—23no—
509C22⋊C4C2×C3×D416yes—
510C42—23no—
511C42—23no—
512C42—26no—
513C42—23no—
514C42—23no—
515C42—23no—
516C42C2×C3×D413yes—
517C42C2×C3×D413yes—
518C42C24×C311yescharacteristic
519C42C24×C313yes—
520C22×C3—23no—
521C22×C3—23no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3C2×C22⋊Q816yes—
526C22×C3C23×D411yescharacteristic
527C3×C4—21no—
528C3×C4—23no—
529C3×C4—21no—
530C3×C4—23no—
531C3×C4—26no—
532C3×C4—43no—
533C3×C4—43no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—43no—
537C3×C4—41no—
538C3×C4C2×C22⋊Q816yes—
539C3×C4C23×Q811yescharacteristic
540C3×C4C22×#4111yescharacteristic
541C23—21no—
542C23C22×C3×D413yes—
543Q8—23no—
544Q8—212no—
545Q8—83no—
546Q8—83no—
547Q8C3×C22⋊Q814yes—
548Q8C22×C3×Q816yes—
549D4—212no—
550D4—21no—
551D4C3×C22⋊Q8112yes—
552D4C22×C3×Q812yes—
553C2×C4—23no—
554C2×C4—212no—
555C2×C4—212no—
556C2×C4—21no—
557C2×C4—23no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—23no—
561C2×C4—23no—
562C2×C4—43no—
563C2×C4—43no—
564C2×C4—42no—
565C2×C4—83no—
566C2×C4—83no—
567C2×C4—83no—
568C2×C4—83no—
569C2×C4C22×C3×D411yescharacteristic
570C2×C4C22×C3×D413yes—
571C2×C4C22×C3×D413yes—
572C2×C4C22×C3×D411yescharacteristic
573C2×C4C22×C3×D413yes—
574C2×C4C22×C3×Q816yes—
575C2×C4C2×C3×#4116yes—
576C2×C4C25×C311yescharacteristic
577C8—21no—
578C8—21no—
579C8—23no—
580C8—23no—
581C2×C3—26no—
582C2×C3—42no—
583C2×C3C22×C22⋊Q811yescharacteristic
584C2×C3#358411yescharacteristic
585C2×C3#358611yescharacteristic
586C22—23no—
587C22—23no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22C2×C3×C22⋊Q816yes—
592C22C23×C3×D411yescharacteristic
593C4—26no—
594C4—21no—
595C4—23no—
596C4—21no—
597C4—23no—
598C4—43no—
599C4—43no—
600C4—41no—
601C4—41no—
602C4—43no—
603C4—41no—
604C4C2×C3×C22⋊Q816yes—
605C4C23×C3×Q811yescharacteristic
606C4C22×C3×#4111yescharacteristic
607C3#6049911yescharacteristic, direct product factor, Sylow
608C2—26no—
609C2—42no—
610C2C22×C3×C22⋊Q811yescharacteristic
611C2C3×#358411yescharacteristic
612C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3504C2semidirect
3. C3×#3084C2semidirect
4. C3×#3083C2semidirect
5. C3×#3082C2semidirect
6. C3×#3082C2semidirect
7. C3×#3080C2semidirect
8. C3×#3078C2semidirect
9. C3×#3076C2semidirect
10. C2×C3×#479C2semidirect
11. C3×#2913C2semidirect
12. C3×#2912C2semidirect
13. C3×#2906C2semidirect
16. #60499C3direct (N × H)
75. #3504C2×C3semidirect
77. #3084C2×C3semidirect
78. #3083C2×C3semidirect
79. #3082C2×C3semidirect
80. #3082C2×C3semidirect
81. #3080C2×C3semidirect
82. #3078C2×C3semidirect
83. #3076C2×C3semidirect
84. C2×#479C2×C3semidirect
85. #2913C2×C3semidirect
86. #2912C2×C3semidirect
87. #2906C2×C3semidirect
91. C3×2−1+4Q8semidirect
315. 2−1+4C3×Q8semidirect
607. C3#60499direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35042
C3×C4×2−1+42
C3×#30842
C3×#30832
C3×#30822
C3×#30802
C3×#30782
C3×#30762
C2×C3×#4792
C3×#29132
C3×#29122
C3×#29062
C3×#28982
#604993