Finite Group Databasebeta

← C3×#61846 #10680453

Subgroups — C3×#61846 #10680453

Order 768.

1664 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3502C211yesmaximal, characteristic, complemented
2C3×Q8×#41C211yesmaximal, characteristic
3C3×#3387C213yesmaximal, complemented
4C3×#3385C213yesmaximal, complemented
5C3×Q8×Q16C211yesmaximal, characteristic, complemented
6C3×Q8×SD16C211yesmaximal, characteristic, complemented
7C3×#3374C213yesmaximal, complemented
8C3×#3372C213yesmaximal, complemented
9C3×#3363C211yesmaximal, characteristic, complemented
10C3×#3359C211yesmaximal, characteristic, complemented
11C3×#3237C213yesmaximal, complemented
12C3×#3231C213yesmaximal, complemented
13C3×#3116C213yesmaximal
14C3×#3107C213yesmaximal
15C3×#2984C211yesmaximal, characteristic, complemented
16#61846C311yesmaximal, characteristic, direct product factor, Sylow
17C3×Q82C2213yes—
18C3×Q82C2211yescharacteristic
19C3×Q82C2211yescharacteristic
20C3×#557—21no—
21C3×#557—23no—
22C3×#557C2211yescharacteristic
23C3×#557C2213yes—
24C3×#557C2213yes—
25C3×#554—23no—
26C3×#554C2213yes—
27C3×D4×Q8—21no—
28C3×D4×Q8C2211yescharacteristic
29C3×#544—23no—
30C3×#533—23no—
31C3×#533C2213yes—
32C3×#533C2213yes—
33C3×#533C2213yes—
34C3×#519—21no—
35C3×C4×#41C2213yes—
36C2×C3×C4×Q8—21no—
37C2×C3×C4×Q8C2211yescharacteristic
38C3×#501C2216yes—
39C3×#500C2213yes—
40C3×#498C2213yes—
41C3×#491C2216yes—
42C3×#487C2213yes—
43C3×#487C2213yes—
44C3×#485C2213yes—
45C3×#485C2216yes—
46C3×#485C2213yes—
47C3×#479C2216yes—
48C3×#479C2216yes—
49C3×#477C2213yes—
50C3×#477C2213yes—
51C3×#477C2213yes—
52C3×#476C2213yes—
53C3×#475C2213yes—
54C3×#475C2213yes—
55C3×#475C2213yes—
56C3×#474C2213yes—
57C3×#462C2213yes—
58C3×#462C2213yes—
59C3×#461C2213yes—
60C3×#461C2213yes—
61C3×#452C2211yescharacteristic
62C3×#452C2213yes—
63C3×#446C2213yes—
64C3×#446C2213yes—
65C3×C8×Q8C2211yescharacteristic
66C3×C8×Q8C2211yescharacteristic
67C3×#441C2216yes—
68C3×#440C2216yes—
69C3×C4×Q16C2213yes—
70C3×C4×Q16C2213yes—
71C3×C4×SD16C2213yes—
72C3×C4×SD16C2213yes—
73C3×#432C2213yes—
74C3×#431C2213yes—
75#3502C2×C311yescharacteristic, complemented
76Q8×#41C2×C311yescharacteristic
77#3387C2×C313yescomplemented
78#3385C2×C313yescomplemented
79Q8×Q16C2×C311yescharacteristic, complemented
80Q8×SD16C2×C311yescharacteristic, complemented
81#3374C2×C313yescomplemented
82#3372C2×C313yescomplemented
83#3363C2×C311yescharacteristic, complemented
84#3359C2×C311yescharacteristic, complemented
85#3237C2×C313yescomplemented
86#3231C2×C313yescomplemented
87#3116C2×C313yes—
88#3107C2×C313yes—
89#2984C2×C311yescharacteristic, complemented
90C2×C3×#41C2311yescharacteristic
91C22×C3×Q8—21no—
92C22×C3×Q8D411yescharacteristic
93C2×C3×Q16C2311yescharacteristic
94C2×C3×Q16C2313yes—
95C2×C3×SD16C2311yescharacteristic
96C2×C3×SD16C2313yes—
97C3×C4⋊Q8—23no—
98C3×C4⋊Q8—23no—
99C3×C4⋊Q8—23no—
100C3×C4⋊Q8—26no—
101C3×C4⋊Q8—23no—
102C3×C4⋊Q8C2313yes—
103C3×C4⋊Q8C2313yes—
104C3×C4⋊Q8C2313yes—
105C3×C4⋊Q8C2313yes—
106C3×C4⋊Q8C2313yes—
107C3×C4⋊Q8C2313yes—
108C3×#125—26no—
109C3×#125—23no—
110C3×#125—26no—
111C3×#125—23no—
112C3×#125—23no—
113C3×#125—23no—
114C3×#125C2316yes—
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⋊Q8C2311yescharacteristic
123C3×C22⋊Q8C2313yes—
124C3×C4×Q8—21no—
125C3×C4×Q8—23no—
126C3×C4×Q8—21no—
127C3×C4×Q8—23no—
128C3×C4×Q8—21no—
129C3×C4×Q8—23no—
130C3×C4×Q8—23no—
131C3×C4×Q8—23no—
132C3×C4×Q8—23no—
133C3×C4×Q8—23no—
134C3×C4×Q8D411yescharacteristic
135C3×C4×Q8D413yes—
136C3×C4×Q8C2313yes—
137C3×C4×Q8C2313yes—
138C3×C4×Q8C2313yes—
139C3×C4×Q8C2316yes—
140C3×C4×Q8C2313yes—
141C3×C4×Q8C2311yescharacteristic
142C3×C4×Q8C2313yes—
143C3×C4×D4—23no—
144C3×C4×D4—23no—
145C3×C4×D4C2313yes—
146C3×C4×D4C2313yes—
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×#107C2313yes—
157C3×#107C2316yes—
158C3×#106C2313yes—
159C3×#106C2313yes—
160C3×#106C2316yes—
161C3×C4⋊C8C2313yes—
162C3×C4⋊C8C2313yes—
163C3×C4⋊C8C2313yes—
164C3×C4⋊C8C2313yes—
165C3×#103C2313yes—
166C3×#103C2313yes—
167C3×#103C2311yescharacteristic
168C3×#103C2313yes—
169C3×#103C2313yes—
170C3×#103C2313yes—
171C3×#103C2316yes—
172C3×#103C2316yes—
173C3×#102C2311yescharacteristic
174C3×#102C2313yes—
175C3×C22⋊C8C2311yescharacteristic
176C3×C22⋊C8C2313yes—
177C3×#97C2316yes—
178C3×C4×C8C2313yes—
179C3×C4×C8C2313yes—
180Q82C22×C311yescharacteristic
181Q82C22×C311yescharacteristic
182Q82C22×C313yes—
183#557—21no—
184#557—23no—
185#557C22×C311yescharacteristic
186#557C22×C313yes—
187#557C22×C313yes—
188#554—23no—
189#554C22×C313yes—
190D4×Q8—21no—
191D4×Q8C22×C311yescharacteristic
192#544—23no—
193#533—23no—
194#533C22×C313yes—
195#533C22×C313yes—
196#533C22×C313yes—
197#519—21no—
198C4×#41C22×C313yes—
199C2×C4×Q8—21no—
200C2×C4×Q8C22×C311yescharacteristic
201#501C22×C316yes—
202#500C22×C313yes—
203#498C22×C313yes—
204#491C22×C316yes—
205#487C22×C313yes—
206#487C22×C313yes—
207#485C22×C313yes—
208#485C22×C313yes—
209#485C22×C316yes—
210#479C22×C316yes—
211#479C22×C316yes—
212#477C22×C313yes—
213#477C22×C313yes—
214#477C22×C313yes—
215#476C22×C313yes—
216#475C22×C313yes—
217#475C22×C313yes—
218#475C22×C313yes—
219#474C22×C313yes—
220#462C22×C313yes—
221#462C22×C313yes—
222#461C22×C313yes—
223#461C22×C313yes—
224#452C22×C311yescharacteristic
225#452C22×C313yes—
226#446C22×C313yes—
227#446C22×C313yes—
228C8×Q8C22×C311yescharacteristic
229C8×Q8C22×C311yescharacteristic
230#441C22×C316yes—
231#440C22×C316yes—
232C4×Q16C22×C313yes—
233C4×Q16C22×C313yes—
234C4×SD16C22×C313yes—
235C4×SD16C22×C313yes—
236#432C22×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—23no—
245C2×C3×Q8—44no—
246C2×C3×Q8C2×D413yes—
247C2×C3×Q8C2×D411yescharacteristic
248C2×C3×Q8C2411yescharacteristic
249C2×C3×Q8C2413yes—
250C2×C3×Q8C2411yescharacteristic
251C2×C3×D4—21no—
252C2×C3×D4C2411yescharacteristic
253C22×C3×C4—21no—
254C22×C3×C4—23no—
255C22×C3×C4—23no—
256C22×C3×C4C2×D413yes—
257C22×C3×C4C2×D413yes—
258C22×C3×C4C2411yescharacteristic
259C3×Q16—22no—
260C3×Q16—26no—
261C3×SD16—22no—
262C3×SD16—26no—
263C2×C3×C8C2411yescharacteristic
264C2×C3×C8C2411yescharacteristic
265C2×C3×C8C2413yes—
266C2×C3×C8C2413yes—
267C3×C4⋊C4—26no—
268C3×C4⋊C4—23no—
269C3×C4⋊C4—23no—
270C3×C4⋊C4—23no—
271C3×C4⋊C4—23no—
272C3×C4⋊C4—23no—
273C3×C4⋊C4—23no—
274C3×C4⋊C4—23no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—23no—
277C3×C4⋊C4—23no—
278C3×C4⋊C4—26no—
279C3×C4⋊C4—23no—
280C3×C4⋊C4—23no—
281C3×C4⋊C4—26no—
282C3×C4⋊C4—23no—
283C3×C4⋊C4—23no—
284C3×C4⋊C4C2×D413yes—
285C3×C4⋊C4C2×D416yes—
286C3×C4⋊C4C2×D413yes—
287C3×C4⋊C4C2411yescharacteristic
288C3×C4⋊C4C2413yes—
289C3×C4⋊C4C2413yes—
290C3×C4⋊C4C2413yes—
291C3×C22⋊C4—23no—
292C3×C22⋊C4—21no—
293C3×C22⋊C4—23no—
294C3×C22⋊C4—23no—
295C3×C22⋊C4—23no—
296C3×C42—23no—
297C3×C42—23no—
298C3×C42—23no—
299C3×C42—26no—
300C3×C42—23no—
301C3×C42—23no—
302C3×C42C2×D413yes—
303C3×C42C2×D413yes—
304C3×C42C2413yes—
305C3×C42C2413yes—
306C2×#41C23×C311yescharacteristic
307C22×Q8—21no—
308C22×Q8C3×D411yescharacteristic
309C2×Q16C23×C311yescharacteristic
310C2×Q16C23×C313yes—
311C2×SD16C23×C311yescharacteristic
312C2×SD16C23×C313yes—
313C4⋊Q8—23no—
314C4⋊Q8—23no—
315C4⋊Q8—23no—
316C4⋊Q8—26no—
317C4⋊Q8—23no—
318C4⋊Q8C23×C313yes—
319C4⋊Q8C23×C313yes—
320C4⋊Q8C23×C313yes—
321C4⋊Q8C23×C313yes—
322C4⋊Q8C23×C313yes—
323C4⋊Q8C23×C313yes—
324#125—26no—
325#125—23no—
326#125—26no—
327#125—23no—
328#125—23no—
329#125—23no—
330#125C23×C316yes—
331C22⋊Q8—23no—
332C22⋊Q8—23no—
333C22⋊Q8—23no—
334C22⋊Q8—23no—
335C22⋊Q8—23no—
336C22⋊Q8—23no—
337C22⋊Q8—26no—
338C22⋊Q8C23×C311yescharacteristic
339C22⋊Q8C23×C313yes—
340C4×Q8—21no—
341C4×Q8—23no—
342C4×Q8—21no—
343C4×Q8—23no—
344C4×Q8—21no—
345C4×Q8—23no—
346C4×Q8—23no—
347C4×Q8—23no—
348C4×Q8—23no—
349C4×Q8—23no—
350C4×Q8C3×D411yescharacteristic
351C4×Q8C3×D413yes—
352C4×Q8C23×C313yes—
353C4×Q8C23×C313yes—
354C4×Q8C23×C313yes—
355C4×Q8C23×C313yes—
356C4×Q8C23×C316yes—
357C4×Q8C23×C311yescharacteristic
358C4×Q8C23×C313yes—
359C4×D4—23no—
360C4×D4—23no—
361C4×D4C23×C313yes—
362C4×D4C23×C313yes—
363#117—23no—
364#117—23no—
365#117—23no—
366#117C23×C313yes—
367C2×C4⋊C4—23no—
368C2×C4⋊C4C3×D413yes—
369C2×C42—23no—
370C2×C42C23×C313yes—
371#107C23×C313yes—
372#107C23×C313yes—
373#107C23×C316yes—
374#106C23×C313yes—
375#106C23×C313yes—
376#106C23×C316yes—
377C4⋊C8C23×C313yes—
378C4⋊C8C23×C313yes—
379C4⋊C8C23×C313yes—
380C4⋊C8C23×C313yes—
381#103C23×C313yes—
382#103C23×C313yes—
383#103C23×C311yescharacteristic
384#103C23×C313yes—
385#103C23×C313yes—
386#103C23×C313yes—
387#103C23×C316yes—
388#103C23×C316yes—
389#102C23×C311yescharacteristic
390#102C23×C313yes—
391C22⋊C8C23×C311yescharacteristic
392C22⋊C8C23×C313yes—
393#97C23×C316yes—
394C4×C8C23×C313yes—
395C4×C8C23×C313yes—
396C23×C3—21no—
397C23×C3C22×D411yescharacteristic
398C3×Q8—26no—
399C3×Q8—21no—
400C3×Q8—21no—
401C3×Q8—23no—
402C3×Q8—21no—
403C3×Q8—23no—
404C3×Q8—46no—
405C3×Q8—83no—
406C3×Q8—83no—
407C3×Q8—83no—
408C3×Q8#13714yes—
409C3×Q82−1+412yes—
410C3×D4—24no—
411C3×D4—21no—
412C3×D42−1+412yes—
413C2×C3×C4—24no—
414C2×C3×C4—21no—
415C2×C3×C4—26no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—23no—
419C2×C3×C4—21no—
420C2×C3×C4—23no—
421C2×C3×C4—21no—
422C2×C3×C4—23no—
423C2×C3×C4—23no—
424C2×C3×C4—43no—
425C2×C3×C4—46no—
426C2×C3×C4—83no—
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D413yes—
429C2×C3×C4C22×D413yes—
430C2×C3×C4C22×D413yes—
431C2×C3×C4C22×D413yes—
432C2×C3×C4C2511yescharacteristic
433C3×C8—21no—
434C3×C8—23no—
435C3×C8—21no—
436C3×C8—23no—
437#41—22no—
438#41—22no—
439C2×Q8—24no—
440C2×Q8—23no—
441C2×Q8—23no—
442C2×Q8—23no—
443C2×Q8—23no—
444C2×Q8—44no—
445C2×Q8C2×C3×D413yes—
446C2×Q8C2×C3×D411yescharacteristic
447C2×Q8C24×C311yescharacteristic
448C2×Q8C24×C313yes—
449C2×Q8C24×C311yescharacteristic
450C2×D4—21no—
451C2×D4C24×C311yescharacteristic
452C22×C4—21no—
453C22×C4—23no—
454C22×C4—23no—
455C22×C4C2×C3×D413yes—
456C22×C4C2×C3×D413yes—
457C22×C4C24×C311yescharacteristic
458Q16—22no—
459Q16—26no—
460SD16—22no—
461SD16—26no—
462C2×C8C24×C311yescharacteristic
463C2×C8C24×C311yescharacteristic
464C2×C8C24×C313yes—
465C2×C8C24×C313yes—
466C4⋊C4—26no—
467C4⋊C4—23no—
468C4⋊C4—23no—
469C4⋊C4—23no—
470C4⋊C4—23no—
471C4⋊C4—23no—
472C4⋊C4—23no—
473C4⋊C4—23no—
474C4⋊C4—21no—
475C4⋊C4—23no—
476C4⋊C4—23no—
477C4⋊C4—26no—
478C4⋊C4—23no—
479C4⋊C4—23no—
480C4⋊C4—26no—
481C4⋊C4—23no—
482C4⋊C4—23no—
483C4⋊C4C2×C3×D413yes—
484C4⋊C4C2×C3×D416yes—
485C4⋊C4C2×C3×D413yes—
486C4⋊C4C24×C311yescharacteristic
487C4⋊C4C24×C313yes—
488C4⋊C4C24×C313yes—
489C4⋊C4C24×C313yes—
490C22⋊C4—21no—
491C22⋊C4—23no—
492C22⋊C4—23no—
493C22⋊C4—23no—
494C22⋊C4—23no—
495C42—23no—
496C42—23no—
497C42—23no—
498C42—26no—
499C42—23no—
500C42—23no—
501C42C2×C3×D413yes—
502C42C2×C3×D413yes—
503C42C24×C313yes—
504C42C24×C313yes—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3C23×D411yescharacteristic
512C3×C4—23no—
513C3×C4—23no—
514C3×C4—21no—
515C3×C4—22no—
516C3×C4—23no—
517C3×C4—43no—
518C3×C4—41no—
519C3×C4—43no—
520C3×C4—43no—
521C3×C4—41no—
522C3×C4—43no—
523C3×C4C2×#13716yes—
524C3×C4C2×2−1+411yescharacteristic
525C3×C4C2×2−1+411yescharacteristic
526C23—21no—
527C23C22×C3×D411yescharacteristic
528Q8—26no—
529Q8—21no—
530Q8—21no—
531Q8—23no—
532Q8—21no—
533Q8—23no—
534Q8—46no—
535Q8—83no—
536Q8—83no—
537Q8—83no—
538Q8C3×#13714yes—
539Q8C3×2−1+412yes—
540D4—24no—
541D4—21no—
542D4C3×2−1+412yes—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—26no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—23no—
549C2×C4—21no—
550C2×C4—23no—
551C2×C4—21no—
552C2×C4—23no—
553C2×C4—23no—
554C2×C4—43no—
555C2×C4—46no—
556C2×C4—83no—
557C2×C4C22×C3×D411yescharacteristic
558C2×C4C22×C3×D413yes—
559C2×C4C22×C3×D413yes—
560C2×C4C22×C3×D413yes—
561C2×C4C22×C3×D413yes—
562C2×C4C25×C311yescharacteristic
563C8—21no—
564C8—23no—
565C8—21no—
566C8—23no—
567C2×C3—41no—
568C2×C3—42no—
569C2×C3#348611yescharacteristic
570C2×C3C22×#13711yescharacteristic
571C2×C3#358411yescharacteristic
572C22—21no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22C23×C3×D411yescharacteristic
579C4—21no—
580C4—22no—
581C4—23no—
582C4—23no—
583C4—23no—
584C4—43no—
585C4—41no—
586C4—43no—
587C4—43no—
588C4—41no—
589C4—43no—
590C4C2×C3×#13716yes—
591C4C2×C3×2−1+411yescharacteristic
592C4C2×C3×2−1+411yescharacteristic
593C3#6184611yescharacteristic, direct product factor, Sylow
594C2—41no—
595C2—42no—
596C2C3×#348611yescharacteristic
597C2C22×C3×#13711yescharacteristic
598C2C3×#358411yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#41C23
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×#3502C2semidirect
3. C3×#3387C2semidirect
4. C3×#3385C2semidirect
5. C3×Q8×Q16C2semidirect
6. C3×Q8×SD16C2semidirect
7. C3×#3374C2semidirect
8. C3×#3372C2semidirect
9. C3×#3363C2semidirect
10. C3×#3359C2semidirect
11. C3×#3237C2semidirect
12. C3×#3231C2semidirect
15. C3×#2984C2semidirect
16. #61846C3direct (N × H)
75. #3502C2×C3semidirect
77. #3387C2×C3semidirect
78. #3385C2×C3semidirect
79. Q8×Q16C2×C3semidirect
80. Q8×SD16C2×C3semidirect
81. #3374C2×C3semidirect
82. #3372C2×C3semidirect
83. #3363C2×C3semidirect
84. #3359C2×C3semidirect
85. #3237C2×C3semidirect
86. #3231C2×C3semidirect
89. #2984C2×C3semidirect
593. C3#61846direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35022
C3×Q8×#412
C3×#33872
C3×#33852
C3×Q8×Q162
C3×Q8×SD162
C3×#33742
C3×#33722
C3×#33632
C3×#33592
C3×#32372
C3×#32312
C3×#31162
C3×#31072
C3×#29842
#618463