Finite Group Databasebeta

← C3×#32663 #10651270

Subgroups — C3×#32663 #10651270

Order 768.

698 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×#2581C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic
3C3×#2097C211yesmaximal, characteristic, complemented
4C3×#2092C211yesmaximal, characteristic
5C3×#2085C211yesmaximal, characteristic, complemented
6C3×#2085C211yesmaximal, characteristic, complemented
7C3×#2066C211yesmaximal, characteristic, complemented
8C3×#2065C211yesmaximal, characteristic
9C3×#1985C211yesmaximal, characteristic, complemented
10C3×#1973C211yesmaximal, characteristic
11C3×#1943C211yesmaximal, characteristic, complemented
12C3×#1880C211yesmaximal, characteristic
13C3×#1877C211yesmaximal, characteristic
14C3×#1865C211yesmaximal, characteristic, complemented
15C3×#1864C211yesmaximal, characteristic
16#32663C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#97C2211yescharacteristic
34C3×#400—21no—
35C3×#399—21no—
36C3×#395—21no—
37C3×#393—21no—
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#387—21no—
47C3×#387—21no—
48C3×#385—21no—
49C3×#385—21no—
50C3×#382C2211yescharacteristic
51C3×#382C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2581C2×C311yescharacteristic, complemented
61#2580C2×C311yescharacteristic
62#2097C2×C311yescharacteristic, complemented
63#2092C2×C311yescharacteristic
64#2085C2×C311yescharacteristic, complemented
65#2085C2×C311yescharacteristic, complemented
66#2066C2×C311yescharacteristic, complemented
67#2065C2×C311yescharacteristic
68#1985C2×C311yescharacteristic, complemented
69#1973C2×C311yescharacteristic
70#1943C2×C311yescharacteristic, complemented
71#1880C2×C311yescharacteristic
72#1877C2×C311yescharacteristic
73#1865C2×C311yescharacteristic, complemented
74#1864C2×C311yescharacteristic
75C22×C3×Q8D411yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C2×C3×SD16—41no—
80C2×C3×SD16—41no—
81C2×C3×SD16Q814yescomplemented
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—41no—
87C3×C4×Q8—41no—
88C3×C4×Q8—41no—
89C3×C4×Q8—41no—
90C3×C4×D4—41no—
91C3×C4×D4—41no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×#107—41no—
117C3×#106—41no—
118C3×#106—41no—
119C3×#106—41no—
120C3×C4⋊C8—41no—
121C3×#103—22no—
122C3×#103—41no—
123C3×#103—41no—
124C3×#102—22no—
125C3×#102—41no—
126C3×#102—41no—
127C3×#97—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C22×SD16C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#102C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#97C22×C311yescharacteristic
157#400—21no—
158#399—21no—
159#395—21no—
160#393—21no—
161#391C22×C311yescharacteristic
162#391C22×C311yescharacteristic
163#390C22×C311yescharacteristic
164#390C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#389C22×C311yescharacteristic
167#389C22×C311yescharacteristic
168#389C22×C311yescharacteristic
169#387—21no—
170#387—21no—
171#385—21no—
172#385—21no—
173#382C22×C311yescharacteristic
174#382C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—41no—
190C2×C3×Q8—41no—
191C2×C3×Q8—41no—
192C2×C3×Q8C2×Q812yes—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C2×Q812yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C3×SD16—44no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8C2×Q812yes—
228C2×C3×C8#4112yes—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4—81no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—42no—
261C22×Q8C3×D411yescharacteristic
262C22×Q8C23×C311yescharacteristic
263C22×D4C23×C311yescharacteristic
264C23×C4—21no—
265C2×SD16—41no—
266C2×SD16—41no—
267C2×SD16C3×Q814yescomplemented
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C4×Q8—41no—
273C4×Q8—41no—
274C4×Q8—41no—
275C4×Q8—41no—
276C4×D4—41no—
277C4×D4—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—21no—
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#107—22no—
302#107—41no—
303#106—41no—
304#106—41no—
305#106—41no—
306C4⋊C8—41no—
307#103—22no—
308#103—41no—
309#103—41no—
310#102—22no—
311#102—41no—
312#102—41no—
313#97—41no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3C22×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—44no—
337C3×Q8—41no—
338C3×Q8—41no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—41no—
342C3×D4—41no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—84no—
383C2×C3×C4C22×Q811yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C3×C8—22no—
390C3×C8—22no—
391C3×C8—41no—
392C3×C8—41no—
393C24—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8C2×C3×Q812yes—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4C2×C3×Q812yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C24×C311yescharacteristic
426SD16—44no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8C2×C3×Q812yes—
438C2×C8C3×#4112yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3#53411yescharacteristic
479C22×C3D4×Q811yescharacteristic
480C22×C3#55011yescharacteristic
481C22×C3#57511yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57711yescharacteristic
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#53911yescharacteristic
499C3×C4#54311yescharacteristic
500C3×C4#55411yescharacteristic
501C3×C4#56211yescharacteristic
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C3×D411yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—44no—
513Q8—41no—
514Q8—41no—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4C22×C3×Q811yescharacteristic
560C2×C4C2×C3×#4111yescharacteristic
561C2×C4C3×H(5,𝔽2)11yescharacteristic
562C2×C4C3×H(5,𝔽2)11yescharacteristic
563C2×C4C3×H(5,𝔽2)11yescharacteristic
564C2×C4C3×2−1+411yescharacteristic
565C8—22no—
566C8—22no—
567C8—41no—
568C8—41no—
569C2×C3—82no—
570C2×C3#273211yescharacteristic
571C2×C3#319711yescharacteristic
572C2×C3#320011yescharacteristic
573C2×C3#338811yescharacteristic
574C2×C3#338811yescharacteristic
575C2×C3#340111yescharacteristic
576C2×C3#340111yescharacteristic
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C3×#53411yescharacteristic
585C22C3×D4×Q811yescharacteristic
586C22C3×#55011yescharacteristic
587C22C3×#57511yescharacteristic
588C22C3×#57511yescharacteristic
589C22C3×#57711yescharacteristic
590C22C3×#57711yescharacteristic
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4C3×#53911yescharacteristic
605C4C3×#54311yescharacteristic
606C4C3×#55411yescharacteristic
607C4C3×#56211yescharacteristic
608C3#3266311yescharacteristic, direct product factor, Sylow
609C2—82no—
610C2C3×#273211yescharacteristic
611C2C3×#319711yescharacteristic
612C2C3×#320011yescharacteristic
613C2C3×#338811yescharacteristic
614C2C3×#338811yescharacteristic
615C2C3×#340111yescharacteristic
616C2C3×#340111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2581C2semidirect
3. C3×#2097C2semidirect
5. C3×#2085C2semidirect
6. C3×#2085C2semidirect
7. C3×#2066C2semidirect
9. C3×#1985C2semidirect
11. C3×#1943C2semidirect
14. C3×#1865C2semidirect
16. #32663C3direct (N × H)
60. #2581C2×C3semidirect
62. #2097C2×C3semidirect
64. #2085C2×C3semidirect
65. #2085C2×C3semidirect
66. #2066C2×C3semidirect
68. #1985C2×C3semidirect
70. #1943C2×C3semidirect
73. #1865C2×C3semidirect
81. C2×C3×SD16Q8semidirect
267. C2×SD16C3×Q8semidirect
608. C3#32663direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25812
C3×#25802
C3×#20972
C3×#20922
C3×#20852
C3×#20662
C3×#20652
C3×#19852
C3×#19732
C3×#19432
C3×#18802
C3×#18772
C3×#18652
C3×#18642
#326633