Finite Group Databasebeta

← C3×#27066 #10645673

Subgroups — C3×#27066 #10645673

Order 768.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2512C211yesmaximal, characteristic
2C3×#2365C211yesmaximal, characteristic, complemented
3C3×#2096C211yesmaximal, characteristic, complemented
4C3×#2055C211yesmaximal, characteristic
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2013C211yesmaximal, characteristic
7C3×#1968C211yesmaximal, characteristic
8C3×#1927C211yesmaximal, characteristic, complemented
9C3×#1926C211yesmaximal, characteristic
10C3×#1919C211yesmaximal, characteristic, complemented
11C3×#1916C211yesmaximal, characteristic, complemented
12C3×#1880C211yesmaximal, characteristic
13C3×#1880C211yesmaximal, characteristic
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×C4×#103C211yesmaximal, characteristic, complemented
16#27066C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#397—21no—
36C3×#396—21no—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389—21no—
43C3×#389C2211yescharacteristic
44C3×#388—21no—
45C3×#387—21no—
46C3×#387—21no—
47C3×#384C2211yescharacteristic
48C3×C4×C4⋊C4—21no—
49C3×C4×C4⋊C4C2211yescharacteristic
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2512C2×C311yescharacteristic
61#2365C2×C311yescharacteristic, complemented
62#2096C2×C311yescharacteristic, complemented
63#2055C2×C311yescharacteristic
64#2043C2×C311yescharacteristic, complemented
65#2013C2×C311yescharacteristic
66#1968C2×C311yescharacteristic
67#1927C2×C311yescharacteristic, complemented
68#1926C2×C311yescharacteristic
69#1919C2×C311yescharacteristic, complemented
70#1916C2×C311yescharacteristic, complemented
71#1880C2×C311yescharacteristic
72#1880C2×C311yescharacteristic
73#1803C2×C311yescharacteristic, complemented
74C4×#103C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16—41no—
82C2×C3×SD16—41no—
83C2×C3×SD16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8—22no—
90C3×#126—44no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—41no—
94C3×C4×Q8—41no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C3×#106—22no—
123C3×#106—41no—
124C3×C4⋊C8—41no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#103—41no—
129C3×#103—41no—
130C3×#103—41no—
131C3×#102—22no—
132C3×#102—41no—
133C3×C4×C8—41no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95D411yescharacteristic
142C3×#95D411yescharacteristic
143C22×SD16C22×C311yescharacteristic
144C22×SD16C22×C311yescharacteristic
145C2×C4⋊Q8C22×C311yescharacteristic
146C2×#126—21no—
147C2×C4×Q8C22×C311yescharacteristic
148C2×C4×Q8C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×#106C22×C311yescharacteristic
151C2×C4⋊C8C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×#103C22×C311yescharacteristic
156C2×#103C22×C311yescharacteristic
157C2×#103C22×C311yescharacteristic
158C2×#102C22×C311yescharacteristic
159C2×#102C22×C311yescharacteristic
160C2×C4×C8C22×C311yescharacteristic
161#397—21no—
162#396—21no—
163#391C22×C311yescharacteristic
164#391C22×C311yescharacteristic
165#390C22×C311yescharacteristic
166#390C22×C311yescharacteristic
167#390C22×C311yescharacteristic
168#389—21no—
169#389C22×C311yescharacteristic
170#388—21no—
171#387—21no—
172#387—21no—
173#384C22×C311yescharacteristic
174C4×C4⋊C4—21no—
175C4×C4⋊C4C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177C4×C4⋊C4C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183#336C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185#336C22×C311yescharacteristic
186C24×C3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8#4112yes—
198C2×C3×Q8#4112yes—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C3×SD16—44no—
220C3×SD16—44no—
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×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8#4112yes—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—24no—
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—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—81no—
250C3×C4⋊C4SD1614yescomplemented
251C3×C4⋊C4#4112yes—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—81no—
257C3×C22⋊C4—81no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—42no—
265C3×C42—42no—
266C3×C42—42no—
267C3×C42—42no—
268C22×Q8C23×C311yescharacteristic
269C22×Q8C23×C311yescharacteristic
270C22×D4C23×C311yescharacteristic
271C2×SD16—22no—
272C2×SD16—22no—
273C2×SD16—22no—
274C2×SD16—41no—
275C2×SD16—41no—
276C2×SD16—41no—
277C22×C8C23×C311yescharacteristic
278C22×C8C23×C311yescharacteristic
279C22×C8C23×C311yescharacteristic
280C22×C8C23×C311yescharacteristic
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283#126—44no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—41no—
287C4×Q8—41no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C3×D411yescharacteristic
296C2×C4⋊C4C3×D411yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C23×C311yescharacteristic
313C2×C42C23×C311yescharacteristic
314C2×C42C23×C311yescharacteristic
315#106—22no—
316#106—41no—
317C4⋊C8—41no—
318#103—22no—
319#103—22no—
320#103—22no—
321#103—41no—
322#103—41no—
323#103—41no—
324#102—22no—
325#102—41no—
326C4×C8—41no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95C3×D411yescharacteristic
335#95C3×D411yescharacteristic
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3C22×D411yescharacteristic
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
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—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
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—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4C2×SD1612yes—
393C2×C3×C4C2×SD1612yes—
394C2×C3×C4C2×SD1612yes—
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C3×C8—22no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C24—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C3×#4112yes—
417C2×Q8C3×#4112yes—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438SD16—44no—
439SD16—44no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8C3×#4112yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4C3×SD1614yescomplemented
470C4⋊C4C3×#4112yes—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3#52511yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#55011yescharacteristic
497C22×C3C22×SD1611yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#53811yescharacteristic
516C3×C4#55111yescharacteristic
517C3×C4#55311yescharacteristic
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C3×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4C2×C3×SD1612yes—
575C2×C4C2×C3×SD1612yes—
576C2×C4C2×C3×SD1612yes—
577C2×C4C2×C3×#4111yescharacteristic
578C2×C4C2×C3×#4111yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C3×H(5,𝔽2)11yescharacteristic
582C2×C4C3×H(5,𝔽2)11yescharacteristic
583C8—22no—
584C8—41no—
585C8—41no—
586C8—41no—
587C2×C3—161no—
588C2×C3#246711yescharacteristic
589C2×C3#310311yescharacteristic
590C2×C3#312111yescharacteristic
591C2×C3#329811yescharacteristic
592C2×C3#330711yescharacteristic
593C2×C3#339211yescharacteristic
594C2×C3#340211yescharacteristic
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C3×#52511yescharacteristic
603C22C3×#54611yescharacteristic
604C22C3×#55011yescharacteristic
605C22C22×C3×SD1611yescharacteristic
606C22C3×#57511yescharacteristic
607C22C3×#57711yescharacteristic
608C22C3×#57811yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4C3×#52911yescharacteristic
623C4C3×#53811yescharacteristic
624C4C3×#55111yescharacteristic
625C4C3×#55311yescharacteristic
626C3#2706611yescharacteristic, direct product factor, Sylow
627C2—161no—
628C2C3×#246711yescharacteristic
629C2C3×#310311yescharacteristic
630C2C3×#312111yescharacteristic
631C2C3×#329811yescharacteristic
632C2C3×#330711yescharacteristic
633C2C3×#339211yescharacteristic
634C2C3×#340211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2365C2semidirect
3. C3×#2096C2semidirect
5. C3×#2043C2semidirect
8. C3×#1927C2semidirect
10. C3×#1919C2semidirect
11. C3×#1916C2semidirect
14. C3×#1803C2semidirect
15. C3×C4×#103C2semidirect
16. #27066C3direct (N × H)
61. #2365C2×C3semidirect
62. #2096C2×C3semidirect
64. #2043C2×C3semidirect
67. #1927C2×C3semidirect
69. #1919C2×C3semidirect
70. #1916C2×C3semidirect
73. #1803C2×C3semidirect
74. C4×#103C2×C3semidirect
250. C3×C4⋊C4SD16semidirect
469. C4⋊C4C3×SD16semidirect
626. C3#27066direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25122
C3×#23652
C3×#20962
C3×#20552
C3×#20432
C3×#20132
C3×#19682
C3×#19272
C3×#19262
C3×#19192
C3×#19162
C3×#18802
C3×#18032
C3×C4×#1032
#270663