Finite Group Databasebeta

← C3×#29487 #10648094

Subgroups — C3×#29487 #10648094

Order 768.

700 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2625C211yesmaximal, characteristic
2C3×#2323C211yesmaximal, characteristic
3C3×#2077C211yesmaximal, characteristic
4C3×#2036C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic
6C3×#2001C211yesmaximal, characteristic
7C3×#1971C211yesmaximal, characteristic, complemented
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1933C211yesmaximal, characteristic, complemented
11C3×#1928C211yesmaximal, characteristic, complemented
12C3×#1920C211yesmaximal, characteristic, complemented
13C3×#1878C211yesmaximal, characteristic, complemented
14C3×#1865C211yesmaximal, characteristic, complemented
15C3×#1750C211yesmaximal, characteristic
16#29487C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×#97C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#400—21no—
36C3×#398—21no—
37C3×#397—21no—
38C3×#397—21no—
39C3×#391C2211yescharacteristic
40C3×#391C2211yescharacteristic
41C3×#391C2211yescharacteristic
42C3×#389—21no—
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#388—21no—
47C3×#388—21no—
48C3×#387—21no—
49C3×#380C2211yescharacteristic
50C3×#380C2211yescharacteristic
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2625C2×C311yescharacteristic
61#2323C2×C311yescharacteristic
62#2077C2×C311yescharacteristic
63#2036C2×C311yescharacteristic
64#2035C2×C311yescharacteristic
65#2001C2×C311yescharacteristic
66#1971C2×C311yescharacteristic, complemented
67#1943C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1933C2×C311yescharacteristic, complemented
70#1928C2×C311yescharacteristic, complemented
71#1920C2×C311yescharacteristic, complemented
72#1878C2×C311yescharacteristic, complemented
73#1865C2×C311yescharacteristic, complemented
74#1750C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C2×C3×Q16—22no—
79C2×C3×Q16—22no—
80C2×C3×Q16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—41no—
88C3×C22⋊Q8—41no—
89C3×C4×Q8—41no—
90C3×C4×Q8—41no—
91C3×#117—41no—
92C3×#117—41no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4D411yescharacteristic
112C2×C3×C22⋊C4D411yescharacteristic
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×#106—41no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—41no—
125C3×#103—41no—
126C3×C22⋊C8—22no—
127C3×C22⋊C8—22no—
128C3×#97—41no—
129C3×C4×C8—41no—
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—
140C3×#95—21no—
141C3×#95—21no—
142C22×Q16C22×C311yescharacteristic
143C2×C22⋊Q8C22×C311yescharacteristic
144C2×C22⋊Q8C22×C311yescharacteristic
145C2×C4×Q8C22×C311yescharacteristic
146C2×#117C22×C311yescharacteristic
147C2×#107C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×#103C22×C311yescharacteristic
156C2×C22⋊C8C22×C311yescharacteristic
157C2×C22⋊C8C22×C311yescharacteristic
158C2×#97C22×C311yescharacteristic
159C2×C4×C8C22×C311yescharacteristic
160#400—21no—
161#398—21no—
162#397—21no—
163#397—21no—
164#391C22×C311yescharacteristic
165#391C22×C311yescharacteristic
166#391C22×C311yescharacteristic
167#389—21no—
168#389C22×C311yescharacteristic
169#389C22×C311yescharacteristic
170#389C22×C311yescharacteristic
171#388—21no—
172#388—21no—
173#387—21no—
174#380C22×C311yescharacteristic
175#380C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#336C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185C24×C3C2×D411yescharacteristic
186C2×C3×Q8—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—
198C22×C3×C4—21no—
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—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×C4C2411yescharacteristic
218C3×Q16—44no—
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×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8#4112yes—
230C2×C3×C8#4112yes—
231C3×C4⋊C4—24no—
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—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4—81no—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—41no—
253C3×C22⋊C4—41no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—81no—
256C3×C22⋊C4—81no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—81no—
263C22×Q8C23×C311yescharacteristic
264C22×Q8C23×C311yescharacteristic
265C23×C4C23×C311yescharacteristic
266C2×Q16—22no—
267C2×Q16—22no—
268C2×Q16—41no—
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—41no—
276C22⋊Q8—41no—
277C4×Q8—41no—
278C4×Q8—41no—
279#117—41no—
280#117—41no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C3×D411yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C22⋊C4C3×D411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305#107—22no—
306#106—22no—
307#106—41no—
308#103—22no—
309#103—22no—
310#103—22no—
311#103—22no—
312#103—41no—
313#103—41no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316#97—41no—
317C4×C8—41no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
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—21no—
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—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×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—41no—
398C3×C8—41no—
399C24C2×C3×D411yescharacteristic
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C3×#4112yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C24×C311yescharacteristic
432Q16—44no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8C3×#4112yes—
444C2×C8C3×#4112yes—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4C3×#4112yes—
465C22⋊C4—24no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—42no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—81no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3#52511yescharacteristic
485C22×C3#53211yescharacteristic
486C22×C3#53411yescharacteristic
487C22×C3#57711yescharacteristic
488C22×C3#57711yescharacteristic
489C22×C3#57811yescharacteristic
490C22×C3#57811yescharacteristic
491C3×C4—41no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#55111yescharacteristic
505C3×C4#55311yescharacteristic
506C3×C4#55311yescharacteristic
507C3×C4#55911yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23C22×C3×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
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—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
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—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×C4C2×C3×#4111yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C2×C3×#4111yescharacteristic
570C2×C4C2×C3×#4111yescharacteristic
571C2×C4C3×H(5,𝔽2)11yescharacteristic
572C2×C4C3×H(5,𝔽2)11yescharacteristic
573C8—22no—
574C8—22no—
575C8—41no—
576C8—41no—
577C2×C3—81no—
578C2×C3#260211yescharacteristic
579C2×C3#312511yescharacteristic
580C2×C3#312711yescharacteristic
581C2×C3#317811yescharacteristic
582C2×C3#317811yescharacteristic
583C2×C3#320111yescharacteristic
584C2×C3#320111yescharacteristic
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C3×#52511yescharacteristic
593C22C3×#53211yescharacteristic
594C22C3×#53411yescharacteristic
595C22C3×#57711yescharacteristic
596C22C3×#57711yescharacteristic
597C22C3×#57811yescharacteristic
598C22C3×#57811yescharacteristic
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4C3×#55111yescharacteristic
613C4C3×#55311yescharacteristic
614C4C3×#55311yescharacteristic
615C4C3×#55911yescharacteristic
616C3#2948711yescharacteristic, direct product factor, Sylow
617C2—81no—
618C2C3×#260211yescharacteristic
619C2C3×#312511yescharacteristic
620C2C3×#312711yescharacteristic
621C2C3×#317811yescharacteristic
622C2C3×#317811yescharacteristic
623C2C3×#320111yescharacteristic
624C2C3×#320111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
7. C3×#1971C2semidirect
8. C3×#1943C2semidirect
9. C3×#1933C2semidirect
10. C3×#1933C2semidirect
11. C3×#1928C2semidirect
12. C3×#1920C2semidirect
13. C3×#1878C2semidirect
14. C3×#1865C2semidirect
16. #29487C3direct (N × H)
66. #1971C2×C3semidirect
67. #1943C2×C3semidirect
68. #1933C2×C3semidirect
69. #1933C2×C3semidirect
70. #1928C2×C3semidirect
71. #1920C2×C3semidirect
72. #1878C2×C3semidirect
73. #1865C2×C3semidirect
616. C3#29487direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26252
C3×#23232
C3×#20772
C3×#20362
C3×#20352
C3×#20012
C3×#19712
C3×#19432
C3×#19332
C3×#19282
C3×#19202
C3×#18782
C3×#18652
C3×#17502
#294873