Finite Group Databasebeta

← C3×#29462 #10648069

Subgroups — C3×#29462 #10648069

Order 768.

766 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2855C211yesmaximal, characteristic
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C3×#2076C211yesmaximal, characteristic
4C3×#2036C211yesmaximal, characteristic
5C3×#2036C211yesmaximal, characteristic
6C3×#2001C211yesmaximal, characteristic
7C3×#1969C211yesmaximal, characteristic, complemented
8C3×#1942C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1933C211yesmaximal, characteristic, complemented
11C3×#1928C211yesmaximal, characteristic, complemented
12C3×#1921C211yesmaximal, characteristic, complemented
13C3×#1878C211yesmaximal, characteristic, complemented
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×C4×C22⋊C8C211yesmaximal, characteristic
16#29462C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
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×C4×C8C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#391C2211yescharacteristic
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#387—21no—
42C3×#387—21no—
43C3×#387—21no—
44C3×#385—21no—
45C3×#385—21no—
46C3×#381—21no—
47C3×#380C2211yescharacteristic
48C3×#380C2211yescharacteristic
49C3×C4×C4⋊C4—21no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×C4×C22⋊C4—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2855C2×C311yescharacteristic
61C4×C22⋊Q8C2×C311yescharacteristic
62#2076C2×C311yescharacteristic
63#2036C2×C311yescharacteristic
64#2036C2×C311yescharacteristic
65#2001C2×C311yescharacteristic
66#1969C2×C311yescharacteristic, complemented
67#1942C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1933C2×C311yescharacteristic, complemented
70#1928C2×C311yescharacteristic, complemented
71#1921C2×C311yescharacteristic, complemented
72#1878C2×C311yescharacteristic, complemented
73#1863C2×C311yescharacteristic, complemented
74C4×C22⋊C8C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C23×C3×C4D411yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—41no—
90C3×C22⋊Q8—41no—
91C3×C4×Q8—41no—
92C3×C4×Q8—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⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—41no—
112C2×C3×C42—41no—
113C2×C3×C42D411yescharacteristic
114C2×C3×C42D411yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#107—22no—
119C3×#107—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×C4×C8—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
146C22×C42C22×C311yescharacteristic
147C2×#107C22×C311yescharacteristic
148C2×#107C22×C311yescharacteristic
149C2×#107C22×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×C4×C8C22×C311yescharacteristic
159C2×C4×C8C22×C311yescharacteristic
160#391C22×C311yescharacteristic
161#391C22×C311yescharacteristic
162#391C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#389C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#387—21no—
167#387—21no—
168#387—21no—
169#385—21no—
170#385—21no—
171#381—21no—
172#380C22×C311yescharacteristic
173#380C22×C311yescharacteristic
174C4×C4⋊C4—21no—
175C4×C4⋊C4C22×C311yescharacteristic
176C4×C22⋊C4—21no—
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—22no—
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×C4—21no—
213C22×C3×C4—41no—
214C22×C3×C4—41no—
215C22×C3×C4—41no—
216C22×C3×C4—41no—
217C22×C3×C4Q1614yescomplemented
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C3×Q16—44no—
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—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C2×C3×C8#4112yes—
237C3×C4⋊C4—24no—
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—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4#4112yes—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—42no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—41no—
264C3×C42—41no—
265C3×C42—42no—
266C3×C42—42no—
267C22×Q8C23×C311yescharacteristic
268C22×Q8C23×C311yescharacteristic
269C23×C4C3×D411yescharacteristic
270C23×C4C3×D411yescharacteristic
271C23×C4C23×C311yescharacteristic
272C2×Q16—22no—
273C2×Q16—22no—
274C2×Q16—41no—
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C311yescharacteristic
277C22×C8C23×C311yescharacteristic
278C22×C8C23×C311yescharacteristic
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—41no—
282C22⋊Q8—41no—
283C4×Q8—41no—
284C4×Q8—41no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42C3×D411yescharacteristic
306C2×C42C3×D411yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#107—22no—
310#107—22no—
311#107—41no—
312#103—22no—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—41no—
317#103—41no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C4×C8—41no—
321C4×C8—41no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3C2×Q1612yes—
341C23×C3C22×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
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—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—44no—
365C2×C3×C4—44no—
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×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4C2×Q1612yes—
393C2×C3×C4C2×Q1612yes—
394C2×C3×C4#13512yes—
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—22no—
403C3×C8—41no—
404C3×C8—41no—
405C24C2×C3×D411yescharacteristic
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×Q8—21no—
417C2×Q8C3×#4112yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4C3×Q1614yescomplemented
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C24×C311yescharacteristic
444Q16—44no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8C3×#4112yes—
456C2×C8C3×#4112yes—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4C3×#4112yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—42no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3#52511yescharacteristic
495C22×C3#53211yescharacteristic
496C22×C3#53411yescharacteristic
497C22×C3C22×Q1611yescharacteristic
498C22×C3C2×#13511yescharacteristic
499C22×C3#57511yescharacteristic
500C22×C3#57511yescharacteristic
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—42no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#55111yescharacteristic
515C3×C4#55311yescharacteristic
516C3×C4#55311yescharacteristic
517C3×C4#55911yescharacteristic
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C2×C3×Q1612yes—
525C23C22×C3×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—44no—
549C2×C4—44no—
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×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4C2×C3×Q1612yes—
577C2×C4C2×C3×Q1612yes—
578C2×C4C3×#13512yes—
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C2×C3×#4111yescharacteristic
582C2×C4C2×C3×#4111yescharacteristic
583C2×C4C3×H(5,𝔽2)11yescharacteristic
584C2×C4C3×H(5,𝔽2)11yescharacteristic
585C8—22no—
586C8—22no—
587C8—41no—
588C8—41no—
589C2×C3—42no—
590C2×C3#260211yescharacteristic
591C2×C3#310211yescharacteristic
592C2×C3#310411yescharacteristic
593C2×C3#317011yescharacteristic
594C2×C3#318311yescharacteristic
595C2×C3#318911yescharacteristic
596C2×C3#319111yescharacteristic
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22C3×#52511yescharacteristic
605C22C3×#53211yescharacteristic
606C22C3×#53411yescharacteristic
607C22C22×C3×Q1611yescharacteristic
608C22C2×C3×#13511yescharacteristic
609C22C3×#57511yescharacteristic
610C22C3×#57511yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C3×#55111yescharacteristic
625C4C3×#55311yescharacteristic
626C4C3×#55311yescharacteristic
627C4C3×#55911yescharacteristic
628C3#2946211yescharacteristic, direct product factor, Sylow
629C2—42no—
630C2C3×#260211yescharacteristic
631C2C3×#310211yescharacteristic
632C2C3×#310411yescharacteristic
633C2C3×#317011yescharacteristic
634C2C3×#318311yescharacteristic
635C2C3×#318911yescharacteristic
636C2C3×#319111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
7. C3×#1969C2semidirect
8. C3×#1942C2semidirect
9. C3×#1933C2semidirect
10. C3×#1933C2semidirect
11. C3×#1928C2semidirect
12. C3×#1921C2semidirect
13. C3×#1878C2semidirect
14. C3×#1863C2semidirect
16. #29462C3direct (N × H)
66. #1969C2×C3semidirect
67. #1942C2×C3semidirect
68. #1933C2×C3semidirect
69. #1933C2×C3semidirect
70. #1928C2×C3semidirect
71. #1921C2×C3semidirect
72. #1878C2×C3semidirect
73. #1863C2×C3semidirect
217. C22×C3×C4Q16semidirect
437. C22×C4C3×Q16semidirect
628. C3#29462direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28552
C3×C4×C22⋊Q82
C3×#20762
C3×#20362
C3×#20012
C3×#19692
C3×#19422
C3×#19332
C3×#19282
C3×#19212
C3×#18782
C3×#18632
C3×C4×C22⋊C82
#294623