Finite Group Databasebeta

← C3×#29571 #10648178

Subgroups — C3×#29571 #10648178

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2640C211yesmaximal, characteristic
2C3×#2320C211yesmaximal, characteristic, complemented
3C3×#2058C211yesmaximal, characteristic, complemented
4C3×#2042C211yesmaximal, characteristic
5C3×#2024C211yesmaximal, characteristic, complemented
6C3×#2015C211yesmaximal, characteristic
7C3×#1971C211yesmaximal, characteristic, complemented
8C3×#1968C211yesmaximal, characteristic
9C3×#1936C211yesmaximal, characteristic
10C3×#1933C211yesmaximal, characteristic, complemented
11C3×#1927C211yesmaximal, characteristic, complemented
12C3×#1926C211yesmaximal, characteristic
13C3×#1880C211yesmaximal, characteristic
14C3×#1878C211yesmaximal, characteristic, complemented
15C3×#1746C211yesmaximal, characteristic, complemented
16#29571C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33C2×C3×C4×C8C2211yescharacteristic
34C3×#397—21no—
35C3×#397—21no—
36C3×#393—21no—
37C3×#393—21no—
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#391C2211yescharacteristic
41C3×#391C2211yescharacteristic
42C3×#390C2211yescharacteristic
43C3×#390C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#387—21no—
47C3×#387—21no—
48C3×#383—21no—
49C3×#382—21no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#376C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2640C2×C311yescharacteristic
61#2320C2×C311yescharacteristic, complemented
62#2058C2×C311yescharacteristic, complemented
63#2042C2×C311yescharacteristic
64#2024C2×C311yescharacteristic, complemented
65#2015C2×C311yescharacteristic
66#1971C2×C311yescharacteristic, complemented
67#1968C2×C311yescharacteristic
68#1936C2×C311yescharacteristic
69#1933C2×C311yescharacteristic, complemented
70#1927C2×C311yescharacteristic, complemented
71#1926C2×C311yescharacteristic
72#1880C2×C311yescharacteristic
73#1878C2×C311yescharacteristic, complemented
74#1746C2×C311yescharacteristic, complemented
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C2×C3×Q16—22no—
80C2×C3×Q16—22no—
81C2×C3×Q16—41no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4⋊Q8—41no—
90C3×C4⋊Q8—41no—
91C3×#124—41no—
92C3×#124—41no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
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×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×#106—41no—
116C3×#106—41no—
117C3×#103—22no—
118C3×#103—22no—
119C3×#103—41no—
120C3×#103—41no—
121C3×#103—41no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—41no—
125C3×#97—22no—
126C3×C4×C8—22no—
127C3×#95—21no—
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×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140C3×#95D411yescharacteristic
141C3×#95D411yescharacteristic
142C22×Q16C22×C311yescharacteristic
143C22×SD16C22×C311yescharacteristic
144C2×C4⋊Q8C22×C311yescharacteristic
145C2×#124C22×C311yescharacteristic
146C2×C4×Q8C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×#103C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#97C22×C311yescharacteristic
158C2×C4×C8C22×C311yescharacteristic
159#397—21no—
160#397—21no—
161#393—21no—
162#393—21no—
163#391C22×C311yescharacteristic
164#391C22×C311yescharacteristic
165#391C22×C311yescharacteristic
166#391C22×C311yescharacteristic
167#390C22×C311yescharacteristic
168#390C22×C311yescharacteristic
169#389C22×C311yescharacteristic
170#389C22×C311yescharacteristic
171#387—21no—
172#387—21no—
173#383—21no—
174#382—21no—
175C4×C4⋊C4C22×C311yescharacteristic
176#376C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183#336C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185C24×C3—21no—
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—42no—
198C2×C3×Q8—81no—
199C2×C3×Q8—81no—
200C2×C3×Q8#4112yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—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—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C3×Q16—44no—
222C3×SD16—44no—
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—21no—
233C2×C3×C8#4112yes—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—24no—
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—21no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C22⋊C4—81no—
255C3×C42—24no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—81no—
263C22×Q8—21no—
264C22×Q8C23×C311yescharacteristic
265C22×Q8C23×C311yescharacteristic
266C22×D4C23×C311yescharacteristic
267C2×Q16—22no—
268C2×Q16—22no—
269C2×Q16—41no—
270C2×SD16—22no—
271C2×SD16—22no—
272C2×SD16—41no—
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C311yescharacteristic
277C4⋊Q8—41no—
278C4⋊Q8—41no—
279#124—41no—
280#124—41no—
281C4×Q8—22no—
282C4×Q8—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
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×C22⋊C4—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42C23×C311yescharacteristic
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#106—41no—
304#106—41no—
305#103—22no—
306#103—22no—
307#103—41no—
308#103—41no—
309#103—41no—
310#102—22no—
311#102—22no—
312#102—41no—
313#97—22no—
314C4×C8—22no—
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—
326#95C3×D411yescharacteristic
327#95C3×D411yescharacteristic
328#95C3×D411yescharacteristic
329#95C3×D411yescharacteristic
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
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—
346C3×Q8—84no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
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—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
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×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C3×C8—22no—
397C3×C8—22no—
398C3×C8—41no—
399C3×C8—41no—
400C24—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×Q8—21no—
412C2×Q8—42no—
413C2×Q8—81no—
414C2×Q8—81no—
415C2×Q8C3×#4112yes—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C24×C311yescharacteristic
436Q16—44no—
437SD16—44no—
438C2×C8—21no—
439C2×C8—21no—
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×C8C3×#4112yes—
449C2×C8C3×#4112yes—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4C3×#4112yes—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C42—24no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3#52511yescharacteristic
486C22×C3#53211yescharacteristic
487C22×C3#53511yescharacteristic
488C22×C3#57711yescharacteristic
489C22×C3#57711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4—81no—
505C3×C4#55111yescharacteristic
506C3×C4#55111yescharacteristic
507C3×C4#55511yescharacteristic
508C3×C4#55511yescharacteristic
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C22×C3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—84no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
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—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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×C4C2×C3×#4111yescharacteristic
570C2×C4C2×C3×#4111yescharacteristic
571C2×C4C2×C3×#4111yescharacteristic
572C2×C4C2×C3×#4111yescharacteristic
573C2×C4C3×H(5,𝔽2)11yescharacteristic
574C2×C4C3×H(5,𝔽2)11yescharacteristic
575C8—22no—
576C8—22no—
577C8—41no—
578C8—41no—
579C2×C3—161no—
580C2×C3#261011yescharacteristic
581C2×C3#312111yescharacteristic
582C2×C3#312511yescharacteristic
583C2×C3#317311yescharacteristic
584C2×C3#317811yescharacteristic
585C2×C3#322211yescharacteristic
586C2×C3#322511yescharacteristic
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C3×#52511yescharacteristic
595C22C3×#53211yescharacteristic
596C22C3×#53511yescharacteristic
597C22C3×#57711yescharacteristic
598C22C3×#57711yescharacteristic
599C22C3×#57711yescharacteristic
600C22C3×#57811yescharacteristic
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4C3×#55111yescharacteristic
615C4C3×#55111yescharacteristic
616C4C3×#55511yescharacteristic
617C4C3×#55511yescharacteristic
618C3#2957111yescharacteristic, direct product factor, Sylow
619C2—161no—
620C2C3×#261011yescharacteristic
621C2C3×#312111yescharacteristic
622C2C3×#312511yescharacteristic
623C2C3×#317311yescharacteristic
624C2C3×#317811yescharacteristic
625C2C3×#322211yescharacteristic
626C2C3×#322511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2320C2semidirect
3. C3×#2058C2semidirect
5. C3×#2024C2semidirect
7. C3×#1971C2semidirect
10. C3×#1933C2semidirect
11. C3×#1927C2semidirect
14. C3×#1878C2semidirect
15. C3×#1746C2semidirect
16. #29571C3direct (N × H)
61. #2320C2×C3semidirect
62. #2058C2×C3semidirect
64. #2024C2×C3semidirect
66. #1971C2×C3semidirect
69. #1933C2×C3semidirect
70. #1927C2×C3semidirect
73. #1878C2×C3semidirect
74. #1746C2×C3semidirect
618. C3#29571direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26402
C3×#23202
C3×#20582
C3×#20422
C3×#20242
C3×#20152
C3×#19712
C3×#19682
C3×#19362
C3×#19332
C3×#19272
C3×#19262
C3×#18802
C3×#18782
C3×#17462
#295713