Finite Group Databasebeta

← C3×#26743 #10645350

Subgroups — C3×#26743 #10645350

Order 768.

758 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×#2492C211yesmaximal, characteristic
2C3×#2360C211yesmaximal, characteristic
3C3×#2056C211yesmaximal, characteristic, complemented
4C3×#2036C211yesmaximal, characteristic
5C3×#2024C211yesmaximal, characteristic, complemented
6C3×#2001C211yesmaximal, characteristic
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1935C211yesmaximal, characteristic, complemented
9C3×#1928C211yesmaximal, characteristic, complemented
10C3×#1924C211yesmaximal, characteristic
11C3×#1918C211yesmaximal, characteristic, complemented
12C3×#1878C211yesmaximal, characteristic, complemented
13C3×#1872C211yesmaximal, characteristic
14C3×#1823C211yesmaximal, characteristic
15C3×#1763C211yesmaximal, characteristic, complemented
16#26743C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
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
34C3×#400—21no—
35C3×#393—21no—
36C3×#391—21no—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#389—21no—
41C3×#389C2211yescharacteristic
42C3×#388—21no—
43C3×#387—21no—
44C3×#385—21no—
45C3×#380C2211yescharacteristic
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C22⋊C4—21no—
49C3×C4×C22⋊C4C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2492C2×C311yescharacteristic
61#2360C2×C311yescharacteristic
62#2056C2×C311yescharacteristic, complemented
63#2036C2×C311yescharacteristic
64#2024C2×C311yescharacteristic, complemented
65#2001C2×C311yescharacteristic
66#1988C2×C311yescharacteristic, complemented
67#1935C2×C311yescharacteristic, complemented
68#1928C2×C311yescharacteristic, complemented
69#1924C2×C311yescharacteristic
70#1918C2×C311yescharacteristic, complemented
71#1878C2×C311yescharacteristic, complemented
72#1872C2×C311yescharacteristic
73#1823C2×C311yescharacteristic
74#1763C2×C311yescharacteristic, complemented
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—41no—
82C2×C3×Q16—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8—41no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—41no—
93C3×C4×Q8—22no—
94C3×C4×Q8—41no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
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⋊C4—41no—
103C2×C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#107—22no—
120C3×C4⋊C8—41no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—41no—
126C3×#103—41no—
127C3×C22⋊C8—41no—
128C3×C22⋊C8—41no—
129C3×#97—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×#95D411yescharacteristic
141C3×#95D411yescharacteristic
142C22×Q16C22×C311yescharacteristic
143C2×C4⋊Q8C22×C311yescharacteristic
144C2×C22⋊Q8C22×C311yescharacteristic
145C2×C22⋊Q8C22×C311yescharacteristic
146C2×C4×Q8C22×C311yescharacteristic
147C22×C4⋊C4C22×C311yescharacteristic
148C2×#107C22×C311yescharacteristic
149C2×C4⋊C8C22×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
159#400—21no—
160#393—21no—
161#391—21no—
162#391C22×C311yescharacteristic
163#391C22×C311yescharacteristic
164#391C22×C311yescharacteristic
165#389—21no—
166#389C22×C311yescharacteristic
167#388—21no—
168#387—21no—
169#385—21no—
170#380C22×C311yescharacteristic
171C4×C4⋊C4C22×C311yescharacteristic
172C4×C4⋊C4C22×C311yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×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—42no—
198C2×C3×Q8—42no—
199C2×C3×Q8—81no—
200C2×C3×Q8#4112yes—
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—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4—81no—
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—21no—
236C2×C3×C8—21no—
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—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4#4112yes—
253C3×C4⋊C4#4112yes—
254C3×C4⋊C4#4112yes—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—42no—
259C3×C22⋊C4—81no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—42no—
265C3×C42—81no—
266C3×C42—81no—
267C22×Q8—21no—
268C22×Q8C23×C311yescharacteristic
269C22×Q8C23×C311yescharacteristic
270C23×C4—21no—
271C23×C4C23×C311yescharacteristic
272C2×Q16—22no—
273C2×Q16—41no—
274C2×Q16—41no—
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C311yescharacteristic
277C22×C8C23×C311yescharacteristic
278C22×C8C23×C311yescharacteristic
279C4⋊Q8—22no—
280C4⋊Q8—41no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—41no—
285C4×Q8—22no—
286C4×Q8—41no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4C3×D411yescharacteristic
305C2×C22⋊C4C3×D411yescharacteristic
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C23×C311yescharacteristic
310C2×C42C23×C311yescharacteristic
311#107—22no—
312C4⋊C8—41no—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#103—41no—
318#103—41no—
319C22⋊C8—41no—
320C22⋊C8—41no—
321#97—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#95C3×D411yescharacteristic
333#95C3×D411yescharacteristic
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3#13712yes—
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—
350C3×Q8—84no—
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—22no—
360C2×C3×C4—22no—
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—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×C4—41no—
393C2×C3×C4—84no—
394C2×C3×C4#13712yes—
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—41no—
402C3×C8—41no—
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×Q8—42no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×Q8C3×#4112yes—
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—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
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×C8—21no—
456C2×C8—21no—
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—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4C3×#4112yes—
473C4⋊C4C3×#4112yes—
474C4⋊C4C3×#4112yes—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—81no—
486C42—81no—
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#54611yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3C2×#13711yescharacteristic
498C22×C3C2×#13711yescharacteristic
499C22×C3#57611yescharacteristic
500C22×C3#57811yescharacteristic
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#53811yescharacteristic
516C3×C4#55511yescharacteristic
517C3×C4#55911yescharacteristic
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C3×#13712yes—
525C23C22×C3×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—84no—
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—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
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×C4—41no—
577C2×C4—84no—
578C2×C4C3×#13712yes—
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—41no—
586C8—41no—
587C8—41no—
588C8—41no—
589C2×C3—42no—
590C2×C3#245911yescharacteristic
591C2×C3#311311yescharacteristic
592C2×C3#312411yescharacteristic
593C2×C3#331211yescharacteristic
594C2×C3#331411yescharacteristic
595C2×C3#331411yescharacteristic
596C2×C3#332811yescharacteristic
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22C3×#52511yescharacteristic
605C22C3×#54611yescharacteristic
606C22C3×#54611yescharacteristic
607C22C2×C3×#13711yescharacteristic
608C22C2×C3×#13711yescharacteristic
609C22C3×#57611yescharacteristic
610C22C3×#57811yescharacteristic
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C3×#52911yescharacteristic
625C4C3×#53811yescharacteristic
626C4C3×#55511yescharacteristic
627C4C3×#55911yescharacteristic
628C3#2674311yescharacteristic, direct product factor, Sylow
629C2—42no—
630C2C3×#245911yescharacteristic
631C2C3×#311311yescharacteristic
632C2C3×#312411yescharacteristic
633C2C3×#331211yescharacteristic
634C2C3×#331411yescharacteristic
635C2C3×#331411yescharacteristic
636C2C3×#332811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2056C2semidirect
5. C3×#2024C2semidirect
7. C3×#1988C2semidirect
8. C3×#1935C2semidirect
9. C3×#1928C2semidirect
11. C3×#1918C2semidirect
12. C3×#1878C2semidirect
15. C3×#1763C2semidirect
16. #26743C3direct (N × H)
62. #2056C2×C3semidirect
64. #2024C2×C3semidirect
66. #1988C2×C3semidirect
67. #1935C2×C3semidirect
68. #1928C2×C3semidirect
70. #1918C2×C3semidirect
71. #1878C2×C3semidirect
74. #1763C2×C3semidirect
628. C3#26743direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24922
C3×#23602
C3×#20562
C3×#20362
C3×#20242
C3×#20012
C3×#19882
C3×#19352
C3×#19282
C3×#19242
C3×#19182
C3×#18782
C3×#18722
C3×#18232
C3×#17632
#267433