Finite Group Databasebeta

← C3×#26735 #10645342

Subgroups — C3×#26735 #10645342

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2492C211yesmaximal, characteristic
2C3×#2362C211yesmaximal, characteristic
3C3×#2056C211yesmaximal, characteristic, complemented
4C3×#2035C211yesmaximal, characteristic
5C3×#2024C211yesmaximal, characteristic, complemented
6C3×#2001C211yesmaximal, characteristic
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1933C211yesmaximal, characteristic, complemented
9C3×#1927C211yesmaximal, characteristic, complemented
10C3×#1923C211yesmaximal, characteristic
11C3×#1919C211yesmaximal, characteristic, complemented
12C3×#1878C211yesmaximal, characteristic, complemented
13C3×#1873C211yesmaximal, characteristic
14C3×#1825C211yesmaximal, characteristic
15C3×C4×#103C211yesmaximal, characteristic, complemented
16#26735C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
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×C4×C8C2211yescharacteristic
34C3×#400—21no—
35C3×#393—21no—
36C3×#391—21no—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#388—21no—
42C3×#388—21no—
43C3×#388—21no—
44C3×#387—21no—
45C3×#380C2211yescharacteristic
46C3×C4×C4⋊C4—21no—
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
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#2362C2×C311yescharacteristic
62#2056C2×C311yescharacteristic, complemented
63#2035C2×C311yescharacteristic
64#2024C2×C311yescharacteristic, complemented
65#2001C2×C311yescharacteristic
66#1988C2×C311yescharacteristic, complemented
67#1933C2×C311yescharacteristic, complemented
68#1927C2×C311yescharacteristic, complemented
69#1923C2×C311yescharacteristic
70#1919C2×C311yescharacteristic, complemented
71#1878C2×C311yescharacteristic, complemented
72#1873C2×C311yescharacteristic
73#1825C2×C311yescharacteristic
74C4×#103C2×C311yescharacteristic, complemented
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C2×C3×Q16—22no—
80C2×C3×Q16—41no—
81C2×C3×Q16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—41no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—41no—
92C3×C4×Q8—22no—
93C3×C4×Q8—41no—
94C3×#117—22no—
95C3×#117—41no—
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⋊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×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#106—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×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×#95D411yescharacteristic
141C3×#95D411yescharacteristic
142C22×Q16C22×C311yescharacteristic
143C2×C4⋊Q8C22×C311yescharacteristic
144C2×C22⋊Q8C22×C311yescharacteristic
145C2×C22⋊Q8C22×C311yescharacteristic
146C2×C4×Q8C22×C311yescharacteristic
147C2×#117C22×C311yescharacteristic
148C2×#106C22×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×C4×C8C22×C311yescharacteristic
159#400—21no—
160#393—21no—
161#391—21no—
162#391C22×C311yescharacteristic
163#391C22×C311yescharacteristic
164#391C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#388—21no—
167#388—21no—
168#388—21no—
169#387—21no—
170#380C22×C311yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4C22×C311yescharacteristic
173C4×C4⋊C4C22×C311yescharacteristic
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×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C3×Q16—44no—
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—21no—
233C2×C3×C8—21no—
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—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4#4112yes—
249C3×C4⋊C4#4112yes—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—81no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—42no—
263C3×C42—42no—
264C3×C42—81no—
265C22×Q8—21no—
266C22×Q8C23×C311yescharacteristic
267C22×Q8C23×C311yescharacteristic
268C23×C4C23×C311yescharacteristic
269C2×Q16—22no—
270C2×Q16—41no—
271C2×Q16—41no—
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C4⋊Q8—22no—
277C4⋊Q8—41no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—41no—
282C4×Q8—22no—
283C4×Q8—41no—
284#117—22no—
285#117—41no—
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×C22⋊C4C3×D411yescharacteristic
302C2×C22⋊C4C3×D411yescharacteristic
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#106—22no—
310C4⋊C8—41no—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315#103—41no—
316#103—41no—
317C22⋊C8—41no—
318C22⋊C8—41no—
319C4×C8—41no—
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—
330#95C3×D411yescharacteristic
331#95C3×D411yescharacteristic
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—84no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
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—22no—
359C2×C3×C4—21no—
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—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#13512yes—
391C2×C3×C4#13712yes—
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C24C2×C3×D411yescharacteristic
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—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×Q8C3×#4112yes—
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×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438Q16—44no—
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×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C4⋊C4—24no—
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—21no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4C3×#4112yes—
466C4⋊C4C3×#4112yes—
467C4⋊C4C3×#4112yes—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—81no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3#52511yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54611yescharacteristic
492C22×C3C2×#13511yescharacteristic
493C22×C3C2×#13711yescharacteristic
494C22×C3#57711yescharacteristic
495C22×C3#57811yescharacteristic
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—41no—
504C3×C4—42no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4#52911yescharacteristic
510C3×C4#53811yescharacteristic
511C3×C4#55511yescharacteristic
512C3×C4#55911yescharacteristic
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23C22×C3×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—84no—
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—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
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×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4C3×#13512yes—
572C2×C4C3×#13712yes—
573C2×C4C2×C3×#4111yescharacteristic
574C2×C4C2×C3×#4111yescharacteristic
575C2×C4C2×C3×#4111yescharacteristic
576C2×C4C2×C3×#4111yescharacteristic
577C2×C4C3×H(5,𝔽2)11yescharacteristic
578C2×C4C3×H(5,𝔽2)11yescharacteristic
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C2×C3—81no—
584C2×C3#245911yescharacteristic
585C2×C3#310711yescharacteristic
586C2×C3#312111yescharacteristic
587C2×C3#329611yescharacteristic
588C2×C3#330211yescharacteristic
589C2×C3#331611yescharacteristic
590C2×C3#332811yescharacteristic
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C3×#52511yescharacteristic
599C22C3×#54611yescharacteristic
600C22C3×#54611yescharacteristic
601C22C2×C3×#13511yescharacteristic
602C22C2×C3×#13711yescharacteristic
603C22C3×#57711yescharacteristic
604C22C3×#57811yescharacteristic
605C4—22no—
606C4—22no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4C3×#52911yescharacteristic
619C4C3×#53811yescharacteristic
620C4C3×#55511yescharacteristic
621C4C3×#55911yescharacteristic
622C3#2673511yescharacteristic, direct product factor, Sylow
623C2—81no—
624C2C3×#245911yescharacteristic
625C2C3×#310711yescharacteristic
626C2C3×#312111yescharacteristic
627C2C3×#329611yescharacteristic
628C2C3×#330211yescharacteristic
629C2C3×#331611yescharacteristic
630C2C3×#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×#1933C2semidirect
9. C3×#1927C2semidirect
11. C3×#1919C2semidirect
12. C3×#1878C2semidirect
15. C3×C4×#103C2semidirect
16. #26735C3direct (N × H)
62. #2056C2×C3semidirect
64. #2024C2×C3semidirect
66. #1988C2×C3semidirect
67. #1933C2×C3semidirect
68. #1927C2×C3semidirect
70. #1919C2×C3semidirect
71. #1878C2×C3semidirect
74. C4×#103C2×C3semidirect
622. C3#26735direct (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×#23622
C3×#20562
C3×#20352
C3×#20242
C3×#20012
C3×#19882
C3×#19332
C3×#19272
C3×#19232
C3×#19192
C3×#18782
C3×#18732
C3×#18252
C3×C4×#1032
#267353