Finite Group Databasebeta

← C3×#26792 #10645399

Subgroups — C3×#26792 #10645399

Order 768.

782 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
1C2×C3×#485C211yesmaximal, characteristic
2C2×C3×#479C211yesmaximal, characteristic, complemented
3C3×#2489C211yesmaximal, characteristic
4C3×#2362C211yesmaximal, characteristic
5C3×#2068C211yesmaximal, characteristic, complemented
6C3×#2059C211yesmaximal, characteristic, complemented
7C3×#2036C211yesmaximal, characteristic
8C3×#2001C211yesmaximal, characteristic
9C3×#1969C211yesmaximal, characteristic, complemented
10C3×#1927C211yesmaximal, characteristic, complemented
11C3×#1927C211yesmaximal, characteristic, complemented
12C3×#1923C211yesmaximal, characteristic
13C3×#1878C211yesmaximal, characteristic, complemented
14C3×#1854C211yesmaximal, characteristic
15C3×C4×#107C211yesmaximal, characteristic, complemented
16#26792C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C3×#485—24no—
24C3×#479—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C2×C3×C22⋊C8C2211yescharacteristic
39C2×C3×C4×C8C2211yescharacteristic
40C3×#400—21no—
41C3×#397—21no—
42C3×#391C2211yescharacteristic
43C3×#391C2211yescharacteristic
44C3×#391C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#388—21no—
47C3×#388—21no—
48C3×#388—21no—
49C3×#387—21no—
50C3×#382—21no—
51C3×#380C2211yescharacteristic
52C3×C4×C4⋊C4—21no—
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×C4×C4⋊C4C2211yescharacteristic
55C3×C4×C22⋊C4C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#485C2×C311yescharacteristic
63C2×#479C2×C311yescharacteristic, complemented
64#2489C2×C311yescharacteristic
65#2362C2×C311yescharacteristic
66#2068C2×C311yescharacteristic, complemented
67#2059C2×C311yescharacteristic, complemented
68#2036C2×C311yescharacteristic
69#2001C2×C311yescharacteristic
70#1969C2×C311yescharacteristic, complemented
71#1927C2×C311yescharacteristic, complemented
72#1927C2×C311yescharacteristic, complemented
73#1923C2×C311yescharacteristic
74#1878C2×C311yescharacteristic, complemented
75#1854C2×C311yescharacteristic
76C4×#107C2×C311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
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×#125—22no—
88C3×#125—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—
95C3×#117—22no—
96C3×#117—41no—
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—21no—
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⋊C4—21no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#107—41no—
123C3×#107D414yes—
124C3×#106—22no—
125C3×#106—22no—
126C3×C4⋊C8—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#103—22no—
133C3×C22⋊C8—22no—
134C3×C22⋊C8—22no—
135C3×C4×C8—22no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—21no—
145C3×#95—21no—
146C3×#95D411yescharacteristic
147C3×#95D411yescharacteristic
148C22×Q16C22×C311yescharacteristic
149C2×#125C22×C311yescharacteristic
150C2×C22⋊Q8C22×C311yescharacteristic
151C2×C22⋊Q8C22×C311yescharacteristic
152C2×C4×Q8C22×C311yescharacteristic
153C2×#117C22×C311yescharacteristic
154#485—24no—
155#479—24no—
156C2×#107C22×C311yescharacteristic
157C2×#107C22×C311yescharacteristic
158C2×#107C22×C311yescharacteristic
159C2×#106C22×C311yescharacteristic
160C2×#106C22×C311yescharacteristic
161C2×C4⋊C8C22×C311yescharacteristic
162C2×#103C22×C311yescharacteristic
163C2×#103C22×C311yescharacteristic
164C2×#103C22×C311yescharacteristic
165C2×#103C22×C311yescharacteristic
166C2×#103C22×C311yescharacteristic
167C2×#103C22×C311yescharacteristic
168C2×C22⋊C8C22×C311yescharacteristic
169C2×C22⋊C8C22×C311yescharacteristic
170C2×C4×C8C22×C311yescharacteristic
171#400—21no—
172#397—21no—
173#391C22×C311yescharacteristic
174#391C22×C311yescharacteristic
175#391C22×C311yescharacteristic
176#389C22×C311yescharacteristic
177#388—21no—
178#388—21no—
179#388—21no—
180#387—21no—
181#382—21no—
182#380C22×C311yescharacteristic
183C4×C4⋊C4—21no—
184C4×C4⋊C4C22×C311yescharacteristic
185C4×C4⋊C4C22×C311yescharacteristic
186C4×C22⋊C4C22×C311yescharacteristic
187#340C22×C311yescharacteristic
188#340C22×C311yescharacteristic
189#340C22×C311yescharacteristic
190#340C22×C311yescharacteristic
191#336C22×C311yescharacteristic
192#336C22×C311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8#4112yes—
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—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C3×Q16—44no—
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—
237C2×C3×C8C2×D412yes—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—24no—
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⋊C4C2×D412yes—
255C3×C4⋊C4C2×D412yes—
256C3×C4⋊C4#4112yes—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—41no—
260C3×C22⋊C4—42no—
261C3×C22⋊C4—42no—
262C3×C22⋊C4—42no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—42no—
268C3×C42—42no—
269C3×C42—42no—
270C3×C42—42no—
271C22×Q8C23×C311yescharacteristic
272C22×Q8C23×C311yescharacteristic
273C23×C4C23×C311yescharacteristic
274C2×Q16—22no—
275C2×Q16—22no—
276C2×Q16—41no—
277C22×C8C23×C311yescharacteristic
278C22×C8C23×C311yescharacteristic
279C22×C8C23×C311yescharacteristic
280C22×C8C23×C311yescharacteristic
281#125—22no—
282#125—41no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—41no—
287C4×Q8—22no—
288C4×Q8—41no—
289#117—22no—
290#117—41no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C4⋊C4C23×C311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4C3×D411yescharacteristic
308C2×C22⋊C4C3×D411yescharacteristic
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C23×C311yescharacteristic
314C2×C42C23×C311yescharacteristic
315#107—22no—
316#107—41no—
317#107C3×D414yes—
318#106—22no—
319#106—22no—
320C4⋊C8—22no—
321#103—22no—
322#103—22no—
323#103—22no—
324#103—22no—
325#103—22no—
326#103—22no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C4×C8—22no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95C3×D411yescharacteristic
341#95C3×D411yescharacteristic
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
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—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4#13512yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C3×C8—22no—
408C3×C8—22no—
409C3×C8—41no—
410C3×C8—41no—
411C24C2×C3×D411yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C3×#4112yes—
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—21no—
436C22×C4—21no—
437C22×C4—21no—
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×C8C2×C3×D412yes—
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—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4C2×C3×D412yes—
473C4⋊C4C2×C3×D412yes—
474C4⋊C4C3×#4112yes—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3#52511yescharacteristic
497C22×C3D4211yescharacteristic
498C22×C3#54711yescharacteristic
499C22×C3C2×#13511yescharacteristic
500C22×C3#57511yescharacteristic
501C22×C3#57811yescharacteristic
502C22×C3#57811yescharacteristic
503C3×C4—22no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4#53211yescharacteristic
517C3×C4#54211yescharacteristic
518C3×C4#54811yescharacteristic
519C3×C4#55111yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23C22×C3×D411yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
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—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
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—41no—
578C2×C4C3×#13512yes—
579C2×C4C22×C3×D411yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C2×C3×#4111yescharacteristic
582C2×C4C2×C3×#4111yescharacteristic
583C2×C4C3×H(5,𝔽2)11yescharacteristic
584C2×C4C3×2−1+411yescharacteristic
585C8—22no—
586C8—22no—
587C8—41no—
588C8—41no—
589C2×C3—81no—
590C2×C3#246111yescharacteristic
591C2×C3#310211yescharacteristic
592C2×C3#312711yescharacteristic
593C2×C3#327711yescharacteristic
594C2×C3#328311yescharacteristic
595C2×C3#333611yescharacteristic
596C2×C3#334911yescharacteristic
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22C3×#52511yescharacteristic
605C22C3×D4211yescharacteristic
606C22C3×#54711yescharacteristic
607C22C2×C3×#13511yescharacteristic
608C22C3×#57511yescharacteristic
609C22C3×#57811yescharacteristic
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×#53211yescharacteristic
625C4C3×#54211yescharacteristic
626C4C3×#54811yescharacteristic
627C4C3×#55111yescharacteristic
628C3#2679211yescharacteristic, direct product factor, Sylow
629C2—81no—
630C2C3×#246111yescharacteristic
631C2C3×#310211yescharacteristic
632C2C3×#312711yescharacteristic
633C2C3×#327711yescharacteristic
634C2C3×#328311yescharacteristic
635C2C3×#333611yescharacteristic
636C2C3×#334911yescharacteristic
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
2. C2×C3×#479C2semidirect
5. C3×#2068C2semidirect
6. C3×#2059C2semidirect
9. C3×#1969C2semidirect
10. C3×#1927C2semidirect
11. C3×#1927C2semidirect
13. C3×#1878C2semidirect
15. C3×C4×#107C2semidirect
16. #26792C3direct (N × H)
63. C2×#479C2×C3semidirect
66. #2068C2×C3semidirect
67. #2059C2×C3semidirect
70. #1969C2×C3semidirect
71. #1927C2×C3semidirect
72. #1927C2×C3semidirect
74. #1878C2×C3semidirect
76. C4×#107C2×C3semidirect
628. C3#26792direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4852
C2×C3×#4792
C3×#24892
C3×#23622
C3×#20682
C3×#20592
C3×#20362
C3×#20012
C3×#19692
C3×#19272
C3×#19232
C3×#18782
C3×#18542
C3×C4×#1072
#267923