Finite Group Databasebeta

← C3×#29261 #10647868

Subgroups — C3×#29261 #10647868

Order 768.

748 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#125C211yesmaximal, characteristic, complemented
2C3×C4×#124C211yesmaximal, characteristic
3C3×#2097C211yesmaximal, characteristic, complemented
4C3×#2092C211yesmaximal, characteristic
5C3×#2026C211yesmaximal, characteristic, complemented
6C3×#2025C211yesmaximal, characteristic
7C3×#1983C211yesmaximal, characteristic
8C3×#1983C211yesmaximal, characteristic
9C3×#1949C211yesmaximal, characteristic, complemented
10C3×#1949C211yesmaximal, characteristic, complemented
11C3×#1865C211yesmaximal, characteristic, complemented
12C3×#1864C211yesmaximal, characteristic
13C3×#1863C211yesmaximal, characteristic, complemented
14C3×#1862C211yesmaximal, characteristic
15C3×#1831C211yesmaximal, characteristic, complemented
16#29261C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#387—21no—
42C3×#387—21no—
43C3×#382C2211yescharacteristic
44C3×#382C2211yescharacteristic
45C3×C4×C4⋊C4—21no—
46C3×C4×C4⋊C4—21no—
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C22⋊C4—21no—
50C3×C4×C22⋊C4—21no—
51C3×C43C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C4×#125C2×C311yescharacteristic, complemented
61C4×#124C2×C311yescharacteristic
62#2097C2×C311yescharacteristic, complemented
63#2092C2×C311yescharacteristic
64#2026C2×C311yescharacteristic, complemented
65#2025C2×C311yescharacteristic
66#1983C2×C311yescharacteristic
67#1983C2×C311yescharacteristic
68#1949C2×C311yescharacteristic, complemented
69#1949C2×C311yescharacteristic, complemented
70#1865C2×C311yescharacteristic, complemented
71#1864C2×C311yescharacteristic
72#1863C2×C311yescharacteristic, complemented
73#1862C2×C311yescharacteristic
74#1831C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—22no—
83C3×#125—22no—
84C3×#124—22no—
85C3×#124Q814yes—
86C3×C4×Q8—41no—
87C3×C4×Q8—41no—
88C3×C4×D4—41no—
89C3×C4×D4—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42D411yescharacteristic
112C2×C3×C42D411yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#107—41no—
117C3×#107—41no—
118C3×#106—41no—
119C3×#106—41no—
120C3×C4⋊C8—22no—
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—41no—
125C3×#103—41no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#102—41no—
129C3×#102—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—
140C2×#125C22×C311yescharacteristic
141C2×#124C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158#391C22×C311yescharacteristic
159#390C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#388—21no—
163#388—21no—
164#387—21no—
165#387—21no—
166#382C22×C311yescharacteristic
167#382C22×C311yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C22×C311yescharacteristic
171C4×C4⋊C4C22×C311yescharacteristic
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C43C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×Q812yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4#4112yes—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—41no—
253C3×C42—41no—
254C3×C42—41no—
255C3×C42—42no—
256C3×C42—42no—
257C3×C42—42no—
258C3×C42C2×Q812yes—
259C3×C42#4112yes—
260C22×Q8C23×C311yescharacteristic
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267#125—22no—
268#125—22no—
269#124—22no—
270#124C3×Q814yes—
271C4×Q8—41no—
272C4×Q8—41no—
273C4×D4—41no—
274C4×D4—41no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C3×D411yescharacteristic
295C2×C42C3×D411yescharacteristic
296C2×C42C3×D411yescharacteristic
297C2×C42C3×D411yescharacteristic
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#107—41no—
302#107—41no—
303#106—41no—
304#106—41no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307#103—22no—
308#103—22no—
309#103—41no—
310#103—41no—
311#102—22no—
312#102—22no—
313#102—41no—
314#102—41no—
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—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
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—84no—
378C2×C3×C4C22⋊Q812yes—
379C2×C3×C4C22⋊Q812yes—
380C2×C3×C4#13512yes—
381C2×C3×C4C22×Q811yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C24—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C2×C3×Q812yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4C2×C3×Q812yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C24×C311yescharacteristic
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4C3×#4112yes—
449C4⋊C4C3×#4112yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C42—22no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42C2×C3×Q812yes—
467C42C3×#4112yes—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3C2×C22⋊Q811yescharacteristic
476C22×C3#52511yescharacteristic
477C22×C3#53511yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3#57511yescharacteristic
480C22×C3#57511yescharacteristic
481C22×C3#57511yescharacteristic
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4#55411yescharacteristic
496C3×C4#55411yescharacteristic
497C3×C4#55911yescharacteristic
498C3×C4#55911yescharacteristic
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4C3×C22⋊Q812yes—
553C2×C4C3×C22⋊Q812yes—
554C2×C4C3×#13512yes—
555C2×C4C22×C3×Q811yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C2×C3×#4111yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C3×H(5,𝔽2)11yescharacteristic
560C2×C4C3×H(5,𝔽2)11yescharacteristic
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C2×C3—82no—
566C2×C3#258011yescharacteristic
567C2×C3#307611yescharacteristic
568C2×C3#308011yescharacteristic
569C2×C3#310211yescharacteristic
570C2×C3#311111yescharacteristic
571C2×C3#321111yescharacteristic
572C2×C3#321911yescharacteristic
573C22—42no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C2×C3×C22⋊Q811yescharacteristic
581C22C3×#52511yescharacteristic
582C22C3×#53511yescharacteristic
583C22C2×C3×#13511yescharacteristic
584C22C3×#57511yescharacteristic
585C22C3×#57511yescharacteristic
586C22C3×#57511yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4C3×#55411yescharacteristic
601C4C3×#55411yescharacteristic
602C4C3×#55911yescharacteristic
603C4C3×#55911yescharacteristic
604C3#2926111yescharacteristic, direct product factor, Sylow
605C2—82no—
606C2C3×#258011yescharacteristic
607C2C3×#307611yescharacteristic
608C2C3×#308011yescharacteristic
609C2C3×#310211yescharacteristic
610C2C3×#311111yescharacteristic
611C2C3×#321111yescharacteristic
612C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C4×#125C2semidirect
3. C3×#2097C2semidirect
5. C3×#2026C2semidirect
9. C3×#1949C2semidirect
10. C3×#1949C2semidirect
11. C3×#1865C2semidirect
13. C3×#1863C2semidirect
15. C3×#1831C2semidirect
16. #29261C3direct (N × H)
60. C4×#125C2×C3semidirect
62. #2097C2×C3semidirect
64. #2026C2×C3semidirect
68. #1949C2×C3semidirect
69. #1949C2×C3semidirect
70. #1865C2×C3semidirect
72. #1863C2×C3semidirect
74. #1831C2×C3semidirect
604. C3#29261direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1252
C3×C4×#1242
C3×#20972
C3×#20922
C3×#20262
C3×#20252
C3×#19832
C3×#19492
C3×#18652
C3×#18642
C3×#18632
C3×#18622
C3×#18312
#292613