Finite Group Databasebeta

← C3×#24659 #10643266

Subgroups — C3×#24659 #10643266

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic, complemented
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×C4×#121C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1973C212yesmaximal, complemented
9C3×#1947C211yesmaximal, characteristic, complemented
10C3×#1935C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24659C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#461—24no—
21C3×#432—24no—
22C2×C3×#107C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×#97C2212yes—
31C3×#391C2212yes—
32C3×#390C2211yescharacteristic
33C3×#389—21no—
34C3×#389—21no—
35C3×#389C2211yescharacteristic
36C3×#388—21no—
37C3×#386—21no—
38C3×#384C2211yescharacteristic
39C3×#380—21no—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#461C2×C311yescharacteristic, complemented
47C2×#432C2×C311yescharacteristic, complemented
48#2857C2×C311yescharacteristic, complemented
49C4×#121C2×C311yescharacteristic
50#2036C2×C312yescomplemented
51#2000C2×C312yes—
52#1981C2×C311yescharacteristic, complemented
53#1973C2×C312yescomplemented
54#1947C2×C311yescharacteristic, complemented
55#1935C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C22×C3×Q8C2312yes—
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C22⋊Q8—42no—
72C3×C22⋊Q8—42no—
73C3×#121—22no—
74C3×#121—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—44no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—21no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C3×#107—42no—
102C3×C4⋊C8—22no—
103C3×C4⋊C8D414yescomplemented
104C3×#103—24no—
105C3×#103—24no—
106C3×#103—42no—
107C3×#102—42no—
108C3×C22⋊C8—24no—
109C3×#97—24no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—22no—
114C3×#95—22no—
115C22×SD16C22×C312yes—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×C22⋊Q8C22×C312yes—
118C2×#121C22×C311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4C22×C311yescharacteristic
121C22×C42C22×C311yescharacteristic
122#461—24no—
123#432—24no—
124C2×#107C22×C312yes—
125C2×C4⋊C8C22×C311yescharacteristic
126C2×C4⋊C8C22×C311yescharacteristic
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×#103C22×C312yes—
130C2×#102C22×C312yes—
131C2×C22⋊C8C22×C312yes—
132C2×#97C22×C312yes—
133#391C22×C312yes—
134#390C22×C311yescharacteristic
135#389—21no—
136#389—21no—
137#389C22×C311yescharacteristic
138#388—21no—
139#386—21no—
140#384C22×C311yescharacteristic
141#380—21no—
142#380—21no—
143#380C22×C311yescharacteristic
144C4×C4⋊C4C22×C311yescharacteristic
145C4×C22⋊C4—21no—
146#340C22×C312yes—
147#340C22×C312yes—
148C24×C3—21no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×D4—24no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4#4112yes—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—41no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C3×SD16—48no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8C2×D414yes—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—44no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4#4112yes—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—42no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—42no—
223C3×C42C2×D412yes—
224C3×C42#4112yes—
225C22×Q8C23×C312yes—
226C22×D4—21no—
227C22×D4C23×C311yescharacteristic
228C23×C4—21no—
229C23×C4C3×D411yescharacteristic
230C23×C4C3×D411yescharacteristic
231C23×C4C23×C311yescharacteristic
232C2×SD16—24no—
233C2×SD16—24no—
234C2×SD16—42no—
235C22×C8C23×C312yes—
236C22×C8C23×C312yes—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241#121—22no—
242#121—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—44no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42C3×D411yescharacteristic
266C2×C42C3×D411yescharacteristic
267C2×C42C23×C311yescharacteristic
268C2×C42C23×C311yescharacteristic
269#107—42no—
270C4⋊C8—22no—
271C4⋊C8C3×D414yescomplemented
272#103—24no—
273#103—24no—
274#103—42no—
275#102—42no—
276C22⋊C8—24no—
277#97—24no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C3×Q8—42no—
299C3×Q8—42no—
300C3×Q8—42no—
301C3×Q8—42no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—48no—
305C3×D4—41no—
306C3×D4—41no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—44no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—44no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—84no—
340C2×C3×C4#12112yes—
341C2×C3×C4#12112yes—
342C2×C3×C4AGL(1,ℤ/8)12yes—
343C2×C3×C4#13712yes—
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C3×C8—24no—
351C3×C8—42no—
352C24—21no—
353C24C2×C3×D411yescharacteristic
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×D4—24no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4C3×#4112yes—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C24×C311yescharacteristic
395SD16—48no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8C2×C3×D414yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C3×#4112yes—
416C22⋊C4—24no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C42—22no—
423C42—22no—
424C42—21no—
425C42—21no—
426C42—42no—
427C42C2×C3×D412yes—
428C42C3×#4112yes—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3C2×#12111yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3#53511yescharacteristic
446C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
447C22×C3C2×#13711yescharacteristic
448C22×C3C2×#13711yescharacteristic
449C22×C3#57511yescharacteristic
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—82no—
456C3×C4—82no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4#46814yes—
460C3×C4#54611yescharacteristic
461C3×C4#55011yescharacteristic
462C3×C4#55311yescharacteristic
463C3×C4#55511yescharacteristic
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C3×#13712yes—
478C23C22×C3×D411yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483D4—22no—
484D4—22no—
485D4—48no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4C3×#12112yes—
522C2×C4C3×#12112yes—
523C2×C4C3×AGL(1,ℤ/8)12yes—
524C2×C4C3×#13712yes—
525C2×C4C22×C3×D411yescharacteristic
526C2×C4C2×C3×#4111yescharacteristic
527C2×C4C2×C3×#4111yescharacteristic
528C2×C4C2×C3×#4111yescharacteristic
529C2×C4C3×H(5,𝔽2)11yescharacteristic
530C2×C4C3×H(5,𝔽2)11yescharacteristic
531C8—24no—
532C8—42no—
533C2×C3—42no—
534C2×C3—82no—
535C2×C3#241411yescharacteristic
536C2×C3C2×#46811yescharacteristic
537C2×C3#305611yescharacteristic
538C2×C3#311011yescharacteristic
539C2×C3#311611yescharacteristic
540C2×C3#321411yescharacteristic
541C2×C3#321811yescharacteristic
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C2×C3×#12111yescharacteristic
557C22C3×#52511yescharacteristic
558C22C3×#53511yescharacteristic
559C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
560C22C2×C3×#13711yescharacteristic
561C22C2×C3×#13711yescharacteristic
562C22C3×#57511yescharacteristic
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4—82no—
571C4—81no—
572C4C3×#46814yes—
573C4C3×#54611yescharacteristic
574C4C3×#55011yescharacteristic
575C4C3×#55311yescharacteristic
576C4C3×#55511yescharacteristic
577C3#2465911yescharacteristic, direct product factor, Sylow
578C2—42no—
579C2—82no—
580C2C3×#241411yescharacteristic
581C2C2×C3×#46811yescharacteristic
582C2C3×#305611yescharacteristic
583C2C3×#311011yescharacteristic
584C2C3×#311611yescharacteristic
585C2C3×#321411yescharacteristic
586C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#461C2semidirect
2. C2×C3×#432C2semidirect
3. C3×#2857C2semidirect
5. C3×#2036C2semidirect
7. C3×#1981C2semidirect
8. C3×#1973C2semidirect
9. C3×#1947C2semidirect
10. C3×#1935C2semidirect
11. C3×#1802C2semidirect
12. #24659C3direct (N × H)
46. C2×#461C2×C3semidirect
47. C2×#432C2×C3semidirect
48. #2857C2×C3semidirect
50. #2036C2×C3semidirect
52. #1981C2×C3semidirect
53. #1973C2×C3semidirect
54. #1947C2×C3semidirect
55. #1935C2×C3semidirect
56. #1802C2×C3semidirect
103. C3×C4⋊C8D4semidirect
271. C4⋊C8C3×D4semidirect
577. C3#24659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×C3×#4322
C3×#28572
C3×C4×#1212
C3×#20362
C3×#20002
C3×#19812
C3×#19732
C3×#19472
C3×#19352
C3×#18022
#246593