Finite Group Databasebeta

← C3×#29814 #10648421

Subgroups — C3×#29814 #10648421

Order 768.

1060 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#496C211yesmaximal, characteristic, complemented
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×#2856C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2054C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1961C211yesmaximal, characteristic, complemented
10C3×#1959C211yesmaximal, characteristic, complemented
11C3×#1836C211yesmaximal, characteristic, complemented
12#29814C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#496—24no—
24C3×#432—24no—
25C2×C3×C4⋊C8C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C2×C3×#97C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#390—21no—
35C3×#390—21no—
36C3×#390C2211yescharacteristic
37C3×#390C2212yes—
38C3×#388—21no—
39C3×#388—21no—
40C3×#386—21no—
41C3×#386—21no—
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#496C2×C311yescharacteristic, complemented
47C2×#432C2×C311yescharacteristic, complemented
48#2856C2×C311yescharacteristic, complemented
49#2850C2×C311yescharacteristic, complemented
50#2055C2×C312yescomplemented
51#2054C2×C312yescomplemented
52#2000C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1961C2×C311yescharacteristic, complemented
55#1959C2×C311yescharacteristic, complemented
56#1836C2×C311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2312yes—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—42no—
69C2×C3×SD16—42no—
70C2×C3×D8—24no—
71C2×C3×D8—42no—
72C2×C3×D8—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×#127—22no—
76C3×#127—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×C22⋊Q8—42no—
80C3×#121—42no—
81C3×#121—42no—
82C3×#121—42no—
83C3×#121—44no—
84C3×#121—44no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C42—22no—
101C2×C3×C42—22no—
102C2×C3×C42D411yescharacteristic
103C2×C3×C42D411yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C3×C4⋊C8—24no—
107C3×#103—42no—
108C3×#102—42no—
109C3×#102—42no—
110C3×#102—42no—
111C3×C22⋊C8—24no—
112C3×#97—22no—
113C3×#97D414yescomplemented
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C22×SD16C22×C312yes—
119C22×D8C22×C312yes—
120C2×#127C22×C311yescharacteristic
121C2×#124C22×C311yescharacteristic
122C2×C22⋊Q8C22×C311yescharacteristic
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C22×C311yescharacteristic
126C2×#121C22×C312yes—
127C22×C42C22×C311yescharacteristic
128#496—24no—
129#432—24no—
130C2×C4⋊C8C22×C312yes—
131C2×#103C22×C312yes—
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×#97C22×C311yescharacteristic
137C2×#97C22×C311yescharacteristic
138#391C22×C311yescharacteristic
139#390—21no—
140#390—21no—
141#390C22×C311yescharacteristic
142#390C22×C312yes—
143#388—21no—
144#388—21no—
145#386—21no—
146#386—21no—
147#384C22×C311yescharacteristic
148#382C22×C311yescharacteristic
149#340C22×C312yes—
150#340C22×C312yes—
151C24×C3—21no—
152C24×C3—22no—
153C24×C3C2×D411yescharacteristic
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—42no—
176C2×C3×D4—44no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—41no—
193C22×C3×C4—41no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×SD16—48no—
201C3×D8—48no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C2×D414yes—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—44no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42C2×D412yes—
229C3×C42#4112yes—
230C22×Q8C23×C311yescharacteristic
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C23×C311yescharacteristic
236C22×D4C23×C312yes—
237C23×C4C3×D411yescharacteristic
238C23×C4C3×D411yescharacteristic
239C23×C4C23×C311yescharacteristic
240C2×SD16—24no—
241C2×SD16—42no—
242C2×SD16—42no—
243C2×D8—24no—
244C2×D8—42no—
245C2×D8—42no—
246C22×C8C23×C312yes—
247C22×C8C23×C312yes—
248#127—22no—
249#127—22no—
250#124—22no—
251#124—22no—
252C22⋊Q8—42no—
253#121—42no—
254#121—42no—
255#121—42no—
256#121—44no—
257#121—44no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C312yes—
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42C3×D411yescharacteristic
276C2×C42C3×D411yescharacteristic
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C311yescharacteristic
279C4⋊C8—24no—
280#103—42no—
281#102—42no—
282#102—42no—
283#102—42no—
284C22⋊C8—24no—
285#97—22no—
286#97C3×D414yescomplemented
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3AGL(1,ℤ/8)12yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—42no—
314C3×Q8—42no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—84no—
322C3×D4—84no—
323C3×D4—84no—
324C3×D4—88no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4#12112yes—
354C2×C3×C4#12112yes—
355C2×C3×C4AGL(1,ℤ/8)12yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)12yes—
359C2×C3×C4H(5,𝔽2)12yes—
360C3×C8—24no—
361C3×C8—42no—
362C24—21no—
363C24—22no—
364C24C2×C3×D411yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C24×C311yescharacteristic
411SD16—48no—
412D8—48no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8C2×C3×D414yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42C2×C3×D412yes—
440C42C3×#4112yes—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—81no—
449C22×C3—82no—
450C22×C3—81no—
451C22×C3—82no—
452C22×C3—81no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—81no—
456C22×C3—82no—
457C22×C3—81no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3C2×#12111yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3#53511yescharacteristic
465C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
467C22×C3#57511yescharacteristic
468C22×C3#57511yescharacteristic
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—82no—
478C3×C4#53512yes—
479C3×C4#54012yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23C3×AGL(1,ℤ/8)12yes—
501C23C22×C3×D411yescharacteristic
502Q8—42no—
503Q8—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—84no—
511D4—84no—
512D4—84no—
513D4—88no—
514C2×C4—21no—
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—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4C3×#12112yes—
543C2×C4C3×#12112yes—
544C2×C4C3×AGL(1,ℤ/8)12yes—
545C2×C4C22×C3×D411yescharacteristic
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C3×H(5,𝔽2)12yes—
548C2×C4C3×H(5,𝔽2)12yes—
549C8—24no—
550C8—42no—
551C2×C3—42no—
552C2×C3—161no—
553C2×C3—162no—
554C2×C3#261911yescharacteristic
555C2×C3#305511yescharacteristic
556C2×C3#305511yescharacteristic
557C2×C3#321311yescharacteristic
558C2×C3#321411yescharacteristic
559C2×C3#321711yescharacteristic
560C2×C3#321911yescharacteristic
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—82no—
570C22—81no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—81no—
581C22—81no—
582C22C2×C3×#12111yescharacteristic
583C22C3×#53511yescharacteristic
584C22C3×#53511yescharacteristic
585C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
586C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
587C22C3×#57511yescharacteristic
588C22C3×#57511yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—82no—
598C4C3×#53512yes—
599C4C3×#54012yes—
600C3#2981411yescharacteristic, direct product factor, Sylow
601C2—42no—
602C2—161no—
603C2—162no—
604C2C3×#261911yescharacteristic
605C2C3×#305511yescharacteristic
606C2C3×#305511yescharacteristic
607C2C3×#321311yescharacteristic
608C2C3×#321411yescharacteristic
609C2C3×#321711yescharacteristic
610C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#496C2semidirect
2. C2×C3×#432C2semidirect
3. C3×#2856C2semidirect
4. C3×#2850C2semidirect
5. C3×#2055C2semidirect
6. C3×#2054C2semidirect
7. C3×#2000C2semidirect
8. C3×#1999C2semidirect
9. C3×#1961C2semidirect
10. C3×#1959C2semidirect
11. C3×#1836C2semidirect
12. #29814C3direct (N × H)
46. C2×#496C2×C3semidirect
47. C2×#432C2×C3semidirect
48. #2856C2×C3semidirect
49. #2850C2×C3semidirect
50. #2055C2×C3semidirect
51. #2054C2×C3semidirect
52. #2000C2×C3semidirect
53. #1999C2×C3semidirect
54. #1961C2×C3semidirect
55. #1959C2×C3semidirect
56. #1836C2×C3semidirect
113. C3×#97D4semidirect
286. #97C3×D4semidirect
600. C3#29814direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4962
C2×C3×#4322
C3×#28562
C3×#28502
C3×#20552
C3×#20542
C3×#20002
C3×#19992
C3×#19612
C3×#19592
C3×#18362
#298143