Finite Group Databasebeta

← C3×#14213 #10632820

Subgroups — C3×#14213 #10632820

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#393C211yesmaximal, characteristic
2C2×C3×#391C212yesmaximal, complemented
3C2×C3×C4×C22⋊C4C212yesmaximal
4C3×#1439C211yesmaximal, characteristic
5C3×#1439C212yesmaximal
6C3×#1436C212yesmaximal, complemented
7C3×#1436C212yesmaximal, complemented
8C3×#1434C212yesmaximal, complemented
9C3×C22⋊#95C211yesmaximal, characteristic
10#14213C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C42C2212yes—
20C22×C3×C42C2212yes—
21C3×#393C418yescomplemented
22C3×#391—28no—
23C3×C4×C22⋊C4—28no—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×#393C2×C311yescharacteristic
36C2×#391C2×C312yescomplemented
37C2×C4×C22⋊C4C2×C312yes—
38#1439C2×C311yescharacteristic
39#1439C2×C312yes—
40#1436C2×C312yescomplemented
41#1436C2×C312yescomplemented
42#1434C2×C312yescomplemented
43C22⋊#95C2×C311yescharacteristic
44C25×C3C2311yescharacteristic
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—41no—
48C22×C3×Q8—41no—
49C22×C3×Q8C2×C414yescomplemented
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—41no—
53C23×C3×C4—41no—
54C23×C3×C4Q814yescomplemented
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—41no—
75C2×C3×C22⋊C4—41no—
76C2×C3×C22⋊C4—41no—
77C2×C3×C22⋊C4—41no—
78C2×C3×C22⋊C4C2×C418yes—
79C2×C3×C22⋊C4C2×C414yes—
80C2×C3×C42—24no—
81C2×C3×C42—24no—
82C2×C3×C42—24no—
83C2×C3×C42—24no—
84C2×C3×C42—42no—
85C2×C3×C42—42no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—42no—
93C3×#95—41no—
94C3×#95C2×C414yescomplemented
95C3×#95C2×C418yescomplemented
96C3×#95D418yes—
97C23×Q8C22×C311yescharacteristic
98C24×C4C22×C311yescharacteristic
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C312yes—
101C22×C22⋊C4C22×C312yes—
102C22×C22⋊C4C22×C312yes—
103C22×C22⋊C4C22×C311yescharacteristic
104C22×C22⋊C4C22×C311yescharacteristic
105C22×C42C22×C312yes—
106C22×C42C22×C312yes—
107#393C3×C418yescomplemented
108#391—28no—
109C4×C22⋊C4—28no—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3—21no—
128C24×C3—21no—
129C24×C3C22×C412yes—
130C24×C3C2×Q812yes—
131C24×C3#4112yes—
132C24×C3C2411yescharacteristic
133C2×C3×Q8—216no—
134C2×C3×Q8—44no—
135C2×C3×Q8—44no—
136C2×C3×Q8—48no—
137C2×C3×Q8—44no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—216no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4—48no—
183C22×C3×C4—44no—
184C22×C3×C4C22×C414yes—
185C22×C3×C4C22×C412yes—
186C22×C3×C4C22×C412yes—
187C22×C3×C4C22×C414yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×Q814yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4114yes—
194C22×C3×C4#4114yes—
195C3×C4⋊C4—48no—
196C3×C4⋊C4—44no—
197C3×C22⋊C4—216no—
198C3×C22⋊C4—216no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—44no—
201C3×C42—48no—
202C3×C42—48no—
203C25C23×C311yescharacteristic
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—41no—
207C22×Q8—41no—
208C22×Q8C2×C3×C414yescomplemented
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—41no—
212C23×C4—41no—
213C23×C4C3×Q814yescomplemented
214C23×C4C23×C312yes—
215C23×C4C23×C311yescharacteristic
216C23×C4C23×C311yescharacteristic
217C23×C4C23×C312yes—
218C23×C4C23×C312yes—
219C23×C4C23×C312yes—
220C23×C4C23×C312yes—
221C23×C4C23×C312yes—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4C2×C3×C418yes—
238C2×C22⋊C4C2×C3×C414yes—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—42no—
252#95—41no—
253#95C2×C3×C414yescomplemented
254#95C2×C3×C418yescomplemented
255#95C3×D418yes—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3C4×Q814yes—
279C23×C3C23×C411yescharacteristic
280C23×C3C22×D412yes—
281C23×C3C22×Q811yescharacteristic
282C23×C3C2×#4112yes—
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4112yes—
285C23×C3C2×#4112yes—
286C23×C3H(5,𝔽2)11yescharacteristic
287C23×C3H(5,𝔽2)11yescharacteristic
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C32−1+411yescharacteristic
290C3×Q8—416no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—416no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4C4×D418yes—
343C2×C3×C4C22⋊Q818yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24C22×C3×C412yes—
353C24C2×C3×Q812yes—
354C24C3×#4112yes—
355C24C24×C311yescharacteristic
356C2×Q8—216no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4C22×C3×C414yes—
408C22×C4C22×C3×C412yes—
409C22×C4C22×C3×C412yes—
410C22×C4C22×C3×C414yes—
411C22×C4C2×C3×D414yes—
412C22×C4C2×C3×D414yes—
413C22×C4C2×C3×D414yes—
414C22×C4C2×C3×Q814yes—
415C22×C4C3×#4114yes—
416C22×C4C3×#4114yes—
417C22×C4C3×#4114yes—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C22⋊C4—216no—
421C22⋊C4—216no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C42—48no—
425C42—48no—
426C22×C3—24no—
427C22×C3—28no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—44no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3C2×C4×D412yes—
438C22×C3C2×C4×Q811yescharacteristic
439C22×C3C4×#4112yes—
440C22×C3#51811yescharacteristic
441C22×C3#52011yescharacteristic
442C22×C3C2×C22⋊Q812yes—
443C22×C3#52512yes—
444C22×C3#52912yes—
445C22×C3#53312yes—
446C22×C3#53811yescharacteristic
447C22×C3#53911yescharacteristic
448C22×C3#54311yescharacteristic
449C22×C3#54411yescharacteristic
450C22×C3D4211yescharacteristic
451C22×C3#54612yes—
452C22×C3#54612yes—
453C22×C3#54812yes—
454C22×C3#55111yescharacteristic
455C22×C3#55111yescharacteristic
456C22×C3#55111yescharacteristic
457C22×C3#55512yes—
458C22×C3#55612yes—
459C22×C3#55912yes—
460C3×C4—28no—
461C3×C4—28no—
462C3×C4—42no—
463C3×C4—44no—
464C3×C4—42no—
465C3×C4—44no—
466C3×C4—44no—
467C3×C4—82no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23C3×C4×Q814yes—
491C23C23×C3×C411yescharacteristic
492C23C22×C3×D412yes—
493C23C22×C3×Q811yescharacteristic
494C23C2×C3×#4112yes—
495C23C2×C3×#4112yes—
496C23C2×C3×#4111yescharacteristic
497C23C2×C3×#4112yes—
498C23C3×H(5,𝔽2)11yescharacteristic
499C23C3×H(5,𝔽2)11yescharacteristic
500C23C3×H(5,𝔽2)11yescharacteristic
501C23C3×2−1+411yescharacteristic
502Q8—416no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4C3×C4×D418yes—
555C2×C4C3×C22⋊Q818yes—
556C2×C3—44no—
557C2×C3C4×C22⋊Q812yes—
558C2×C3#232911yescharacteristic
559C2×C3#235811yescharacteristic
560C2×C3#236712yes—
561C2×C3#237211yescharacteristic
562C2×C3#243711yescharacteristic
563C2×C3#243912yes—
564C2×C3#245912yes—
565C2×C3#246311yescharacteristic
566C2×C3#252311yescharacteristic
567C2×C3#260711yescharacteristic
568C22—28no—
569C22—24no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C3×C4×D412yes—
580C22C2×C3×C4×Q811yescharacteristic
581C22C3×C4×#4112yes—
582C22C3×#51811yescharacteristic
583C22C3×#52011yescharacteristic
584C22C2×C3×C22⋊Q812yes—
585C22C3×#52512yes—
586C22C3×#52912yes—
587C22C3×#53312yes—
588C22C3×#53811yescharacteristic
589C22C3×#53911yescharacteristic
590C22C3×#54311yescharacteristic
591C22C3×#54411yescharacteristic
592C22C3×D4211yescharacteristic
593C22C3×#54612yes—
594C22C3×#54612yes—
595C22C3×#54812yes—
596C22C3×#55111yescharacteristic
597C22C3×#55111yescharacteristic
598C22C3×#55111yescharacteristic
599C22C3×#55512yes—
600C22C3×#55612yes—
601C22C3×#55912yes—
602C4—28no—
603C4—28no—
604C4—44no—
605C4—42no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—82no—
610C3#1421311yescharacteristic, direct product factor, Sylow
611C2—44no—
612C2C3×C4×C22⋊Q812yes—
613C2C3×#232911yescharacteristic
614C2C3×#235811yescharacteristic
615C2C3×#236712yes—
616C2C3×#237211yescharacteristic
617C2C3×#243711yescharacteristic
618C2C3×#243912yes—
619C2C3×#245912yes—
620C2C3×#246311yescharacteristic
621C2C3×#252311yescharacteristic
622C2C3×#260711yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#391C2semidirect
6. C3×#1436C2semidirect
7. C3×#1436C2semidirect
8. C3×#1434C2semidirect
10. #14213C3direct (N × H)
21. C3×#393C4semidirect
36. C2×#391C2×C3semidirect
40. #1436C2×C3semidirect
41. #1436C2×C3semidirect
42. #1434C2×C3semidirect
49. C22×C3×Q8C2×C4semidirect
54. C23×C3×C4Q8semidirect
94. C3×#95C2×C4semidirect
95. C3×#95C2×C4semidirect
107. #393C3×C4semidirect
208. C22×Q8C2×C3×C4semidirect
213. C23×C4C3×Q8semidirect
253. #95C2×C3×C4semidirect
254. #95C2×C3×C4semidirect
610. C3#14213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3932
C2×C3×#3912
C2×C3×C4×C22⋊C42
C3×#14392
C3×#14362
C3×#14342
C3×C22⋊#952
#142133