Finite Group Databasebeta

← C3×#14215 #10632822

Subgroups — C3×#14215 #10632822

Order 768.

1738 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal, complemented
3C2×C3×C4×C22⋊C4C212yesmaximal
4C3×#1439C212yesmaximal
5C3×#1439C211yesmaximal, characteristic
6C3×#1436C212yesmaximal, complemented
7C3×#1436C212yesmaximal, complemented
8C3×C4⋊#95C212yesmaximal, complemented
9C3×C22⋊#95C211yesmaximal, characteristic
10#14215C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C42C2212yes—
21C22×C3×C42C2212yes—
22C3×#399C418yescomplemented
23C3×#387—28no—
24C3×C4×C22⋊C4—28no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×#399C2×C311yescharacteristic
36C2×#387C2×C312yescomplemented
37C2×C4×C22⋊C4C2×C312yes—
38#1439C2×C311yescharacteristic
39#1439C2×C312yes—
40#1436C2×C312yescomplemented
41#1436C2×C312yescomplemented
42C4⋊#95C2×C312yescomplemented
43C22⋊#95C2×C311yescharacteristic
44C25×C3C2311yescharacteristic
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—41no—
48C23×C3×C4—41no—
49C23×C3×C4Q814yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C4⋊C4C2×C418yescomplemented
68C2×C3×C4⋊C4C2×C414yescomplemented
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—41no—
76C2×C3×C22⋊C4—41no—
77C2×C3×C22⋊C4—41no—
78C2×C3×C22⋊C4—41no—
79C2×C3×C22⋊C4C2×C418yes—
80C2×C3×C22⋊C4C2×C414yes—
81C2×C3×C42—24no—
82C2×C3×C42—24no—
83C2×C3×C42—24no—
84C2×C3×C42—24no—
85C2×C3×C42—42no—
86C2×C3×C42—42no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—41no—
94C3×#95—42no—
95C3×#95C2×C414yescomplemented
96C3×#95D418yes—
97C24×C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C312yes—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C312yes—
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C22⋊C4C22×C312yes—
103C22×C22⋊C4C22×C312yes—
104C22×C22⋊C4C22×C311yescharacteristic
105C22×C22⋊C4C22×C311yescharacteristic
106C22×C42C22×C312yes—
107C22×C42C22×C312yes—
108#399C3×C418yescomplemented
109#387—28no—
110C4×C22⋊C4—28no—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C312yes—
119C2×#95C22×C311yescharacteristic
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
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
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—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—216no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4—48no—
178C22×C3×C4—44no—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C414yes—
181C22×C3×C4C22×C412yes—
182C22×C3×C4C22×C414yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×Q814yes—
187C22×C3×C4#4114yes—
188C22×C3×C4#4114yes—
189C22×C3×C4#4114yes—
190C3×C4⋊C4—216no—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—48no—
193C3×C4⋊C4—44no—
194C3×C22⋊C4—216no—
195C3×C22⋊C4—216no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C42—48no—
199C3×C42—48no—
200C25C23×C311yescharacteristic
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4C3×Q814yes—
206C23×C4C23×C312yes—
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C312yes—
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C312yes—
211C23×C4C23×C312yes—
212C23×C4C23×C312yes—
213C23×C4C23×C312yes—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×C3×C418yescomplemented
224C2×C4⋊C4C2×C3×C414yescomplemented
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4C2×C3×C418yes—
236C2×C22⋊C4C2×C3×C414yes—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—41no—
250#95—42no—
251#95C2×C3×C414yescomplemented
252#95C3×D418yes—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—41no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3C4×Q814yes—
276C23×C3C23×C411yescharacteristic
277C23×C3C22×D412yes—
278C23×C3C22×Q811yescharacteristic
279C23×C3C2×#4112yes—
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4112yes—
282C23×C3C2×#4112yes—
283C23×C3H(5,𝔽2)11yescharacteristic
284C23×C3H(5,𝔽2)11yescharacteristic
285C23×C3H(5,𝔽2)11yescharacteristic
286C23×C32−1+411yescharacteristic
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
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—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—416no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
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×C4C4×D418yes—
339C2×C3×C4C22⋊Q818yes—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24C22×C3×C412yes—
349C24C2×C3×Q812yes—
350C24C3×#4112yes—
351C24C24×C311yescharacteristic
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
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—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4C22×C3×C412yes—
399C22×C4C22×C3×C414yes—
400C22×C4C22×C3×C412yes—
401C22×C4C22×C3×C414yes—
402C22×C4C2×C3×D414yes—
403C22×C4C2×C3×D414yes—
404C22×C4C2×C3×D414yes—
405C22×C4C2×C3×Q814yes—
406C22×C4C3×#4114yes—
407C22×C4C3×#4114yes—
408C22×C4C3×#4114yes—
409C4⋊C4—216no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C22⋊C4—216no—
414C22⋊C4—216no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C42—48no—
418C42—48no—
419C22×C3—24no—
420C22×C3—28no—
421C22×C3—42no—
422C22×C3—44no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—44no—
429C22×C3—42no—
430C22×C3C2×C4×D412yes—
431C22×C3C2×C4×Q811yescharacteristic
432C22×C3C4×#4112yes—
433C22×C3#51811yescharacteristic
434C22×C3#52011yescharacteristic
435C22×C3C2×C22⋊Q812yes—
436C22×C3#52512yes—
437C22×C3#52912yes—
438C22×C3#53312yes—
439C22×C3#53811yescharacteristic
440C22×C3#53911yescharacteristic
441C22×C3#54311yescharacteristic
442C22×C3#54411yescharacteristic
443C22×C3D4211yescharacteristic
444C22×C3#54612yes—
445C22×C3#54712yes—
446C22×C3#55012yes—
447C22×C3#55112yes—
448C22×C3#55111yescharacteristic
449C22×C3#55212yes—
450C22×C3#55312yes—
451C22×C3#55911yescharacteristic
452C22×C3#55911yescharacteristic
453C3×C4—28no—
454C3×C4—28no—
455C3×C4—44no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—44no—
460C3×C4—82no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C3×C4×Q814yes—
484C23C23×C3×C411yescharacteristic
485C23C22×C3×D412yes—
486C23C22×C3×Q811yescharacteristic
487C23C2×C3×#4112yes—
488C23C2×C3×#4112yes—
489C23C2×C3×#4111yescharacteristic
490C23C2×C3×#4112yes—
491C23C3×H(5,𝔽2)11yescharacteristic
492C23C3×H(5,𝔽2)11yescharacteristic
493C23C3×H(5,𝔽2)11yescharacteristic
494C23C3×2−1+411yescharacteristic
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
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—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—416no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4C3×C4×D418yes—
547C2×C4C3×C22⋊Q818yes—
548C2×C3—44no—
549C2×C3C4×C22⋊Q812yes—
550C2×C3#232911yescharacteristic
551C2×C3#235912yes—
552C2×C3#236611yescharacteristic
553C2×C3#237411yescharacteristic
554C2×C3#243711yescharacteristic
555C2×C3#244112yes—
556C2×C3#245511yescharacteristic
557C2×C3#246712yes—
558C2×C3#251711yescharacteristic
559C2×C3#260711yescharacteristic
560C22—28no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22C2×C3×C4×D412yes—
572C22C2×C3×C4×Q811yescharacteristic
573C22C3×C4×#4112yes—
574C22C3×#51811yescharacteristic
575C22C3×#52011yescharacteristic
576C22C2×C3×C22⋊Q812yes—
577C22C3×#52512yes—
578C22C3×#52912yes—
579C22C3×#53312yes—
580C22C3×#53811yescharacteristic
581C22C3×#53911yescharacteristic
582C22C3×#54311yescharacteristic
583C22C3×#54411yescharacteristic
584C22C3×D4211yescharacteristic
585C22C3×#54612yes—
586C22C3×#54712yes—
587C22C3×#55012yes—
588C22C3×#55111yescharacteristic
589C22C3×#55112yes—
590C22C3×#55212yes—
591C22C3×#55312yes—
592C22C3×#55911yescharacteristic
593C22C3×#55911yescharacteristic
594C4—28no—
595C4—28no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—44no—
601C4—82no—
602C3#1421511yescharacteristic, direct product factor, Sylow
603C2—44no—
604C2C3×C4×C22⋊Q812yes—
605C2C3×#232911yescharacteristic
606C2C3×#235912yes—
607C2C3×#236611yescharacteristic
608C2C3×#237411yescharacteristic
609C2C3×#243711yescharacteristic
610C2C3×#244112yes—
611C2C3×#245511yescharacteristic
612C2C3×#246712yes—
613C2C3×#251711yescharacteristic
614C2C3×#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×#387C2semidirect
6. C3×#1436C2semidirect
7. C3×#1436C2semidirect
8. C3×C4⋊#95C2semidirect
10. #14215C3direct (N × H)
22. C3×#399C4semidirect
36. C2×#387C2×C3semidirect
40. #1436C2×C3semidirect
41. #1436C2×C3semidirect
42. C4⋊#95C2×C3semidirect
67. C2×C3×C4⋊C4C2×C4semidirect
68. C2×C3×C4⋊C4C2×C4semidirect
95. C3×#95C2×C4semidirect
108. #399C3×C4semidirect
223. C2×C4⋊C4C2×C3×C4semidirect
224. C2×C4⋊C4C2×C3×C4semidirect
251. #95C2×C3×C4semidirect
602. C3#14215direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3872
C2×C3×C4×C22⋊C42
C3×#14392
C3×#14362
C3×C4⋊#952
C3×C22⋊#952
#142153