Finite Group Databasebeta

← C3×#14188 #10632795

Subgroups — C3×#14188 #10632795

Order 768.

1894 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#393C211yesmaximal, characteristic
3C2×C3×#388C211yesmaximal, characteristic
4C2×C3×#388C212yesmaximal
5C3×#1446C211yesmaximal, characteristic, complemented
6C3×#1442C211yesmaximal, characteristic, complemented
7C3×#1439C211yesmaximal, characteristic
8C3×#1437C212yesmaximal
9C3×#1436C212yesmaximal, complemented
10C3×#1436C212yesmaximal, complemented
11C3×C4×#95C211yesmaximal, characteristic, complemented
12#14188C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×C4×Q8—28no—
15C2×C3×C4×Q8—24no—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C42C2212yes—
25C22×C3×C42C2212yes—
26C22×C3×C42C2211yescharacteristic
27C3×#393C418yescomplemented
28C3×#388—28no—
29C3×#388—24no—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×C3×#95C2212yes—
41C22×C4×Q8C2×C311yescharacteristic, complemented
42C2×#393C2×C311yescharacteristic
43C2×#388C2×C311yescharacteristic
44C2×#388C2×C312yes—
45#1446C2×C311yescharacteristic, complemented
46#1442C2×C311yescharacteristic, complemented
47#1439C2×C311yescharacteristic
48#1437C2×C312yes—
49#1436C2×C312yescomplemented
50#1436C2×C312yescomplemented
51C4×#95C2×C311yescharacteristic, complemented
52C25×C3C2311yescharacteristic
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—24no—
57C22×C3×Q8C2×C414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C3×C4×Q8—232no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4D418yes—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—42no—
87C2×C3×C22⋊C4C2×C418yes—
88C2×C3×C22⋊C4C2×C414yes—
89C2×C3×C42—24no—
90C2×C3×C42—24no—
91C2×C3×C42—24no—
92C2×C3×C42—24no—
93C2×C3×C42—24no—
94C2×C3×C42—24no—
95C2×C3×C42—42no—
96C2×C3×C42D414yes—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—22no—
103C3×#95—24no—
104C3×#95—24no—
105C3×#95—42no—
106C3×#95C2×C414yes—
107C3×#95C2×C418yes—
108C23×Q8C22×C311yescharacteristic
109C2×C4×Q8—28no—
110C2×C4×Q8—24no—
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C4⋊C4C22×C311yescharacteristic
113C22×C4⋊C4C22×C312yes—
114C22×C22⋊C4C22×C311yescharacteristic
115C22×C22⋊C4C22×C311yescharacteristic
116C22×C22⋊C4C22×C312yes—
117C22×C22⋊C4C22×C312yes—
118C22×C22⋊C4C22×C311yescharacteristic
119C22×C42C22×C312yes—
120C22×C42C22×C312yes—
121C22×C42C22×C311yescharacteristic
122#393C3×C418yescomplemented
123#388—28no—
124#388—24no—
125C2×#95C22×C312yes—
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C312yes—
128C2×#95C22×C311yescharacteristic
129C2×#95C22×C312yes—
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C2×#95C22×C311yescharacteristic
133C2×#95C22×C312yes—
134C2×#95C22×C311yescharacteristic
135C2×#95C22×C312yes—
136C24×C3—21no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3C22×C412yes—
147C24×C3C2411yescharacteristic
148C2×C3×Q8—28no—
149C2×C3×Q8—28no—
150C2×C3×Q8—216no—
151C2×C3×Q8—28no—
152C2×C3×Q8—216no—
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—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
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—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C22×C412yes—
198C22×C3×C4C22×C414yes—
199C22×C3×C4C22×C414yes—
200C22×C3×C4C22×C412yes—
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2×D414yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4#4114yes—
208C22×C3×C4#4112yes—
209C22×C3×C4#4114yes—
210C22×C3×C4#4114yes—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—216no—
213C3×C4⋊C4—44no—
214C3×C22⋊C4—216no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—48no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—44no—
219C3×C42—216no—
220C3×C42—28no—
221C3×C42—48no—
222C25C23×C311yescharacteristic
223C22×Q8—24no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—22no—
227C22×Q8C2×C3×C414yes—
228C23×C4C23×C311yescharacteristic
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C312yes—
233C23×C4C23×C312yes—
234C23×C4C23×C312yes—
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C311yescharacteristic
238C4×Q8—232no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4C3×D418yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C2×C3×C418yes—
258C2×C22⋊C4C2×C3×C414yes—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42C3×D414yes—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—42no—
276#95C2×C3×C414yes—
277#95C2×C3×C418yes—
278C23×C3—22no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3C23×C411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D412yes—
304C23×C3C2×#4111yescharacteristic
305C23×C3C2×#4112yes—
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C3H(5,𝔽2)11yescharacteristic
309C23×C32−1+411yescharacteristic
310C23×C32−1+411yescharacteristic
311C23×C32−1+411yescharacteristic
312C3×Q8—232no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4C4×D414yes—
368C2×C3×C4C4×D418yes—
369C2×C3×C4#12014yes—
370C2×C3×C4#12414yes—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24C22×C3×C412yes—
382C24C24×C311yescharacteristic
383C2×Q8—28no—
384C2×Q8—216no—
385C2×Q8—28no—
386C2×Q8—216no—
387C2×Q8—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C22×C3×C412yes—
433C22×C4C22×C3×C414yes—
434C22×C4C22×C3×C412yes—
435C22×C4C22×C3×C414yes—
436C22×C4C2×C3×D414yes—
437C22×C4C2×C3×D414yes—
438C22×C4C2×C3×D412yes—
439C22×C4C2×C3×D412yes—
440C22×C4C2×C3×D414yes—
441C22×C4C2×C3×D412yes—
442C22×C4C3×#4114yes—
443C22×C4C3×#4114yes—
444C22×C4C3×#4112yes—
445C22×C4C3×#4114yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—44no—
449C22⋊C4—216no—
450C22⋊C4—28no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—216no—
455C42—28no—
456C42—48no—
457C22×C3—42no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3C2×C4×D411yescharacteristic
469C22×C3C2×C4×D412yes—
470C22×C3C4×#4112yes—
471C22×C3#51911yescharacteristic
472C22×C3#52011yescharacteristic
473C22×C3C2×#12011yescharacteristic
474C22×C3#52511yescharacteristic
475C22×C3#52512yes—
476C22×C3C2×#12411yescharacteristic
477C22×C3#52912yes—
478C22×C3#53211yescharacteristic
479C22×C3#53611yescharacteristic
480C22×C3#53711yescharacteristic
481C22×C3#54111yescharacteristic
482C22×C3#54211yescharacteristic
483C22×C3#54612yes—
484C22×C3#54712yes—
485C22×C3#54812yes—
486C22×C3#54812yes—
487C22×C3#55212yes—
488C22×C3#55612yes—
489C22×C3#55612yes—
490C22×C3#55912yes—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—28no—
494C3×C4—24no—
495C3×C4—44no—
496C3×C4—44no—
497C3×C4—44no—
498C3×C4—42no—
499C3×C4—81no—
500C3×C4—81no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C23×C3×C411yescharacteristic
525C23C22×C3×D411yescharacteristic
526C23C22×C3×D412yes—
527C23C2×C3×#4112yes—
528C23C2×C3×#4112yes—
529C23C2×C3×#4111yescharacteristic
530C23C2×C3×#4112yes—
531C23C3×H(5,𝔽2)11yescharacteristic
532C23C3×2−1+411yescharacteristic
533C23C3×2−1+411yescharacteristic
534C23C3×2−1+411yescharacteristic
535Q8—232no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4C3×C4×D418yes—
591C2×C4C3×C4×D414yes—
592C2×C4C3×#12014yes—
593C2×C4C3×#12414yes—
594C2×C3—82no—
595C2×C3C4×#12011yescharacteristic
596C2×C3C4×#12411yescharacteristic
597C2×C3#232311yescharacteristic
598C2×C3#236212yes—
599C2×C3#236712yes—
600C2×C3#243011yescharacteristic
601C2×C3#243611yescharacteristic
602C2×C3#245812yes—
603C2×C3#246912yes—
604C2×C3#259211yescharacteristic
605C2×C3#260411yescharacteristic
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22C2×C3×C4×D412yes—
618C22C2×C3×C4×D411yescharacteristic
619C22C3×C4×#4112yes—
620C22C3×#51911yescharacteristic
621C22C3×#52011yescharacteristic
622C22C2×C3×#12011yescharacteristic
623C22C3×#52512yes—
624C22C3×#52511yescharacteristic
625C22C2×C3×#12411yescharacteristic
626C22C3×#52912yes—
627C22C3×#53211yescharacteristic
628C22C3×#53611yescharacteristic
629C22C3×#53711yescharacteristic
630C22C3×#54111yescharacteristic
631C22C3×#54211yescharacteristic
632C22C3×#54612yes—
633C22C3×#54712yes—
634C22C3×#54812yes—
635C22C3×#54812yes—
636C22C3×#55212yes—
637C22C3×#55612yes—
638C22C3×#55612yes—
639C22C3×#55912yes—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—28no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—81no—
649C4—81no—
650C3#1418811yescharacteristic, direct product factor, Sylow
651C2—82no—
652C2C3×C4×#12011yescharacteristic
653C2C3×C4×#12411yescharacteristic
654C2C3×#232311yescharacteristic
655C2C3×#236212yes—
656C2C3×#236712yes—
657C2C3×#243011yescharacteristic
658C2C3×#243611yescharacteristic
659C2C3×#245812yes—
660C2C3×#246912yes—
661C2C3×#259211yescharacteristic
662C2C3×#260411yescharacteristic
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
1. C22×C3×C4×Q8C2semidirect
5. C3×#1446C2semidirect
6. C3×#1442C2semidirect
9. C3×#1436C2semidirect
10. C3×#1436C2semidirect
11. C3×C4×#95C2semidirect
12. #14188C3direct (N × H)
27. C3×#393C4semidirect
41. C22×C4×Q8C2×C3semidirect
45. #1446C2×C3semidirect
46. #1442C2×C3semidirect
49. #1436C2×C3semidirect
50. #1436C2×C3semidirect
51. C4×#95C2×C3semidirect
122. #393C3×C4semidirect
650. C3#14188direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#3932
C2×C3×#3882
C3×#14462
C3×#14422
C3×#14392
C3×#14372
C3×#14362
C3×C4×#952
#141883