Finite Group Databasebeta

← C3×#14143 #10632750

Subgroups — C3×#14143 #10632750

Order 768.

1702 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#387C212yesmaximal, complemented
2C2×C3×#385C211yesmaximal, characteristic
3C2×C3×C4×C22⋊C4C212yesmaximal
4C3×#1440C212yesmaximal
5C3×#1439C211yesmaximal, characteristic
6C3×#1439C211yesmaximal, characteristic
7C3×#1436C212yesmaximal, complemented
8C3×#1436C212yesmaximal, complemented
9C3×#1434C212yesmaximal, complemented
10#14143C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C42C2212yes—
20C22×C3×C42C2212yes—
21C3×#387—28no—
22C3×#385C418yescomplemented
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×#387C2×C312yescomplemented
36C2×#385C2×C311yescharacteristic
37C2×C4×C22⋊C4C2×C312yes—
38#1440C2×C312yes—
39#1439C2×C311yescharacteristic
40#1439C2×C311yescharacteristic
41#1436C2×C312yescomplemented
42#1436C2×C312yescomplemented
43#1434C2×C312yescomplemented
44C25×C3C2311yescharacteristic
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—41no—
48C23×C3×C4—41no—
49C23×C3×C4C2×C414yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
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—41no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4C2×C418yes—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—41no—
74C2×C3×C22⋊C4—41no—
75C2×C3×C22⋊C4C2×C414yes—
76C2×C3×C22⋊C4C2×C414yes—
77C2×C3×C42—24no—
78C2×C3×C42—24no—
79C2×C3×C42—24no—
80C2×C3×C42—24no—
81C2×C3×C42—42no—
82C2×C3×C42—42no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—41no—
91C3×#95—41no—
92C3×#95C2×C418yes—
93C3×#95D418yes—
94C24×C4C22×C311yescharacteristic
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C4⋊C4C22×C311yescharacteristic
97C22×C4⋊C4C22×C312yes—
98C22×C22⋊C4C22×C312yes—
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C311yescharacteristic
101C22×C22⋊C4C22×C311yescharacteristic
102C22×C42C22×C312yes—
103C22×C42C22×C312yes—
104#387—28no—
105#385C3×C418yescomplemented
106C4×C22⋊C4—28no—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3C22×C412yes—
127C24×C3#4112yes—
128C24×C3#4112yes—
129C24×C3C2411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
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—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—216no—
173C22×C3×C4—44no—
174C22×C3×C4—44no—
175C22×C3×C4—48no—
176C22×C3×C4—44no—
177C22×C3×C4C22×C412yes—
178C22×C3×C4C22×C414yes—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C414yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4114yes—
186C22×C3×C4#4114yes—
187C3×C4⋊C4—216no—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—44no—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—216no—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—44no—
194C3×C42—48no—
195C3×C42—48no—
196C25C23×C311yescharacteristic
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4C2×C3×C414yes—
202C23×C4C23×C311yescharacteristic
203C23×C4C23×C312yes—
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C312yes—
206C23×C4C23×C312yes—
207C23×C4C23×C312yes—
208C23×C4C23×C312yes—
209C23×C4C23×C312yes—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×C3×C418yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4C2×C3×C414yes—
228C2×C22⋊C4C2×C3×C414yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—42no—
234C2×C42—42no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—41no—
243#95—41no—
244#95C2×C3×C418yes—
245#95C3×D418yes—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—41no—
268C23×C3#11714yes—
269C23×C3C23×C411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4112yes—
274C23×C3C2×#4112yes—
275C23×C3C2×#4112yes—
276C23×C3H(5,𝔽2)11yescharacteristic
277C23×C3H(5,𝔽2)11yescharacteristic
278C23×C32−1+411yescharacteristic
279C23×C32−1+411yescharacteristic
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
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—416no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
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—41no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4C4×D418yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24C22×C3×C412yes—
342C24C3×#4112yes—
343C24C3×#4112yes—
344C24C24×C311yescharacteristic
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
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—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—216no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4C22×C3×C412yes—
393C22×C4C22×C3×C414yes—
394C22×C4C22×C3×C412yes—
395C22×C4C22×C3×C414yes—
396C22×C4C2×C3×D414yes—
397C22×C4C2×C3×D414yes—
398C22×C4C2×C3×D414yes—
399C22×C4C3×#4114yes—
400C22×C4C3×#4114yes—
401C22×C4C3×#4114yes—
402C4⋊C4—216no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C22⋊C4—216no—
406C22⋊C4—216no—
407C22⋊C4—28no—
408C22⋊C4—44no—
409C42—48no—
410C42—48no—
411C22×C3—24no—
412C22×C3—28no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3C2×#11711yescharacteristic
423C22×C3C2×C4×D412yes—
424C22×C3C4×#4112yes—
425C22×C3#51811yescharacteristic
426C22×C3#51911yescharacteristic
427C22×C3#52512yes—
428C22×C3#52512yes—
429C22×C3#52912yes—
430C22×C3#52912yes—
431C22×C3#53911yescharacteristic
432C22×C3#53911yescharacteristic
433C22×C3#53911yescharacteristic
434C22×C3#53911yescharacteristic
435C22×C3D4211yescharacteristic
436C22×C3#54612yes—
437C22×C3#54712yes—
438C22×C3#54812yes—
439C22×C3#55111yescharacteristic
440C22×C3#55111yescharacteristic
441C22×C3#55212yes—
442C22×C3#55312yes—
443C22×C3#55612yes—
444C22×C3#55911yescharacteristic
445C3×C4—28no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—44no—
450C3×C4—44no—
451C3×C4—44no—
452C3×C4—44no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23C3×#11714yes—
476C23C23×C3×C411yescharacteristic
477C23C22×C3×D412yes—
478C23C2×C3×#4112yes—
479C23C2×C3×#4111yescharacteristic
480C23C2×C3×#4111yescharacteristic
481C23C2×C3×#4112yes—
482C23C2×C3×#4112yes—
483C23C3×H(5,𝔽2)11yescharacteristic
484C23C3×H(5,𝔽2)11yescharacteristic
485C23C3×2−1+411yescharacteristic
486C23C3×2−1+411yescharacteristic
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
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—24no—
512C2×C4—416no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4C3×C4×D418yes—
540C2×C3—44no—
541C2×C3#229012yes—
542C2×C3#231311yescharacteristic
543C2×C3#235811yescharacteristic
544C2×C3#236212yes—
545C2×C3#237411yescharacteristic
546C2×C3#245511yescharacteristic
547C2×C3#246012yes—
548C2×C3#246311yescharacteristic
549C2×C3#246612yes—
550C2×C3#251711yescharacteristic
551C2×C3#252311yescharacteristic
552C22—28no—
553C22—24no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C2×C3×#11711yescharacteristic
564C22C2×C3×C4×D412yes—
565C22C3×C4×#4112yes—
566C22C3×#51811yescharacteristic
567C22C3×#51911yescharacteristic
568C22C3×#52512yes—
569C22C3×#52512yes—
570C22C3×#52912yes—
571C22C3×#52912yes—
572C22C3×#53911yescharacteristic
573C22C3×#53911yescharacteristic
574C22C3×#53911yescharacteristic
575C22C3×#53911yescharacteristic
576C22C3×D4211yescharacteristic
577C22C3×#54612yes—
578C22C3×#54712yes—
579C22C3×#54812yes—
580C22C3×#55111yescharacteristic
581C22C3×#55111yescharacteristic
582C22C3×#55212yes—
583C22C3×#55312yes—
584C22C3×#55612yes—
585C22C3×#55911yescharacteristic
586C4—28no—
587C4—42no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—42no—
594C3#1414311yescharacteristic, direct product factor, Sylow
595C2—44no—
596C2C3×#229012yes—
597C2C3×#231311yescharacteristic
598C2C3×#235811yescharacteristic
599C2C3×#236212yes—
600C2C3×#237411yescharacteristic
601C2C3×#245511yescharacteristic
602C2C3×#246012yes—
603C2C3×#246311yescharacteristic
604C2C3×#246612yes—
605C2C3×#251711yescharacteristic
606C2C3×#252311yescharacteristic
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. C2×C3×#387C2semidirect
7. C3×#1436C2semidirect
8. C3×#1436C2semidirect
9. C3×#1434C2semidirect
10. #14143C3direct (N × H)
22. C3×#385C4semidirect
35. C2×#387C2×C3semidirect
41. #1436C2×C3semidirect
42. #1436C2×C3semidirect
43. #1434C2×C3semidirect
105. #385C3×C4semidirect
594. C3#14143direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3872
C2×C3×#3852
C2×C3×C4×C22⋊C42
C3×#14402
C3×#14392
C3×#14362
C3×#14342
#141433