Finite Group Databasebeta

← C3×#14913 #10633520

Subgroups — C3×#14913 #10633520

Order 768.

2638 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
1C22×C3×#121C211yesmaximal, characteristic
2C2×C3×#386C212yesmaximal, complemented
3C2×C3×#386C212yesmaximal, complemented
4C22×C3×#95C211yesmaximal, characteristic, complemented
5C3×#1442C212yesmaximal, complemented
6C3×#1440C214yesmaximal, complemented
7C3×#1438C212yesmaximal, complemented
8C3×C22⋊#95C211yesmaximal, characteristic, complemented
9#14913C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2212yes—
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×#121—28no—
16C2×C3×#121—24no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C22⋊C4C2214yes—
22C3×#386—28no—
23C3×#386—28no—
24C2×C3×#95—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2214yes—
28C2×C3×#95C2214yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C22×#121C2×C311yescharacteristic
32C2×#386C2×C312yescomplemented
33C2×#386C2×C312yescomplemented
34C22×#95C2×C311yescharacteristic, complemented
35#1442C2×C312yescomplemented
36#1440C2×C314yescomplemented
37#1438C2×C312yescomplemented
38C22⋊#95C2×C311yescharacteristic, complemented
39C25×C3C2312yes—
40C25×C3C2311yescharacteristic
41C22×C3×D4—24no—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—22no—
49C22×C3×D4—24no—
50C22×C3×D4—42no—
51C22×C3×D4D418yes—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4—22no—
56C23×C3×C4—24no—
57C23×C3×C4—24no—
58C23×C3×C4—24no—
59C23×C3×C4—42no—
60C23×C3×C4—42no—
61C23×C3×C4—42no—
62C23×C3×C4—42no—
63C23×C3×C4D414yes—
64C23×C3×C4D414yescomplemented
65C23×C3×C4C2314yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C3×#121—232no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—28no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—28no—
83C2×C3×C22⋊C4—44no—
84C2×C3×C22⋊C4—44no—
85C2×C3×C22⋊C4—44no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—28no—
89C3×#95—28no—
90C3×#95—24no—
91C3×#95—42no—
92C23×D4C22×C312yes—
93C23×D4C22×C311yescharacteristic
94C24×C4C22×C311yescharacteristic
95C24×C4C22×C311yescharacteristic
96C24×C4C22×C311yescharacteristic
97C2×#121—28no—
98C2×#121—24no—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C22⋊C4C22×C312yes—
101C22×C22⋊C4C22×C312yes—
102C22×C22⋊C4C22×C314yes—
103C22×C22⋊C4C22×C314yes—
104#386—28no—
105#386—28no—
106C2×#95—24no—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C314yes—
111C2×#95C22×C314yes—
112C2×#95C22×C312yes—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—24no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—24no—
125C24×C3—24no—
126C24×C3—24no—
127C24×C3C2×D414yes—
128C24×C3C2×D414yes—
129C24×C3C2×D412yes—
130C24×C3C2×D412yes—
131C24×C3#4112yes—
132C24×C3#4114yes—
133C24×C3C2411yescharacteristic
134C2×C3×D4—28no—
135C2×C3×D4—216no—
136C2×C3×D4—216no—
137C2×C3×D4—216no—
138C2×C3×D4—416no—
139C2×C3×D4—44no—
140C2×C3×D4—48no—
141C2×C3×D4—48no—
142C2×C3×D4—48no—
143C2×C3×D4—48no—
144C2×C3×D4—44no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—28no—
148C22×C3×C4—28no—
149C22×C3×C4—28no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
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—24no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—28no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—44no—
176C22×C3×C4—48no—
177C22×C3×C4—416no—
178C22×C3×C4—416no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4—48no—
183C22×C3×C4—44no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4#4114yes—
189C22×C3×C4#4114yes—
190C3×C4⋊C4—28no—
191C3×C22⋊C4—216no—
192C3×C22⋊C4—416no—
193C3×C22⋊C4—416no—
194C3×C22⋊C4—48no—
195C25C23×C312yes—
196C25C23×C311yescharacteristic
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—42no—
207C22×D4C3×D418yes—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C3×D414yes—
220C23×C4C3×D414yescomplemented
221C23×C4C23×C314yes—
222C23×C4C23×C312yes—
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C312yes—
225C23×C4C23×C312yes—
226C23×C4C23×C311yescharacteristic
227#121—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242#95—24no—
243#95—24no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—42no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—28no—
257C23×C3—28no—
258C23×C3—28no—
259C23×C3—28no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—28no—
264C23×C3—28no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3#12014yes—
284C23×C3#12118yes—
285C23×C3#12114yes—
286C23×C3#12318yes—
287C23×C3#12414yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D412yes—
290C23×C3C22×D411yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4112yes—
293C23×C3C2×#4112yes—
294C23×C3C2×#4112yes—
295C23×C3H(5,𝔽2)12yes—
296C23×C3H(5,𝔽2)12yes—
297C3×D4—416no—
298C3×D4—432no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—28no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—28no—
310C2×C3×C4—22no—
311C2×C3×C4—28no—
312C2×C3×C4—416no—
313C2×C3×C4—416no—
314C2×C3×C4—416no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—44no—
332C2×C3×C4—44no—
333C2×C3×C4#12314yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24C2×C3×D414yes—
349C24C2×C3×D414yes—
350C24C2×C3×D412yes—
351C24C2×C3×D412yes—
352C24C3×#4112yes—
353C24C3×#4114yes—
354C24C24×C311yescharacteristic
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
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—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D414yes—
407C22×C4C2×C3×D414yes—
408C22×C4C2×C3×D412yes—
409C22×C4C3×#4114yes—
410C22×C4C3×#4114yes—
411C4⋊C4—28no—
412C22⋊C4—216no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—48no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—216no—
422C22×C3—24no—
423C22×C3—216no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—48no—
428C22×C3—48no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—84no—
433C22×C3—82no—
434C22×C3—84no—
435C22×C3#39418yes—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12111yescharacteristic
438C22×C3C2×#12112yes—
439C22×C3C2×#12311yescharacteristic
440C22×C3C2×#12312yes—
441C22×C3#52511yescharacteristic
442C22×C3C2×#12411yescharacteristic
443C22×C3#52912yes—
444C22×C3#53212yes—
445C22×C3#53411yescharacteristic
446C22×C3#53412yes—
447C22×C3#53511yescharacteristic
448C22×C3#53812yes—
449C22×C3#54614yes—
450C22×C3#54614yes—
451C22×C3#55314yes—
452C22×C3#55314yes—
453C3×C4—24no—
454C3×C4—44no—
455C3×C4—44no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—84no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23C3×#12014yes—
495C23C3×#12118yes—
496C23C3×#12114yes—
497C23C3×#12318yes—
498C23C3×#12414yes—
499C23C22×C3×D411yescharacteristic
500C23C22×C3×D412yes—
501C23C22×C3×D411yescharacteristic
502C23C2×C3×#4111yescharacteristic
503C23C2×C3×#4112yes—
504C23C2×C3×#4112yes—
505C23C2×C3×#4112yes—
506C23C3×H(5,𝔽2)12yes—
507C23C3×H(5,𝔽2)12yes—
508D4—416no—
509D4—432no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4C3×#12314yes—
545C2×C3—28no—
546C2×C3—84no—
547C2×C3C2×#39411yescharacteristic
548C2×C3#240411yescharacteristic
549C2×C3#240511yescharacteristic
550C2×C3#244414yes—
551C2×C3#250411yescharacteristic
552C2×C3#254314yes—
553C2×C3#259111yescharacteristic
554C2×C3#261711yescharacteristic
555C2×C3#285411yescharacteristic
556C22—24no—
557C22—24no—
558C22—24no—
559C22—216no—
560C22—24no—
561C22—216no—
562C22—24no—
563C22—24no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—48no—
568C22—48no—
569C22—82no—
570C22—84no—
571C22—82no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22C3×#39418yes—
576C22C2×C3×#12011yescharacteristic
577C22C2×C3×#12111yescharacteristic
578C22C2×C3×#12112yes—
579C22C2×C3×#12312yes—
580C22C2×C3×#12311yescharacteristic
581C22C3×#52511yescharacteristic
582C22C2×C3×#12411yescharacteristic
583C22C3×#52912yes—
584C22C3×#53212yes—
585C22C3×#53411yescharacteristic
586C22C3×#53412yes—
587C22C3×#53511yescharacteristic
588C22C3×#53812yes—
589C22C3×#54614yes—
590C22C3×#54614yes—
591C22C3×#55314yes—
592C22C3×#55314yes—
593C4—24no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—42no—
598C4—84no—
599C3#1491311yescharacteristic, direct product factor, Sylow
600C2—28no—
601C2—84no—
602C2C2×C3×#39411yescharacteristic
603C2C3×#240411yescharacteristic
604C2C3×#240511yescharacteristic
605C2C3×#244414yes—
606C2C3×#250411yescharacteristic
607C2C3×#254314yes—
608C2C3×#259111yescharacteristic
609C2C3×#261711yescharacteristic
610C2C3×#285411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#386C2semidirect
3. C2×C3×#386C2semidirect
4. C22×C3×#95C2semidirect
5. C3×#1442C2semidirect
6. C3×#1440C2semidirect
7. C3×#1438C2semidirect
8. C3×C22⋊#95C2semidirect
9. #14913C3direct (N × H)
32. C2×#386C2×C3semidirect
33. C2×#386C2×C3semidirect
34. C22×#95C2×C3semidirect
35. #1442C2×C3semidirect
36. #1440C2×C3semidirect
37. #1438C2×C3semidirect
38. C22⋊#95C2×C3semidirect
64. C23×C3×C4D4semidirect
220. C23×C4C3×D4semidirect
599. C3#14913direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#3862
C22×C3×#952
C3×#14422
C3×#14402
C3×#14382
C3×C22⋊#952
#149133