Finite Group Databasebeta

← C3×#24352 #10642959

Subgroups — C3×#24352 #10642959

Order 768.

1198 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
1C3×#3501C211yesmaximal, characteristic, complemented
2C3×#2626C212yesmaximal, complemented
3C3×#2321C211yesmaximal, characteristic
4C3×#2062C212yesmaximal, complemented
5C3×#2029C212yesmaximal, complemented
6C3×#1996C212yesmaximal, complemented
7C3×#1889C212yesmaximal
8C3×#1867C212yesmaximal, complemented
9C3×#1799C211yesmaximal, characteristic, complemented
10#24352C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—22no—
13C3×D4×Q8—22no—
14C3×#544—21no—
15C3×#543—21no—
16C3×#543C2212yes—
17C3×#533—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#120C2211yescharacteristic
21C3×#518C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C22⋊C4C2211yescharacteristic
24C2×C3×#99C2212yes—
25C3×#399—22no—
26C3×#396—24no—
27C3×#393—22no—
28C3×#390—21no—
29C3×#388—21no—
30C3×#388—21no—
31C3×#386—21no—
32C3×#386C2211yescharacteristic
33C3×#385C2214yes—
34C3×#385C2211yescharacteristic
35C3×#380C2212yes—
36C3×#379C2214yes—
37C3×#379C2211yescharacteristic
38C3×C4×C22⋊C4C2211yescharacteristic
39C3×#342C2214yes—
40C3×#342C2214yes—
41C3×#342C2212yes—
42#3501C2×C311yescharacteristic, complemented
43#2626C2×C312yescomplemented
44#2321C2×C311yescharacteristic
45#2062C2×C312yescomplemented
46#2029C2×C312yescomplemented
47#1996C2×C312yescomplemented
48#1889C2×C312yes—
49#1867C2×C312yescomplemented
50#1799C2×C311yescharacteristic, complemented
51C25×C3C2311yescharacteristic
52C22×C3×Q8—21no—
53C22×C3×Q8D412yescomplemented
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—24no—
62C3×#125—22no—
63C3×#125—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×#120—22no—
77C3×#120Q814yescomplemented
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D414yescomplemented
92C2×C3×C4⋊C4D412yescomplemented
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—24no—
103C2×C3×C22⋊C4—24no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2311yescharacteristic
110C2×C3×C42—21no—
111C2×C3×C42D411yescharacteristic
112C3×#99—24no—
113C3×#95—21no—
114C3×#95—22no—
115C3×#95—21no—
116C3×#95—24no—
117C3×#95—21no—
118#554—24no—
119#554—22no—
120D4×Q8—22no—
121#544—21no—
122#543—21no—
123#543C22×C312yes—
124#533—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C22×C312yes—
127C2×#120C22×C311yescharacteristic
128#518C22×C312yes—
129C2×C4×D4C22×C311yescharacteristic
130C22×C22⋊C4C22×C311yescharacteristic
131C2×#99C22×C312yes—
132#399—22no—
133#396—24no—
134#393—22no—
135#390—21no—
136#388—21no—
137#388—21no—
138#386—21no—
139#386C22×C311yescharacteristic
140#385C22×C311yescharacteristic
141#385C22×C314yes—
142#380C22×C312yes—
143#379C22×C311yescharacteristic
144#379C22×C314yes—
145C4×C22⋊C4C22×C311yescharacteristic
146#342C22×C312yes—
147#342C22×C314yes—
148#342C22×C314yes—
149C24×C3—22no—
150C24×C3—21no—
151C24×C3—21no—
152C24×C3—22no—
153C24×C3—81no—
154C24×C3—81no—
155C24×C3C2×D411yescharacteristic
156C24×C3C2×Q812yescomplemented
157C24×C3C2411yescharacteristic
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—82no—
163C2×C3×Q8—82no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—81no—
168C2×C3×D4—81no—
169C2×C3×D4C2×Q812yes—
170C2×C3×D4#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—81no—
181C22×C3×C4—81no—
182C22×C3×C4—81no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—82no—
204C3×C4⋊C4—81no—
205C3×C4⋊C4—84no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—41no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—81no—
220C3×C22⋊C4—84no—
221C3×C22⋊C4—81no—
222C3×C22⋊C4—84no—
223C3×C22⋊C4—81no—
224C3×C22⋊C4C2×Q812yescomplemented
225C3×C42—24no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—81no—
229C25C23×C311yescharacteristic
230C22×Q8—21no—
231C22×Q8C3×D412yescomplemented
232C22×D4—21no—
233C22×D4C23×C311yescharacteristic
234C23×C4C3×D411yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C311yescharacteristic
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—24no—
240#125—22no—
241#125—24no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254#120—22no—
255#120C3×Q814yescomplemented
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—24no—
261C4×D4—24no—
262#117—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4C3×D411yescharacteristic
269C2×C4⋊C4C3×D414yescomplemented
270C2×C4⋊C4C3×D412yescomplemented
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4C3×D411yescharacteristic
283C2×C22⋊C4C23×C311yescharacteristic
284C2×C22⋊C4C23×C314yes—
285C2×C22⋊C4C23×C311yescharacteristic
286C2×C22⋊C4C23×C314yes—
287C2×C22⋊C4C23×C311yescharacteristic
288C2×C42—21no—
289C2×C42C3×D411yescharacteristic
290#99—24no—
291#95—21no—
292#95—22no—
293#95—21no—
294#95—24no—
295#95—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—44no—
305C23×C3—41no—
306C23×C3—44no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3—82no—
314C23×C3—82no—
315C23×C3—84no—
316C23×C3C22⋊Q812yes—
317C23×C3C22×D411yescharacteristic
318C23×C3C22×D412yes—
319C23×C3C22×Q811yescharacteristic
320C23×C3C2×#4111yescharacteristic
321C23×C32−1+412yes—
322C3×Q8—44no—
323C3×Q8—82no—
324C3×Q8—82no—
325C3×Q8—82no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—84no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—44no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—44no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—44no—
350C2×C3×C4—44no—
351C2×C3×C4—41no—
352C2×C3×C4—44no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—44no—
357C2×C3×C4—81no—
358C2×C3×C4—82no—
359C2×C3×C4—82no—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4#12012yes—
363C2×C3×C4C22⋊Q812yes—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—81no—
369C24—81no—
370C24C2×C3×D411yescharacteristic
371C24C2×C3×Q812yescomplemented
372C24C24×C311yescharacteristic
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4C2×C3×Q812yes—
385C2×D4C3×#4112yes—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D414yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D414yes—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D412yes—
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D412yes—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—82no—
419C4⋊C4—81no—
420C4⋊C4—84no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—21no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—81no—
435C22⋊C4—84no—
436C22⋊C4—81no—
437C22⋊C4—84no—
438C22⋊C4—81no—
439C22⋊C4C2×C3×Q812yescomplemented
440C42—24no—
441C42—21no—
442C42—21no—
443C42—81no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—48no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—81no—
460C22×C3—84no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×C22⋊Q811yescharacteristic
467C22×C3#53611yescharacteristic
468C22×C3#54812yes—
469C22×C3D4×Q812yes—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—44no—
477C3×C4—44no—
478C3×C4—41no—
479C3×C4—84no—
480C3×C4—81no—
481C3×C4—81no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—24no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—41no—
492C23—44no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—82no—
500C23—82no—
501C23—84no—
502C23C3×C22⋊Q812yes—
503C23C22×C3×D411yescharacteristic
504C23C22×C3×D412yes—
505C23C22×C3×Q811yescharacteristic
506C23C2×C3×#4111yescharacteristic
507C23C3×2−1+412yes—
508Q8—44no—
509Q8—82no—
510Q8—82no—
511Q8—82no—
512D4—24no—
513D4—24no—
514D4—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C3×#12011yescharacteristic
547C2×C4C3×#12011yescharacteristic
548C2×C4C3×#12012yes—
549C2×C4C3×C22⋊Q812yes—
550C2×C3—21no—
551C2×C3—22no—
552C2×C3—21no—
553C2×C3—22no—
554C2×C3—42no—
555C2×C3—82no—
556C2×C3#240911yescharacteristic
557C2×C3#302611yescharacteristic
558C2×C3#302711yescharacteristic
559C22—22no—
560C22—21no—
561C22—22no—
562C22—21no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—48no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—81no—
575C22—84no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C2×C3×#12011yescharacteristic
581C22C2×C3×C22⋊Q811yescharacteristic
582C22C3×#53611yescharacteristic
583C22C3×#54812yes—
584C22C3×D4×Q812yes—
585C4—22no—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—44no—
592C4—44no—
593C4—41no—
594C4—84no—
595C4—81no—
596C4—81no—
597C3#2435211yescharacteristic, direct product factor, Sylow
598C2—21no—
599C2—22no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2C3×#240911yescharacteristic
605C2C3×#302611yescharacteristic
606C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3501C2semidirect
2. C3×#2626C2semidirect
4. C3×#2062C2semidirect
5. C3×#2029C2semidirect
6. C3×#1996C2semidirect
8. C3×#1867C2semidirect
9. C3×#1799C2semidirect
10. #24352C3direct (N × H)
42. #3501C2×C3semidirect
43. #2626C2×C3semidirect
45. #2062C2×C3semidirect
46. #2029C2×C3semidirect
47. #1996C2×C3semidirect
49. #1867C2×C3semidirect
50. #1799C2×C3semidirect
53. C22×C3×Q8D4semidirect
77. C3×#120Q8semidirect
91. C2×C3×C4⋊C4D4semidirect
92. C2×C3×C4⋊C4D4semidirect
156. C24×C3C2×Q8semidirect
224. C3×C22⋊C4C2×Q8semidirect
231. C22×Q8C3×D4semidirect
255. #120C3×Q8semidirect
269. C2×C4⋊C4C3×D4semidirect
270. C2×C4⋊C4C3×D4semidirect
371. C24C2×C3×Q8semidirect
439. C22⋊C4C2×C3×Q8semidirect
597. C3#24352direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#26262
C3×#23212
C3×#20622
C3×#20292
C3×#19962
C3×#18892
C3×#18672
C3×#17992
#243523