Finite Group Databasebeta

← C3×#15671 #10634278

Subgroups — C3×#15671 #10634278

Order 768.

1890 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3C2×C3×#393C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C3×#1446C212yesmaximal, complemented
6C3×#1445C211yesmaximal, characteristic, complemented
7C3×#1441C212yesmaximal, complemented
8C3×#1439C212yesmaximal
9C3×#1437C211yesmaximal, characteristic
10C3×#1436C211yesmaximal, characteristic, complemented
11C3×C22⋊#95C212yesmaximal
12#15671C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C2×C3×C22⋊Q8—24no—
17C2×C3×C22⋊Q8—24no—
18C2×C3×C22⋊Q8—24no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2212yes—
25C22×C3×C22⋊C4C2211yescharacteristic
26C22×C3×C22⋊C4C2212yes—
27C22×C3×C42C2211yescharacteristic
28C3×#395—24no—
29C3×#393—24no—
30C3×#391—24no—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C22×C22⋊Q8C2×C311yescharacteristic
43C2×#395C2×C311yescharacteristic, complemented
44C2×#393C2×C311yescharacteristic
45C2×#391C2×C311yescharacteristic, complemented
46#1446C2×C312yescomplemented
47#1445C2×C311yescharacteristic, complemented
48#1441C2×C312yescomplemented
49#1439C2×C312yes—
50#1437C2×C311yescharacteristic
51#1436C2×C311yescharacteristic, complemented
52C22⋊#95C2×C312yes—
53C25×C3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8—24no—
61C22×C3×Q8—24no—
62C22×C3×Q8—42no—
63C22×C3×Q8—42no—
64C23×C3×C4—22no—
65C23×C3×C4—22no—
66C23×C3×C4—22no—
67C23×C3×C4—24no—
68C23×C3×C4—42no—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C23×C3×C4C2312yes—
75C23×C3×C4C2312yes—
76C23×C3×C4C2312yes—
77C23×C3×C4C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C3×C22⋊Q8—232no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4—42no—
90C2×C3×C4⋊C4D414yes—
91C2×C3×C4⋊C4D414yes—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—24no—
99C2×C3×C22⋊C4—42no—
100C2×C3×C22⋊C4—42no—
101C2×C3×C22⋊C4—42no—
102C2×C3×C22⋊C4D414yes—
103C2×C3×C22⋊C4Q818yescomplemented
104C2×C3×C42—22no—
105C2×C3×C42—42no—
106C3×#95—24no—
107C3×#95—24no—
108C3×#95—24no—
109C3×#95—22no—
110C3×#95—22no—
111C3×#95—24no—
112C3×#95—24no—
113C3×#95—42no—
114C3×#95—42no—
115C3×#95—42no—
116C3×#95—42no—
117C23×Q8C22×C311yescharacteristic
118C23×Q8C22×C311yescharacteristic
119C24×C4C22×C311yescharacteristic
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C22×C4⋊C4C22×C311yescharacteristic
124C22×C4⋊C4C22×C312yes—
125C22×C4⋊C4C22×C311yescharacteristic
126C22×C4⋊C4C22×C311yescharacteristic
127C22×C22⋊C4C22×C311yescharacteristic
128C22×C22⋊C4C22×C312yes—
129C22×C22⋊C4C22×C311yescharacteristic
130C22×C22⋊C4C22×C312yes—
131C22×C42C22×C311yescharacteristic
132#395—24no—
133#393—24no—
134#391—24no—
135C2×#95C22×C311yescharacteristic
136C2×#95C22×C312yes—
137C2×#95C22×C312yes—
138C2×#95C22×C311yescharacteristic
139C2×#95C22×C312yes—
140C2×#95C22×C312yes—
141C2×#95C22×C312yes—
142C2×#95C22×C312yes—
143C2×#95C22×C312yes—
144C2×#95C22×C312yes—
145C2×#95C22×C312yes—
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3C2×D412yes—
154C24×C3C2×Q814yes—
155C24×C3C2411yescharacteristic
156C2×C3×Q8—28no—
157C2×C3×Q8—28no—
158C2×C3×Q8—48no—
159C2×C3×Q8—48no—
160C2×C3×Q8—48no—
161C2×C3×Q8—48no—
162C2×C3×Q8—44no—
163C2×C3×Q8—44no—
164C2×C3×Q8—44no—
165C2×C3×Q8—44no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—28no—
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—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—48no—
214C22×C3×C4—48no—
215C22×C3×C4—44no—
216C22×C3×C4—44no—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×Q814yes—
226C22×C3×C4C2×Q814yes—
227C22×C3×C4#4114yes—
228C22×C3×C4#4114yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—48no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—48no—
236C3×C22⋊C4—48no—
237C3×C42—44no—
238C25C23×C311yescharacteristic
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8—42no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—42no—
254C23×C4C23×C312yes—
255C23×C4C23×C311yescharacteristic
256C23×C4C23×C311yescharacteristic
257C23×C4C23×C311yescharacteristic
258C23×C4C23×C311yescharacteristic
259C23×C4C23×C312yes—
260C23×C4C23×C312yes—
261C23×C4C23×C312yes—
262C23×C4C23×C311yescharacteristic
263C23×C4C23×C311yescharacteristic
264C22⋊Q8—232no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C3×D414yes—
276C2×C4⋊C4C3×D414yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4C3×D414yes—
288C2×C22⋊C4C3×Q818yescomplemented
289C2×C42—22no—
290C2×C42—42no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C4⋊Q814yes—
323C23×C3C22×D411yescharacteristic
324C23×C3C22×D411yescharacteristic
325C23×C3C22×D411yescharacteristic
326C23×C3C22×Q812yes—
327C23×C3C2×#4112yes—
328C23×C3C2×#4112yes—
329C23×C3H(5,𝔽2)12yes—
330C23×C3H(5,𝔽2)12yes—
331C23×C32−1+412yes—
332C3×Q8—416no—
333C3×Q8—416no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—416no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4#12014yes—
392C2×C3×C4#12414yes—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C2×C3×D412yes—
401C24C2×C3×Q814yes—
402C24C24×C311yescharacteristic
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4C2×C3×D412yes—
465C22×C4C2×C3×D412yes—
466C22×C4C2×C3×D412yes—
467C22×C4C2×C3×D412yes—
468C22×C4C2×C3×D412yes—
469C22×C4C2×C3×D412yes—
470C22×C4C2×C3×D412yes—
471C22×C4C2×C3×D412yes—
472C22×C4C2×C3×Q814yes—
473C22×C4C2×C3×Q814yes—
474C22×C4C3×#4114yes—
475C22×C4C3×#4114yes—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C42—44no—
485C22×C3—24no—
486C22×C3—28no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3C2×#12011yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3C2×#12411yescharacteristic
497C22×C3C2×C4⋊Q811yescharacteristic
498C22×C3#53312yes—
499C22×C3#53511yescharacteristic
500C22×C3#53511yescharacteristic
501C22×C3#53611yescharacteristic
502C22×C3#54212yes—
503C22×C3#54212yes—
504C22×C3#54312yes—
505C22×C3#54612yes—
506C22×C3#54612yes—
507C22×C3#54612yes—
508C22×C3#54612yes—
509C22×C3D4×Q812yes—
510C22×C3D4×Q812yes—
511C22×C3#55412yes—
512C22×C3#55412yes—
513C22×C3#55612yes—
514C22×C3#56212yes—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—44no—
523C3×C4—44no—
524C3×C4—82no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C3×C4⋊Q814yes—
546C23C22×C3×D411yescharacteristic
547C23C22×C3×D411yescharacteristic
548C23C22×C3×D411yescharacteristic
549C23C22×C3×Q812yes—
550C23C2×C3×#4112yes—
551C23C2×C3×#4112yes—
552C23C3×H(5,𝔽2)12yes—
553C23C3×H(5,𝔽2)12yes—
554C23C3×2−1+412yes—
555Q8—416no—
556Q8—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4C3×#12014yes—
615C2×C4C3×#12414yes—
616C2×C3—44no—
617C2×C3#240411yescharacteristic
618C2×C3#243411yescharacteristic
619C2×C3#243511yescharacteristic
620C2×C3#246512yes—
621C2×C3#249212yes—
622C2×C3#254912yes—
623C2×C3#269212yes—
624C2×C3#269212yes—
625C2×C3#270012yes—
626C22—24no—
627C22—28no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22C2×C3×#12011yescharacteristic
636C22C3×#52511yescharacteristic
637C22C2×C3×#12411yescharacteristic
638C22C2×C3×C4⋊Q811yescharacteristic
639C22C3×#53312yes—
640C22C3×#53511yescharacteristic
641C22C3×#53511yescharacteristic
642C22C3×#53611yescharacteristic
643C22C3×#54212yes—
644C22C3×#54212yes—
645C22C3×#54312yes—
646C22C3×#54612yes—
647C22C3×#54612yes—
648C22C3×#54612yes—
649C22C3×#54612yes—
650C22C3×D4×Q812yes—
651C22C3×D4×Q812yes—
652C22C3×#55412yes—
653C22C3×#55412yes—
654C22C3×#55612yes—
655C22C3×#56212yes—
656C4—24no—
657C4—24no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—82no—
666C3#1567111yescharacteristic, direct product factor, Sylow
667C2—44no—
668C2C3×#240411yescharacteristic
669C2C3×#243411yescharacteristic
670C2C3×#243511yescharacteristic
671C2C3×#246512yes—
672C2C3×#249212yes—
673C2C3×#254912yes—
674C2C3×#269212yes—
675C2C3×#269212yes—
676C2C3×#270012yes—
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#395C2semidirect
4. C2×C3×#391C2semidirect
5. C3×#1446C2semidirect
6. C3×#1445C2semidirect
7. C3×#1441C2semidirect
10. C3×#1436C2semidirect
12. #15671C3direct (N × H)
43. C2×#395C2×C3semidirect
45. C2×#391C2×C3semidirect
46. #1446C2×C3semidirect
47. #1445C2×C3semidirect
48. #1441C2×C3semidirect
51. #1436C2×C3semidirect
103. C2×C3×C22⋊C4Q8semidirect
288. C2×C22⋊C4C3×Q8semidirect
666. C3#15671direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#3952
C2×C3×#3932
C2×C3×#3912
C3×#14462
C3×#14452
C3×#14412
C3×#14392
C3×#14372
C3×#14362
C3×C22⋊#952
#156713