Finite Group Databasebeta

← C3×#22417 #10641024

Subgroups — C3×#22417 #10641024

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×C8×D4C211yesmaximal, characteristic, complemented
3C2×C3×#416C212yesmaximal, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#1981C211yesmaximal, characteristic, complemented
6C3×#1956C211yesmaximal, characteristic, complemented
7C3×#1892C212yesmaximal, complemented
8C3×#1873C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic, complemented
10C3×#1788C212yesmaximal, complemented
11C3×C4×#102C211yesmaximal, characteristic, complemented
12#22417C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C23×C3×C8C2212yes—
15C3×#547—24no—
16C3×#547C418yescomplemented
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#121—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117C2212yes—
24C22×C3×C4⋊C4—21no—
25C3×C8×D4—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×#416—28no—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C3×#390—21no—
38C3×#388—22no—
39C3×#384C2211yescharacteristic
40C3×#380C2212yes—
41C3×C4×C4⋊C4—21no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2211yescharacteristic
44C3×#340C2211yescharacteristic
45C3×#340C2212yes—
46C2×#547C2×C311yescharacteristic
47C2×C8×D4C2×C311yescharacteristic, complemented
48C2×#416C2×C312yescomplemented
49#2354C2×C311yescharacteristic, complemented
50#1981C2×C311yescharacteristic, complemented
51#1956C2×C311yescharacteristic, complemented
52#1892C2×C312yescomplemented
53#1873C2×C312yescomplemented
54#1802C2×C311yescharacteristic, complemented
55#1788C2×C312yescomplemented
56C4×#102C2×C311yescharacteristic, complemented
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C2×C3×#41C2×C418yes—
63C22×C3×Q8C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C23×C3×C4C2312yes—
70C22×C3×C8—24no—
71C22×C3×C8—24no—
72C22×C3×C8—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8C2×C414yes—
78C3×#123—216no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8C2×C418yes—
81C3×#121—28no—
82C3×C4×D4—22no—
83C3×C4×D4C2×C414yes—
84C3×C4×D4C2×C414yes—
85C3×C4×D4D414yes—
86C3×#117—24no—
87C3×#117—42no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—28no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4D412yes—
104C2×C3×C42—21no—
105C2×C3×C42—22no—
106C2×C3×C42C2311yescharacteristic
107C3×C4⋊C8—22no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#103—42no—
111C3×#103—42no—
112C3×#102—24no—
113C3×#102—22no—
114C3×#102D414yescomplemented
115C3×C22⋊C8—24no—
116C3×C4×C8—22no—
117C3×#95—22no—
118C3×#95—21no—
119C22×#41C22×C312yes—
120C23×C8C22×C312yes—
121#547—24no—
122#547C3×C418yescomplemented
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×#123—22no—
125C2×C22⋊Q8C22×C312yes—
126C2×#121—21no—
127C2×C4×D4C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129C2×#117C22×C312yes—
130C22×C4⋊C4—21no—
131C8×D4—24no—
132C2×C4⋊C8C22×C311yescharacteristic
133#416—28no—
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C311yescharacteristic
141C2×C22⋊C8C22×C312yes—
142C2×C4×C8C22×C311yescharacteristic
143#390—21no—
144#388—22no—
145#384C22×C311yescharacteristic
146#380C22×C312yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C22×C311yescharacteristic
149#340C22×C312yes—
150#340C22×C311yescharacteristic
151#340C22×C311yescharacteristic
152C24×C3—21no—
153C24×C3C2×D412yes—
154C3×#41—216no—
155C3×#41—48no—
156C3×#41—48no—
157C3×#41—48no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C22×C414yes—
164C2×C3×D4—28no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4C22⋊C414yes—
173C2×C3×D4C22×C412yes—
174C2×C3×D4C2×D412yes—
175C2×C3×D4#4112yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—28no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—44no—
188C22×C3×C4—44no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C22×C414yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—22no—
205C2×C3×C8—48no—
206C2×C3×C8C2×D412yes—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—81no—
218C3×C4⋊C4C22⋊C414yes—
219C3×C4⋊C4C22×C412yes—
220C3×C4⋊C4C2×D412yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—82no—
225C3×C22⋊C4C22⋊C418yes—
226C3×C42—44no—
227C3×C42—42no—
228C3×C42C22×C412yes—
229C3×C42C2×D412yes—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—42no—
234C2×#41—42no—
235C2×#41C2×C3×C418yes—
236C22×Q8C23×C312yes—
237C22×D4—22no—
238C22×D4C3×D411yescharacteristic
239C22×D4C23×C311yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C23×C312yes—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—42no—
246C22×C8C23×C312yes—
247C22×C8C23×C311yescharacteristic
248C22×C8C23×C311yescharacteristic
249C4⋊Q8—22no—
250C4⋊Q8C2×C3×C414yes—
251#123—216no—
252C22⋊Q8—24no—
253C22⋊Q8C2×C3×C418yes—
254#121—28no—
255C4×D4—22no—
256C4×D4C2×C3×C414yes—
257C4×D4C2×C3×C414yes—
258C4×D4C3×D414yes—
259#117—24no—
260#117—42no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C3×D412yes—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42C23×C311yescharacteristic
280C4⋊C8—22no—
281#103—24no—
282#103—24no—
283#103—42no—
284#103—42no—
285#102—24no—
286#102—22no—
287#102C3×D414yescomplemented
288C22⋊C8—24no—
289C4×C8—22no—
290#95—22no—
291#95—21no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3C2×C22⋊C414yes—
306C23×C3C22×D411yescharacteristic
307C3×Q8—24no—
308C3×Q8—24no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—28no—
315C3×D4—416no—
316C3×D4C4×D414yes—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—416no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—48no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4C2×C22⋊C414yes—
344C2×C3×C4C2×C22⋊C412yes—
345C2×C3×C4#13512yes—
346C2×C3×C4#13512yes—
347C2×C3×C4C23×C411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C3×C8—22no—
354C3×C8—42no—
355C3×C8—41no—
356C24—21no—
357C24C2×C3×D412yes—
358#41—216no—
359#41—48no—
360#41—48no—
361#41—48no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8C22×C3×C414yes—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4C3×C22⋊C414yes—
377C2×D4C22×C3×C412yes—
378C2×D4C2×C3×D412yes—
379C2×D4C3×#4112yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C22×C3×C414yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D414yes—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C24×C311yescharacteristic
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—48no—
410C2×C8C2×C3×D412yes—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4C3×C22⋊C414yes—
423C4⋊C4C22×C3×C412yes—
424C4⋊C4C2×C3×D412yes—
425C22⋊C4—28no—
426C22⋊C4—24no—
427C22⋊C4—44no—
428C22⋊C4—82no—
429C22⋊C4C3×C22⋊C418yes—
430C42—44no—
431C42—42no—
432C42C22×C3×C412yes—
433C42C2×C3×D412yes—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3C22×C22⋊C411yescharacteristic
449C22×C3D4211yescharacteristic
450C22×C3#54611yescharacteristic
451C22×C3C2×#13511yescharacteristic
452C22×C3C2×#13511yescharacteristic
453C22×C3#57711yescharacteristic
454C22×C3#57811yescharacteristic
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—82no—
461C3×C4—81no—
462C3×C4#41614yescomplemented
463C3×C4C2×C4×D411yescharacteristic
464C3×C4#51811yescharacteristic
465C3×C4D4211yescharacteristic
466C3×C4#54611yescharacteristic
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C2×C3×C22⋊C414yes—
481C23C22×C3×D411yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—42no—
485Q8—42no—
486D4—28no—
487D4—22no—
488D4—22no—
489D4—22no—
490D4—416no—
491D4C3×C4×D414yes—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—416no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4C2×C3×C22⋊C412yes—
518C2×C4C2×C3×C22⋊C414yes—
519C2×C4C2×C3×C22⋊C412yes—
520C2×C4C3×#13512yes—
521C2×C4C3×#13512yes—
522C2×C4C23×C3×C411yescharacteristic
523C2×C4C22×C3×D411yescharacteristic
524C2×C4C22×C3×D411yescharacteristic
525C2×C4C2×C3×#4111yescharacteristic
526C2×C4C3×H(5,𝔽2)11yescharacteristic
527C2×C4C3×H(5,𝔽2)11yescharacteristic
528C8—22no—
529C8—42no—
530C8—41no—
531C2×C3—44no—
532C2×C3—44no—
533C2×C3D4×C22⋊C411yescharacteristic
534C2×C3C2×#41611yescharacteristic
535C2×C3#289811yescharacteristic
536C2×C3#328311yescharacteristic
537C2×C3#328411yescharacteristic
538C2×C3#330211yescharacteristic
539C2×C3#330311yescharacteristic
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22C22×C3×C22⋊C411yescharacteristic
555C22C3×D4211yescharacteristic
556C22C3×#54611yescharacteristic
557C22C2×C3×#13511yescharacteristic
558C22C2×C3×#13511yescharacteristic
559C22C3×#57711yescharacteristic
560C22C3×#57811yescharacteristic
561C4—22no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—81no—
568C4C3×#41614yescomplemented
569C4C2×C3×C4×D411yescharacteristic
570C4C3×#51811yescharacteristic
571C4C3×D4211yescharacteristic
572C4C3×#54611yescharacteristic
573C3#2241711yescharacteristic, direct product factor, Sylow
574C2—44no—
575C2—44no—
576C2C3×D4×C22⋊C411yescharacteristic
577C2C2×C3×#41611yescharacteristic
578C2C3×#289811yescharacteristic
579C2C3×#328311yescharacteristic
580C2C3×#328411yescharacteristic
581C2C3×#330211yescharacteristic
582C2C3×#330311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×D4C2semidirect
3. C2×C3×#416C2semidirect
4. C3×#2354C2semidirect
5. C3×#1981C2semidirect
6. C3×#1956C2semidirect
7. C3×#1892C2semidirect
8. C3×#1873C2semidirect
9. C3×#1802C2semidirect
10. C3×#1788C2semidirect
11. C3×C4×#102C2semidirect
12. #22417C3direct (N × H)
16. C3×#547C4semidirect
47. C2×C8×D4C2×C3semidirect
48. C2×#416C2×C3semidirect
49. #2354C2×C3semidirect
50. #1981C2×C3semidirect
51. #1956C2×C3semidirect
52. #1892C2×C3semidirect
53. #1873C2×C3semidirect
54. #1802C2×C3semidirect
55. #1788C2×C3semidirect
56. C4×#102C2×C3semidirect
114. C3×#102D4semidirect
122. #547C3×C4semidirect
287. #102C3×D4semidirect
462. C3×C4#416semidirect
568. C4C3×#416semidirect
573. C3#22417direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×C8×D42
C2×C3×#4162
C3×#23542
C3×#19812
C3×#19562
C3×#18922
C3×#18732
C3×#18022
C3×#17882
C3×C4×#1022
#224173