Finite Group Databasebeta

← C2×C3×#3092 #10679114

Subgroups — C2×C3×#3092 #10679114

Order 768.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#3092C2116yesmaximal, direct product factor
4C2×C3×#484C212yesmaximal, complemented
5C2×C3×#481C212yesmaximal, complemented
6C2×C3×#483C212yesmaximal, complemented
7C2×C3×#480C212yesmaximal, complemented
8C2×C3×#428C212yesmaximal, complemented
9C2×C3×#417C212yesmaximal, complemented
10C2×C3×#407C211yesmaximal, characteristic, complemented
11C2×#3092C311yesmaximal, characteristic, direct product factor, Sylow
12C22×C3×#41C2211yescharacteristic
13C24×C3×C4—21no—
14C24×C3×C4C2211yescharacteristic
15C22×C3×M16C2212yes—
16C3×#525—24no—
17C3×#525C2218yes—
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×#120—21no—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—21no—
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4—24no—
28C22×C3×C4⋊C4—21no—
29C22×C3×C4⋊C4—21no—
30C22×C3×C4⋊C4—21no—
31C22×C3×C4⋊C4C2212yes—
32C22×C3×C4⋊C4C2212yes—
33C22×C3×C4⋊C4C2212yes—
34C22×C3×C4⋊C4C2218yescomplemented
35C3×#484C22116yescomplemented
36C3×#483C22116yescomplemented
37C3×#481C22116yescomplemented
38C3×#480C22116yescomplemented
39C3×#428C22116yescomplemented
40C2×C3×#107C2214yescomplemented
41C2×C3×#106C2214yescomplemented
42C3×#417C22116yescomplemented
43C2×C3×#103C2214yescomplemented
44C2×C3×#102C2214yescomplemented
45C3×#407C2218yescomplemented
46C2×C3×C22⋊C8C2212yes—
47C2×C3×C22⋊C8C2212yes—
48C2×#525C2×C311yescharacteristic
49C23×C4⋊C4C2×C311yescharacteristic, complemented
50#3092C2×C3116yesdirect product factor
51C2×#484C2×C312yescomplemented
52C2×#481C2×C312yescomplemented
53C2×#483C2×C312yescomplemented
54C2×#480C2×C312yescomplemented
55C2×#428C2×C312yescomplemented
56C2×#417C2×C312yescomplemented
57C2×#407C2×C311yescharacteristic, complemented
58C25×C3D411yescharacteristic
59C2×C3×#41—22no—
60C2×C3×#41—28no—
61C2×C3×#41C2314yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—28no—
68C23×C3×C4—22no—
69C23×C3×C4—22no—
70C23×C3×C4—22no—
71C23×C3×C4—28no—
72C23×C3×C4—21no—
73C23×C3×C4—48no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4D411yescharacteristic
77C23×C3×C4D414yes—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2314yes—
81C23×C3×C4C2311yescharacteristic
82C2×C3×M16—28no—
83C2×C3×M16C2318yes—
84C22×C3×C8C2314yes—
85C3×#123—28no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8C2314yes—
89C3×C22⋊Q8C2314yes—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121C2314yes—
93C3×#121C2314yes—
94C3×#120—28no—
95C3×C4×D4—28no—
96C3×C4×D4—28no—
97C3×#117—22no—
98C3×#117C2314yes—
99C2×C3×C4⋊C4—24no—
100C2×C3×C4⋊C4—24no—
101C2×C3×C4⋊C4—24no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—28no—
106C2×C3×C4⋊C4—28no—
107C2×C3×C4⋊C4—28no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4C2314yes—
111C2×C3×C4⋊C4C2318yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2318yes—
115C2×C3×C4⋊C4C2318yes—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—21no—
121C2×C3×C42—21no—
122C3×#107C23116yescomplemented
123C3×#106C23116yescomplemented
124C3×#103C23116yescomplemented
125C3×#102C23116yescomplemented
126C3×C22⋊C8C2318yes—
127C3×C22⋊C8C2318yes—
128C22×#41C22×C311yescharacteristic
129C24×C4—21no—
130C24×C4C22×C311yescharacteristic
131C22×M16C22×C312yes—
132#525—24no—
133#525C22×C318yes—
134C2×#123—21no—
135C2×C22⋊Q8C22×C311yescharacteristic
136C2×C22⋊Q8C22×C311yescharacteristic
137C2×#121C22×C311yescharacteristic
138C2×#121C22×C311yescharacteristic
139C2×#120—21no—
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×#117C22×C311yescharacteristic
143C22×C4⋊C4—24no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C22×C312yes—
148C22×C4⋊C4C22×C312yes—
149C22×C4⋊C4C22×C312yes—
150C22×C4⋊C4C22×C318yescomplemented
151#484C22×C3116yescomplemented
152#483C22×C3116yescomplemented
153#481C22×C3116yescomplemented
154#480C22×C3116yescomplemented
155#428C22×C3116yescomplemented
156C2×#107C22×C314yescomplemented
157C2×#106C22×C314yescomplemented
158#417C22×C3116yescomplemented
159C2×#103C22×C314yescomplemented
160C2×#102C22×C314yescomplemented
161#407C22×C318yescomplemented
162C2×C22⋊C8C22×C312yes—
163C2×C22⋊C8C22×C312yes—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—21no—
168C24×C3C2×D414yes—
169C24×C3C2×D411yescharacteristic
170C24×C3C2×D411yescharacteristic
171C24×C3C2×D411yescharacteristic
172C3×#41—216no—
173C3×#41—48no—
174C2×C3×Q8—21no—
175C2×C3×Q8—24no—
176C2×C3×Q8C2412yes—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—21no—
181C2×C3×D4—24no—
182C2×C3×D4—44no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4C2412yes—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—216no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—216no—
193C22×C3×C4—28no—
194C22×C3×C4—28no—
195C22×C3×C4—28no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—48no—
202C22×C3×C4—48no—
203C22×C3×C4—432no—
204C22×C3×C4—48no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D414yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4118yes—
216C22×C3×C4C2412yes—
217C22×C3×C4C2412yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2411yescharacteristic
220C3×M16—216no—
221C2×C3×C8—28no—
222C2×C3×C8C2418yes—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—28no—
225C3×C4⋊C4—28no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2418yes—
232C3×C4⋊C4C2412yes—
233C3×C4⋊C4C2412yes—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C42—24no—
240C25C3×D411yescharacteristic
241C2×#41—22no—
242C2×#41—28no—
243C2×#41C23×C314yes—
244C22×Q8C23×C311yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C23×C311yescharacteristic
249C23×C4—28no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—28no—
254C23×C4—21no—
255C23×C4—48no—
256C23×C4C3×D411yescharacteristic
257C23×C4C3×D411yescharacteristic
258C23×C4C3×D411yescharacteristic
259C23×C4C3×D414yes—
260C23×C4C23×C311yescharacteristic
261C23×C4C23×C311yescharacteristic
262C23×C4C23×C314yes—
263C23×C4C23×C311yescharacteristic
264C2×M16—28no—
265C2×M16C23×C318yes—
266C22×C8C23×C314yes—
267#123—28no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8C23×C314yes—
271C22⋊Q8C23×C314yes—
272#121—22no—
273#121—22no—
274#121C23×C314yes—
275#121C23×C314yes—
276#120—28no—
277C4×D4—28no—
278C4×D4—28no—
279#117—22no—
280#117C23×C314yes—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C23×C314yes—
293C2×C4⋊C4C23×C318yes—
294C2×C4⋊C4C23×C318yes—
295C2×C4⋊C4C23×C318yes—
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—21no—
304#107C23×C3116yescomplemented
305#106C23×C3116yescomplemented
306#103C23×C3116yescomplemented
307#102C23×C3116yescomplemented
308C22⋊C8C23×C318yes—
309C22⋊C8C23×C318yes—
310C23×C3—232no—
311C23×C3—28no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—28no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—28no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3#12314yes—
325C23×C3AGL(1,ℤ/8)14yes—
326C23×C3#13714yes—
327C23×C3C22×D412yes—
328C23×C3C22×D412yes—
329C23×C3C22×D412yes—
330C23×C3C22×D411yescharacteristic
331C3×Q8—24no—
332C3×Q8—42no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—48no—
336C3×D4—48no—
337C3×D4—42no—
338C2×C3×C4—28no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—28no—
342C2×C3×C4—28no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—28no—
351C2×C3×C4—416no—
352C2×C3×C4—416no—
353C2×C3×C4—42no—
354C2×C3×C4—416no—
355C2×C3×C4—48no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—48no—
359C2×C3×C4#12314yes—
360C2×C3×C4#12318yes—
361C2×C3×C4C22×D412yes—
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4114yes—
366C2×C3×C4C2×#4118yes—
367C2×C3×C4C2511yescharacteristic
368C3×C8—28no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—21no—
373C24C2×C3×D414yes—
374C24C2×C3×D411yescharacteristic
375C24C2×C3×D411yescharacteristic
376C24C2×C3×D411yescharacteristic
377#41—216no—
378#41—48no—
379C2×Q8—21no—
380C2×Q8—24no—
381C2×Q8C24×C312yes—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4C24×C312yes—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—432no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D414yes—
414C22×C4C2×C3×D414yes—
415C22×C4C2×C3×D412yes—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D414yes—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D412yes—
420C22×C4C3×#4118yes—
421C22×C4C24×C312yes—
422C22×C4C24×C312yes—
423C22×C4C24×C311yescharacteristic
424C22×C4C24×C312yes—
425M16—216no—
426C2×C8—28no—
427C2×C8C24×C318yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4C24×C318yes—
437C4⋊C4C24×C312yes—
438C4⋊C4C24×C312yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C42—24no—
445C22×C3—232no—
446C22×C3—28no—
447C22×C3—28no—
448C22×C3—28no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—84no—
459C22×C3C2×#12314yes—
460C22×C3C2×#12312yes—
461C22×C3C2×AGL(1,ℤ/8)14yes—
462C22×C3C2×AGL(1,ℤ/8)12yes—
463C22×C3C2×#13714yes—
464C22×C3C2×#13712yes—
465C22×C3C23×D411yescharacteristic
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4—48no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4C2×#12314yes—
473C3×C4C2×#12312yes—
474C3×C4C22×#4112yes—
475C23—232no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—22no—
481C23—22no—
482C23—28no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23C3×#12314yes—
490C23C3×AGL(1,ℤ/8)14yes—
491C23C3×#13714yes—
492C23C22×C3×D412yes—
493C23C22×C3×D412yes—
494C23C22×C3×D412yes—
495C23C22×C3×D411yescharacteristic
496Q8—24no—
497Q8—42no—
498D4—24no—
499D4—48no—
500D4—48no—
501D4—48no—
502D4—42no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—416no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4C3×#12314yes—
525C2×C4C3×#12318yes—
526C2×C4C22×C3×D412yes—
527C2×C4C22×C3×D412yes—
528C2×C4C22×C3×D412yes—
529C2×C4C22×C3×D411yescharacteristic
530C2×C4C2×C3×#4114yes—
531C2×C4C2×C3×#4118yes—
532C2×C4C25×C311yescharacteristic
533C8—28no—
534C2×C3—24no—
535C2×C3—24no—
536C2×C3—24no—
537C2×C3—82no—
538C2×C3#309214yesdirect product factor
539C2×C3C22×#12311yescharacteristic
540C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
541C2×C3C22×#13711yescharacteristic
542C22—232no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—28no—
550C22—28no—
551C22—28no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—84no—
556C22C2×C3×#12314yes—
557C22C2×C3×#12312yes—
558C22C2×C3×AGL(1,ℤ/8)14yes—
559C22C2×C3×AGL(1,ℤ/8)12yes—
560C22C2×C3×#13714yes—
561C22C2×C3×#13712yes—
562C22C23×C3×D411yescharacteristic
563C4—22no—
564C4—22no—
565C4—42no—
566C4—48no—
567C4—42no—
568C4—42no—
569C4C2×C3×#12314yes—
570C4C2×C3×#12312yes—
571C4C22×C3×#4112yes—
572C3C2×#309211yescharacteristic, direct product factor, Sylow
573C2—24no—
574C2—24no—
575C2—24no—
576C2—82no—
577C2C3×#309214yesdirect product factor
578C2C22×C3×#12311yescharacteristic
579C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
580C2C22×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4⋊C4C2semidirect
3. C3×#3092C2direct (N × H)
4. C2×C3×#484C2semidirect
5. C2×C3×#481C2semidirect
6. C2×C3×#483C2semidirect
7. C2×C3×#480C2semidirect
8. C2×C3×#428C2semidirect
9. C2×C3×#417C2semidirect
10. C2×C3×#407C2semidirect
11. C2×#3092C3direct (N × H)
34. C22×C3×C4⋊C4C22semidirect
35. C3×#484C22semidirect
36. C3×#483C22semidirect
37. C3×#481C22semidirect
38. C3×#480C22semidirect
39. C3×#428C22semidirect
40. C2×C3×#107C22semidirect
41. C2×C3×#106C22semidirect
42. C3×#417C22semidirect
43. C2×C3×#103C22semidirect
44. C2×C3×#102C22semidirect
45. C3×#407C22semidirect
49. C23×C4⋊C4C2×C3semidirect
50. #3092C2×C3direct (N × H)
51. C2×#484C2×C3semidirect
52. C2×#481C2×C3semidirect
53. C2×#483C2×C3semidirect
54. C2×#480C2×C3semidirect
55. C2×#428C2×C3semidirect
56. C2×#417C2×C3semidirect
57. C2×#407C2×C3semidirect
122. C3×#107C23semidirect
123. C3×#106C23semidirect
124. C3×#103C23semidirect
125. C3×#102C23semidirect
150. C22×C4⋊C4C22×C3semidirect
151. #484C22×C3semidirect
152. #483C22×C3semidirect
153. #481C22×C3semidirect
154. #480C22×C3semidirect
155. #428C22×C3semidirect
156. C2×#107C22×C3semidirect
157. C2×#106C22×C3semidirect
158. #417C22×C3semidirect
159. C2×#103C22×C3semidirect
160. C2×#102C22×C3semidirect
161. #407C22×C3semidirect
304. #107C23×C3semidirect
305. #106C23×C3semidirect
306. #103C23×C3semidirect
307. #102C23×C3semidirect
538. C2×C3#3092direct (N × H)
572. C3C2×#3092direct (N × H)
577. C2C3×#3092direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5252
C23×C3×C4⋊C42
C3×#30922
C2×C3×#4842
C2×C3×#4812
C2×C3×#4832
C2×C3×#4802
C2×C3×#4282
C2×C3×#4172
C2×C3×#4072
C2×#30923