Finite Group Databasebeta

← C2×C3×#3330 #10679757

Subgroups — C2×C3×#3330 #10679757

Order 768.

2374 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, complemented
2C2×C3×#547C211yesmaximal, characteristic, complemented
3C3×#3330C2116yesmaximal, direct product factor
4C2×C3×#487C211yesmaximal, characteristic, complemented
5C2×C3×#484C212yesmaximal, complemented
6C2×C3×#474C211yesmaximal, characteristic, complemented
7C2×C3×#468C212yesmaximal, complemented
8C2×C3×#461C211yesmaximal, characteristic, complemented
9C2×C3×#449C212yesmaximal, complemented
10C2×C3×C4×D8C211yesmaximal, characteristic, complemented
11C2×C3×#436C211yesmaximal, characteristic, complemented
12C2×C3×#417C212yesmaximal, complemented
13C2×#3330C311yesmaximal, characteristic, direct product factor, Sylow
14C22×C3×#41—21no—
15C22×C3×#41C2212yes—
16C22×C3×SD16C2212yes—
17C22×C3×D8C2211yescharacteristic
18C22×C3×M16C2212yes—
19C3×#547—24no—
20C3×#547—24no—
21C3×#547C2218yescomplemented
22C3×#547C2218yescomplemented
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#123—22no—
26C2×C3×#123—22no—
27C2×C3×C22⋊Q8—21no—
28C2×C3×C22⋊Q8C2212yes—
29C2×C3×#121—21no—
30C2×C3×#121C2212yes—
31C2×C3×C4×D4C2211yescharacteristic
32C2×C3×C4×D4C2211yescharacteristic
33C2×C3×C4×D4C2211yescharacteristic
34C22×C3×C4⋊C4—21no—
35C22×C3×C4⋊C4C2212yes—
36C3×#487C2218yescomplemented
37C3×#484C22116yescomplemented
38C3×#474C2218yescomplemented
39C3×#468C22116yescomplemented
40C3×#461C2218yescomplemented
41C3×#449C22116yescomplemented
42C3×C4×D8C2218yescomplemented
43C3×#436C2218yescomplemented
44C2×C3×#107C2211yescharacteristic
45C2×C3×#107C2212yes—
46C2×C3×C4⋊C8C2211yescharacteristic
47C3×#417C22116yescomplemented
48C2×C3×#103C2212yes—
49C2×C3×#103C2212yes—
50C2×C3×#103C2212yes—
51C2×C3×#102C2211yescharacteristic
52C2×C3×#102C2212yes—
53C2×C3×#102C2211yescharacteristic
54C2×C3×C22⋊C8C2212yes—
55C2×C3×C4×C8C2211yescharacteristic
56C2×#547C2×C311yescharacteristic, complemented
57C2×#547C2×C311yescharacteristic, complemented
58#3330C2×C3116yesdirect product factor
59C2×#487C2×C311yescharacteristic, complemented
60C2×#484C2×C312yescomplemented
61C2×#474C2×C311yescharacteristic, complemented
62C2×#468C2×C312yescomplemented
63C2×#461C2×C311yescharacteristic, complemented
64C2×#449C2×C312yescomplemented
65C2×C4×D8C2×C311yescharacteristic, complemented
66C2×#436C2×C311yescharacteristic, complemented
67C2×#417C2×C312yescomplemented
68C2×C3×#41—28no—
69C2×C3×#41—24no—
70C2×C3×#41—28no—
71C2×C3×#41—44no—
72C2×C3×#41—48no—
73C2×C3×#41C2318yes—
74C22×C3×Q8—21no—
75C22×C3×Q8C2312yes—
76C22×C3×D4—22no—
77C22×C3×D4—22no—
78C22×C3×D4D411yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—22no—
82C23×C3×C4—21no—
83C23×C3×C4—22no—
84C23×C3×C4—21no—
85C23×C3×C4C2312yes—
86C2×C3×SD16—44no—
87C2×C3×SD16C2318yes—
88C2×C3×D8—24no—
89C2×C3×D8C2314yes—
90C2×C3×M16—44no—
91C2×C3×M16C2318yes—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2312yes—
94C22×C3×C8C2311yescharacteristic
95C3×C4⋊Q8—22no—
96C3×C4⋊Q8—22no—
97C3×C4⋊Q8C2314yes—
98C3×C4⋊Q8C2314yes—
99C3×#123—216no—
100C3×#123—216no—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8—28no—
103C3×C22⋊Q8C2318yes—
104C3×#121—28no—
105C3×#121—24no—
106C3×#121C2318yes—
107C3×C4×D4—22no—
108C3×C4×D4—22no—
109C3×C4×D4D414yes—
110C3×C4×D4C2314yes—
111C3×C4×D4C2314yes—
112C3×C4×D4C2314yes—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—24no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4—28no—
120C2×C3×C4⋊C4D411yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C4⋊C4C2311yescharacteristic
124C2×C3×C4⋊C4C2318yes—
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4—21no—
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4—22no—
129C2×C3×C22⋊C4D412yes—
130C2×C3×C42C2311yescharacteristic
131C3×#107C2314yes—
132C3×#107C2318yes—
133C3×C4⋊C8C2314yes—
134C3×#103C2318yes—
135C3×#103C2318yes—
136C3×#103C2318yes—
137C3×#102C2314yes—
138C3×#102C2318yes—
139C3×#102C2314yes—
140C3×C22⋊C8C2318yes—
141C3×C4×C8C2314yes—
142C22×#41—21no—
143C22×#41C22×C312yes—
144C22×SD16C22×C312yes—
145C22×D8C22×C311yescharacteristic
146C22×M16C22×C312yes—
147#547—24no—
148#547—24no—
149#547C22×C318yescomplemented
150#547C22×C318yescomplemented
151C2×C4⋊Q8C22×C311yescharacteristic
152C2×C4⋊Q8C22×C311yescharacteristic
153C2×#123—22no—
154C2×#123—22no—
155C2×C22⋊Q8—21no—
156C2×C22⋊Q8C22×C312yes—
157C2×#121—21no—
158C2×#121C22×C312yes—
159C2×C4×D4C22×C311yescharacteristic
160C2×C4×D4C22×C311yescharacteristic
161C2×C4×D4C22×C311yescharacteristic
162C22×C4⋊C4—21no—
163C22×C4⋊C4C22×C312yes—
164#487C22×C318yescomplemented
165#484C22×C3116yescomplemented
166#474C22×C318yescomplemented
167#468C22×C3116yescomplemented
168#461C22×C318yescomplemented
169#449C22×C3116yescomplemented
170C4×D8C22×C318yescomplemented
171#436C22×C318yescomplemented
172C2×#107C22×C311yescharacteristic
173C2×#107C22×C312yes—
174C2×C4⋊C8C22×C311yescharacteristic
175#417C22×C3116yescomplemented
176C2×#103C22×C312yes—
177C2×#103C22×C312yes—
178C2×#103C22×C312yes—
179C2×#102C22×C311yescharacteristic
180C2×#102C22×C312yes—
181C2×#102C22×C311yescharacteristic
182C2×C22⋊C8C22×C312yes—
183C2×C4×C8C22×C311yescharacteristic
184C24×C3—21no—
185C24×C3—21no—
186C24×C3C2×D412yes—
187C3×#41—216no—
188C3×#41—48no—
189C3×#41—416no—
190C3×#41—432no—
191C2×C3×Q8—22no—
192C2×C3×Q8—24no—
193C2×C3×Q8—24no—
194C2×C3×Q8—24no—
195C2×C3×Q8—44no—
196C2×C3×Q8C2414yes—
197C2×C3×D4—28no—
198C2×C3×D4—28no—
199C2×C3×D4—24no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—48no—
206C2×C3×D4—48no—
207C2×C3×D4C2×D414yes—
208C2×C3×D4#4114yescomplemented
209C2×C3×D4C2412yes—
210C2×C3×D4C2412yes—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—28no—
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—24no—
219C22×C3×C4—28no—
220C22×C3×C4—24no—
221C22×C3×C4—48no—
222C22×C3×C4—48no—
223C22×C3×C4—44no—
224C22×C3×C4—44no—
225C22×C3×C4C2×D414yes—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2414yes—
230C22×C3×C4C2411yescharacteristic
231C3×SD16—48no—
232C3×D8—28no—
233C3×M16—48no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—24no—
237C2×C3×C8C2412yes—
238C2×C3×C8C2412yes—
239C2×C3×C8C2414yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—28no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2×D414yes—
249C3×C4⋊C4C2412yes—
250C3×C4⋊C4C2414yes—
251C3×C4⋊C4C2412yes—
252C3×C22⋊C4—28no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—28no—
256C3×C22⋊C4C2×D418yes—
257C3×C42C2×D412yes—
258C3×C42C2412yes—
259C2×#41—28no—
260C2×#41—24no—
261C2×#41—28no—
262C2×#41—44no—
263C2×#41—48no—
264C2×#41C23×C318yes—
265C22×Q8—21no—
266C22×Q8C23×C312yes—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4C3×D411yescharacteristic
270C22×D4C23×C311yescharacteristic
271C22×D4C23×C311yescharacteristic
272C23×C4—22no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4C23×C312yes—
277C2×SD16—44no—
278C2×SD16C23×C318yes—
279C2×D8—24no—
280C2×D8C23×C314yes—
281C2×M16—44no—
282C2×M16C23×C318yes—
283C22×C8C23×C311yescharacteristic
284C22×C8C23×C312yes—
285C22×C8C23×C311yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8C23×C314yes—
289C4⋊Q8C23×C314yes—
290#123—216no—
291#123—216no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8C23×C318yes—
295#121—28no—
296#121—24no—
297#121C23×C318yes—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4C3×D414yes—
301C4×D4C23×C314yes—
302C4×D4C23×C314yes—
303C4×D4C23×C314yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4C3×D411yescharacteristic
312C2×C4⋊C4C23×C311yescharacteristic
313C2×C4⋊C4C23×C312yes—
314C2×C4⋊C4C23×C318yes—
315C2×C4⋊C4C23×C311yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C3×D412yes—
321C2×C42C23×C311yescharacteristic
322#107C23×C314yes—
323#107C23×C318yes—
324C4⋊C8C23×C314yes—
325#103C23×C318yes—
326#103C23×C318yes—
327#103C23×C318yes—
328#102C23×C314yes—
329#102C23×C318yes—
330#102C23×C314yes—
331C22⋊C8C23×C318yes—
332C4×C8C23×C314yes—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3—24no—
336C23×C3—24no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3C22×D414yes—
345C23×C3C22×D411yescharacteristic
346C3×Q8—24no—
347C3×Q8—24no—
348C3×Q8—44no—
349C3×Q8—48no—
350C3×D4—24no—
351C3×D4—28no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—416no—
356C3×D4—416no—
357C3×D4—42no—
358C3×D4C2×#4114yescomplemented
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—28no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—24no—
368C2×C3×C4—416no—
369C2×C3×C4—416no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—44no—
374C2×C3×C4—48no—
375C2×C3×C4—48no—
376C2×C3×C4AGL(1,ℤ/8)14yes—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D414yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D412yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4H(5,𝔽2)12yes—
383C2×C3×C4C2511yescharacteristic
384C3×C8—22no—
385C3×C8—22no—
386C3×C8—24no—
387C24—21no—
388C24—21no—
389C24C2×C3×D412yes—
390#41—216no—
391#41—48no—
392#41—416no—
393#41—432no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—44no—
399C2×Q8C24×C314yes—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4C2×C3×D414yes—
411C2×D4C3×#4114yescomplemented
412C2×D4C24×C312yes—
413C2×D4C24×C312yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D412yes—
430C22×C4C2×C3×D414yes—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C24×C314yes—
433C22×C4C24×C311yescharacteristic
434SD16—48no—
435D8—28no—
436M16—48no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8C24×C312yes—
441C2×C8C24×C312yes—
442C2×C8C24×C314yes—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—22no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4C2×C3×D414yes—
452C4⋊C4C24×C312yes—
453C4⋊C4C24×C314yes—
454C4⋊C4C24×C312yes—
455C22⋊C4—28no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4C2×C3×D418yes—
460C42C2×C3×D412yes—
461C42C24×C312yes—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—48no—
467C22×C3—48no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—84no—
474C22×C3#54612yes—
475C22×C3C2×AGL(1,ℤ/8)12yes—
476C22×C3#57812yes—
477C22×C3C23×D411yescharacteristic
478C3×C4—24no—
479C3×C4—22no—
480C3×C4—44no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—44no—
484C3×C4#54612yes—
485C3×C4C2×AGL(1,ℤ/8)14yes—
486C3×C4C22×#4111yescharacteristic
487C3×C4C2×H(5,𝔽2)11yescharacteristic
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C22×C3×D414yes—
500C23C22×C3×D411yescharacteristic
501Q8—24no—
502Q8—24no—
503Q8—48no—
504Q8—44no—
505D4—28no—
506D4—22no—
507D4—22no—
508D4—22no—
509D4—24no—
510D4—416no—
511D4—416no—
512D4—42no—
513D4C2×C3×#4114yescomplemented
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4C3×AGL(1,ℤ/8)14yes—
532C2×C4C22×C3×D412yes—
533C2×C4C22×C3×D414yes—
534C2×C4C22×C3×D412yes—
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C2×C3×#4112yes—
537C2×C4C3×H(5,𝔽2)12yes—
538C2×C4C25×C311yescharacteristic
539C8—22no—
540C8—22no—
541C8—24no—
542C2×C3—44no—
543C2×C3—44no—
544C2×C3—82no—
545C2×C3#333014yesdirect product factor
546C2×C3C2×#54611yescharacteristic
547C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
548C2×C3C2×#57811yescharacteristic
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—48no—
556C22—48no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—84no—
561C22C3×#54612yes—
562C22C2×C3×AGL(1,ℤ/8)12yes—
563C22C3×#57812yes—
564C22C23×C3×D411yescharacteristic
565C4—24no—
566C4—22no—
567C4—44no—
568C4—42no—
569C4—44no—
570C4—42no—
571C4C3×#54612yes—
572C4C2×C3×AGL(1,ℤ/8)14yes—
573C4C22×C3×#4111yescharacteristic
574C4C2×C3×H(5,𝔽2)11yescharacteristic
575C3C2×#333011yescharacteristic, direct product factor, Sylow
576C2—44no—
577C2—44no—
578C2—82no—
579C2C3×#333014yesdirect product factor
580C2C2×C3×#54611yescharacteristic
581C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
582C2C2×C3×#57811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
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
1. C2×C3×#547C2semidirect
2. C2×C3×#547C2semidirect
3. C3×#3330C2direct (N × H)
4. C2×C3×#487C2semidirect
5. C2×C3×#484C2semidirect
6. C2×C3×#474C2semidirect
7. C2×C3×#468C2semidirect
8. C2×C3×#461C2semidirect
9. C2×C3×#449C2semidirect
10. C2×C3×C4×D8C2semidirect
11. C2×C3×#436C2semidirect
12. C2×C3×#417C2semidirect
13. C2×#3330C3direct (N × H)
21. C3×#547C22semidirect
22. C3×#547C22semidirect
36. C3×#487C22semidirect
37. C3×#484C22semidirect
38. C3×#474C22semidirect
39. C3×#468C22semidirect
40. C3×#461C22semidirect
41. C3×#449C22semidirect
42. C3×C4×D8C22semidirect
43. C3×#436C22semidirect
47. C3×#417C22semidirect
56. C2×#547C2×C3semidirect
57. C2×#547C2×C3semidirect
58. #3330C2×C3direct (N × H)
59. C2×#487C2×C3semidirect
60. C2×#484C2×C3semidirect
61. C2×#474C2×C3semidirect
62. C2×#468C2×C3semidirect
63. C2×#461C2×C3semidirect
64. C2×#449C2×C3semidirect
65. C2×C4×D8C2×C3semidirect
66. C2×#436C2×C3semidirect
67. C2×#417C2×C3semidirect
149. #547C22×C3semidirect
150. #547C22×C3semidirect
164. #487C22×C3semidirect
165. #484C22×C3semidirect
166. #474C22×C3semidirect
167. #468C22×C3semidirect
168. #461C22×C3semidirect
169. #449C22×C3semidirect
170. C4×D8C22×C3semidirect
171. #436C22×C3semidirect
175. #417C22×C3semidirect
208. C2×C3×D4#41semidirect
358. C3×D4C2×#41semidirect
411. C2×D4C3×#41semidirect
513. D4C2×C3×#41semidirect
545. C2×C3#3330direct (N × H)
575. C3C2×#3330direct (N × H)
579. C2C3×#3330direct (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
C3×#33302
C2×C3×#4872
C2×C3×#4842
C2×C3×#4742
C2×C3×#4682
C2×C3×#4612
C2×C3×#4492
C2×C3×C4×D82
C2×C3×#4362
C2×C3×#4172
C2×#33303