Finite Group Databasebeta

← C2×C3×#1932 #10639686

Subgroups — C2×C3×#1932 #10639686

Order 768.

1046 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#430C211yesmaximal, characteristic, complemented
2C2×C3×#421C211yesmaximal, characteristic
3C22×C3×#104C211yesmaximal, characteristic
4C2×C3×#413C211yesmaximal, characteristic
5C2×C3×C4×M16C211yesmaximal, characteristic, complemented
6C3×#1932C218yesmaximal, direct product factor
7C2×C3×#344C211yesmaximal, characteristic, complemented
8C2×C3×#339C211yesmaximal, characteristic, complemented
9C2×#1932C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C3×#430—22no—
18C3×#430C2214yescomplemented
19C2×C3×#108—22no—
20C3×#421—22no—
21C3×#421C2214yes—
22C2×C3×#104—22no—
23C2×C3×#104—21no—
24C2×C3×#104—21no—
25C2×C3×#104—22no—
26C2×C3×#104C2214yes—
27C3×#413C414yescomplemented
28C3×#413C2214yes—
29C2×C3×#101C412yescomplemented
30C2×C3×#100C412yescomplemented
31C3×C4×M16—22no—
32C3×C4×M16C2214yescomplemented
33C2×C3×#97—21no—
34C2×C3×C4×C8—21no—
35C3×#344C2214yescomplemented
36C3×#339C2214yescomplemented
37C2×#430C2×C311yescharacteristic, complemented
38C2×#421C2×C311yescharacteristic
39C22×#104C2×C311yescharacteristic
40C2×#413C2×C311yescharacteristic
41C2×C4×M16C2×C311yescharacteristic, complemented
42#1932C2×C318yesdirect product factor
43C2×#344C2×C311yescharacteristic, complemented
44C2×#339C2×C311yescharacteristic, complemented
45C2×C3×#41—21no—
46C2×C3×#41—21no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C2×C3×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41—21no—
52C2×C3×#41—21no—
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41C2×C412yes—
56C2×C3×#41C2312yes—
57C22×C3×Q8C2×C411yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2×C411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16—21no—
70C2×C3×M16—21no—
71C2×C3×M16—21no—
72C2×C3×M16—21no—
73C2×C3×M16—21no—
74C2×C3×M16—21no—
75C2×C3×M16—21no—
76C2×C3×M16—21no—
77C2×C3×M16C2×C412yes—
78C2×C3×M16C2×C412yes—
79C2×C3×M16C2×C412yes—
80C2×C3×M16C2×C412yes—
81C2×C3×M16D412yes—
82C2×C3×M16D412yes—
83C2×C3×M16C2312yes—
84C2×C3×M16C2312yes—
85C2×C3×M16C2312yes—
86C2×C3×M16C2312yes—
87C22×C3×C8—21no—
88C22×C3×C8—21no—
89C22×C3×C8—21no—
90C22×C3×C8—21no—
91C3×#117—21no—
92C3×#117—21no—
93C3×#117—21no—
94C3×#117C2312yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42D411yescharacteristic
102C2×C3×C42D411yescharacteristic
103C2×C3×C42D412yes—
104C2×C3×C42C2312yes—
105C3×#108—28no—
106C3×#104—28no—
107C3×#104—24no—
108C3×#104—24no—
109C3×#101C2×C418yescomplemented
110C3×#100C2×C418yescomplemented
111C3×#97—24no—
112C3×C4×C8—24no—
113C22×#41C22×C311yescharacteristic
114C22×M16C22×C311yescharacteristic
115C22×M16C22×C311yescharacteristic
116C22×M16C22×C311yescharacteristic
117C22×M16C22×C311yescharacteristic
118C2×#117C22×C311yescharacteristic
119C22×C42C22×C311yescharacteristic
120#430—22no—
121#430C22×C314yescomplemented
122C2×#108—22no—
123#421—22no—
124#421C22×C314yes—
125C2×#104—22no—
126C2×#104—21no—
127C2×#104—21no—
128C2×#104—22no—
129C2×#104C22×C314yes—
130#413C3×C414yescomplemented
131#413C22×C314yes—
132C2×#101C3×C412yescomplemented
133C2×#100C3×C412yescomplemented
134C4×M16—22no—
135C4×M16C22×C314yescomplemented
136C2×#97—21no—
137C2×C4×C8—21no—
138#344C22×C314yescomplemented
139#339C22×C314yescomplemented
140C24×C3—21no—
141C24×C3#4111yescharacteristic
142C3×#41—22no—
143C3×#41—22no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—22no—
149C3×#41—22no—
150C3×#41—41no—
151C3×#41—41no—
152C3×#41—41no—
153C3×#41—41no—
154C3×#41—41no—
155C3×#41—41no—
156C3×#41—41no—
157C3×#41—41no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8C22×C412yes—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—41no—
172C2×C3×D4—41no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4C22×C412yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—44no—
184C22×C3×C4—41no—
185C22×C3×C4—41no—
186C22×C3×C4—41no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4C22×C411yescharacteristic
190C22×C3×C4C22×C411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4111yescharacteristic
196C22×C3×C4#4111yescharacteristic
197C22×C3×C4#4111yescharacteristic
198C22×C3×C4#4111yescharacteristic
199C22×C3×C4#4111yescharacteristic
200C22×C3×C4#4111yescharacteristic
201C22×C3×C4#4111yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C3×M16—22no—
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16—22no—
214C3×M16—41no—
215C3×M16—41no—
216C3×M16C22×C414yes—
217C3×M16C22×C414yes—
218C3×M16C2×D414yes—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—41no—
230C2×C3×C8—41no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—41no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—41no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42—21no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—41no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243C2×#41—21no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41—21no—
247C2×#41—21no—
248C2×#41—21no—
249C2×#41—21no—
250C2×#41—21no—
251C2×#41—21no—
252C2×#41—21no—
253C2×#41C2×C3×C412yes—
254C2×#41C23×C312yes—
255C22×Q8C2×C3×C411yescharacteristic
256C22×D4—21no—
257C22×D4C2×C3×C411yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16—21no—
274C2×M16—21no—
275C2×M16C2×C3×C412yes—
276C2×M16C2×C3×C412yes—
277C2×M16C2×C3×C412yes—
278C2×M16C2×C3×C412yes—
279C2×M16C3×D412yes—
280C2×M16C3×D412yes—
281C2×M16C23×C312yes—
282C2×M16C23×C312yes—
283C2×M16C23×C312yes—
284C2×M16C23×C312yes—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117C23×C312yes—
293C2×C4⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C3×D411yescharacteristic
300C2×C42C3×D411yescharacteristic
301C2×C42C3×D412yes—
302C2×C42C23×C312yes—
303#108—28no—
304#104—28no—
305#104—24no—
306#104—24no—
307#101C2×C3×C418yescomplemented
308#100C2×C3×C418yescomplemented
309#97—24no—
310C4×C8—24no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—82no—
316C23×C3#11711yescharacteristic
317C23×C3#11711yescharacteristic
318C23×C3#12311yescharacteristic
319C23×C3#12311yescharacteristic
320C23×C3#12411yescharacteristic
321C23×C3#12411yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—41no—
334C3×D4—41no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—82no—
350C2×C3×C4#11711yescharacteristic
351C2×C3×C4#11711yescharacteristic
352C2×C3×C4C4×D411yescharacteristic
353C2×C3×C4C4×D411yescharacteristic
354C2×C3×C4C4×D411yescharacteristic
355C2×C3×C4C4×D411yescharacteristic
356C2×C3×C4C4×D411yescharacteristic
357C2×C3×C4C4×D411yescharacteristic
358C2×C3×C4C4×D411yescharacteristic
359C2×C3×C4C4×D411yescharacteristic
360C2×C3×C4#12111yescharacteristic
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12311yescharacteristic
365C2×C3×C4#12311yescharacteristic
366C2×C3×C4#12411yescharacteristic
367C2×C3×C4#12411yescharacteristic
368C2×C3×C4#12611yescharacteristic
369C2×C3×C4#12611yescharacteristic
370C2×C3×C4#12611yescharacteristic
371C2×C3×C4#12611yescharacteristic
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C3×C8—24no—
379C3×C8—24no—
380C3×C8—24no—
381C3×C8—42no—
382C24—21no—
383C24C3×#4111yescharacteristic
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8C22×C3×C412yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4C22×C3×C412yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4C22×C3×C411yescharacteristic
432C22×C4C22×C3×C411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C3×#4111yescharacteristic
438C22×C4C3×#4111yescharacteristic
439C22×C4C3×#4111yescharacteristic
440C22×C4C3×#4111yescharacteristic
441C22×C4C3×#4111yescharacteristic
442C22×C4C3×#4111yescharacteristic
443C22×C4C3×#4111yescharacteristic
444C22×C4C24×C311yescharacteristic
445M16—22no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—41no—
457M16—41no—
458M16C22×C3×C414yes—
459M16C22×C3×C414yes—
460M16C2×C3×D414yes—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—41no—
472C2×C8—41no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C42—21no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—22no—
482C42—41no—
483C42C2×C3×D412yes—
484C42C2×C3×D412yes—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3#38811yescharacteristic
497C22×C3#38811yescharacteristic
498C22×C3#38811yescharacteristic
499C22×C3#38811yescharacteristic
500C22×C3C2×#11711yescharacteristic
501C22×C3C2×#12311yescharacteristic
502C22×C3C2×#12411yescharacteristic
503C3×C4—24no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—82no—
507C3×C4#38811yescharacteristic
508C3×C4#38811yescharacteristic
509C3×C4#38811yescharacteristic
510C3×C4#38811yescharacteristic
511C3×C4C2×C4×D411yescharacteristic
512C3×C4C2×C4×D411yescharacteristic
513C3×C4C2×#12111yescharacteristic
514C3×C4C2×#12611yescharacteristic
515C23—22no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—82no—
520C23C3×#11711yescharacteristic
521C23C3×#11711yescharacteristic
522C23C3×#12311yescharacteristic
523C23C3×#12311yescharacteristic
524C23C3×#12411yescharacteristic
525C23C3×#12411yescharacteristic
526C23C2×C3×#4111yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—22no—
532D4—22no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—41no—
538D4—41no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4C3×#11711yescharacteristic
555C2×C4C3×#11711yescharacteristic
556C2×C4C3×C4×D411yescharacteristic
557C2×C4C3×C4×D411yescharacteristic
558C2×C4C3×C4×D411yescharacteristic
559C2×C4C3×C4×D411yescharacteristic
560C2×C4C3×C4×D411yescharacteristic
561C2×C4C3×C4×D411yescharacteristic
562C2×C4C3×C4×D411yescharacteristic
563C2×C4C3×C4×D411yescharacteristic
564C2×C4C3×#12111yescharacteristic
565C2×C4C3×#12111yescharacteristic
566C2×C4C3×#12111yescharacteristic
567C2×C4C3×#12111yescharacteristic
568C2×C4C3×#12311yescharacteristic
569C2×C4C3×#12311yescharacteristic
570C2×C4C3×#12411yescharacteristic
571C2×C4C3×#12411yescharacteristic
572C2×C4C3×#12611yescharacteristic
573C2×C4C3×#12611yescharacteristic
574C2×C4C3×#12611yescharacteristic
575C2×C4C3×#12611yescharacteristic
576C2×C4C23×C3×C411yescharacteristic
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C2×C3×#4111yescharacteristic
582C8—24no—
583C8—24no—
584C8—24no—
585C8—42no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—82no—
593C2×C3#193212yesdirect product factor
594C2×C3C2×#38811yescharacteristic
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C3×#38811yescharacteristic
607C22C3×#38811yescharacteristic
608C22C3×#38811yescharacteristic
609C22C3×#38811yescharacteristic
610C22C2×C3×#11711yescharacteristic
611C22C2×C3×#12311yescharacteristic
612C22C2×C3×#12411yescharacteristic
613C4—24no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4C3×#38811yescharacteristic
618C4C3×#38811yescharacteristic
619C4C3×#38811yescharacteristic
620C4C3×#38811yescharacteristic
621C4C2×C3×C4×D411yescharacteristic
622C4C2×C3×C4×D411yescharacteristic
623C4C2×C3×#12111yescharacteristic
624C4C2×C3×#12611yescharacteristic
625C3C2×#193211yescharacteristic, direct product factor, Sylow
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2C3×#193212yesdirect product factor
634C2C2×C3×#38811yescharacteristic
subgroupG/H
centerC2×C3×C4#121
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC22×C3#388
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#430C2semidirect
5. C2×C3×C4×M16C2semidirect
6. C3×#1932C2direct (N × H)
7. C2×C3×#344C2semidirect
8. C2×C3×#339C2semidirect
9. C2×#1932C3direct (N × H)
18. C3×#430C22semidirect
27. C3×#413C4semidirect
29. C2×C3×#101C4semidirect
30. C2×C3×#100C4semidirect
32. C3×C4×M16C22semidirect
35. C3×#344C22semidirect
36. C3×#339C22semidirect
37. C2×#430C2×C3semidirect
41. C2×C4×M16C2×C3semidirect
42. #1932C2×C3direct (N × H)
43. C2×#344C2×C3semidirect
44. C2×#339C2×C3semidirect
109. C3×#101C2×C4semidirect
110. C3×#100C2×C4semidirect
121. #430C22×C3semidirect
130. #413C3×C4semidirect
132. C2×#101C3×C4semidirect
133. C2×#100C3×C4semidirect
135. C4×M16C22×C3semidirect
138. #344C22×C3semidirect
139. #339C22×C3semidirect
307. #101C2×C3×C4semidirect
308. #100C2×C3×C4semidirect
593. C2×C3#1932direct (N × H)
625. C3C2×#1932direct (N × H)
633. C2C3×#1932direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4302
C2×C3×#4212
C22×C3×#1042
C2×C3×#4132
C2×C3×C4×M162
C3×#19322
C2×C3×#3442
C2×C3×#3392
C2×#19323