Finite Group Databasebeta

← C3×#10942 #10629549

Subgroups — C3×#10942 #10629549

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#419C211yesmaximal, characteristic
2C22×C3×#103C211yesmaximal, characteristic
3C22×C3×#101C211yesmaximal, characteristic
4C2×C3×#405C211yesmaximal, characteristic
5C2×C3×#343C211yesmaximal, characteristic
6C2×C3×#341C211yesmaximal, characteristic
7C3×#1736C211yesmaximal, characteristic
8#10942C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C3×#419—22no—
17C3×#419C414yescomplemented
18C2×C3×#103—21no—
19C2×C3×#103—21no—
20C2×C3×#103—22no—
21C2×C3×#103—22no—
22C2×C3×#103C414yescomplemented
23C2×C3×#101—22no—
24C2×C3×#101—22no—
25C2×C3×#101C414yescomplemented
26C3×#405—22no—
27C3×#405—22no—
28C2×C3×#97—21no—
29C2×C3×C4×C8—21no—
30C3×#343—22no—
31C3×#341—22no—
32C3×#340—22no—
33C2×#419C2×C311yescharacteristic
34C22×#103C2×C311yescharacteristic
35C22×#101C2×C311yescharacteristic
36C2×#405C2×C311yescharacteristic
37C2×#343C2×C311yescharacteristic
38C2×#341C2×C311yescharacteristic
39#1736C2×C311yescharacteristic
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8C2×C412yes—
52C22×C3×Q8C2×C412yes—
53C22×C3×Q8C2×C412yes—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
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×M16C2×C412yes—
71C2×C3×M16C2×C412yes—
72C2×C3×M16C2×C412yes—
73C2×C3×M16D412yes—
74C2×C3×M16D412yes—
75C22×C3×C8—21no—
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8—21no—
79C22×C3×C8—21no—
80C22×C3×C8—21no—
81C22×C3×C8—21no—
82C22×C3×C8C2×C412yes—
83C22×C3×C8D411yescharacteristic
84C22×C3×C8D411yescharacteristic
85C22×C3×C8D412yes—
86C3×#117—21no—
87C3×#117—21no—
88C3×#117C2×C412yes—
89C3×#117D412yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2×C412yes—
96C2×C3×C4⋊C4D412yes—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C3×#103—24no—
100C3×#103—24no—
101C3×#103—28no—
102C3×#101—28no—
103C3×#97—24no—
104C3×C4×C8—24no—
105C23×Q8C22×C311yescharacteristic
106C22×M16C22×C311yescharacteristic
107C22×M16C22×C311yescharacteristic
108C22×M16C22×C311yescharacteristic
109C23×C8C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112#419—22no—
113#419C3×C414yescomplemented
114C2×#103—22no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—22no—
118C2×#103C3×C414yescomplemented
119C2×#101—22no—
120C2×#101—22no—
121C2×#101C3×C414yescomplemented
122#405—22no—
123#405—22no—
124C2×#97—21no—
125C2×C4×C8—21no—
126#343—22no—
127#341—22no—
128#340—22no—
129C24×C3#4111yescharacteristic
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8—41no—
148C2×C3×Q8—41no—
149C2×C3×Q8—41no—
150C2×C3×Q8—41no—
151C2×C3×Q8—41no—
152C2×C3×Q8—41no—
153C2×C3×Q8—41no—
154C2×C3×Q8—41no—
155C2×C3×Q8—41no—
156C2×C3×Q8—41no—
157C2×C3×Q8C4214yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4—41no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—82no—
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C3×M16—22no—
185C3×M16—22no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—41no—
192C3×M16—41no—
193C3×M16—41no—
194C3×M16—41no—
195C3×M16C22⋊C414yes—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—41no—
210C2×C3×C8—41no—
211C2×C3×C8—41no—
212C2×C3×C8—41no—
213C2×C3×C8C22⋊C412yes—
214C2×C3×C8C22⋊C412yes—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4—41no—
220C3×C4⋊C4C22⋊C414yes—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C42—22no—
224C3×C42—22no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8C2×C3×C412yes—
237C22×Q8C2×C3×C412yes—
238C22×Q8C2×C3×C412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C23×C311yescharacteristic
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16C2×C3×C412yes—
256C2×M16C2×C3×C412yes—
257C2×M16C2×C3×C412yes—
258C2×M16C3×D412yes—
259C2×M16C3×D412yes—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8C2×C3×C412yes—
268C22×C8C3×D412yes—
269C22×C8C3×D411yescharacteristic
270C22×C8C3×D411yescharacteristic
271#117—21no—
272#117—21no—
273#117C2×C3×C412yes—
274#117C3×D412yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C2×C3×C412yes—
281C2×C4⋊C4C3×D412yes—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284#103—24no—
285#103—24no—
286#103—28no—
287#101—28no—
288#97—24no—
289C4×C8—24no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3#11711yescharacteristic
293C23×C3C4×D411yescharacteristic
294C23×C3C4×D411yescharacteristic
295C23×C3#12111yescharacteristic
296C23×C3#12311yescharacteristic
297C23×C3#12411yescharacteristic
298C23×C3#12611yescharacteristic
299C3×Q8—24no—
300C3×Q8—24no—
301C3×Q8—42no—
302C3×Q8—42no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—42no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—82no—
319C2×C3×C4—82no—
320C2×C3×C4—82no—
321C2×C3×C4C2×C4211yescharacteristic
322C2×C3×C4C2×C22⋊C411yescharacteristic
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4#11711yescharacteristic
326C2×C3×C4#11711yescharacteristic
327C2×C3×C4C4×D411yescharacteristic
328C2×C3×C4C4×D411yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12111yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12311yescharacteristic
344C2×C3×C4#12311yescharacteristic
345C2×C3×C4#12411yescharacteristic
346C2×C3×C4#12411yescharacteristic
347C2×C3×C4#12611yescharacteristic
348C2×C3×C4#12711yescharacteristic
349C3×C8—22no—
350C3×C8—22no—
351C3×C8—24no—
352C3×C8—42no—
353C3×C8—42no—
354C24C3×#4111yescharacteristic
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8C3×C4214yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—82no—
395C22×C4C22×C3×C411yescharacteristic
396C22×C4C22×C3×C411yescharacteristic
397C22×C4C22×C3×C411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C3×#4111yescharacteristic
405C22×C4C3×#4111yescharacteristic
406C22×C4C3×#4111yescharacteristic
407C22×C4C3×#4111yescharacteristic
408C22×C4C3×#4111yescharacteristic
409M16—22no—
410M16—22no—
411M16—22no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16C3×C22⋊C414yes—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8C3×C22⋊C412yes—
439C2×C8C3×C22⋊C412yes—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4C3×C22⋊C414yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3C4×C22⋊C411yescharacteristic
459C22×C3#38611yescharacteristic
460C22×C3#38811yescharacteristic
461C22×C3#38811yescharacteristic
462C22×C3#38811yescharacteristic
463C22×C3#38811yescharacteristic
464C22×C3#39011yescharacteristic
465C22×C3#47112yes—
466C22×C3#47312yes—
467C3×C4—44no—
468C3×C4—44no—
469C3×C4—82no—
470C3×C4C4×C22⋊C411yescharacteristic
471C3×C4C4×C22⋊C411yescharacteristic
472C3×C4#38611yescharacteristic
473C3×C4#38611yescharacteristic
474C3×C4#38811yescharacteristic
475C3×C4#38811yescharacteristic
476C3×C4#39011yescharacteristic
477C3×C4#39011yescharacteristic
478C23—22no—
479C23—22no—
480C23C3×#11711yescharacteristic
481C23C3×C4×D411yescharacteristic
482C23C3×C4×D411yescharacteristic
483C23C3×#12111yescharacteristic
484C23C3×#12311yescharacteristic
485C23C3×#12411yescharacteristic
486C23C3×#12611yescharacteristic
487Q8—24no—
488Q8—24no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4C2×C3×C4211yescharacteristic
510C2×C4C2×C3×C22⋊C411yescharacteristic
511C2×C4C2×C3×C22⋊C411yescharacteristic
512C2×C4C2×C3×C22⋊C411yescharacteristic
513C2×C4C3×#11711yescharacteristic
514C2×C4C3×#11711yescharacteristic
515C2×C4C3×C4×D411yescharacteristic
516C2×C4C3×C4×D411yescharacteristic
517C2×C4C3×C4×D411yescharacteristic
518C2×C4C3×C4×D411yescharacteristic
519C2×C4C3×C4×D411yescharacteristic
520C2×C4C3×C4×D411yescharacteristic
521C2×C4C3×C4×D411yescharacteristic
522C2×C4C3×C4×D411yescharacteristic
523C2×C4C3×C4×D411yescharacteristic
524C2×C4C3×C4×D411yescharacteristic
525C2×C4C3×#12011yescharacteristic
526C2×C4C3×#12111yescharacteristic
527C2×C4C3×#12111yescharacteristic
528C2×C4C3×#12111yescharacteristic
529C2×C4C3×#12111yescharacteristic
530C2×C4C3×#12111yescharacteristic
531C2×C4C3×#12311yescharacteristic
532C2×C4C3×#12311yescharacteristic
533C2×C4C3×#12411yescharacteristic
534C2×C4C3×#12411yescharacteristic
535C2×C4C3×#12611yescharacteristic
536C2×C4C3×#12711yescharacteristic
537C8—22no—
538C8—22no—
539C8—24no—
540C8—42no—
541C8—42no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3#143711yescharacteristic
547C2×C3#190812yes—
548C2×C3#193012yes—
549C2×C3#197711yescharacteristic
550C2×C3#197911yescharacteristic
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22C3×C4×C22⋊C411yescharacteristic
560C22C3×#38611yescharacteristic
561C22C3×#38811yescharacteristic
562C22C3×#38811yescharacteristic
563C22C3×#38811yescharacteristic
564C22C3×#38811yescharacteristic
565C22C3×#39011yescharacteristic
566C22C3×#47112yes—
567C22C3×#47312yes—
568C4—44no—
569C4—44no—
570C4—82no—
571C4C3×C4×C22⋊C411yescharacteristic
572C4C3×C4×C22⋊C411yescharacteristic
573C4C3×#38611yescharacteristic
574C4C3×#38611yescharacteristic
575C4C3×#38811yescharacteristic
576C4C3×#38811yescharacteristic
577C4C3×#39011yescharacteristic
578C4C3×#39011yescharacteristic
579C3#1094211yescharacteristic, direct product factor, Sylow
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2C3×#143711yescharacteristic
585C2C3×#190812yes—
586C2C3×#193012yes—
587C2C3×#197711yescharacteristic
588C2C3×#197911yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutatorC2×C4C2×C3×C42
Frattini, cosocleC23×C4C23×C3
prime radicalC24×C3#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
8. #10942C3direct (N × H)
17. C3×#419C4semidirect
22. C2×C3×#103C4semidirect
25. C2×C3×#101C4semidirect
113. #419C3×C4semidirect
118. C2×#103C3×C4semidirect
121. C2×#101C3×C4semidirect
579. C3#10942direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4192
C22×C3×#1032
C22×C3×#1012
C2×C3×#4052
C2×C3×#3432
C2×C3×#3412
C3×#17362
#109423