Finite Group Databasebeta

← C2×C3×#3099 #10679122

Subgroups — C2×C3×#3099 #10679122

Order 768.

2234 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×#537C211yesmaximal, characteristic
2C2×C3×#520C211yesmaximal, characteristic, complemented
3C3×#3099C2116yesmaximal, direct product factor
4C2×C3×#481C212yesmaximal, complemented
5C2×C3×#485C212yesmaximal, complemented
6C2×C3×#482C212yesmaximal, complemented
7C2×C3×#483C212yesmaximal, complemented
8C2×C3×#428C211yesmaximal, characteristic, complemented
9C22×C3×#106C211yesmaximal, characteristic, complemented
10C2×C3×#417C211yesmaximal, characteristic, complemented
11C2×C3×#416C211yesmaximal, characteristic, complemented
12C2×C3×#408C211yesmaximal, characteristic, complemented
13C2×#3099C311yesmaximal, characteristic, direct product factor, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×C3×#41C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C23×C3×C8C2211yescharacteristic
18C3×#537—24no—
19C3×#537C2218yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121—21no—
23C2×C3×#121—22no—
24C2×C3×#121C2212yes—
25C3×#520—24no—
26C3×#520C2218yescomplemented
27C2×C3×C4×Q8—21no—
28C2×C3×C4×D4—22no—
29C2×C3×C4×D4—21no—
30C2×C3×#117C2211yescharacteristic
31C2×C3×#117C2212yes—
32C22×C3×C4⋊C4C2211yescharacteristic
33C22×C3×C4⋊C4C2212yes—
34C22×C3×C4⋊C4C2211yescharacteristic
35C3×#485C22116yescomplemented
36C3×#483C22116yescomplemented
37C3×#482C22116yescomplemented
38C3×#481C22116yescomplemented
39C3×#428C2218yescomplemented
40C2×C3×#107C2212yes—
41C2×C3×#106C2212yes—
42C2×C3×#106C2212yes—
43C2×C3×#106C2212yes—
44C2×C3×#106C2218yescomplemented
45C3×#417C2218yescomplemented
46C3×#416C2218yescomplemented
47C2×C3×#103C2211yescharacteristic
48C2×C3×#103C2211yescharacteristic
49C2×C3×#103C2212yes—
50C2×C3×#102C2211yescharacteristic
51C2×C3×#102C2211yescharacteristic
52C2×C3×#102C2212yes—
53C3×#408C2218yescomplemented
54C2×C3×C22⋊C8C2212yes—
55C2×C3×C22⋊C8C2212yes—
56C2×#537C2×C311yescharacteristic
57C2×#520C2×C311yescharacteristic, complemented
58#3099C2×C3116yesdirect product factor
59C2×#481C2×C312yescomplemented
60C2×#485C2×C312yescomplemented
61C2×#482C2×C312yescomplemented
62C2×#483C2×C312yescomplemented
63C2×#428C2×C311yescharacteristic, complemented
64C22×#106C2×C311yescharacteristic, complemented
65C2×#417C2×C311yescharacteristic, complemented
66C2×#416C2×C311yescharacteristic, complemented
67C2×#408C2×C311yescharacteristic, complemented
68C2×C3×#41—22no—
69C2×C3×#41—24no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41D414yes—
74C2×C3×#41C2314yes—
75C2×C3×#41C2314yes—
76C22×C3×Q8D411yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4—22no—
80C22×C3×D4D412yes—
81C22×C3×D4D411yescharacteristic
82C22×C3×D4C2311yescharacteristic
83C23×C3×C4—22no—
84C23×C3×C4—21no—
85C23×C3×C4—21no—
86C23×C3×C4—21no—
87C23×C3×C4C2312yes—
88C23×C3×C4C2311yescharacteristic
89C2×C3×M16—24no—
90C2×C3×M16C2314yes—
91C22×C3×C8—24no—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2314yes—
95C22×C3×C8C2312yes—
96C3×C22⋊Q8—28no—
97C3×C22⋊Q8—24no—
98C3×C22⋊Q8C2318yes—
99C3×#121—28no—
100C3×#121—216no—
101C3×#121—24no—
102C3×#121C2318yes—
103C3×C4×Q8—28no—
104C3×C4×D4—216no—
105C3×C4×D4—28no—
106C3×#117—24no—
107C3×#117—22no—
108C3×#117C2314yes—
109C3×#117C2318yes—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—24no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2314yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2314yes—
123C2×C3×C4⋊C4C2318yes—
124C2×C3×C22⋊C4—22no—
125C2×C3×C22⋊C4—21no—
126C2×C3×C22⋊C4—22no—
127C2×C3×C22⋊C4—22no—
128C2×C3×C42—22no—
129C2×C3×C42—21no—
130C3×#107C2318yes—
131C3×#106C2318yes—
132C3×#106C2318yes—
133C3×#106C2318yes—
134C3×#103C2314yes—
135C3×#103C2314yes—
136C3×#103C2318yes—
137C3×#102C2314yes—
138C3×#102C2314yes—
139C3×#102C2318yes—
140C3×C22⋊C8C2318yes—
141C3×C22⋊C8C2318yes—
142C22×#41C22×C311yescharacteristic
143C22×#41C22×C311yescharacteristic
144C22×M16C22×C311yescharacteristic
145C23×C8C22×C311yescharacteristic
146#537—24no—
147#537C22×C318yes—
148C2×C22⋊Q8—21no—
149C2×C22⋊Q8C22×C312yes—
150C2×#121—21no—
151C2×#121—22no—
152C2×#121C22×C312yes—
153#520—24no—
154#520C22×C318yescomplemented
155C2×C4×Q8—21no—
156C2×C4×D4—22no—
157C2×C4×D4—21no—
158C2×#117C22×C312yes—
159C2×#117C22×C311yescharacteristic
160C22×C4⋊C4C22×C312yes—
161C22×C4⋊C4C22×C311yescharacteristic
162C22×C4⋊C4C22×C311yescharacteristic
163#485C22×C3116yescomplemented
164#483C22×C3116yescomplemented
165#482C22×C3116yescomplemented
166#481C22×C3116yescomplemented
167#428C22×C318yescomplemented
168C2×#107C22×C312yes—
169C2×#106C22×C312yes—
170C2×#106C22×C312yes—
171C2×#106C22×C312yes—
172C2×#106C22×C318yescomplemented
173#417C22×C318yescomplemented
174#416C22×C318yescomplemented
175C2×#103C22×C311yescharacteristic
176C2×#103C22×C312yes—
177C2×#103C22×C311yescharacteristic
178C2×#102C22×C311yescharacteristic
179C2×#102C22×C312yes—
180C2×#102C22×C311yescharacteristic
181#408C22×C318yescomplemented
182C2×C22⋊C8C22×C312yes—
183C2×C22⋊C8C22×C312yes—
184C24×C3—21no—
185C24×C3C2×D412yes—
186C24×C3C2×D411yescharacteristic
187C3×#41—216no—
188C3×#41—28no—
189C3×#41—28no—
190C3×#41—28no—
191C3×#41—28no—
192C3×#41—48no—
193C2×C3×Q8—24no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8C2×D414yes—
198C2×C3×Q8C2412yes—
199C2×C3×D4—28no—
200C2×C3×D4—24no—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—44no—
207C2×C3×D4—84no—
208C2×C3×D4C2×D418yes—
209C2×C3×D4C2×D414yes—
210C2×C3×D4C2412yes—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—21no—
218C22×C3×C4—28no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—24no—
222C22×C3×C4—44no—
223C22×C3×C4—82no—
224C22×C3×C4—84no—
225C22×C3×C4—82no—
226C22×C3×C4C2×D414yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4#4114yes—
231C22×C3×C4#4114yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C3×M16—28no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—28no—
239C2×C3×C8—24no—
240C2×C3×C8C2412yes—
241C2×C3×C8C2412yes—
242C2×C3×C8C2414yes—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—28no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2412yes—
252C3×C4⋊C4C2414yes—
253C3×C4⋊C4C2412yes—
254C3×C4⋊C4C2414yes—
255C3×C22⋊C4—28no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—28no—
258C3×C22⋊C4—28no—
259C3×C42—28no—
260C3×C42—24no—
261C2×#41—22no—
262C2×#41—24no—
263C2×#41—24no—
264C2×#41—24no—
265C2×#41—24no—
266C2×#41C3×D414yes—
267C2×#41C23×C314yes—
268C2×#41C23×C314yes—
269C22×Q8C3×D411yescharacteristic
270C22×Q8C23×C311yescharacteristic
271C22×D4—21no—
272C22×D4—22no—
273C22×D4C3×D412yes—
274C22×D4C3×D411yescharacteristic
275C22×D4C23×C311yescharacteristic
276C23×C4—22no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C23×C312yes—
281C23×C4C23×C311yescharacteristic
282C2×M16—24no—
283C2×M16C23×C314yes—
284C22×C8—24no—
285C22×C8C23×C311yescharacteristic
286C22×C8C23×C311yescharacteristic
287C22×C8C23×C314yes—
288C22×C8C23×C312yes—
289C22⋊Q8—28no—
290C22⋊Q8—24no—
291C22⋊Q8C23×C318yes—
292#121—28no—
293#121—216no—
294#121—24no—
295#121C23×C318yes—
296C4×Q8—28no—
297C4×D4—216no—
298C4×D4—28no—
299#117—24no—
300#117—22no—
301#117C23×C314yes—
302#117C23×C318yes—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C4⋊C4C23×C312yes—
312C2×C4⋊C4C23×C314yes—
313C2×C4⋊C4C23×C318yes—
314C2×C4⋊C4C23×C312yes—
315C2×C4⋊C4C23×C314yes—
316C2×C4⋊C4C23×C311yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C42—22no—
322C2×C42—21no—
323#107C23×C318yes—
324#106C23×C318yes—
325#106C23×C318yes—
326#106C23×C318yes—
327#103C23×C314yes—
328#103C23×C314yes—
329#103C23×C318yes—
330#102C23×C314yes—
331#102C23×C314yes—
332#102C23×C318yes—
333C22⋊C8C23×C318yes—
334C22⋊C8C23×C318yes—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—24no—
338C23×C3—24no—
339C23×C3—24no—
340C23×C3—22no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3#12314yes—
345C23×C3C22×D414yes—
346C23×C3C22×D412yes—
347C23×C3C22×D411yescharacteristic
348C3×Q8—28no—
349C3×Q8—22no—
350C3×Q8—22no—
351C3×Q8—42no—
352C3×D4—216no—
353C3×D4—28no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—48no—
357C3×D4—42no—
358C3×D4—88no—
359C2×C3×C4—24no—
360C2×C3×C4—28no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—44no—
373C2×C3×C4—48no—
374C2×C3×C4—84no—
375C2×C3×C4—88no—
376C2×C3×C4—84no—
377C2×C3×C4#12314yes—
378C2×C3×C4#12318yes—
379C2×C3×C4C22×D414yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4112yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C4C2×#4114yes—
385C2×C3×C4C2×#4114yes—
386C2×C3×C4C2511yescharacteristic
387C3×C8—22no—
388C3×C8—22no—
389C3×C8—24no—
390C24—21no—
391C24C2×C3×D412yes—
392C24C2×C3×D411yescharacteristic
393#41—216no—
394#41—28no—
395#41—28no—
396#41—28no—
397#41—28no—
398#41—48no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×C3×D414yes—
404C2×Q8C24×C312yes—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4C2×C3×D418yes—
415C2×D4C2×C3×D414yes—
416C2×D4C24×C312yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—84no—
431C22×C4—82no—
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D414yes—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D412yes—
436C22×C4C3×#4114yes—
437C22×C4C3×#4114yes—
438C22×C4C24×C314yes—
439C22×C4C24×C311yescharacteristic
440C22×C4C24×C312yes—
441M16—28no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—28no—
445C2×C8—24no—
446C2×C8C24×C312yes—
447C2×C8C24×C312yes—
448C2×C8C24×C314yes—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—28no—
456C4⋊C4—21no—
457C4⋊C4C24×C312yes—
458C4⋊C4C24×C314yes—
459C4⋊C4C24×C312yes—
460C4⋊C4C24×C314yes—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C42—28no—
466C42—24no—
467C22×C3—28no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—48no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—84no—
478C22×C3C2×#12314yes—
479C22×C3C2×#12312yes—
480C22×C3#57712yes—
481C22×C3#57712yes—
482C22×C3C23×D411yescharacteristic
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—44no—
487C3×C4—42no—
488C3×C4—44no—
489C3×C4C2×#12312yes—
490C3×C4C2×#12314yes—
491C3×C4C22×#4111yescharacteristic
492C3×C4C22×#4111yescharacteristic
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C3×#12314yes—
503C23C22×C3×D414yes—
504C23C22×C3×D412yes—
505C23C22×C3×D411yescharacteristic
506Q8—28no—
507Q8—22no—
508Q8—22no—
509Q8—42no—
510D4—216no—
511D4—22no—
512D4—22no—
513D4—28no—
514D4—48no—
515D4—42no—
516D4—88no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—48no—
532C2×C4—84no—
533C2×C4—88no—
534C2×C4—84no—
535C2×C4C3×#12314yes—
536C2×C4C3×#12318yes—
537C2×C4C22×C3×D414yes—
538C2×C4C22×C3×D412yes—
539C2×C4C22×C3×D411yescharacteristic
540C2×C4C2×C3×#4112yes—
541C2×C4C2×C3×#4112yes—
542C2×C4C2×C3×#4114yes—
543C2×C4C2×C3×#4114yes—
544C2×C4C25×C311yescharacteristic
545C8—22no—
546C8—22no—
547C8—24no—
548C2×C3—24no—
549C2×C3—44no—
550C2×C3—82no—
551C2×C3#309914yesdirect product factor
552C2×C3C22×#12311yescharacteristic
553C2×C3C2×#57711yescharacteristic
554C2×C3C2×#57711yescharacteristic
555C22—22no—
556C22—22no—
557C22—28no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—48no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22C2×C3×#12314yes—
567C22C2×C3×#12312yes—
568C22C3×#57712yes—
569C22C3×#57712yes—
570C22C23×C3×D411yescharacteristic
571C4—24no—
572C4—42no—
573C4—44no—
574C4—42no—
575C4—44no—
576C4—42no—
577C4C2×C3×#12312yes—
578C4C2×C3×#12314yes—
579C4C22×C3×#4111yescharacteristic
580C4C22×C3×#4111yescharacteristic
581C3C2×#309911yescharacteristic, direct product factor, Sylow
582C2—24no—
583C2—44no—
584C2—82no—
585C2C3×#309914yesdirect product factor
586C2C22×C3×#12311yescharacteristic
587C2C2×C3×#57711yescharacteristic
588C2C2×C3×#57711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#520C2semidirect
3. C3×#3099C2direct (N × H)
4. C2×C3×#481C2semidirect
5. C2×C3×#485C2semidirect
6. C2×C3×#482C2semidirect
7. C2×C3×#483C2semidirect
8. C2×C3×#428C2semidirect
9. C22×C3×#106C2semidirect
10. C2×C3×#417C2semidirect
11. C2×C3×#416C2semidirect
12. C2×C3×#408C2semidirect
13. C2×#3099C3direct (N × H)
26. C3×#520C22semidirect
35. C3×#485C22semidirect
36. C3×#483C22semidirect
37. C3×#482C22semidirect
38. C3×#481C22semidirect
39. C3×#428C22semidirect
44. C2×C3×#106C22semidirect
45. C3×#417C22semidirect
46. C3×#416C22semidirect
53. C3×#408C22semidirect
57. C2×#520C2×C3semidirect
58. #3099C2×C3direct (N × H)
59. C2×#481C2×C3semidirect
60. C2×#485C2×C3semidirect
61. C2×#482C2×C3semidirect
62. C2×#483C2×C3semidirect
63. C2×#428C2×C3semidirect
64. C22×#106C2×C3semidirect
65. C2×#417C2×C3semidirect
66. C2×#416C2×C3semidirect
67. C2×#408C2×C3semidirect
154. #520C22×C3semidirect
163. #485C22×C3semidirect
164. #483C22×C3semidirect
165. #482C22×C3semidirect
166. #481C22×C3semidirect
167. #428C22×C3semidirect
172. C2×#106C22×C3semidirect
173. #417C22×C3semidirect
174. #416C22×C3semidirect
181. #408C22×C3semidirect
551. C2×C3#3099direct (N × H)
581. C3C2×#3099direct (N × H)
585. C2C3×#3099direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C2×C3×#5202
C3×#30992
C2×C3×#4812
C2×C3×#4852
C2×C3×#4822
C2×C3×#4832
C2×C3×#4282
C22×C3×#1062
C2×C3×#4172
C2×C3×#4162
C2×C3×#4082
C2×#30993