Finite Group Databasebeta

← C2×C3×#3137 #10679218

Subgroups — C2×C3×#3137 #10679218

Order 768.

2070 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C2×C3×#536C211yesmaximal, characteristic
3C3×#3137C2116yesmaximal, direct product factor
4C2×C3×#491C212yesmaximal, complemented
5C2×C3×#489C212yesmaximal, complemented
6C2×C3×#488C212yesmaximal, complemented
7C2×C3×#487C212yesmaximal, complemented
8C2×C3×#417C211yesmaximal, characteristic
9C2×C3×#419C211yesmaximal, characteristic, complemented
10C2×C3×#416C211yesmaximal, characteristic
11C22×C3×#103C211yesmaximal, characteristic, complemented
12C2×C3×#405C211yesmaximal, characteristic, complemented
13C2×#3137C311yesmaximal, characteristic, direct product factor, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×C3×Q8C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C23×C3×C8C2211yescharacteristic
18C3×#544—24no—
19C3×#544C2218yescomplemented
20C3×#536—24no—
21C3×#536C2218yes—
22C2×C3×C4⋊Q8—21no—
23C2×C3×C4⋊Q8C2212yes—
24C2×C3×C4⋊Q8C2212yes—
25C2×C3×#125—21no—
26C2×C3×#125C2212yes—
27C2×C3×#124C2212yes—
28C2×C3×#123—22no—
29C2×C3×C22⋊Q8—21no—
30C2×C3×C22⋊Q8—21no—
31C2×C3×C22⋊Q8—22no—
32C2×C3×#117C2211yescharacteristic
33C2×C3×#117C2211yescharacteristic
34C22×C3×C4⋊C4C2211yescharacteristic
35C3×#491C22116yescomplemented
36C3×#489C22116yescomplemented
37C3×#488C22116yescomplemented
38C3×#487C22116yescomplemented
39C3×#419C2218yescomplemented
40C3×#417C2218yes—
41C3×#416C2218yes—
42C2×C3×#103C2211yescharacteristic
43C2×C3×#103C2212yes—
44C2×C3×#103C2212yes—
45C2×C3×#103C2212yes—
46C2×C3×#103C2212yes—
47C2×C3×#103C2212yes—
48C2×C3×#103C2211yescharacteristic
49C2×C3×#103C2218yescomplemented
50C2×C3×#102C2211yescharacteristic
51C2×C3×#102C2212yes—
52C2×C3×#102C2211yescharacteristic
53C3×#405C2218yescomplemented
54C2×C3×#97C2212yes—
55C2×C3×C4×C8C2212yes—
56C2×#544C2×C311yescharacteristic, complemented
57C2×#536C2×C311yescharacteristic
58#3137C2×C3116yesdirect product factor
59C2×#491C2×C312yescomplemented
60C2×#489C2×C312yescomplemented
61C2×#488C2×C312yescomplemented
62C2×#487C2×C312yescomplemented
63C2×#417C2×C311yescharacteristic
64C2×#419C2×C311yescharacteristic, complemented
65C2×#416C2×C311yescharacteristic
66C22×#103C2×C311yescharacteristic, complemented
67C2×#405C2×C311yescharacteristic, complemented
68C2×C3×#41—22no—
69C2×C3×#41—24no—
70C2×C3×#41—24no—
71C2×C3×#41C2314yes—
72C22×C3×Q8—21no—
73C22×C3×Q8—22no—
74C22×C3×Q8—22no—
75C22×C3×Q8—24no—
76C22×C3×Q8—24no—
77C22×C3×Q8C2312yes—
78C22×C3×Q8C2314yes—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4—21no—
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4—21no—
83C23×C3×C4—21no—
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C2×C3×M16—24no—
87C2×C3×M16C2314yes—
88C22×C3×C8—24no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2314yes—
92C22×C3×C8C2312yes—
93C3×C4⋊Q8—28no—
94C3×C4⋊Q8—24no—
95C3×C4⋊Q8—24no—
96C3×C4⋊Q8C2318yes—
97C3×C4⋊Q8C2318yes—
98C3×#125—28no—
99C3×#125—24no—
100C3×#125C2318yes—
101C3×#124—24no—
102C3×#124C2318yes—
103C3×#123—216no—
104C3×C22⋊Q8—28no—
105C3×C22⋊Q8—28no—
106C3×C22⋊Q8—216no—
107C3×#117—22no—
108C3×#117D414yes—
109C3×#117C2314yes—
110C3×#117C2314yes—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4D411yescharacteristic
119C2×C3×C4⋊C4D411yescharacteristic
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C4⋊C4C2314yes—
124C2×C3×C22⋊C4—21no—
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4—22no—
127C2×C3×C22⋊C4D412yes—
128C2×C3×C42—21no—
129C2×C3×C42C2312yes—
130C3×#103C2318yes—
131C3×#103C2318yes—
132C3×#103C2314yes—
133C3×#103C2314yes—
134C3×#103C2318yes—
135C3×#103C2318yes—
136C3×#103C2318yes—
137C3×#102C2314yes—
138C3×#102C2314yes—
139C3×#102C2318yes—
140C3×#97C2318yes—
141C3×C4×C8C2318yes—
142C22×#41C22×C311yescharacteristic
143C23×Q8C22×C311yescharacteristic
144C22×M16C22×C311yescharacteristic
145C23×C8C22×C311yescharacteristic
146#544—24no—
147#544C22×C318yescomplemented
148#536—24no—
149#536C22×C318yes—
150C2×C4⋊Q8—21no—
151C2×C4⋊Q8C22×C312yes—
152C2×C4⋊Q8C22×C312yes—
153C2×#125—21no—
154C2×#125C22×C312yes—
155C2×#124C22×C312yes—
156C2×#123—22no—
157C2×C22⋊Q8—21no—
158C2×C22⋊Q8—21no—
159C2×C22⋊Q8—22no—
160C2×#117C22×C311yescharacteristic
161C2×#117C22×C311yescharacteristic
162C22×C4⋊C4C22×C311yescharacteristic
163#491C22×C3116yescomplemented
164#489C22×C3116yescomplemented
165#488C22×C3116yescomplemented
166#487C22×C3116yescomplemented
167#419C22×C318yescomplemented
168#417C22×C318yes—
169#416C22×C318yes—
170C2×#103C22×C311yescharacteristic
171C2×#103C22×C312yes—
172C2×#103C22×C312yes—
173C2×#103C22×C312yes—
174C2×#103C22×C312yes—
175C2×#103C22×C312yes—
176C2×#103C22×C311yescharacteristic
177C2×#103C22×C318yescomplemented
178C2×#102C22×C311yescharacteristic
179C2×#102C22×C312yes—
180C2×#102C22×C311yescharacteristic
181#405C22×C318yescomplemented
182C2×#97C22×C312yes—
183C2×C4×C8C22×C312yes—
184C24×C3—21no—
185C24×C3C2×D411yescharacteristic
186C3×#41—28no—
187C3×#41—28no—
188C3×#41—48no—
189C2×C3×Q8—24no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—28no—
193C2×C3×Q8—24no—
194C2×C3×Q8—24no—
195C2×C3×Q8—28no—
196C2×C3×Q8—28no—
197C2×C3×Q8—22no—
198C2×C3×Q8—22no—
199C2×C3×Q8—44no—
200C2×C3×Q8—48no—
201C2×C3×Q8—84no—
202C2×C3×Q8C2414yes—
203C2×C3×Q8C2412yes—
204C2×C3×D4—24no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—44no—
209C2×C3×D4C2412yes—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—44no—
219C22×C3×C4—44no—
220C22×C3×C4—82no—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4114yes—
226C22×C3×C4#4114yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2411yescharacteristic
229C3×M16—28no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—28no—
233C2×C3×C8—24no—
234C2×C3×C8C2412yes—
235C2×C3×C8C2414yes—
236C2×C3×C8C2412yes—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—28no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D414yes—
247C3×C4⋊C4C2×D414yes—
248C3×C4⋊C4C2414yes—
249C3×C4⋊C4C2412yes—
250C3×C4⋊C4C2412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—28no—
253C3×C22⋊C4—28no—
254C3×C22⋊C4C2×D418yes—
255C3×C42—24no—
256C3×C42C2×D414yes—
257C3×C42C2414yes—
258C2×#41—22no—
259C2×#41—24no—
260C2×#41—24no—
261C2×#41C23×C314yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—24no—
266C22×Q8—24no—
267C22×Q8C23×C312yes—
268C22×Q8C23×C311yescharacteristic
269C22×Q8C23×C314yes—
270C22×D4—21no—
271C22×D4C23×C311yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4C23×C311yescharacteristic
276C2×M16—24no—
277C2×M16C23×C314yes—
278C22×C8—24no—
279C22×C8C23×C311yescharacteristic
280C22×C8C23×C311yescharacteristic
281C22×C8C23×C314yes—
282C22×C8C23×C312yes—
283C4⋊Q8—28no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8C23×C318yes—
287C4⋊Q8C23×C318yes—
288#125—28no—
289#125—24no—
290#125C23×C318yes—
291#124—24no—
292#124C23×C318yes—
293#123—216no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—216no—
297#117—22no—
298#117C3×D414yes—
299#117C23×C314yes—
300#117C23×C314yes—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C3×D411yescharacteristic
309C2×C4⋊C4C3×D411yescharacteristic
310C2×C4⋊C4C23×C312yes—
311C2×C4⋊C4C23×C314yes—
312C2×C4⋊C4C23×C311yescharacteristic
313C2×C4⋊C4C23×C311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C3×D412yes—
318C2×C42—21no—
319C2×C42C23×C312yes—
320#103C23×C318yes—
321#103C23×C318yes—
322#103C23×C314yes—
323#103C23×C314yes—
324#103C23×C318yes—
325#103C23×C318yes—
326#103C23×C318yes—
327#102C23×C314yes—
328#102C23×C314yes—
329#102C23×C318yes—
330#97C23×C318yes—
331C4×C8C23×C318yes—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3#12414yes—
339C23×C3C22×D412yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—24no—
342C3×Q8—24no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—48no—
346C3×Q8—44no—
347C3×Q8—42no—
348C3×Q8—88no—
349C3×D4—22no—
350C3×D4—22no—
351C3×D4—48no—
352C3×D4—42no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—42no—
361C2×C3×C4—48no—
362C2×C3×C4—44no—
363C2×C3×C4—44no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—48no—
367C2×C3×C4—84no—
368C2×C3×C4#12414yes—
369C2×C3×C4#12418yes—
370C2×C3×C4C22×D414yes—
371C2×C3×C4C22×D414yes—
372C2×C3×C4C22×D412yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4112yes—
375C2×C3×C4C2×#4112yes—
376C2×C3×C4C2×#4114yes—
377C2×C3×C4C2×#4114yes—
378C2×C3×C4C2511yescharacteristic
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—24no—
382C24—21no—
383C24C2×C3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—48no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—84no—
400C2×Q8C24×C314yes—
401C2×Q8C24×C312yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—44no—
407C2×D4C24×C312yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—82no—
419C22×C4C2×C3×D412yes—
420C22×C4C2×C3×D412yes—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D412yes—
423C22×C4C3×#4114yes—
424C22×C4C3×#4114yes—
425C22×C4C24×C311yescharacteristic
426C22×C4C24×C312yes—
427M16—28no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—28no—
431C2×C8—24no—
432C2×C8C24×C312yes—
433C2×C8C24×C314yes—
434C2×C8C24×C312yes—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4C2×C3×D414yes—
445C4⋊C4C2×C3×D414yes—
446C4⋊C4C24×C314yes—
447C4⋊C4C24×C312yes—
448C4⋊C4C24×C312yes—
449C22⋊C4—24no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4C2×C3×D418yes—
453C42—24no—
454C42C2×C3×D414yes—
455C42C24×C314yes—
456C22×C3—28no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—84no—
463C22×C3C2×#12414yes—
464C22×C3C2×#12412yes—
465C22×C3#57712yes—
466C22×C3#57812yes—
467C22×C3C23×D411yescharacteristic
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4—44no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4C2×#12414yes—
476C3×C4C2×#12412yes—
477C3×C4C22×#4111yescharacteristic
478C3×C4C22×#4111yescharacteristic
479C23—22no—
480C23—22no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C3×#12414yes—
486C23C22×C3×D412yes—
487C23C22×C3×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—22no—
491Q8—22no—
492Q8—48no—
493Q8—44no—
494Q8—42no—
495Q8—88no—
496D4—22no—
497D4—22no—
498D4—48no—
499D4—42no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—84no—
515C2×C4C3×#12414yes—
516C2×C4C3×#12418yes—
517C2×C4C22×C3×D414yes—
518C2×C4C22×C3×D414yes—
519C2×C4C22×C3×D412yes—
520C2×C4C22×C3×D411yescharacteristic
521C2×C4C2×C3×#4112yes—
522C2×C4C2×C3×#4112yes—
523C2×C4C2×C3×#4114yes—
524C2×C4C2×C3×#4114yes—
525C2×C4C25×C311yescharacteristic
526C8—22no—
527C8—22no—
528C8—24no—
529C2×C3—24no—
530C2×C3—82no—
531C2×C3#313714yesdirect product factor
532C2×C3C22×#12411yescharacteristic
533C2×C3C2×#57711yescharacteristic
534C2×C3C2×#57811yescharacteristic
535C22—28no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—84no—
542C22C2×C3×#12414yes—
543C22C2×C3×#12412yes—
544C22C3×#57712yes—
545C22C3×#57812yes—
546C22C23×C3×D411yescharacteristic
547C4—24no—
548C4—24no—
549C4—42no—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4C2×C3×#12414yes—
555C4C2×C3×#12412yes—
556C4C22×C3×#4111yescharacteristic
557C4C22×C3×#4111yescharacteristic
558C3C2×#313711yescharacteristic, direct product factor, Sylow
559C2—24no—
560C2—82no—
561C2C3×#313714yesdirect product factor
562C2C22×C3×#12411yescharacteristic
563C2C2×C3×#57711yescharacteristic
564C2C2×C3×#57811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
3. C3×#3137C2direct (N × H)
4. C2×C3×#491C2semidirect
5. C2×C3×#489C2semidirect
6. C2×C3×#488C2semidirect
7. C2×C3×#487C2semidirect
9. C2×C3×#419C2semidirect
11. C22×C3×#103C2semidirect
12. C2×C3×#405C2semidirect
13. C2×#3137C3direct (N × H)
19. C3×#544C22semidirect
35. C3×#491C22semidirect
36. C3×#489C22semidirect
37. C3×#488C22semidirect
38. C3×#487C22semidirect
39. C3×#419C22semidirect
49. C2×C3×#103C22semidirect
53. C3×#405C22semidirect
56. C2×#544C2×C3semidirect
58. #3137C2×C3direct (N × H)
59. C2×#491C2×C3semidirect
60. C2×#489C2×C3semidirect
61. C2×#488C2×C3semidirect
62. C2×#487C2×C3semidirect
64. C2×#419C2×C3semidirect
66. C22×#103C2×C3semidirect
67. C2×#405C2×C3semidirect
147. #544C22×C3semidirect
163. #491C22×C3semidirect
164. #489C22×C3semidirect
165. #488C22×C3semidirect
166. #487C22×C3semidirect
167. #419C22×C3semidirect
177. C2×#103C22×C3semidirect
181. #405C22×C3semidirect
531. C2×C3#3137direct (N × H)
558. C3C2×#3137direct (N × H)
561. C2C3×#3137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#5362
C3×#31372
C2×C3×#4912
C2×C3×#4892
C2×C3×#4882
C2×C3×#4872
C2×C3×#4172
C2×C3×#4192
C2×C3×#4162
C22×C3×#1032
C2×C3×#4052
C2×#31373