Finite Group Databasebeta

← C2×C3×#3084 #10679095

Subgroups — C2×C3×#3084 #10679095

Order 768.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C2×C3×#520C211yesmaximal, characteristic
3C3×#3084C2116yesmaximal, direct product factor
4C2×C3×#479C212yesmaximal, complemented
5C2×C3×#475C212yesmaximal, complemented
6C2×C3×#478C212yesmaximal, complemented
7C2×C3×#476C212yesmaximal, complemented
8C2×C3×#428C211yesmaximal, characteristic, complemented
9C22×C3×#106C211yesmaximal, characteristic, complemented
10C2×C3×#424C211yesmaximal, characteristic, complemented
11C2×C3×#417C211yesmaximal, characteristic
12C2×C3×#416C211yesmaximal, characteristic
13C2×#3084C311yesmaximal, characteristic, direct product factor, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C3×#544—24no—
18C3×#544C2218yescomplemented
19C2×C3×C4⋊Q8—21no—
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×#125—21no—
22C2×C3×#125C2212yes—
23C2×C3×C22⋊Q8—22no—
24C3×#520—24no—
25C3×#520C2218yes—
26C2×C3×C4×Q8C2212yes—
27C2×C3×C4×D4—22no—
28C2×C3×C4×D4C2212yes—
29C2×C3×#117—21no—
30C2×C3×#117C2211yescharacteristic
31C2×C3×#117C2211yescharacteristic
32C22×C3×C4⋊C4—21no—
33C22×C3×C4⋊C4C2211yescharacteristic
34C22×C3×C4⋊C4C2211yescharacteristic
35C3×#479C22116yescomplemented
36C3×#478C22116yescomplemented
37C3×#476C22116yescomplemented
38C3×#475C22116yescomplemented
39C3×#428C2218yescomplemented
40C2×C3×#107C2212yes—
41C2×C3×#106C2212yes—
42C2×C3×#106C2212yes—
43C2×C3×#106C2212yes—
44C2×C3×#106C2218yescomplemented
45C3×#424C2218yescomplemented
46C2×C3×C4⋊C8C2212yes—
47C2×C3×C4⋊C8C2212yes—
48C3×#417C2218yes—
49C3×#416C2218yes—
50C2×C3×#103C2211yescharacteristic
51C2×C3×#103C2211yescharacteristic
52C2×C3×#103C2212yes—
53C2×C3×#102C2211yescharacteristic
54C2×C3×#102C2211yescharacteristic
55C2×C3×#102C2212yes—
56C2×#544C2×C311yescharacteristic, complemented
57C2×#520C2×C311yescharacteristic
58#3084C2×C3116yesdirect product factor
59C2×#479C2×C312yescomplemented
60C2×#475C2×C312yescomplemented
61C2×#478C2×C312yescomplemented
62C2×#476C2×C312yescomplemented
63C2×#428C2×C311yescharacteristic, complemented
64C22×#106C2×C311yescharacteristic, complemented
65C2×#424C2×C311yescharacteristic, complemented
66C2×#417C2×C311yescharacteristic
67C2×#416C2×C311yescharacteristic
68C2×C3×#41—22no—
69C2×C3×#41—24no—
70C2×C3×#41Q818yescomplemented
71C2×C3×#41C2314yes—
72C22×C3×Q8—22no—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4—21no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4—22no—
80C23×C3×C4C2311yescharacteristic
81C2×C3×M16—24no—
82C2×C3×M16C2314yes—
83C22×C3×C8—24no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2314yes—
87C22×C3×C8C2312yes—
88C3×C4⋊Q8—28no—
89C3×C4⋊Q8—24no—
90C3×C4⋊Q8C2318yes—
91C3×#125—28no—
92C3×#125—24no—
93C3×#125C2318yes—
94C3×C22⋊Q8—216no—
95C3×C4×Q8—24no—
96C3×C4×Q8C2318yes—
97C3×C4×D4—216no—
98C3×C4×D4—24no—
99C3×C4×D4C2318yes—
100C3×#117—22no—
101C3×#117—28no—
102C3×#117D414yes—
103C3×#117C2314yes—
104C3×#117C2314yes—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—28no—
114C2×C3×C4⋊C4D411yescharacteristic
115C2×C3×C4⋊C4D411yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2314yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2314yes—
122C2×C3×C22⋊C4—21no—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4D412yes—
125C2×C3×C42—22no—
126C2×C3×C42—21no—
127C2×C3×C42C2312yes—
128C3×#107C2318yes—
129C3×#106C2318yes—
130C3×#106C2318yes—
131C3×#106C2318yes—
132C3×C4⋊C8C2318yes—
133C3×C4⋊C8C2318yes—
134C3×#103C2314yes—
135C3×#103C2314yes—
136C3×#103C2318yes—
137C3×#102C2314yes—
138C3×#102C2314yes—
139C3×#102C2318yes—
140C22×#41C22×C311yescharacteristic
141C22×M16C22×C311yescharacteristic
142C23×C8C22×C311yescharacteristic
143#544—24no—
144#544C22×C318yescomplemented
145C2×C4⋊Q8—21no—
146C2×C4⋊Q8C22×C312yes—
147C2×#125—21no—
148C2×#125C22×C312yes—
149C2×C22⋊Q8—22no—
150#520—24no—
151#520C22×C318yes—
152C2×C4×Q8C22×C312yes—
153C2×C4×D4—22no—
154C2×C4×D4C22×C312yes—
155C2×#117—21no—
156C2×#117C22×C311yescharacteristic
157C2×#117C22×C311yescharacteristic
158C22×C4⋊C4—21no—
159C22×C4⋊C4C22×C311yescharacteristic
160C22×C4⋊C4C22×C311yescharacteristic
161#479C22×C3116yescomplemented
162#478C22×C3116yescomplemented
163#476C22×C3116yescomplemented
164#475C22×C3116yescomplemented
165#428C22×C318yescomplemented
166C2×#107C22×C312yes—
167C2×#106C22×C312yes—
168C2×#106C22×C312yes—
169C2×#106C22×C312yes—
170C2×#106C22×C318yescomplemented
171#424C22×C318yescomplemented
172C2×C4⋊C8C22×C312yes—
173C2×C4⋊C8C22×C312yes—
174#417C22×C318yes—
175#416C22×C318yes—
176C2×#103C22×C311yescharacteristic
177C2×#103C22×C312yes—
178C2×#103C22×C311yescharacteristic
179C2×#102C22×C311yescharacteristic
180C2×#102C22×C312yes—
181C2×#102C22×C311yescharacteristic
182C24×C3—21no—
183C24×C3C2×D411yescharacteristic
184C3×#41—216no—
185C3×#41—28no—
186C3×#41C2×Q8116yescomplemented
187C2×C3×Q8—21no—
188C2×C3×Q8—28no—
189C2×C3×Q8—22no—
190C2×C3×Q8—84no—
191C2×C3×Q8C2×Q814yes—
192C2×C3×Q8C2412yes—
193C2×C3×D4—28no—
194C2×C3×D4—24no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4C2×Q814yes—
198C2×C3×D4C2412yes—
199C22×C3×C4—21no—
200C22×C3×C4—28no—
201C22×C3×C4—24no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—28no—
209C22×C3×C4—84no—
210C22×C3×C4—82no—
211C22×C3×C4—82no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×Q814yes—
217C22×C3×C4#4114yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2411yescharacteristic
220C3×M16—28no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—28no—
224C2×C3×C8—24no—
225C2×C3×C8C2412yes—
226C2×C3×C8C2412yes—
227C2×C3×C8C2414yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—28no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—28no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4C2×D414yes—
239C3×C4⋊C4C2×D414yes—
240C3×C4⋊C4C2414yes—
241C3×C4⋊C4C2412yes—
242C3×C4⋊C4C2412yes—
243C3×C4⋊C4C2414yes—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—28no—
246C3×C22⋊C4C2×D418yes—
247C3×C42—28no—
248C3×C42—24no—
249C3×C42C2×D414yes—
250C3×C42C2414yes—
251C2×#41—22no—
252C2×#41—24no—
253C2×#41C3×Q818yescomplemented
254C2×#41C23×C314yes—
255C22×Q8—22no—
256C22×Q8C23×C311yescharacteristic
257C22×D4—21no—
258C22×D4C23×C311yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C23×C311yescharacteristic
264C2×M16—24no—
265C2×M16C23×C314yes—
266C22×C8—24no—
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C314yes—
270C22×C8C23×C312yes—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8C23×C318yes—
274#125—28no—
275#125—24no—
276#125C23×C318yes—
277C22⋊Q8—216no—
278C4×Q8—24no—
279C4×Q8C23×C318yes—
280C4×D4—216no—
281C4×D4—24no—
282C4×D4C23×C318yes—
283#117—22no—
284#117—28no—
285#117C3×D414yes—
286#117C23×C314yes—
287#117C23×C314yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4C3×D411yescharacteristic
298C2×C4⋊C4C3×D411yescharacteristic
299C2×C4⋊C4C23×C312yes—
300C2×C4⋊C4C23×C314yes—
301C2×C4⋊C4C23×C312yes—
302C2×C4⋊C4C23×C314yes—
303C2×C4⋊C4C23×C311yescharacteristic
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C3×D412yes—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42C23×C312yes—
311#107C23×C318yes—
312#106C23×C318yes—
313#106C23×C318yes—
314#106C23×C318yes—
315C4⋊C8C23×C318yes—
316C4⋊C8C23×C318yes—
317#103C23×C314yes—
318#103C23×C314yes—
319#103C23×C318yes—
320#102C23×C314yes—
321#102C23×C314yes—
322#102C23×C318yes—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C22⋊Q814yes—
330C23×C3C22×D412yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—24no—
333C3×Q8—22no—
334C3×Q8—88no—
335C3×Q8C22×Q814yes—
336C3×D4—216no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4C22×Q814yes—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—216no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4—44no—
352C2×C3×C4—42no—
353C2×C3×C4—88no—
354C2×C3×C4—84no—
355C2×C3×C4—84no—
356C2×C3×C4C22⋊Q814yes—
357C2×C3×C4C22⋊Q818yes—
358C2×C3×C4C22×D414yes—
359C2×C3×C4C22×D414yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×Q812yes—
363C2×C3×C4C22×Q814yes—
364C2×C3×C4C2×#4112yes—
365C2×C3×C4C2×#4114yes—
366C2×C3×C4C2511yescharacteristic
367C3×C8—22no—
368C3×C8—22no—
369C3×C8—24no—
370C24—21no—
371C24C2×C3×D411yescharacteristic
372#41—216no—
373#41—28no—
374#41C2×C3×Q8116yescomplemented
375C2×Q8—21no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—84no—
379C2×Q8C2×C3×Q814yes—
380C2×Q8C24×C312yes—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4C2×C3×Q814yes—
386C2×D4C24×C312yes—
387C22×C4—21no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D412yes—
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D412yes—
404C22×C4C2×C3×Q814yes—
405C22×C4C3×#4114yes—
406C22×C4C24×C311yescharacteristic
407C22×C4C24×C312yes—
408M16—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—28no—
412C2×C8—24no—
413C2×C8C24×C312yes—
414C2×C8C24×C312yes—
415C2×C8C24×C314yes—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4C2×C3×D414yes—
427C4⋊C4C2×C3×D414yes—
428C4⋊C4C24×C314yes—
429C4⋊C4C24×C312yes—
430C4⋊C4C24×C312yes—
431C4⋊C4C24×C314yes—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C22⋊C4C2×C3×D418yes—
435C42—28no—
436C42—24no—
437C42C2×C3×D414yes—
438C42C24×C314yes—
439C22×C3—28no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—48no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3C2×C22⋊Q814yes—
447C22×C3C2×C22⋊Q812yes—
448C22×C3#57712yes—
449C22×C3#57712yes—
450C22×C3C23×D411yescharacteristic
451C3×C4—24no—
452C3×C4—24no—
453C3×C4—24no—
454C3×C4—42no—
455C3×C4—44no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4C2×C22⋊Q814yes—
459C3×C4C2×C22⋊Q812yes—
460C3×C4C23×Q811yescharacteristic
461C3×C4C22×#4111yescharacteristic
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23C3×C22⋊Q814yes—
469C23C22×C3×D412yes—
470C23C22×C3×D411yescharacteristic
471Q8—22no—
472Q8—24no—
473Q8—88no—
474Q8C22×C3×Q814yes—
475D4—216no—
476D4—22no—
477D4—24no—
478D4C22×C3×Q814yes—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—216no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—88no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4C3×C22⋊Q814yes—
496C2×C4C3×C22⋊Q818yes—
497C2×C4C22×C3×D414yes—
498C2×C4C22×C3×D414yes—
499C2×C4C22×C3×D412yes—
500C2×C4C22×C3×D411yescharacteristic
501C2×C4C22×C3×Q812yes—
502C2×C4C22×C3×Q814yes—
503C2×C4C2×C3×#4112yes—
504C2×C4C2×C3×#4114yes—
505C2×C4C25×C311yescharacteristic
506C8—22no—
507C8—22no—
508C8—24no—
509C2×C3—24no—
510C2×C3—44no—
511C2×C3#308414yesdirect product factor
512C2×C3C22×C22⋊Q811yescharacteristic
513C2×C3C2×#57711yescharacteristic
514C2×C3C2×#57711yescharacteristic
515C22—22no—
516C22—22no—
517C22—28no—
518C22—24no—
519C22—42no—
520C22—42no—
521C22—48no—
522C22C2×C3×C22⋊Q814yes—
523C22C2×C3×C22⋊Q812yes—
524C22C3×#57712yes—
525C22C3×#57712yes—
526C22C23×C3×D411yescharacteristic
527C4—24no—
528C4—24no—
529C4—24no—
530C4—44no—
531C4—44no—
532C4—42no—
533C4—42no—
534C4C2×C3×C22⋊Q814yes—
535C4C2×C3×C22⋊Q812yes—
536C4C23×C3×Q811yescharacteristic
537C4C22×C3×#4111yescharacteristic
538C3C2×#308411yescharacteristic, direct product factor, Sylow
539C2—24no—
540C2—44no—
541C2C3×#308414yesdirect product factor
542C2C22×C3×C22⋊Q811yescharacteristic
543C2C2×C3×#57711yescharacteristic
544C2C2×C3×#57711yescharacteristic
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×#3084C2direct (N × H)
4. C2×C3×#479C2semidirect
5. C2×C3×#475C2semidirect
6. C2×C3×#478C2semidirect
7. C2×C3×#476C2semidirect
8. C2×C3×#428C2semidirect
9. C22×C3×#106C2semidirect
10. C2×C3×#424C2semidirect
13. C2×#3084C3direct (N × H)
18. C3×#544C22semidirect
35. C3×#479C22semidirect
36. C3×#478C22semidirect
37. C3×#476C22semidirect
38. C3×#475C22semidirect
39. C3×#428C22semidirect
44. C2×C3×#106C22semidirect
45. C3×#424C22semidirect
56. C2×#544C2×C3semidirect
58. #3084C2×C3direct (N × H)
59. C2×#479C2×C3semidirect
60. C2×#475C2×C3semidirect
61. C2×#478C2×C3semidirect
62. C2×#476C2×C3semidirect
63. C2×#428C2×C3semidirect
64. C22×#106C2×C3semidirect
65. C2×#424C2×C3semidirect
70. C2×C3×#41Q8semidirect
144. #544C22×C3semidirect
161. #479C22×C3semidirect
162. #478C22×C3semidirect
163. #476C22×C3semidirect
164. #475C22×C3semidirect
165. #428C22×C3semidirect
170. C2×#106C22×C3semidirect
171. #424C22×C3semidirect
186. C3×#41C2×Q8semidirect
253. C2×#41C3×Q8semidirect
374. #41C2×C3×Q8semidirect
511. C2×C3#3084direct (N × H)
538. C3C2×#3084direct (N × H)
541. C2C3×#3084direct (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×#5202
C3×#30842
C2×C3×#4792
C2×C3×#4752
C2×C3×#4782
C2×C3×#4762
C2×C3×#4282
C22×C3×#1062
C2×C3×#4242
C2×C3×#4172
C2×C3×#4162
C2×#30843