Finite Group Databasebeta

← C2×C3×#1792 #10639220

Subgroups — C2×C3×#1792 #10639220

Order 768.

4114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic
2C22×C3×#104C212yesmaximal, complemented
3C2×C3×#417C212yesmaximal, complemented
4C22×C3×#100C211yesmaximal, characteristic, complemented
5C2×C3×#390C211yesmaximal, characteristic, complemented
6C3×#1792C218yesmaximal, direct product factor
7C2×#1792C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×H(5,𝔽2)—28no—
9C2×C3×H(5,𝔽2)—24no—
10C2×C3×H(5,𝔽2)—24no—
11C2×C3×H(5,𝔽2)—22no—
12C2×C3×H(5,𝔽2)C418yescomplemented
13C2×C3×H(5,𝔽2)C2214yes—
14C22×C3×#41—22no—
15C22×C3×#41C2212yes—
16C23×C3×D4—22no—
17C23×C3×D4—22no—
18C23×C3×D4C2211yescharacteristic
19C22×C3×M16C2212yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic, complemented
21C22×C3×C22⋊C4—22no—
22C22×C3×C42C2211yescharacteristic, complemented
23C2×C3×#104—28no—
24C2×C3×#104C2218yescomplemented
25C3×#417—24no—
26C3×#417C2218yescomplemented
27C2×C3×#103—22no—
28C2×C3×#102—22no—
29C2×C3×#100—24no—
30C2×C3×#100C2214yescomplemented
31C3×#390—22no—
32C3×#390C2214yescomplemented
33C22×H(5,𝔽2)C2×C311yescharacteristic
34C22×#104C2×C312yescomplemented
35C2×#417C2×C312yescomplemented
36C22×#100C2×C311yescharacteristic, complemented
37C2×#390C2×C311yescharacteristic, complemented
38#1792C2×C318yesdirect product factor
39C25×C3—22no—
40C25×C3D412yes—
41C3×H(5,𝔽2)—216no—
42C3×H(5,𝔽2)—232no—
43C3×H(5,𝔽2)—232no—
44C3×H(5,𝔽2)—216no—
45C3×H(5,𝔽2)—216no—
46C3×H(5,𝔽2)—28no—
47C3×H(5,𝔽2)C2×C4116yescomplemented
48C2×C3×#41—28no—
49C2×C3×#41—216no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—24no—
53C2×C3×#41—22no—
54C2×C3×#41—24no—
55C2×C3×#41—22no—
56C2×C3×#41C2×C418yes—
57C2×C3×#41D418yes—
58C2×C3×#41D414yes—
59C2×C3×#41C2314yes—
60C22×C3×Q8D412yes—
61C22×C3×D4—24no—
62C22×C3×D4—28no—
63C22×C3×D4—216no—
64C22×C3×D4—28no—
65C22×C3×D4—216no—
66C22×C3×D4—216no—
67C22×C3×D4—22no—
68C22×C3×D4—28no—
69C22×C3×D4—216no—
70C22×C3×D4—28no—
71C22×C3×D4—22no—
72C22×C3×D4—24no—
73C22×C3×D4—22no—
74C22×C3×D4C2×C414yes—
75C22×C3×D4D412yes—
76C22×C3×D4D414yes—
77C22×C3×D4D412yes—
78C22×C3×D4C2312yes—
79C23×C3×C4—22no—
80C23×C3×C4—22no—
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C23×C3×C4C2311yescharacteristic
84C2×C3×M16—22no—
85C2×C3×M16—24no—
86C2×C3×M16—24no—
87C2×C3×M16C2314yescomplemented
88C22×C3×C8—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2312yescomplemented
93C2×C3×C22⋊C4—28no—
94C2×C3×C22⋊C4—48no—
95C2×C3×C42—24no—
96C2×C3×C42—21no—
97C2×C3×C42C2312yescomplemented
98C3×#104—216no—
99C3×#103—28no—
100C3×#102—28no—
101C3×#100—28no—
102C2×H(5,𝔽2)—28no—
103C2×H(5,𝔽2)—24no—
104C2×H(5,𝔽2)—24no—
105C2×H(5,𝔽2)—22no—
106C2×H(5,𝔽2)C3×C418yescomplemented
107C2×H(5,𝔽2)C22×C314yes—
108C22×#41—22no—
109C22×#41C22×C312yes—
110C23×D4—22no—
111C23×D4—22no—
112C23×D4C22×C311yescharacteristic
113C22×M16C22×C312yescomplemented
114C22×C4⋊C4C22×C311yescharacteristic, complemented
115C22×C22⋊C4—22no—
116C22×C42C22×C311yescharacteristic, complemented
117C2×#104—28no—
118C2×#104C22×C318yescomplemented
119#417—24no—
120#417C22×C318yescomplemented
121C2×#103—22no—
122C2×#102—22no—
123C2×#100—24no—
124C2×#100C22×C314yescomplemented
125#390—22no—
126#390C22×C314yescomplemented
127C24×C3—28no—
128C24×C3—216no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—24no—
133C24×C3—44no—
134C24×C3—44no—
135C24×C3—84no—
136C24×C3—84no—
137C24×C3C22⋊C418yes—
138C24×C3C2×D411yescharacteristic
139C24×C3C2×D414yes—
140C3×#41—232no—
141C3×#41—264no—
142C3×#41—232no—
143C3×#41—28no—
144C3×#41—28no—
145C3×#41—28no—
146C3×#41—28no—
147C3×#41—28no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41C22⋊C4116yescomplemented
151C3×#41C22×C418yes—
152C3×#41C2×D418yes—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C2×C3×Q8C22⋊C418yescomplemented
156C2×C3×Q8C2×D414yes—
157C2×C3×D4—216no—
158C2×C3×D4—232no—
159C2×C3×D4—232no—
160C2×C3×D4—264no—
161C2×C3×D4—232no—
162C2×C3×D4—264no—
163C2×C3×D4—28no—
164C2×C3×D4—232no—
165C2×C3×D4—24no—
166C2×C3×D4—28no—
167C2×C3×D4—24no—
168C2×C3×D4—232no—
169C2×C3×D4—28no—
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—22no—
175C2×C3×D4—28no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4C22⋊C418yes—
180C2×C3×D4C22⋊C418yescomplemented
181C2×C3×D4C22×C414yes—
182C2×C3×D4C2×D414yes—
183C2×C3×D4C2×D414yes—
184C22×C3×C4—28no—
185C22×C3×C4—216no—
186C22×C3×C4—22no—
187C22×C3×C4—28no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—44no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4C22×C412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2411yescharacteristic
206C3×M16—24no—
207C3×M16—24no—
208C3×M16—44no—
209C2×C3×C8—24no—
210C2×C3×C8—44no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—42no—
213C3×C22⋊C4—416no—
214C3×C42—24no—
215C3×C42—42no—
216C25—22no—
217C25C3×D412yes—
218H(5,𝔽2)—232no—
219H(5,𝔽2)—216no—
220H(5,𝔽2)—232no—
221H(5,𝔽2)—216no—
222H(5,𝔽2)—216no—
223H(5,𝔽2)—28no—
224H(5,𝔽2)C2×C3×C4116yescomplemented
225C2×#41—216no—
226C2×#41—28no—
227C2×#41—216no—
228C2×#41—28no—
229C2×#41—24no—
230C2×#41—22no—
231C2×#41—24no—
232C2×#41—22no—
233C2×#41C2×C3×C418yes—
234C2×#41C3×D418yes—
235C2×#41C3×D414yes—
236C2×#41C23×C314yes—
237C22×Q8C3×D412yes—
238C22×D4—24no—
239C22×D4—216no—
240C22×D4—28no—
241C22×D4—216no—
242C22×D4—28no—
243C22×D4—216no—
244C22×D4—22no—
245C22×D4—28no—
246C22×D4—216no—
247C22×D4—28no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—22no—
251C22×D4C2×C3×C414yes—
252C22×D4C3×D412yes—
253C22×D4C3×D414yes—
254C22×D4C3×D412yes—
255C22×D4C23×C312yes—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261C2×M16—22no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16C23×C314yescomplemented
265C22×C8—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C23×C312yescomplemented
270C2×C22⋊C4—28no—
271C2×C22⋊C4—48no—
272C2×C42—24no—
273C2×C42—21no—
274C2×C42C23×C312yescomplemented
275#104—216no—
276#103—28no—
277#102—28no—
278#100—28no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—232no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—28no—
285C23×C3—28no—
286C23×C3—24no—
287C23×C3—28no—
288C23×C3—44no—
289C23×C3—416no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—88no—
296C23×C3—82no—
297C23×C3—82no—
298C23×C3—832no—
299C23×C3—88no—
300C23×C3—88no—
301C23×C3C2×C22⋊C412yes—
302C23×C3C2×C22⋊C418yes—
303C23×C3#12012yes—
304C23×C3#12011yescharacteristic
305C23×C3#12011yescharacteristic
306C23×C3C22×D411yescharacteristic
307C3×Q8—24no—
308C3×Q8—28no—
309C3×Q8C2×C22⋊C418yescomplemented
310C3×D4—264no—
311C3×D4—28no—
312C3×D4—28no—
313C3×D4—216no—
314C3×D4—28no—
315C3×D4—28no—
316C3×D4—24no—
317C3×D4—216no—
318C3×D4—28no—
319C3×D4C2×C22⋊C418yescomplemented
320C2×C3×C4—232no—
321C2×C3×C4—24no—
322C2×C3×C4—28no—
323C2×C3×C4—28no—
324C2×C3×C4—28no—
325C2×C3×C4—216no—
326C2×C3×C4—24no—
327C2×C3×C4—48no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—82no—
333C2×C3×C4—82no—
334C2×C3×C4C2×C22⋊C414yes—
335C2×C3×C4C2×C22⋊C414yes—
336C2×C3×C4C2×C22⋊C412yes—
337C2×C3×C4#12014yes—
338C2×C3×C4#12014yes—
339C2×C3×C4#12012yes—
340C2×C3×C4#12012yes—
341C2×C3×C4C23×C411yescharacteristic
342C2×C3×C4C22×D412yes—
343C2×C3×C4C22×D412yes—
344C2×C3×C4C22×D411yescharacteristic
345C3×C8—44no—
346C24—216no—
347C24—28no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—44no—
353C24—44no—
354C24—84no—
355C24—84no—
356C24C3×C22⋊C418yes—
357C24C2×C3×D411yescharacteristic
358C24C2×C3×D414yes—
359#41—264no—
360#41—232no—
361#41—232no—
362#41—28no—
363#41—28no—
364#41—28no—
365#41—28no—
366#41—28no—
367#41—24no—
368#41—24no—
369#41C3×C22⋊C4116yescomplemented
370#41C22×C3×C418yes—
371#41C2×C3×D418yes—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8C3×C22⋊C418yescomplemented
375C2×Q8C2×C3×D414yes—
376C2×D4—232no—
377C2×D4—216no—
378C2×D4—264no—
379C2×D4—232no—
380C2×D4—264no—
381C2×D4—232no—
382C2×D4—28no—
383C2×D4—232no—
384C2×D4—28no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—232no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4C3×C22⋊C418yes—
399C2×D4C3×C22⋊C418yescomplemented
400C2×D4C22×C3×C414yes—
401C2×D4C2×C3×D414yes—
402C2×D4C2×C3×D414yes—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C22×C3×C412yes—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D414yes—
421C22×C4C2×C3×D412yes—
422C22×C4C2×C3×D412yes—
423C22×C4C2×C3×D412yes—
424C22×C4C24×C311yescharacteristic
425M16—24no—
426M16—24no—
427M16—44no—
428C2×C8—24no—
429C2×C8—44no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C22⋊C4—416no—
433C42—24no—
434C42—42no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—28no—
442C22×C3—28no—
443C22×C3—48no—
444C22×C3—416no—
445C22×C3—416no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—84no—
451C22×C3—88no—
452C22×C3—832no—
453C22×C3#37912yes—
454C22×C3#37912yes—
455C22×C3#45312yes—
456C22×C3#45512yes—
457C22×C3C22×C22⋊C411yescharacteristic
458C22×C3C2×#12011yescharacteristic
459C22×C3C2×#12011yescharacteristic
460C3×C4—28no—
461C3×C4—44no—
462C3×C4—82no—
463C3×C4#37914yes—
464C3×C4C22×C22⋊C412yes—
465C3×C4C2×#12012yes—
466C23—22no—
467C23—22no—
468C23—232no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—24no—
475C23—44no—
476C23—416no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—88no—
483C23—82no—
484C23—82no—
485C23—832no—
486C23—88no—
487C23—88no—
488C23C2×C3×C22⋊C412yes—
489C23C2×C3×C22⋊C418yes—
490C23C3×#12012yes—
491C23C3×#12011yescharacteristic
492C23C3×#12011yescharacteristic
493C23C22×C3×D411yescharacteristic
494Q8—28no—
495Q8—24no—
496Q8C2×C3×C22⋊C418yescomplemented
497D4—264no—
498D4—216no—
499D4—28no—
500D4—28no—
501D4—216no—
502D4—28no—
503D4—28no—
504D4—28no—
505D4—24no—
506D4C2×C3×C22⋊C418yescomplemented
507C2×C4—232no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—216no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4C2×C3×C22⋊C412yes—
522C2×C4C2×C3×C22⋊C414yes—
523C2×C4C2×C3×C22⋊C414yes—
524C2×C4C3×#12012yes—
525C2×C4C3×#12012yes—
526C2×C4C3×#12014yes—
527C2×C4C3×#12014yes—
528C2×C4C23×C3×C411yescharacteristic
529C2×C4C22×C3×D412yes—
530C2×C4C22×C3×D412yes—
531C2×C4C22×C3×D411yescharacteristic
532C8—44no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—48no—
537C2×C3—48no—
538C2×C3#179214yesdirect product factor
539C2×C3C2×#37911yescharacteristic
540C2×C3C2×#45311yescharacteristic
541C2×C3C2×#45511yescharacteristic
542C22—24no—
543C22—24no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—28no—
549C22—28no—
550C22—48no—
551C22—416no—
552C22—416no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—84no—
558C22—88no—
559C22—832no—
560C22C3×#37912yes—
561C22C3×#37912yes—
562C22C3×#45312yes—
563C22C3×#45512yes—
564C22C22×C3×C22⋊C411yescharacteristic
565C22C2×C3×#12011yescharacteristic
566C22C2×C3×#12011yescharacteristic
567C4—28no—
568C4—44no—
569C4—82no—
570C4C3×#37914yes—
571C4C22×C3×C22⋊C412yes—
572C4C2×C3×#12012yes—
573C3C2×#179211yescharacteristic, direct product factor, Sylow
574C2—21no—
575C2—21no—
576C2—22no—
577C2—48no—
578C2—48no—
579C2C3×#179214yesdirect product factor
580C2C2×C3×#37911yescharacteristic
581C2C2×C3×#45311yescharacteristic
582C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×H(5,𝔽2)C2
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
2. C22×C3×#104C2semidirect
3. C2×C3×#417C2semidirect
4. C22×C3×#100C2semidirect
5. C2×C3×#390C2semidirect
6. C3×#1792C2direct (N × H)
7. C2×#1792C3direct (N × H)
12. C2×C3×H(5,𝔽2)C4semidirect
19. C22×C3×M16C22semidirect
20. C22×C3×C4⋊C4C22semidirect
22. C22×C3×C42C22semidirect
24. C2×C3×#104C22semidirect
26. C3×#417C22semidirect
30. C2×C3×#100C22semidirect
32. C3×#390C22semidirect
34. C22×#104C2×C3semidirect
35. C2×#417C2×C3semidirect
36. C22×#100C2×C3semidirect
37. C2×#390C2×C3semidirect
38. #1792C2×C3direct (N × H)
47. C3×H(5,𝔽2)C2×C4semidirect
87. C2×C3×M16C23semidirect
92. C2×C3×C4⋊C4C23semidirect
97. C2×C3×C42C23semidirect
106. C2×H(5,𝔽2)C3×C4semidirect
113. C22×M16C22×C3semidirect
114. C22×C4⋊C4C22×C3semidirect
116. C22×C42C22×C3semidirect
118. C2×#104C22×C3semidirect
120. #417C22×C3semidirect
124. C2×#100C22×C3semidirect
126. #390C22×C3semidirect
150. C3×#41C22⋊C4semidirect
155. C2×C3×Q8C22⋊C4semidirect
180. C2×C3×D4C22⋊C4semidirect
224. H(5,𝔽2)C2×C3×C4semidirect
264. C2×M16C23×C3semidirect
269. C2×C4⋊C4C23×C3semidirect
274. C2×C42C23×C3semidirect
309. C3×Q8C2×C22⋊C4semidirect
319. C3×D4C2×C22⋊C4semidirect
369. #41C3×C22⋊C4semidirect
374. C2×Q8C3×C22⋊C4semidirect
399. C2×D4C3×C22⋊C4semidirect
496. Q8C2×C3×C22⋊C4semidirect
506. D4C2×C3×C22⋊C4semidirect
538. C2×C3#1792direct (N × H)
573. C3C2×#1792direct (N × H)
579. C2C3×#1792direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C22×C3×#1042
C2×C3×#4172
C22×C3×#1002
C2×C3×#3902
C3×#17922
C2×#17923