Finite Group Databasebeta

← C3×#60425 #10679032

Subgroups — C3×#60425 #10679032

Order 768.

2750 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C212yesmaximal, complemented
2C3×#3449C211yesmaximal, characteristic, complemented
3C2×C3×#520C211yesmaximal, characteristic, complemented
4C3×#3046C218yesmaximal, complemented
5C3×#3045C218yesmaximal, complemented
6C3×#3037C214yesmaximal, complemented
7C3×#3036C214yesmaximal, complemented
8C3×#2905C211yesmaximal, characteristic, complemented
9C2×C3×#417C212yesmaximal, complemented
10#60425C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×#41C2211yescharacteristic
13C3×#575—216no—
14C2×C3×#137C2214yescomplemented
15C2×C3×#137C2214yescomplemented
16C2×C3×AGL(1,ℤ/8)C2214yescomplemented
17C2×C3×AGL(1,ℤ/8)C2214yescomplemented
18C2×C3×#135C2218yes—
19C22×C3×M16C2212yes—
20C3×#537—21no—
21C3×#537—21no—
22C3×#536C2214yescomplemented
23C3×#535C2214yescomplemented
24C3×#534—22no—
25C3×#532C2214yes—
26C3×#525—24no—
27C3×#520—24no—
28C3×#520C2218yescomplemented
29C2×C3×C4×Q8C2212yescomplemented
30C2×C3×C4×D4—22no—
31C2×C3×C4×D4C2212yescomplemented
32C2×C3×#117—21no—
33C2×C3×#117C2211yescharacteristic
34C22×C3×C4⋊C4—21no—
35C22×C3×C4⋊C4C2211yescharacteristic
36C3×#464C22116yescomplemented
37C3×#463C22116yescomplemented
38C3×#462C2218yescomplemented
39C3×#461C2218yescomplemented
40C3×#460C2218yescomplemented
41C3×#459C2218yescomplemented
42C3×#424C2218yescomplemented
43C3×#423C2214yescomplemented
44C3×#417C2218yescomplemented
45C3×#417C2218yescomplemented
46C2×C3×#103C2214yescomplemented
47C2×C3×#102C2214yescomplemented
48C2×#575C2×C312yescomplemented
49#3449C2×C311yescharacteristic, complemented
50C2×#520C2×C311yescharacteristic, complemented
51#3046C2×C318yescomplemented
52#3045C2×C318yescomplemented
53#3037C2×C314yescomplemented
54#3036C2×C314yescomplemented
55#2905C2×C311yescharacteristic, complemented
56C2×#417C2×C312yescomplemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C2×C3×#41—28no—
62C2×C3×#41—28no—
63C2×C3×#41D418yescomplemented
64C2×C3×#41C2314yes—
65C2×C3×#41C2314yes—
66C2×C3×#41C2314yes—
67C22×C3×Q8C2312yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4C2312yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4—21no—
75C23×C3×C4—22no—
76C23×C3×C4—22no—
77C23×C3×C4C2311yescharacteristic
78C3×#137—216no—
79C3×#137—216no—
80C3×AGL(1,ℤ/8)—216no—
81C3×AGL(1,ℤ/8)—216no—
82C3×#135—232no—
83C2×C3×Q16C2318yes—
84C2×C3×SD16C2318yes—
85C2×C3×SD16C2318yes—
86C2×C3×D8C2318yes—
87C2×C3×M16—28no—
88C2×C3×M16C2314yes—
89C2×C3×M16C2314yes—
90C22×C3×C8C2314yes—
91C3×C4⋊Q8C2314yes—
92C3×#127C2314yes—
93C3×#124C2318yes—
94C3×#123—28no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—24no—
97C3×#121—24no—
98C3×#121—24no—
99C3×#121—24no—
100C3×#121—24no—
101C3×#120—28no—
102C3×C4×Q8—24no—
103C3×C4×Q8C2318yes—
104C3×C4×D4—216no—
105C3×C4×D4—24no—
106C3×C4×D4—24no—
107C3×C4×D4—24no—
108C3×C4×D4C2318yes—
109C3×#117—28no—
110C3×#117D414yes—
111C3×#117C2314yes—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4—28no—
116C2×C3×C4⋊C4D411yescharacteristic
117C2×C3×C4⋊C4D411yescharacteristic
118C2×C3×C4⋊C4C2314yes—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4D412yes—
122C2×C3×C42—22no—
123C2×C3×C42C2312yes—
124C3×C4⋊C8C2318yes—
125C3×C4⋊C8C2318yes—
126C3×#103C23116yes—
127C3×#102C23116yes—
128C22×#41C22×C312yes—
129C22×#41C22×C311yescharacteristic
130#575—216no—
131C2×#137C22×C314yescomplemented
132C2×#137C22×C314yescomplemented
133C2×AGL(1,ℤ/8)C22×C314yescomplemented
134C2×AGL(1,ℤ/8)C22×C314yescomplemented
135C2×#135C22×C318yes—
136C22×M16C22×C312yes—
137#537—21no—
138#537—21no—
139#536C22×C314yescomplemented
140#535C22×C314yescomplemented
141#534—22no—
142#532C22×C314yes—
143#525—24no—
144#520—24no—
145#520C22×C318yescomplemented
146C2×C4×Q8C22×C312yescomplemented
147C2×C4×D4—22no—
148C2×C4×D4C22×C312yescomplemented
149C2×#117—21no—
150C2×#117C22×C311yescharacteristic
151C22×C4⋊C4—21no—
152C22×C4⋊C4C22×C311yescharacteristic
153#464C22×C3116yescomplemented
154#463C22×C3116yescomplemented
155#462C22×C318yescomplemented
156#461C22×C318yescomplemented
157#460C22×C318yescomplemented
158#459C22×C318yescomplemented
159#424C22×C318yescomplemented
160#423C22×C314yescomplemented
161#417C22×C318yescomplemented
162#417C22×C318yescomplemented
163C2×#103C22×C314yescomplemented
164C2×#102C22×C314yescomplemented
165C24×C3—21no—
166C24×C3—22no—
167C24×C3C2×D411yescharacteristic
168C3×#41—216no—
169C3×#41—28no—
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—48no—
175C3×#41—48no—
176C3×#41C2×D4116yes—
177C2×C3×Q8—21no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—24no—
181C2×C3×Q8—24no—
182C2×C3×Q8C2×D414yes—
183C2×C3×Q8C2414yes—
184C2×C3×Q8C2412yes—
185C2×C3×D4—28no—
186C2×C3×D4—24no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—21no—
190C2×C3×D4—24no—
191C2×C3×D4—22no—
192C2×C3×D4—24no—
193C2×C3×D4—24no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—48no—
197C2×C3×D4C2×D414yes—
198C2×C3×D4C2414yes—
199C2×C3×D4C2412yes—
200C22×C3×C4—21no—
201C22×C3×C4—28no—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—28no—
209C22×C3×C4—48no—
210C22×C3×C4—84no—
211C22×C3×C4—82no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D414yes—
217C22×C3×C4#4114yes—
218C22×C3×C4C2411yescharacteristic
219C22×C3×C4C2412yes—
220C3×Q16—216no—
221C3×SD16—216no—
222C3×SD16—216no—
223C3×D8—216no—
224C3×M16—28no—
225C3×M16—28no—
226C2×C3×C8—28no—
227C2×C3×C8C2418yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4C2×D414yes—
234C3×C4⋊C4C2×D414yes—
235C3×C4⋊C4C2414yes—
236C3×C22⋊C4—28no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—28no—
239C3×C22⋊C4C2×D418yes—
240C3×C42—28no—
241C3×C42C2×D414yes—
242C3×C42C2414yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—24no—
247C2×#41—28no—
248C2×#41—28no—
249C2×#41C3×D418yescomplemented
250C2×#41C23×C314yes—
251C2×#41C23×C314yes—
252C2×#41C23×C314yes—
253C22×Q8C23×C312yes—
254C22×Q8C23×C311yescharacteristic
255C22×D4—21no—
256C22×D4—22no—
257C22×D4C23×C312yes—
258C22×D4C23×C311yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C23×C311yescharacteristic
264#137—216no—
265#137—216no—
266AGL(1,ℤ/8)—216no—
267AGL(1,ℤ/8)—216no—
268#135—232no—
269C2×Q16C23×C318yes—
270C2×SD16C23×C318yes—
271C2×SD16C23×C318yes—
272C2×D8C23×C318yes—
273C2×M16—28no—
274C2×M16C23×C314yes—
275C2×M16C23×C314yes—
276C22×C8C23×C314yes—
277C4⋊Q8C23×C314yes—
278#127C23×C314yes—
279#124C23×C318yes—
280#123—28no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#121—24no—
284#121—24no—
285#121—24no—
286#121—24no—
287#120—28no—
288C4×Q8—24no—
289C4×Q8C23×C318yes—
290C4×D4—216no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4C23×C318yes—
295#117—28no—
296#117C3×D414yes—
297#117C23×C314yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4C3×D411yescharacteristic
303C2×C4⋊C4C3×D411yescharacteristic
304C2×C4⋊C4C23×C312yes—
305C2×C4⋊C4C23×C314yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C3×D412yes—
308C2×C42—22no—
309C2×C42C23×C312yes—
310C4⋊C8C23×C318yes—
311C4⋊C8C23×C318yes—
312#103C23×C3116yes—
313#102C23×C3116yes—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—44no—
322C23×C3—82no—
323C23×C3—82no—
324C23×C3#12114yes—
325C23×C3C22×D412yes—
326C23×C3C22×D411yescharacteristic
327C3×Q8—24no—
328C3×Q8—22no—
329C3×Q8—24no—
330C3×Q8—24no—
331C3×Q8—44no—
332C3×Q8C22×D414yes—
333C3×D4—216no—
334C3×D4—24no—
335C3×D4—24no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—48no—
339C3×D4—48no—
340C3×D4—44no—
341C3×D4—84no—
342C3×D4—84no—
343C3×D4C22×D414yes—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—216no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4—48no—
354C2×C3×C4—48no—
355C2×C3×C4—88no—
356C2×C3×C4—84no—
357C2×C3×C4#12114yes—
358C2×C3×C4#12118yes—
359C2×C3×C4C22×D414yes—
360C2×C3×C4C22×D414yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D414yes—
364C2×C3×C4C22×D412yes—
365C2×C3×C4C2×#4114yes—
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2511yescharacteristic
368C3×C8—28no—
369C24—21no—
370C24—22no—
371C24C2×C3×D411yescharacteristic
372#41—216no—
373#41—28no—
374#41—28no—
375#41—28no—
376#41—28no—
377#41—28no—
378#41—48no—
379#41—48no—
380#41C2×C3×D4116yes—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8C2×C3×D414yes—
387C2×Q8C24×C314yes—
388C2×Q8C24×C312yes—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4C2×C3×D414yes—
402C2×D4C24×C314yes—
403C2×D4C24×C312yes—
404C22×C4—21no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—48no—
414C22×C4—84no—
415C22×C4—82no—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D412yes—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D412yes—
420C22×C4C2×C3×D414yes—
421C22×C4C3×#4114yes—
422C22×C4C24×C311yescharacteristic
423C22×C4C24×C312yes—
424Q16—216no—
425SD16—216no—
426SD16—216no—
427D8—216no—
428M16—28no—
429M16—28no—
430C2×C8—28no—
431C2×C8C24×C318yes—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4C2×C3×D414yes—
438C4⋊C4C2×C3×D414yes—
439C4⋊C4C24×C314yes—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4C2×C3×D418yes—
444C42—28no—
445C42C2×C3×D414yes—
446C42C24×C314yes—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—44no—
453C22×C3—48no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—88no—
460C22×C3C2×#12112yes—
461C22×C3C2×#12114yes—
462C22×C3C23×D411yescharacteristic
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4C2×#12114yes—
469C3×C4C2×#12112yes—
470C3×C4C23×D411yescharacteristic
471C3×C4C22×#4111yescharacteristic
472C23—22no—
473C23—24no—
474C23—24no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—82no—
481C23—82no—
482C23C3×#12114yes—
483C23C22×C3×D412yes—
484C23C22×C3×D411yescharacteristic
485Q8—22no—
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—44no—
490Q8C22×C3×D414yes—
491D4—216no—
492D4—24no—
493D4—22no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—84no—
500D4—84no—
501D4C22×C3×D414yes—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—216no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4C3×#12114yes—
516C2×C4C3×#12118yes—
517C2×C4C22×C3×D414yes—
518C2×C4C22×C3×D414yes—
519C2×C4C22×C3×D414yes—
520C2×C4C22×C3×D412yes—
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C22×C3×D412yes—
523C2×C4C2×C3×#4114yes—
524C2×C4C2×C3×#4112yes—
525C2×C4C25×C311yescharacteristic
526C8—28no—
527C2×C3—22no—
528C2×C3—24no—
529C2×C3—44no—
530C2×C3—84no—
531C2×C3C22×#12111yescharacteristic
532C2×C3#358511yescharacteristic
533C2×C3#358611yescharacteristic
534C22—21no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—24no—
539C22—44no—
540C22—48no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—88no—
547C22C2×C3×#12112yes—
548C22C2×C3×#12114yes—
549C22C23×C3×D411yescharacteristic
550C4—24no—
551C4—24no—
552C4—24no—
553C4—44no—
554C4—44no—
555C4C2×C3×#12114yes—
556C4C2×C3×#12112yes—
557C4C23×C3×D411yescharacteristic
558C4C22×C3×#4111yescharacteristic
559C3#6042511yescharacteristic, direct product factor, Sylow
560C2—22no—
561C2—24no—
562C2—44no—
563C2—84no—
564C2C22×C3×#12111yescharacteristic
565C2C3×#358511yescharacteristic
566C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#575C2semidirect
2. C3×#3449C2semidirect
3. C2×C3×#520C2semidirect
4. C3×#3046C2semidirect
5. C3×#3045C2semidirect
6. C3×#3037C2semidirect
7. C3×#3036C2semidirect
8. C3×#2905C2semidirect
9. C2×C3×#417C2semidirect
10. #60425C3direct (N × H)
14. C2×C3×#137C22semidirect
15. C2×C3×#137C22semidirect
16. C2×C3×AGL(1,ℤ/8)C22semidirect
17. C2×C3×AGL(1,ℤ/8)C22semidirect
22. C3×#536C22semidirect
23. C3×#535C22semidirect
28. C3×#520C22semidirect
29. C2×C3×C4×Q8C22semidirect
31. C2×C3×C4×D4C22semidirect
36. C3×#464C22semidirect
37. C3×#463C22semidirect
38. C3×#462C22semidirect
39. C3×#461C22semidirect
40. C3×#460C22semidirect
41. C3×#459C22semidirect
42. C3×#424C22semidirect
43. C3×#423C22semidirect
44. C3×#417C22semidirect
45. C3×#417C22semidirect
46. C2×C3×#103C22semidirect
47. C2×C3×#102C22semidirect
48. C2×#575C2×C3semidirect
49. #3449C2×C3semidirect
50. C2×#520C2×C3semidirect
51. #3046C2×C3semidirect
52. #3045C2×C3semidirect
53. #3037C2×C3semidirect
54. #3036C2×C3semidirect
55. #2905C2×C3semidirect
56. C2×#417C2×C3semidirect
63. C2×C3×#41D4semidirect
131. C2×#137C22×C3semidirect
132. C2×#137C22×C3semidirect
133. C2×AGL(1,ℤ/8)C22×C3semidirect
134. C2×AGL(1,ℤ/8)C22×C3semidirect
139. #536C22×C3semidirect
140. #535C22×C3semidirect
145. #520C22×C3semidirect
146. C2×C4×Q8C22×C3semidirect
148. C2×C4×D4C22×C3semidirect
153. #464C22×C3semidirect
154. #463C22×C3semidirect
155. #462C22×C3semidirect
156. #461C22×C3semidirect
157. #460C22×C3semidirect
158. #459C22×C3semidirect
159. #424C22×C3semidirect
160. #423C22×C3semidirect
161. #417C22×C3semidirect
162. #417C22×C3semidirect
163. C2×#103C22×C3semidirect
164. C2×#102C22×C3semidirect
249. C2×#41C3×D4semidirect
559. C3#60425direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C3×#34492
C2×C3×#5202
C3×#30462
C3×#30452
C3×#30372
C3×#30362
C3×#29052
C2×C3×#4172
#604253