Finite Group Databasebeta

← C3×#33540 #10652147

Subgroups — C3×#33540 #10652147

Order 768.

1078 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#578C211yesmaximal, characteristic, complemented
2C3×#3414C211yesmaximal, characteristic, complemented
3C2×C3×#510C211yesmaximal, characteristic, complemented
4C3×#3067C211yesmaximal, characteristic, complemented
5C3×#2195C214yesmaximal, complemented
6C3×#2193C214yesmaximal, complemented
7C2×C3×#362C211yesmaximal, characteristic, complemented
8C3×#2143C211yesmaximal, characteristic, complemented
9C3×#2117C211yesmaximal, characteristic, complemented
10#33540C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4—21no—
12C3×#578—22no—
13C3×#578—22no—
14C3×#578—22no—
15C3×#578C2214yes—
16C3×#575—21no—
17C2×C3×#137—22no—
18C2×C3×#137—21no—
19C2×C3×#135—21no—
20C2×C3×#135C2211yescharacteristic
21C2×C3×#135C2211yescharacteristic
22C22×C3×Q16—21no—
23C22×C3×Q16C2211yescharacteristic
24C2×C3×#131C2211yescharacteristic
25C3×#510—22no—
26C3×#510—22no—
27C3×#510—21no—
28C3×#510C2212yescomplemented
29C3×#509—21no—
30C3×#509C2212yescomplemented
31C3×#508—22no—
32C2×C3×Q32C2212yescomplemented
33C2×C3×SD32C2212yescomplemented
34C2×C3×M32C2211yescharacteristic
35C2×C3×M32C2211yescharacteristic
36C3×#473C2212yescomplemented
37C3×#472C2212yescomplemented
38C3×#471—22no—
39C2×C3×#108C2211yescharacteristic
40C3×#413—21no—
41C3×#362C2214yescomplemented
42C3×#359C2214yescomplemented
43C3×#350C2214yescomplemented
44C2×#578C2×C311yescharacteristic, complemented
45#3414C2×C311yescharacteristic, complemented
46C2×#510C2×C311yescharacteristic, complemented
47#3067C2×C311yescharacteristic, complemented
48#2195C2×C314yescomplemented
49#2193C2×C314yescomplemented
50C2×#362C2×C311yescharacteristic, complemented
51#2143C2×C311yescharacteristic, complemented
52#2117C2×C311yescharacteristic, complemented
53C3×2−1+4—24no—
54C3×2−1+4—24no—
55C3×2−1+4—42no—
56C3×2−1+4—42no—
57C2×C3×#41—22no—
58C2×C3×#41—21no—
59C2×C3×#41—21no—
60C2×C3×#41—42no—
61C2×C3×#41—41no—
62C2×C3×#41D411yescharacteristic
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×Q8—41no—
66C3×#137—24no—
67C3×#137—28no—
68C3×#137—28no—
69C3×#137—22no—
70C3×#137—24no—
71C3×#137—41no—
72C3×AGL(1,ℤ/8)—22no—
73C3×AGL(1,ℤ/8)—41no—
74C3×#135—24no—
75C3×#135—24no—
76C3×#135—21no—
77C3×#135—21no—
78C3×#135—21no—
79C3×#135—21no—
80C3×#135—21no—
81C3×#135—21no—
82C3×#135—42no—
83C3×#135D412yes—
84C3×#135C2312yes—
85C2×C3×Q16—22no—
86C2×C3×Q16—24no—
87C2×C3×Q16—24no—
88C2×C3×Q16—21no—
89C2×C3×Q16—21no—
90C2×C3×Q16—21no—
91C2×C3×Q16D411yescharacteristic
92C2×C3×Q16D412yes—
93C2×C3×Q16D412yes—
94C2×C3×Q16C2311yescharacteristic
95C2×C3×Q16C2312yes—
96C2×C3×SD16—22no—
97C2×C3×SD16—21no—
98C2×C3×SD16—21no—
99C2×C3×D8D411yescharacteristic
100C2×C3×D8C2311yescharacteristic
101C3×#131—21no—
102C3×#131—21no—
103C3×#131D412yes—
104C3×#131C2312yes—
105C2×C3×M16—21no—
106C2×C3×M16—21no—
107C2×C3×M16D411yescharacteristic
108C22×C3×C8—21no—
109C22×C3×C8C2311yescharacteristic
110C3×Q32—24no—
111C3×Q32—22no—
112C3×SD32—24no—
113C3×SD32—22no—
114C3×SD32—22no—
115C3×D16—22no—
116C3×M32—21no—
117C3×M32—21no—
118C3×M32—21no—
119C3×M32C2312yes—
120C2×C3×C16—21no—
121C2×C3×C16C2312yes—
122C3×#108—21no—
123C3×#108C2312yes—
124C3×#101—22no—
125C3×#100—22no—
126C2×2−1+4—21no—
127#578—22no—
128#578—22no—
129#578—22no—
130#578C22×C314yes—
131#575—21no—
132C2×#137—22no—
133C2×#137—21no—
134C2×#135—21no—
135C2×#135C22×C311yescharacteristic
136C2×#135C22×C311yescharacteristic
137C22×Q16—21no—
138C22×Q16C22×C311yescharacteristic
139C2×#131C22×C311yescharacteristic
140#510—22no—
141#510—22no—
142#510—21no—
143#510C22×C312yescomplemented
144#509—21no—
145#509C22×C312yescomplemented
146#508—22no—
147C2×Q32C22×C312yescomplemented
148C2×SD32C22×C312yescomplemented
149C2×M32C22×C311yescharacteristic
150C2×M32C22×C311yescharacteristic
151#473C22×C312yescomplemented
152#472C22×C312yescomplemented
153#471—22no—
154C2×#108C22×C311yescharacteristic
155#413—21no—
156#362C22×C314yescomplemented
157#359C22×C314yescomplemented
158#350C22×C314yescomplemented
159C3×#41—21no—
160C3×#41—21no—
161C3×#41—28no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—44no—
165C3×#41—48no—
166C3×#41—48no—
167C3×#41—44no—
168C3×#41—44no—
169C3×#41—41no—
170C3×#41—41no—
171C3×#41—41no—
172C3×#41—41no—
173C3×#41—41no—
174C3×#41—41no—
175C3×#41D812yes—
176C3×#41C2×D412yes—
177C2×C3×Q8—28no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—42no—
184C2×C3×Q8—44no—
185C2×C3×Q8—44no—
186C2×C3×Q8—44no—
187C2×C3×Q8—41no—
188C2×C3×Q8—41no—
189C2×C3×Q8—41no—
190C2×C3×Q8D811yescharacteristic
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—42no—
195C2×C3×D4—41no—
196C2×C3×D4—41no—
197C2×C3×D4D811yescharacteristic
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—41no—
202C22×C3×C4—41no—
203C22×C3×C4—41no—
204C22×C3×C4C2×D411yescharacteristic
205C3×Q16—28no—
206C3×Q16—21no—
207C3×Q16—22no—
208C3×Q16—22no—
209C3×Q16—22no—
210C3×Q16—21no—
211C3×Q16—21no—
212C3×Q16—42no—
213C3×Q16C2×D412yes—
214C3×Q16C2×D414yes—
215C3×SD16—28no—
216C3×SD16—22no—
217C3×SD16—22no—
218C3×SD16—42no—
219C3×SD16—42no—
220C3×SD16—41no—
221C3×SD16—41no—
222C3×D8—21no—
223C3×D8—21no—
224C3×D8—21no—
225C3×D8—42no—
226C3×D8C2×D412yes—
227C3×M16—21no—
228C3×M16—22no—
229C3×M16—22no—
230C3×M16—22no—
231C3×M16—41no—
232C3×M16C2×D412yes—
233C2×C3×C8—21no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—41no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2411yescharacteristic
243C3×C16—22no—
244C3×C16—22no—
2452−1+4—24no—
2462−1+4—24no—
2472−1+4—42no—
2482−1+4—42no—
249C2×#41—22no—
250C2×#41—21no—
251C2×#41—21no—
252C2×#41—42no—
253C2×#41—41no—
254C2×#41C3×D411yescharacteristic
255C22×Q8—22no—
256C22×Q8—21no—
257C22×Q8—41no—
258#137—24no—
259#137—28no—
260#137—28no—
261#137—22no—
262#137—24no—
263#137—41no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)—41no—
266#135—24no—
267#135—24no—
268#135—21no—
269#135—21no—
270#135—21no—
271#135—21no—
272#135—21no—
273#135—21no—
274#135—42no—
275#135C3×D412yes—
276#135C23×C312yes—
277C2×Q16—22no—
278C2×Q16—24no—
279C2×Q16—24no—
280C2×Q16—21no—
281C2×Q16—21no—
282C2×Q16—21no—
283C2×Q16C3×D411yescharacteristic
284C2×Q16C3×D412yes—
285C2×Q16C3×D412yes—
286C2×Q16C23×C311yescharacteristic
287C2×Q16C23×C312yes—
288C2×SD16—22no—
289C2×SD16—21no—
290C2×SD16—21no—
291C2×D8C3×D411yescharacteristic
292C2×D8C23×C311yescharacteristic
293#131—21no—
294#131—21no—
295#131C3×D412yes—
296#131C23×C312yes—
297C2×M16—21no—
298C2×M16—21no—
299C2×M16C3×D411yescharacteristic
300C22×C8—21no—
301C22×C8C23×C311yescharacteristic
302Q32—24no—
303Q32—22no—
304SD32—24no—
305SD32—22no—
306SD32—22no—
307D16—22no—
308M32—21no—
309M32—21no—
310M32—21no—
311M32C23×C312yes—
312C2×C16—21no—
313C2×C16C23×C312yes—
314#108—21no—
315#108C23×C312yes—
316#101—22no—
317#100—22no—
318C23×C3—21no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3AGL(1,ℤ/8)11yescharacteristic
322C3×Q8—21no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—48no—
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8C2×D812yes—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—21no—
337C3×D4—48no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—41no—
343C3×D4—81no—
344C3×D4—81no—
345C3×D4C2×D812yes—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—48no—
350C2×C3×C4—48no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—81no—
359C2×C3×C4—81no—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4C2×D811yescharacteristic
363C2×C3×C4C2×D811yescharacteristic
364C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C3×C8—22no—
367C3×C8—42no—
368C3×C8#12011yescharacteristic
369C3×C8#12011yescharacteristic
370C3×C8C22×D411yescharacteristic
371C3×C8C22×D411yescharacteristic
372#41—21no—
373#41—21no—
374#41—28no—
375#41—22no—
376#41—22no—
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41C3×D812yes—
389#41C2×C3×D412yes—
390C2×Q8—28no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8C3×D811yescharacteristic
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4C3×D811yescharacteristic
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4C2×C3×D411yescharacteristic
418Q16—28no—
419Q16—21no—
420Q16—22no—
421Q16—22no—
422Q16—22no—
423Q16—21no—
424Q16—21no—
425Q16—42no—
426Q16C2×C3×D412yes—
427Q16C2×C3×D414yes—
428SD16—28no—
429SD16—22no—
430SD16—22no—
431SD16—42no—
432SD16—42no—
433SD16—41no—
434SD16—41no—
435D8—21no—
436D8—21no—
437D8—21no—
438D8—42no—
439D8C2×C3×D412yes—
440M16—21no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—41no—
445M16C2×C3×D412yes—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—41no—
450C2×C8C2×C3×D411yescharacteristic
451C2×C8C2×C3×D411yescharacteristic
452C2×C8C2×C3×D411yescharacteristic
453C2×C8C2×C3×D411yescharacteristic
454C2×C8C2×C3×D411yescharacteristic
455C2×C8C24×C311yescharacteristic
456C16—22no—
457C16—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—82no—
463C22×C3—81no—
464C22×C3—161no—
465C22×C3#44711yescharacteristic
466C22×C3#44711yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C3×C4—22no—
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—81no—
472C3×C4#44711yescharacteristic
473C3×C4#44711yescharacteristic
474C3×C4C2×#12011yescharacteristic
475C3×C4C22×D811yescharacteristic
476C23—21no—
477C23—41no—
478C23—41no—
479C23C3×AGL(1,ℤ/8)11yescharacteristic
480Q8—21no—
481Q8—24no—
482Q8—22no—
483Q8—48no—
484Q8—42no—
485Q8—41no—
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8C2×C3×D812yes—
491D4—22no—
492D4—22no—
493D4—21no—
494D4—22no—
495D4—48no—
496D4—42no—
497D4—42no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—81no—
502D4—81no—
503D4C2×C3×D812yes—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4C3×#12011yescharacteristic
519C2×C4C3×#12011yescharacteristic
520C2×C4C2×C3×D811yescharacteristic
521C2×C4C2×C3×D811yescharacteristic
522C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
523C2×C4C22×C3×D411yescharacteristic
524C8—22no—
525C8—42no—
526C8C3×#12011yescharacteristic
527C8C3×#12011yescharacteristic
528C8C22×C3×D411yescharacteristic
529C8C22×C3×D411yescharacteristic
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—42no—
534C2×C3—82no—
535C2×C3—161no—
536C2×C3C2×#44711yescharacteristic
537C22—22no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—81no—
543C22—161no—
544C22C3×#44711yescharacteristic
545C22C3×#44711yescharacteristic
546C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
547C4—22no—
548C4—44no—
549C4—42no—
550C4—81no—
551C4C3×#44711yescharacteristic
552C4C3×#44711yescharacteristic
553C4C2×C3×#12011yescharacteristic
554C4C22×C3×D811yescharacteristic
555C3#3354011yescharacteristic, direct product factor, Sylow
556C2—21no—
557C2—21no—
558C2—21no—
559C2—42no—
560C2—82no—
561C2—161no—
562C2C2×C3×#44711yescharacteristic
subgroupG/H
center, socleC2×C3C2×#447
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#578C2semidirect
2. C3×#3414C2semidirect
3. C2×C3×#510C2semidirect
4. C3×#3067C2semidirect
5. C3×#2195C2semidirect
6. C3×#2193C2semidirect
7. C2×C3×#362C2semidirect
8. C3×#2143C2semidirect
9. C3×#2117C2semidirect
10. #33540C3direct (N × H)
28. C3×#510C22semidirect
30. C3×#509C22semidirect
32. C2×C3×Q32C22semidirect
33. C2×C3×SD32C22semidirect
36. C3×#473C22semidirect
37. C3×#472C22semidirect
41. C3×#362C22semidirect
42. C3×#359C22semidirect
43. C3×#350C22semidirect
44. C2×#578C2×C3semidirect
45. #3414C2×C3semidirect
46. C2×#510C2×C3semidirect
47. #3067C2×C3semidirect
48. #2195C2×C3semidirect
49. #2193C2×C3semidirect
50. C2×#362C2×C3semidirect
51. #2143C2×C3semidirect
52. #2117C2×C3semidirect
143. #510C22×C3semidirect
145. #509C22×C3semidirect
147. C2×Q32C22×C3semidirect
148. C2×SD32C22×C3semidirect
151. #473C22×C3semidirect
152. #472C22×C3semidirect
156. #362C22×C3semidirect
157. #359C22×C3semidirect
158. #350C22×C3semidirect
555. C3#33540direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5782
C3×#34142
C2×C3×#5102
C3×#30672
C3×#21952
C3×#21932
C2×C3×#3622
C3×#21432
C3×#21172
#335403