Finite Group Databasebeta

← C3×#18642 #10637249

Subgroups — C3×#18642 #10637249

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic, complemented
2C3×#3208C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#3038C211yesmaximal, characteristic
5C3×#1627C214yesmaximal, complemented
6C3×#1626C214yesmaximal, complemented
7C3×#1549C211yesmaximal, characteristic, complemented
8C3×#1493C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18642C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)—21no—
12C3×Q82—21no—
13C3×#557C2212yescomplemented
14C3×#554—21no—
15C3×#554—22no—
16C3×D4×Q8—22no—
17C3×D4×Q8C2212yescomplemented
18C3×#544—22no—
19C3×#544—21no—
20C3×#533—21no—
21C3×#533C2211yescharacteristic
22C2×C3×C4⋊Q8—21no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×#127C2211yescharacteristic
25C3×#520—21no—
26C3×C4×#41C2211yescharacteristic
27C3×#492—21no—
28C3×#486—22no—
29C3×#486—21no—
30C3×#486C2214yes—
31C3×#460—21no—
32C3×#460C2214yes—
33C3×#460C2212yes—
34C3×#459—21no—
35C3×#459C2212yes—
36C3×#450—22no—
37C3×#431C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C3×#418—21no—
40C2×C3×#102—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#332C2214yescomplemented
43C3×Q8⋊C8C2214yescomplemented
44C3×Q8⋊C8C2212yescomplemented
45C3×D4⋊C8C2212yescomplemented
46#3503C2×C311yescharacteristic, complemented
47#3208C2×C311yescharacteristic, complemented
48C2×#486C2×C311yescharacteristic, complemented
49#3038C2×C311yescharacteristic
50#1627C2×C314yescomplemented
51#1626C2×C314yescomplemented
52#1549C2×C311yescharacteristic, complemented
53#1493C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic, complemented
55C2×C3×#41D411yescharacteristic
56C22×C3×Q8—22no—
57C22×C3×Q8—21no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C3×AGL(1,ℤ/8)—44no—
62C2×C3×SD16—22no—
63C2×C3×SD16—24no—
64C2×C3×D8—22no—
65C2×C3×M16—21no—
66C22×C3×C8—21no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8D411yescharacteristic
75C3×C4⋊Q8D412yes—
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8C2312yes—
78C3×#127—21no—
79C3×#127—21no—
80C3×#127—21no—
81C3×#127C2312yes—
82C3×#125—22no—
83C3×#125—24no—
84C3×#125—22no—
85C3×#125—21no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8D412yes—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—21no—
97C3×C4×Q8D411yescharacteristic
98C3×C4×Q8C2311yescharacteristic
99C3×C4×Q8C2312yes—
100C3×C4×D4—24no—
101C3×C4×D4—22no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4D411yescharacteristic
105C3×C4×D4C2311yescharacteristic
106C3×#117—22no—
107C3×#117—21no—
108C3×#117—21no—
109C3×#117D411yescharacteristic
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4D412yes—
114C2×C3×C42—21no—
115C2×C3×C42C2311yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2312yes—
118C3×C4⋊C8C2312yes—
119C3×#102—28no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—24no—
123C3×C22⋊C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8—21no—
126C3×C4×C8C2312yes—
127C2×AGL(1,ℤ/8)—21no—
128Q82—21no—
129#557C22×C312yescomplemented
130#554—21no—
131#554—22no—
132D4×Q8—22no—
133D4×Q8C22×C312yescomplemented
134#544—22no—
135#544—21no—
136#533—21no—
137#533C22×C311yescharacteristic
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×#127C22×C311yescharacteristic
141#520—21no—
142C4×#41C22×C311yescharacteristic
143#492—21no—
144#486—22no—
145#486—21no—
146#486C22×C314yes—
147#460—21no—
148#460C22×C312yes—
149#460C22×C314yes—
150#459—21no—
151#459C22×C312yes—
152#450—22no—
153#431C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155#418—21no—
156C2×#102—22no—
157C2×C4×C8C22×C311yescharacteristic
158#332C22×C314yescomplemented
159Q8⋊C8C22×C312yescomplemented
160Q8⋊C8C22×C314yescomplemented
161D4⋊C8C22×C312yescomplemented
162C24×C3—41no—
163C3×#41—22no—
164C3×#41SD1614yescomplemented
165C2×C3×Q8—24no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—82no—
170C2×C3×Q8—84no—
171C2×C3×Q8C2×D412yes—
172C2×C3×Q8C2×D411yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—44no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—82no—
184C2×C3×D4C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C3×SD16—44no—
194C3×SD16—84no—
195C3×D8—44no—
196C3×M16—42no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—42no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4C2×D411yescharacteristic
217C3×C4⋊C4C2×D411yescharacteristic
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—21no—
224C3×C42—24no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—22no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2411yescharacteristic
233C2×#41C3×D411yescharacteristic
234C22×Q8—22no—
235C22×Q8—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239AGL(1,ℤ/8)—44no—
240C2×SD16—22no—
241C2×SD16—24no—
242C2×D8—22no—
243C2×M16—21no—
244C22×C8—21no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C3×D411yescharacteristic
253C4⋊Q8C3×D412yes—
254C4⋊Q8C3×D411yescharacteristic
255C4⋊Q8C23×C312yes—
256#127—21no—
257#127—21no—
258#127—21no—
259#127C23×C312yes—
260#125—22no—
261#125—24no—
262#125—22no—
263#125—21no—
264C22⋊Q8—22no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—21no—
268C22⋊Q8—22no—
269C22⋊Q8—24no—
270C22⋊Q8C3×D412yes—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—21no—
275C4×Q8C3×D411yescharacteristic
276C4×Q8C23×C311yescharacteristic
277C4×Q8C23×C312yes—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4C3×D411yescharacteristic
283C4×D4C23×C311yescharacteristic
284#117—22no—
285#117—21no—
286#117—21no—
287#117C3×D411yescharacteristic
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C3×D412yes—
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8C23×C312yes—
296C4⋊C8C23×C312yes—
297#102—28no—
298#102—22no—
299#102—22no—
300#102—24no—
301C22⋊C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8C23×C312yes—
305C23×C3—21no—
306C23×C3—42no—
307C23×C3—82no—
308C23×C3—82no—
309C23×C3—82no—
310C23×C3#12011yescharacteristic
311C3×Q8—21no—
312C3×Q8—24no—
313C3×Q8—44no—
314C3×Q8—42no—
315C3×Q8—82no—
316C3×Q8—84no—
317C3×Q8C2×SD1612yes—
318C3×D4—24no—
319C3×D4—21no—
320C3×D4—44no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—44no—
324C3×D4—82no—
325C3×D4—82no—
326C3×D4—82no—
327C3×D4C2×SD1612yes—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—82no—
340C2×C3×C4—84no—
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4C2×SD1612yes—
345C2×C3×C4#13712yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C3×C8—42no—
350C3×C8—42no—
351C3×C8—42no—
352C24—41no—
353#41—22no—
354#41C3×SD1614yescomplemented
355C2×Q8—24no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—82no—
360C2×Q8—84no—
361C2×Q8C2×C3×D412yes—
362C2×Q8C2×C3×D411yescharacteristic
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4C2×C3×D411yescharacteristic
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383SD16—44no—
384SD16—84no—
385D8—44no—
386M16—42no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4C2×C3×D414yes—
406C4⋊C4C2×C3×D411yescharacteristic
407C4⋊C4C2×C3×D411yescharacteristic
408C4⋊C4C2×C3×D412yes—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C42—24no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—22no—
419C42C2×C3×D411yescharacteristic
420C42C2×C3×D411yescharacteristic
421C42C2×C3×D411yescharacteristic
422C42C24×C311yescharacteristic
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—164no—
432C22×C3#44812yes—
433C22×C3C2×#12011yescharacteristic
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—42no—
439C3×C4—44no—
440C3×C4—41no—
441C3×C4—42no—
442C3×C4#44812yes—
443C3×C4#45412yes—
444C3×C4C22×SD1611yescharacteristic
445C3×C4C2×#13711yescharacteristic
446C3×C4#57612yes—
447C23—21no—
448C23—42no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23C3×#12011yescharacteristic
453Q8—21no—
454Q8—24no—
455Q8—44no—
456Q8—42no—
457Q8—82no—
458Q8—84no—
459Q8C2×C3×SD1612yes—
460D4—24no—
461D4—21no—
462D4—44no—
463D4—42no—
464D4—42no—
465D4—44no—
466D4—82no—
467D4—82no—
468D4—82no—
469D4C2×C3×SD1612yes—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4C3×#12011yescharacteristic
484C2×C4C3×#12011yescharacteristic
485C2×C4C3×#12011yescharacteristic
486C2×C4C2×C3×SD1612yes—
487C2×C4C3×#13712yes—
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C22×C3×D411yescharacteristic
491C8—42no—
492C8—42no—
493C8—42no—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3—162no—
497C2×C3C2×#44811yescharacteristic
498C2×C3#300811yescharacteristic
499C2×C3C2×#45411yescharacteristic
500C22—21no—
501C22—21no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—164no—
509C22C3×#44812yes—
510C22C2×C3×#12011yescharacteristic
511C4—22no—
512C4—21no—
513C4—21no—
514C4—41no—
515C4—42no—
516C4—44no—
517C4—41no—
518C4—42no—
519C4C3×#44812yes—
520C4C3×#45412yes—
521C4C22×C3×SD1611yescharacteristic
522C4C2×C3×#13711yescharacteristic
523C4C3×#57612yes—
524C3#1864211yescharacteristic, direct product factor, Sylow
525C2—22no—
526C2—42no—
527C2—162no—
528C2C2×C3×#44811yescharacteristic
529C2C3×#300811yescharacteristic
530C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3038C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3503C2semidirect
2. C3×#3208C2semidirect
3. C2×C3×#486C2semidirect
5. C3×#1627C2semidirect
6. C3×#1626C2semidirect
7. C3×#1549C2semidirect
8. C3×#1493C2semidirect
9. C3×#1476C2semidirect
10. #18642C3direct (N × H)
13. C3×#557C22semidirect
17. C3×D4×Q8C22semidirect
42. C3×#332C22semidirect
43. C3×Q8⋊C8C22semidirect
44. C3×Q8⋊C8C22semidirect
45. C3×D4⋊C8C22semidirect
46. #3503C2×C3semidirect
47. #3208C2×C3semidirect
48. C2×#486C2×C3semidirect
50. #1627C2×C3semidirect
51. #1626C2×C3semidirect
52. #1549C2×C3semidirect
53. #1493C2×C3semidirect
54. #1476C2×C3semidirect
129. #557C22×C3semidirect
133. D4×Q8C22×C3semidirect
158. #332C22×C3semidirect
159. Q8⋊C8C22×C3semidirect
160. Q8⋊C8C22×C3semidirect
161. D4⋊C8C22×C3semidirect
164. C3×#41SD16semidirect
354. #41C3×SD16semidirect
524. C3#18642direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#32082
C2×C3×#4862
C3×#30382
C3×#16272
C3×#16262
C3×#15492
C3×#14932
C3×#14762
#186423