Finite Group Databasebeta

← C3×#18634 #10637241

Subgroups — C3×#18634 #10637241

Order 768.

1024 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2C3×#3229C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#3039C211yesmaximal, characteristic, complemented
5C3×#1642C214yesmaximal, complemented
6C3×#1640C214yesmaximal, complemented
7C3×#1545C211yesmaximal, characteristic, complemented
8C3×#1493C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18634C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137—21no—
12C3×#559—21no—
13C3×#559C2212yes—
14C3×#550—21no—
15C3×#550C2212yes—
16C3×#546—22no—
17C3×#546—22no—
18C3×#540—22no—
19C3×#535—21no—
20C3×#533C2211yescharacteristic
21C3×#532—21no—
22C3×#532—21no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×#127C2211yescharacteristic
25C3×#518—21no—
26C3×C4×#41C2211yescharacteristic
27C3×#498—21no—
28C3×#487—21no—
29C3×#486—22no—
30C3×#486C2214yes—
31C3×#483—22no—
32C3×#475C2214yescomplemented
33C3×#462—21no—
34C3×#462C2212yescomplemented
35C3×#461—21no—
36C3×#461C2212yescomplemented
37C3×#431C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C3×#419—21no—
40C2×C3×#102—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#331C2214yes—
43C3×Q8⋊C8C2214yescomplemented
44C3×Q8⋊C8C2212yescomplemented
45C3×D4⋊C8C2212yescomplemented
46#3508C2×C311yescharacteristic
47#3229C2×C311yescharacteristic, complemented
48C2×#486C2×C311yescharacteristic, complemented
49#3039C2×C311yescharacteristic, complemented
50#1642C2×C314yescomplemented
51#1640C2×C314yescomplemented
52#1545C2×C311yescharacteristic, complemented
53#1493C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D411yescharacteristic
58C22×C3×Q8—21no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4D412yes—
62C3×#137—44no—
63C2×C3×Q16—22no—
64C2×C3×SD16—22no—
65C2×C3×M16—21no—
66C22×C3×C8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8D411yescharacteristic
71C3×C4⋊Q8D411yescharacteristic
72C3×C4⋊Q8C2312yes—
73C3×#127—22no—
74C3×#127—21no—
75C3×#127—21no—
76C3×#127D412yes—
77C3×#127C2312yes—
78C3×#125—21no—
79C3×#124—22no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#123—24no—
83C3×#123—24no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8D412yes—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—22no—
92C3×#120—24no—
93C3×C4×Q8—21no—
94C3×C4×Q8D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×Q8C2312yes—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4D411yescharacteristic
106C3×C4×D4C2311yescharacteristic
107C3×#117—21no—
108C3×#117—21no—
109C3×#117D411yescharacteristic
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C3×#106—24no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2312yes—
118C3×C4⋊C8C2312yes—
119C3×#103—24no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—28no—
123C3×C22⋊C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8—21no—
126C3×C4×C8C2312yes—
127C2×#137—21no—
128#559—21no—
129#559C22×C312yes—
130#550—21no—
131#550C22×C312yes—
132#546—22no—
133#546—22no—
134#540—22no—
135#535—21no—
136#533C22×C311yescharacteristic
137#532—21no—
138#532—21no—
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×#127C22×C311yescharacteristic
141#518—21no—
142C4×#41C22×C311yescharacteristic
143#498—21no—
144#487—21no—
145#486—22no—
146#486C22×C314yes—
147#483—22no—
148#475C22×C314yescomplemented
149#462—21no—
150#462C22×C312yescomplemented
151#461—21no—
152#461C22×C312yescomplemented
153#431C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155#419—21no—
156C2×#102—22no—
157C2×C4×C8C22×C311yescharacteristic
158#331C22×C314yes—
159Q8⋊C8C22×C312yescomplemented
160Q8⋊C8C22×C314yescomplemented
161D4⋊C8C22×C312yescomplemented
162C24×C3—22no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—84no—
166C3×#41—84no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—41no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8C2×D412yes—
173C2×C3×Q8C2×D411yescharacteristic
174C2×C3×D4—21no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—44no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—82no—
186C2×C3×D4C2×D411yescharacteristic
187C2×C3×D4C2×D414yes—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—41no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C3×Q16—44no—
198C3×SD16—44no—
199C3×M16—42no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—42no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4C2×D411yescharacteristic
216C3×C4⋊C4C2×D411yescharacteristic
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—22no—
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2411yescharacteristic
234C2×#41—22no—
235C2×#41—22no—
236C2×#41C3×D411yescharacteristic
237C22×Q8—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C3×D412yes—
241#137—44no—
242C2×Q16—22no—
243C2×SD16—22no—
244C2×M16—21no—
245C22×C8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C3×D411yescharacteristic
250C4⋊Q8C3×D411yescharacteristic
251C4⋊Q8C23×C312yes—
252#127—22no—
253#127—21no—
254#127—21no—
255#127C3×D412yes—
256#127C23×C312yes—
257#125—21no—
258#124—22no—
259#124—22no—
260#124—22no—
261#123—24no—
262#123—24no—
263C22⋊Q8—21no—
264C22⋊Q8C3×D412yes—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—24no—
269#121—24no—
270#121—22no—
271#120—24no—
272C4×Q8—21no—
273C4×Q8C3×D411yescharacteristic
274C4×Q8C23×C311yescharacteristic
275C4×Q8C23×C312yes—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4C3×D411yescharacteristic
285C4×D4C23×C311yescharacteristic
286#117—21no—
287#117—21no—
288#117C3×D411yescharacteristic
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C22⋊C4—24no—
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294#106—24no—
295C4⋊C8—21no—
296C4⋊C8C23×C312yes—
297C4⋊C8C23×C312yes—
298#103—24no—
299#103—22no—
300#103—22no—
301#102—28no—
302C22⋊C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8C23×C312yes—
306C23×C3—24no—
307C23×C3—21no—
308C23×C3—44no—
309C23×C3—82no—
310C23×C3—82no—
311C23×C3#12011yescharacteristic
312C3×Q8—21no—
313C3×Q8—21no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—82no—
319C3×D4—21no—
320C3×D4—21no—
321C3×D4—44no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—44no—
328C3×D4—82no—
329C3×D4—84no—
330C3×D4—84no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—84no—
343C2×C3×C4—84no—
344C2×C3×C4—82no—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#13512yes—
349C2×C3×C4AGL(1,ℤ/8)12yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C3×C8—42no—
354C3×C8—42no—
355C3×C8—42no—
356C24—22no—
357#41—22no—
358#41—22no—
359#41—84no—
360#41—84no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8C2×C3×D412yes—
367C2×Q8C2×C3×D411yescharacteristic
368C2×D4—21no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—82no—
380C2×D4C2×C3×D411yescharacteristic
381C2×D4C2×C3×D414yes—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391Q16—44no—
392SD16—44no—
393M16—42no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—42no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4C2×C3×D411yescharacteristic
410C4⋊C4C2×C3×D411yescharacteristic
411C4⋊C4C2×C3×D412yes—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—21no—
417C22⋊C4—22no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—22no—
424C42C2×C3×D411yescharacteristic
425C42C2×C3×D411yescharacteristic
426C42C2×C3×D411yescharacteristic
427C42C24×C311yescharacteristic
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3—44no—
432C22×C3—41no—
433C22×C3—44no—
434C22×C3—41no—
435C22×C3—44no—
436C22×C3—164no—
437C22×C3#44912yes—
438C22×C3C2×#12011yescharacteristic
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4—82no—
447C3×C4#44912yes—
448C3×C4#45512yes—
449C3×C4C2×#13511yescharacteristic
450C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C3×C4#57712yes—
452C23—24no—
453C23—21no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23C3×#12011yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—24no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—82no—
465D4—21no—
466D4—21no—
467D4—44no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—82no—
475D4—84no—
476D4—84no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4C3×#12011yescharacteristic
492C2×C4C3×#12011yescharacteristic
493C2×C4C3×#12011yescharacteristic
494C2×C4C3×#13512yes—
495C2×C4C3×AGL(1,ℤ/8)12yes—
496C2×C4C22×C3×D411yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C22×C3×D411yescharacteristic
499C8—42no—
500C8—42no—
501C8—42no—
502C2×C3—22no—
503C2×C3—81no—
504C2×C3—84no—
505C2×C3C2×#44911yescharacteristic
506C2×C3#300911yescharacteristic
507C2×C3C2×#45511yescharacteristic
508C22—21no—
509C22—21no—
510C22—41no—
511C22—44no—
512C22—41no—
513C22—44no—
514C22—41no—
515C22—44no—
516C22—164no—
517C22C3×#44912yes—
518C22C2×C3×#12011yescharacteristic
519C4—21no—
520C4—21no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—82no—
527C4C3×#44912yes—
528C4C3×#45512yes—
529C4C2×C3×#13511yescharacteristic
530C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
531C4C3×#57712yes—
532C3#1863411yescharacteristic, direct product factor, Sylow
533C2—22no—
534C2—84no—
535C2—81no—
536C2C2×C3×#44911yescharacteristic
537C2C3×#300911yescharacteristic
538C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3508C2
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
2. C3×#3229C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#3039C2semidirect
5. C3×#1642C2semidirect
6. C3×#1640C2semidirect
7. C3×#1545C2semidirect
8. C3×#1493C2semidirect
9. C3×#1476C2semidirect
10. #18634C3direct (N × H)
32. C3×#475C22semidirect
34. C3×#462C22semidirect
36. C3×#461C22semidirect
43. C3×Q8⋊C8C22semidirect
44. C3×Q8⋊C8C22semidirect
45. C3×D4⋊C8C22semidirect
47. #3229C2×C3semidirect
48. C2×#486C2×C3semidirect
49. #3039C2×C3semidirect
50. #1642C2×C3semidirect
51. #1640C2×C3semidirect
52. #1545C2×C3semidirect
53. #1493C2×C3semidirect
54. #1476C2×C3semidirect
148. #475C22×C3semidirect
150. #462C22×C3semidirect
152. #461C22×C3semidirect
159. Q8⋊C8C22×C3semidirect
160. Q8⋊C8C22×C3semidirect
161. D4⋊C8C22×C3semidirect
532. C3#18634direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35082
C3×#32292
C2×C3×#4862
C3×#30392
C3×#16422
C3×#16402
C3×#15452
C3×#14932
C3×#14762
#186343