Finite Group Databasebeta

← C3×#18635 #10637242

Subgroups — C3×#18635 #10637242

Order 768.

1004 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×#3508C211yesmaximal, characteristic
2C3×#3209C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#3079C211yesmaximal, characteristic, complemented
5C3×#1641C214yesmaximal, complemented
6C3×#1640C214yesmaximal, complemented
7C3×#1545C211yesmaximal, characteristic, complemented
8C3×#1493C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18635C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—21no—
12C3×#559C2212yes—
13C3×#550—21no—
14C3×#550C2212yes—
15C3×#546—22no—
16C3×#546—22no—
17C3×#540—22no—
18C3×#535—21no—
19C3×#533C2211yescharacteristic
20C3×#532—21no—
21C3×#532—21no—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#127C2211yescharacteristic
24C3×#518—21no—
25C3×C4×#41C2211yescharacteristic
26C3×#494—21no—
27C3×#487—21no—
28C3×#486—22no—
29C3×#486C2214yes—
30C3×#477—21no—
31C3×#476—21no—
32C3×#475C2212yescomplemented
33C3×#474C2212yescomplemented
34C3×#462C2214yescomplemented
35C3×#451—22no—
36C3×#431C2211yescharacteristic
37C3×#428—21no—
38C3×#423C2211yescharacteristic
39C3×#417—21no—
40C2×C3×#102—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#331C2214yes—
43C3×Q8⋊C8C2214yescomplemented
44C3×Q8⋊C8C2212yescomplemented
45C3×D4⋊C8C2212yescomplemented
46#3508C2×C311yescharacteristic
47#3209C2×C311yescharacteristic, complemented
48C2×#486C2×C311yescharacteristic, complemented
49#3079C2×C311yescharacteristic, complemented
50#1641C2×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—
62C2×C3×Q16—24no—
63C2×C3×M16—21no—
64C22×C3×C8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8C2312yes—
71C3×#127—22no—
72C3×#127—21no—
73C3×#127—21no—
74C3×#127D412yes—
75C3×#127C2312yes—
76C3×#125—21no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124—22no—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8D412yes—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—24no—
88C3×#121—24no—
89C3×#121—22no—
90C3×#120—24no—
91C3×C4×Q8—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×C4×Q8C2312yes—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4D411yescharacteristic
104C3×C4×D4C2311yescharacteristic
105C3×#117—21no—
106C3×#117—21no—
107C3×#117D411yescharacteristic
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C22⋊C4—24no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C3×#107—22no—
114C3×#106—22no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2312yes—
118C3×#103—22no—
119C3×#103—24no—
120C3×#102—28no—
121C3×#102—22no—
122C3×C22⋊C8—21no—
123C3×C4×C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8C2312yes—
126#559—21no—
127#559C22×C312yes—
128#550—21no—
129#550C22×C312yes—
130#546—22no—
131#546—22no—
132#540—22no—
133#535—21no—
134#533C22×C311yescharacteristic
135#532—21no—
136#532—21no—
137C2×C4⋊Q8C22×C311yescharacteristic
138C2×#127C22×C311yescharacteristic
139#518—21no—
140C4×#41C22×C311yescharacteristic
141#494—21no—
142#487—21no—
143#486—22no—
144#486C22×C314yes—
145#477—21no—
146#476—21no—
147#475C22×C312yescomplemented
148#474C22×C312yescomplemented
149#462C22×C314yescomplemented
150#451—22no—
151#431C22×C311yescharacteristic
152#428—21no—
153#423C22×C311yescharacteristic
154#417—21no—
155C2×#102—22no—
156C2×C4×C8C22×C311yescharacteristic
157#331C22×C314yes—
158Q8⋊C8C22×C312yescomplemented
159Q8⋊C8C22×C314yescomplemented
160D4⋊C8C22×C312yescomplemented
161C24×C3—22no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—84no—
165C3×#41—84no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—41no—
169C2×C3×Q8—82no—
170C2×C3×Q8C2×D411yescharacteristic
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—21no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—44no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—82no—
184C2×C3×D4C2×D411yescharacteristic
185C2×C3×D4C2×D414yes—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—41no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C3×Q16—84no—
196C3×M16—42no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—42no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D411yescharacteristic
214C3×C4⋊C4C2×D411yescharacteristic
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—22no—
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2411yescharacteristic
231C2×#41—22no—
232C2×#41—22no—
233C2×#41C3×D411yescharacteristic
234C22×Q8—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C3×D412yes—
238C2×Q16—24no—
239C2×M16—21no—
240C22×C8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C3×D411yescharacteristic
245C4⋊Q8C3×D411yescharacteristic
246C4⋊Q8C23×C312yes—
247#127—22no—
248#127—21no—
249#127—21no—
250#127C3×D412yes—
251#127C23×C312yes—
252#125—21no—
253#124—22no—
254#124—22no—
255#124—22no—
256#123—24no—
257#123—24no—
258C22⋊Q8—21no—
259C22⋊Q8C3×D412yes—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—24no—
264#121—24no—
265#121—22no—
266#120—24no—
267C4×Q8—21no—
268C4×Q8C3×D411yescharacteristic
269C4×Q8C23×C311yescharacteristic
270C4×Q8C23×C312yes—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4C3×D411yescharacteristic
280C4×D4C23×C311yescharacteristic
281#117—21no—
282#117—21no—
283#117C3×D411yescharacteristic
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C22⋊C4—24no—
287C2×C42—21no—
288C2×C42C23×C311yescharacteristic
289#107—22no—
290#106—22no—
291C4⋊C8—21no—
292C4⋊C8C23×C312yes—
293C4⋊C8C23×C312yes—
294#103—22no—
295#103—24no—
296#102—28no—
297#102—22no—
298C22⋊C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8C23×C312yes—
302C23×C3—24no—
303C23×C3—21no—
304C23×C3—44no—
305C23×C3—82no—
306C23×C3—82no—
307C23×C3#12011yescharacteristic
308C3×Q8—21no—
309C3×Q8—21no—
310C3×Q8—24no—
311C3×Q8—42no—
312C3×Q8—44no—
313C3×Q8—82no—
314C3×D4—21no—
315C3×D4—21no—
316C3×D4—44no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—44no—
321C3×D4—44no—
322C3×D4—44no—
323C3×D4—82no—
324C3×D4—84no—
325C3×D4—84no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—84no—
338C2×C3×C4—84no—
339C2×C3×C4—82no—
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#13512yes—
344C2×C3×C4#13712yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C3×C8—42no—
349C3×C8—42no—
350C3×C8—42no—
351C24—22no—
352#41—22no—
353#41—22no—
354#41—84no—
355#41—84no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×Q8C2×C3×D411yescharacteristic
361C2×Q8C2×C3×D412yes—
362C2×D4—21no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4C2×C3×D411yescharacteristic
375C2×D4C2×C3×D414yes—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385Q16—84no—
386M16—42no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—42no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4C2×C3×D412yes—
403C4⋊C4C2×C3×D411yescharacteristic
404C4⋊C4C2×C3×D411yescharacteristic
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—21no—
410C22⋊C4—22no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—22no—
417C42C2×C3×D411yescharacteristic
418C42C2×C3×D411yescharacteristic
419C42C2×C3×D411yescharacteristic
420C42C24×C311yescharacteristic
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—44no—
425C22×C3—41no—
426C22×C3—44no—
427C22×C3—41no—
428C22×C3—44no—
429C22×C3—164no—
430C22×C3#45212yes—
431C22×C3C2×#12011yescharacteristic
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—42no—
439C3×C4—82no—
440C3×C4#45212yes—
441C3×C4#45512yes—
442C3×C4C2×#13511yescharacteristic
443C3×C4C2×#13711yescharacteristic
444C3×C4#57612yes—
445C23—24no—
446C23—21no—
447C23—44no—
448C23—82no—
449C23—82no—
450C23C3×#12011yescharacteristic
451Q8—21no—
452Q8—21no—
453Q8—24no—
454Q8—42no—
455Q8—44no—
456Q8—82no—
457D4—21no—
458D4—21no—
459D4—44no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—82no—
467D4—84no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—82no—
483C2×C4C3×#12011yescharacteristic
484C2×C4C3×#12011yescharacteristic
485C2×C4C3×#12011yescharacteristic
486C2×C4C3×#13512yes—
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—81no—
496C2×C3—84no—
497C2×C3C2×#45211yescharacteristic
498C2×C3#300811yescharacteristic
499C2×C3C2×#45511yescharacteristic
500C22—21no—
501C22—21no—
502C22—41no—
503C22—44no—
504C22—41no—
505C22—44no—
506C22—41no—
507C22—44no—
508C22—164no—
509C22C3×#45212yes—
510C22C2×C3×#12011yescharacteristic
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—82no—
519C4C3×#45212yes—
520C4C3×#45512yes—
521C4C2×C3×#13511yescharacteristic
522C4C2×C3×#13711yescharacteristic
523C4C3×#57612yes—
524C3#1863511yescharacteristic, direct product factor, Sylow
525C2—22no—
526C2—84no—
527C2—81no—
528C2C2×C3×#45211yescharacteristic
529C2C3×#300811yescharacteristic
530C2C2×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×#3209C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#3079C2semidirect
5. C3×#1641C2semidirect
6. C3×#1640C2semidirect
7. C3×#1545C2semidirect
8. C3×#1493C2semidirect
9. C3×#1476C2semidirect
10. #18635C3direct (N × H)
32. C3×#475C22semidirect
33. C3×#474C22semidirect
34. C3×#462C22semidirect
43. C3×Q8⋊C8C22semidirect
44. C3×Q8⋊C8C22semidirect
45. C3×D4⋊C8C22semidirect
47. #3209C2×C3semidirect
48. C2×#486C2×C3semidirect
49. #3079C2×C3semidirect
50. #1641C2×C3semidirect
51. #1640C2×C3semidirect
52. #1545C2×C3semidirect
53. #1493C2×C3semidirect
54. #1476C2×C3semidirect
147. #475C22×C3semidirect
148. #474C22×C3semidirect
149. #462C22×C3semidirect
158. Q8⋊C8C22×C3semidirect
159. Q8⋊C8C22×C3semidirect
160. D4⋊C8C22×C3semidirect
524. C3#18635direct (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×#32092
C2×C3×#4862
C3×#30792
C3×#16412
C3×#16402
C3×#15452
C3×#14932
C3×#14762
#186353