Finite Group Databasebeta

← C3×#29644 #10648251

Subgroups — C3×#29644 #10648251

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#482C212yesmaximal
3C3×#2854C211yesmaximal, characteristic
4C3×#2059C212yesmaximal, complemented
5C3×#2058C211yesmaximal, characteristic, complemented
6C3×#2057C211yesmaximal, characteristic, complemented
7C3×#2025C212yesmaximal, complemented
8C3×#2023C211yesmaximal, characteristic, complemented
9C3×#2023C211yesmaximal, characteristic, complemented
10C3×#1825C212yesmaximal, complemented
11C3×#1817C211yesmaximal, characteristic, complemented
12#29644C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×#123—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8—21no—
23C2×C3×#117C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#482—28no—
26C2×C3×#107C2212yes—
27C2×C3×#106C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#390C2211yescharacteristic
32C3×#390C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#388—21no—
36C3×#388—21no—
37C3×#386—21no—
38C3×#380C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#533C2×C311yescharacteristic, complemented
47C2×#482C2×C312yes—
48#2854C2×C311yescharacteristic
49#2059C2×C312yescomplemented
50#2058C2×C311yescharacteristic, complemented
51#2057C2×C311yescharacteristic, complemented
52#2025C2×C312yescomplemented
53#2023C2×C311yescharacteristic, complemented
54#2023C2×C311yescharacteristic, complemented
55#1825C2×C312yescomplemented
56#1817C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—24no—
68C3×#125—24no—
69C3×#125—24no—
70C3×#123—44no—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—28no—
73C3×#121—24no—
74C3×C4×Q8—28no—
75C3×C4×Q8—28no—
76C3×#117—24no—
77C3×#117D418yes—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—24no—
97C2×C3×C42—22no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C3×#107—24no—
103C3×#106—24no—
104C3×#102—24no—
105C3×#102—24no—
106C3×C22⋊C8—24no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110#533—28no—
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×#125C22×C312yes—
114C2×#123—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×#121C22×C311yescharacteristic
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×#117C22×C312yes—
121C22×C42C22×C311yescharacteristic
122#482—28no—
123C2×#107C22×C312yes—
124C2×#106C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×C22⋊C8C22×C312yes—
128#390C22×C311yescharacteristic
129#390C22×C311yescharacteristic
130#389C22×C311yescharacteristic
131#389C22×C311yescharacteristic
132#388—21no—
133#388—21no—
134#386—21no—
135#380C22×C311yescharacteristic
136#340C22×C312yes—
137#340C22×C312yes—
138#340C22×C312yes—
139#340C22×C312yes—
140#336C22×C311yescharacteristic
141#336C22×C311yescharacteristic
142#336C22×C312yes—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C2×C3×Q8—24no—
146C2×C3×Q8—24no—
147C2×C3×Q8—42no—
148C2×C3×Q8—42no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4C2×D414yes—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—42no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—28no—
209C3×C42—22no—
210C3×C42—22no—
211C22×Q8—21no—
212C22×Q8—21no—
213C22×D4—21no—
214C22×D4C23×C311yescharacteristic
215C23×C4C3×D411yescharacteristic
216C23×C4C3×D411yescharacteristic
217C23×C4C23×C311yescharacteristic
218C22×C8C23×C312yes—
219C22×C8C23×C312yes—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222#125—24no—
223#125—24no—
224#123—44no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227#121—24no—
228C4×Q8—28no—
229C4×Q8—28no—
230#117—24no—
231#117C3×D418yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C4⋊C4C23×C312yes—
242C2×C4⋊C4C23×C312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42C3×D411yescharacteristic
253C2×C42C3×D411yescharacteristic
254C2×C42C23×C311yescharacteristic
255C2×C42C23×C311yescharacteristic
256#107—24no—
257#106—24no—
258#102—24no—
259#102—24no—
260C22⋊C8—24no—
261#95—21no—
262#95—21no—
263#95—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3#12012yes—
278C23×C3C22×D411yescharacteristic
279C3×Q8—48no—
280C3×Q8—48no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—84no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—44no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C42−1+412yes—
320C2×C3×C42−1+411yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322C3×C8—42no—
323C3×C8—42no—
324C24—21no—
325C24C2×C3×D411yescharacteristic
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D414yes—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C24×C311yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4C2×C3×D414yes—
379C4⋊C4C2×C3×D414yes—
380C22⋊C4—28no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—22no—
388C42—22no—
389C42—28no—
390C42—22no—
391C42—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3C2×#12011yescharacteristic
407C22×C3#53611yescharacteristic
408C22×C3#53611yescharacteristic
409C22×C3C2×#13511yescharacteristic
410C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C3#57511yescharacteristic
412C22×C3#57511yescharacteristic
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—44no—
418C3×C4—41no—
419C3×C4—44no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4#53612yes—
424C3×C4#53612yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C3×#12012yes—
439C23C22×C3×D411yescharacteristic
440Q8—48no—
441Q8—48no—
442D4—42no—
443D4—42no—
444D4—84no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4C3×#12012yes—
477C2×C4C3×#13512yes—
478C2×C4C3×AGL(1,ℤ/8)12yes—
479C2×C4C22×C3×D412yes—
480C2×C4C3×2−1+412yes—
481C2×C4C3×2−1+411yescharacteristic
482C2×C4C3×2−1+411yescharacteristic
483C8—42no—
484C8—42no—
485C2×C3—42no—
486C2×C3—161no—
487C2×C3#261411yescharacteristic
488C2×C3#300211yescharacteristic
489C2×C3#300611yescharacteristic
490C2×C3#322811yescharacteristic
491C2×C3#323311yescharacteristic
492C2×C3#323511yescharacteristic
493C2×C3#324011yescharacteristic
494C22—22no—
495C22—22no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×C3×#12011yescharacteristic
509C22C3×#53611yescharacteristic
510C22C3×#53611yescharacteristic
511C22C2×C3×#13511yescharacteristic
512C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
513C22C3×#57511yescharacteristic
514C22C3×#57511yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—44no—
520C4—41no—
521C4—44no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4C3×#53612yes—
526C4C3×#53612yes—
527C3#2964411yescharacteristic, direct product factor, Sylow
528C2—42no—
529C2—161no—
530C2C3×#261411yescharacteristic
531C2C3×#300211yescharacteristic
532C2C3×#300611yescharacteristic
533C2C3×#322811yescharacteristic
534C2C3×#323311yescharacteristic
535C2C3×#323511yescharacteristic
536C2C3×#324011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
4. C3×#2059C2semidirect
5. C3×#2058C2semidirect
6. C3×#2057C2semidirect
7. C3×#2025C2semidirect
8. C3×#2023C2semidirect
9. C3×#2023C2semidirect
10. C3×#1825C2semidirect
11. C3×#1817C2semidirect
12. #29644C3direct (N × H)
46. C2×#533C2×C3semidirect
49. #2059C2×C3semidirect
50. #2058C2×C3semidirect
51. #2057C2×C3semidirect
52. #2025C2×C3semidirect
53. #2023C2×C3semidirect
54. #2023C2×C3semidirect
55. #1825C2×C3semidirect
56. #1817C2×C3semidirect
527. C3#29644direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4822
C3×#28542
C3×#20592
C3×#20582
C3×#20572
C3×#20252
C3×#20232
C3×#18252
C3×#18172
#296443