Finite Group Databasebeta

← C3×#25655 #10644262

Subgroups — C3×#25655 #10644262

Order 768.

1324 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#490C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic, complemented
4C3×#2584C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2041C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1982C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25655C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C3×#550—24no—
15C3×#550—24no—
16C2×C3×#127C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×#121—22no—
20C2×C3×#121—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C3×#490—24no—
25C3×#478—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#398—21no—
38C3×#395—21no—
39C3×#389—22no—
40C3×#389C2212yes—
41C3×#389C2212yes—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic
47C2×#490C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic, complemented
49#2584C2×C311yescharacteristic, complemented
50#2041C2×C312yescomplemented
51#2041C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic, complemented
54#1864C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#127—24no—
72C3×#127—24no—
73C3×#127—22no—
74C3×#127D414yes—
75C3×#125—22no—
76C3×#125—22no—
77C3×#121—28no—
78C3×#121—216no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4D414yes—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×#107—22no—
101C3×#106—22no—
102C3×C4⋊C8—22no—
103C3×#102—22no—
104C3×#102—22no—
105C3×#102—24no—
106C3×#102—24no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#97—22no—
111C3×#95—21no—
112C3×#95—22no—
113C3×#95—22no—
114C22×#41—21no—
115#550—24no—
116#550—24no—
117C2×#127C22×C312yes—
118C2×#127C22×C311yescharacteristic
119C2×#125C22×C311yescharacteristic
120C2×#121—22no—
121C2×#121—21no—
122C2×C4×Q8C22×C311yescharacteristic
123C2×C4×D4C22×C312yes—
124C2×C4×D4C22×C311yescharacteristic
125#490—24no—
126#478—24no—
127C2×#107C22×C311yescharacteristic
128C2×#106C22×C311yescharacteristic
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#102C22×C311yescharacteristic
131C2×#102C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×#97C22×C311yescharacteristic
138#398—21no—
139#395—21no—
140#389—22no—
141#389C22×C312yes—
142#389C22×C312yes—
143C4×C4⋊C4C22×C311yescharacteristic
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—22no—
148C24×C3—21no—
149C3×#41—432no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×D4—28no—
153C2×C3×D4—28no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—81no—
168C2×C3×D4—81no—
169C2×C3×D4—82no—
170C2×C3×D4—82no—
171C2×C3×D4C2×D414yes—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—28no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—82no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4#4114yescomplemented
208C3×C22⋊C4—28no—
209C3×C22⋊C4—24no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—81no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215C2×#41—28no—
216C2×#41—48no—
217C22×Q8C3×D411yescharacteristic
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4C23×C312yes—
223C22×D4C23×C311yescharacteristic
224C23×C4—22no—
225C23×C4—21no—
226C22×C8C23×C312yes—
227C22×C8C23×C311yescharacteristic
228C22×C8C23×C311yescharacteristic
229#127—24no—
230#127—24no—
231#127—22no—
232#127C3×D414yes—
233#125—22no—
234#125—22no—
235#121—28no—
236#121—216no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4C3×D414yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C3×D412yes—
248C2×C4⋊C4C3×D411yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C312yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42C23×C312yes—
257C2×C42C23×C311yescharacteristic
258#107—22no—
259#106—22no—
260C4⋊C8—22no—
261#102—22no—
262#102—22no—
263#102—24no—
264#102—24no—
265#102—24no—
266#102—24no—
267#102—24no—
268#97—22no—
269#95—21no—
270#95—22no—
271#95—22no—
272C23×C3—24no—
273C23×C3—22no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3C22×D411yescharacteristic
285C3×Q8—28no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—48no—
291C3×D4—416no—
292C3×D4—42no—
293C3×D4—42no—
294C3×D4—42no—
295C3×D4—84no—
296C3×D4—88no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—48no—
308C2×C3×C4—416no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4#12012yes—
318C2×C3×C4#12012yes—
319C2×C3×C4AGL(1,ℤ/8)14yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C42−1+412yes—
324C3×C8—41no—
325C3×C8—41no—
326C3×C8—42no—
327C24—22no—
328C24—21no—
329#41—432no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4C2×C3×D414yes—
352C2×D4C2×C3×D412yes—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D412yes—
366C22×C4C24×C311yescharacteristic
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—82no—
386C4⋊C4C2×C3×D412yes—
387C4⋊C4C3×#4114yescomplemented
388C22⋊C4—28no—
389C22⋊C4—24no—
390C42—22no—
391C42—22no—
392C42—81no—
393C42C2×C3×D412yes—
394C42C2×C3×D412yes—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—42no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#54812yes—
409C22×C3C2×AGL(1,ℤ/8)12yes—
410C22×C3#57611yescharacteristic
411C22×C3#57711yescharacteristic
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—44no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4#53211yescharacteristic
421C3×C4#53611yescharacteristic
422C3×C4#54712yes—
423C23—24no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C22×C3×D411yescharacteristic
436Q8—28no—
437D4—22no—
438D4—22no—
439D4—24no—
440D4—24no—
441D4—48no—
442D4—416no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—84no—
447D4—88no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—48no—
459C2×C4—416no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4C3×#12012yes—
469C2×C4C3×#12012yes—
470C2×C4C3×AGL(1,ℤ/8)14yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C2×C3×#4112yes—
474C2×C4C3×2−1+412yes—
475C8—41no—
476C8—41no—
477C8—42no—
478C2×C3—82no—
479C2×C3—84no—
480C2×C3#243211yescharacteristic
481C2×C3#300311yescharacteristic
482C2×C3#301211yescharacteristic
483C2×C3#336612yes—
484C2×C3#336812yes—
485C22—42no—
486C22—44no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22C2×C3×#12011yescharacteristic
498C22C3×#54812yes—
499C22C2×C3×AGL(1,ℤ/8)12yes—
500C22C3×#57611yescharacteristic
501C22C3×#57711yescharacteristic
502C4—22no—
503C4—24no—
504C4—22no—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—81no—
509C4—81no—
510C4C3×#53211yescharacteristic
511C4C3×#53611yescharacteristic
512C4C3×#54712yes—
513C3#2565511yescharacteristic, direct product factor, Sylow
514C2—82no—
515C2—84no—
516C2C3×#243211yescharacteristic
517C2C3×#300311yescharacteristic
518C2C3×#301211yescharacteristic
519C2C3×#336612yes—
520C2C3×#336812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#490C2semidirect
3. C2×C3×#478C2semidirect
4. C3×#2584C2semidirect
5. C3×#2041C2semidirect
6. C3×#2041C2semidirect
7. C3×#1985C2semidirect
8. C3×#1982C2semidirect
9. C3×#1864C2semidirect
10. C3×#1862C2semidirect
11. C3×#1762C2semidirect
12. #25655C3direct (N × H)
47. C2×#490C2×C3semidirect
48. C2×#478C2×C3semidirect
49. #2584C2×C3semidirect
50. #2041C2×C3semidirect
51. #2041C2×C3semidirect
52. #1985C2×C3semidirect
53. #1982C2×C3semidirect
54. #1864C2×C3semidirect
55. #1862C2×C3semidirect
56. #1762C2×C3semidirect
207. C3×C4⋊C4#41semidirect
387. C4⋊C4C3×#41semidirect
513. C3#25655direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4902
C2×C3×#4782
C3×#25842
C3×#20412
C3×#19852
C3×#19822
C3×#18642
C3×#18622
C3×#17622
#256553