Finite Group Databasebeta

← C3×#25611 #10644218

Subgroups — C3×#25611 #10644218

Order 768.

1222 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#488C211yesmaximal, characteristic, complemented
4C3×#2620C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25611C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C3×#540—24no—
16C3×#540—24no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121—22no—
23C2×C3×#121C2212yes—
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×D4C2212yes—
26C2×C3×#117C2211yescharacteristic
27C3×#496—24no—
28C3×#488—24no—
29C2×C3×#103C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C3×#396—22no—
37C3×#394—22no—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#540C2×C311yescharacteristic, complemented
45C2×#496C2×C311yescharacteristic, complemented
46C2×#488C2×C311yescharacteristic, complemented
47#2620C2×C311yescharacteristic, complemented
48#2000C2×C312yescomplemented
49#1999C2×C312yescomplemented
50#1992C2×C311yescharacteristic, complemented
51#1957C2×C311yescharacteristic, complemented
52#1879C2×C312yescomplemented
53#1877C2×C312yescomplemented
54#1750C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2312yes—
61C23×C3×C4—22no—
62C23×C3×C4C2311yescharacteristic
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C2×C3×D8—24no—
67C2×C3×D8—24no—
68C2×C3×D8—42no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×#127—22no—
72C3×#127D414yescomplemented
73C3×#125—22no—
74C3×#125D414yescomplemented
75C3×#124—22no—
76C3×#124—22no—
77C3×#123—216no—
78C3×C22⋊Q8—42no—
79C3×#121—216no—
80C3×#121—24no—
81C3×#121—24no—
82C3×#121—42no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C3×#103—24no—
104C3×#102—24no—
105C3×#102—42no—
106C3×#102—42no—
107C3×C22⋊C8—24no—
108C3×#97—22no—
109C3×C4×C8—22no—
110C3×#95—22no—
111C22×SD16C22×C312yes—
112C22×D8C22×C312yes—
113#540—24no—
114#540—24no—
115C2×#127C22×C311yescharacteristic
116C2×#125C22×C311yescharacteristic
117C2×#124C22×C311yescharacteristic
118C2×#123—22no—
119C2×C22⋊Q8C22×C311yescharacteristic
120C2×#121—22no—
121C2×#121C22×C312yes—
122C2×#121C22×C311yescharacteristic
123C2×C4×D4C22×C312yes—
124C2×#117C22×C311yescharacteristic
125#496—24no—
126#488—24no—
127C2×#103C22×C312yes—
128C2×#102C22×C312yes—
129C2×#102C22×C312yes—
130C2×#102C22×C312yes—
131C2×C22⋊C8C22×C312yes—
132C2×#97C22×C311yescharacteristic
133C2×C4×C8C22×C311yescharacteristic
134#396—22no—
135#394—22no—
136#391C22×C311yescharacteristic
137#390C22×C311yescharacteristic
138#384C22×C311yescharacteristic
139#340C22×C312yes—
140#340C22×C312yes—
141#336C22×C312yes—
142C24×C3—22no—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×D4—28no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—42no—
161C2×C3×D4—82no—
162C2×C3×D4—82no—
163C2×C3×D4—82no—
164C2×C3×D4—82no—
165C2×C3×D4—82no—
166C2×C3×D4C2×D414yes—
167C22×C3×C4—22no—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—82no—
177C22×C3×C4—82no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C3×SD16—48no—
183C3×D8—48no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8#4114yes—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—82no—
197C3×C4⋊C4C2×D414yes—
198C3×C22⋊C4—28no—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—82no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209C22×Q8C23×C311yescharacteristic
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4C23×C311yescharacteristic
214C22×D4C23×C312yes—
215C23×C4—22no—
216C23×C4C23×C311yescharacteristic
217C2×SD16—24no—
218C2×SD16—24no—
219C2×SD16—42no—
220C2×D8—24no—
221C2×D8—24no—
222C2×D8—42no—
223C22×C8C23×C312yes—
224C22×C8C23×C312yes—
225#127—22no—
226#127C3×D414yescomplemented
227#125—22no—
228#125C3×D414yescomplemented
229#124—22no—
230#124—22no—
231#123—216no—
232C22⋊Q8—42no—
233#121—216no—
234#121—24no—
235#121—24no—
236#121—42no—
237C4×D4—24no—
238C4×D4—24no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C3×D412yes—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C312yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4C3×D412yes—
255C2×C42C23×C311yescharacteristic
256C2×C42C23×C311yescharacteristic
257#103—24no—
258#102—24no—
259#102—42no—
260#102—42no—
261C22⋊C8—24no—
262#97—22no—
263C4×C8—22no—
264#95—22no—
265C23×C3—21no—
266C23×C3—24no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3C22×D411yescharacteristic
283C3×Q8—42no—
284C3×Q8—42no—
285C3×D4—24no—
286C3×D4—24no—
287C3×D4—42no—
288C3×D4—42no—
289C3×D4—42no—
290C3×D4—42no—
291C3×D4—88no—
292C3×D4—88no—
293C3×D4—84no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—88no—
315C2×C3×C4#12014yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4112yes—
319C2×C3×C4H(5,𝔽2)12yes—
320C3×C8—24no—
321C3×C8—42no—
322C24—22no—
323C24—21no—
324C24C2×C3×D411yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—42no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C2×D4C2×C3×D414yes—
347C22×C4—22no—
348C22×C4—28no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D412yes—
361C22×C4C24×C311yescharacteristic
362SD16—48no—
363D8—48no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8C3×#4114yes—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C4⋊C4C2×C3×D414yes—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42C2×C3×D412yes—
388C42C2×C3×D412yes—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—44no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—82no—
406C22×C3C2×#12011yescharacteristic
407C22×C3#53211yescharacteristic
408C22×C3#53511yescharacteristic
409C22×C3#57612yes—
410C22×C3#57712yes—
411C3×C4—24no—
412C3×C4—41no—
413C3×C4—42no—
414C3×C4—44no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4#54612yes—
419C3×C4#55012yes—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—24no—
424C23—21no—
425C23—21no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23C22×C3×D411yescharacteristic
438Q8—42no—
439Q8—42no—
440D4—24no—
441D4—24no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—88no—
447D4—88no—
448D4—84no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—42no—
459C2×C4—42no—
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×C4—41no—
469C2×C4—88no—
470C2×C4C3×#12014yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C2×C3×#4112yes—
474C2×C4C3×H(5,𝔽2)12yes—
475C8—24no—
476C8—42no—
477C2×C3—81no—
478C2×C3—84no—
479C2×C3—161no—
480C2×C3#243111yescharacteristic
481C2×C3#301212yes—
482C2×C3#317912yes—
483C2×C3#322011yescharacteristic
484C2×C3#322111yescharacteristic
485C22—42no—
486C22—41no—
487C22—44no—
488C22—41no—
489C22—41no—
490C22—44no—
491C22—42no—
492C22—44no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—82no—
502C22C2×C3×#12011yescharacteristic
503C22C3×#53211yescharacteristic
504C22C3×#53511yescharacteristic
505C22C3×#57612yes—
506C22C3×#57712yes—
507C4—24no—
508C4—41no—
509C4—42no—
510C4—44no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4C3×#54612yes—
515C4C3×#55012yes—
516C3#2561111yescharacteristic, direct product factor, Sylow
517C2—81no—
518C2—84no—
519C2—161no—
520C2C3×#243111yescharacteristic
521C2C3×#301212yes—
522C2C3×#317912yes—
523C2C3×#322011yescharacteristic
524C2C3×#322111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#540C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#488C2semidirect
4. C3×#2620C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1992C2semidirect
8. C3×#1957C2semidirect
9. C3×#1879C2semidirect
10. C3×#1877C2semidirect
11. C3×#1750C2semidirect
12. #25611C3direct (N × H)
44. C2×#540C2×C3semidirect
45. C2×#496C2×C3semidirect
46. C2×#488C2×C3semidirect
47. #2620C2×C3semidirect
48. #2000C2×C3semidirect
49. #1999C2×C3semidirect
50. #1992C2×C3semidirect
51. #1957C2×C3semidirect
52. #1879C2×C3semidirect
53. #1877C2×C3semidirect
54. #1750C2×C3semidirect
72. C3×#127D4semidirect
74. C3×#125D4semidirect
226. #127C3×D4semidirect
228. #125C3×D4semidirect
516. C3#25611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4962
C2×C3×#4882
C3×#26202
C3×#20002
C3×#19992
C3×#19922
C3×#19572
C3×#18792
C3×#18772
C3×#17502
#256113