Finite Group Databasebeta

← C3×#25754 #10644361

Subgroups — C3×#25754 #10644361

Order 768.

1682 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C2×C3×#449C214yesmaximal, complemented
5C3×#2541C211yesmaximal, characteristic, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1954C211yesmaximal, characteristic, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25754C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×SD16C2214yes—
13C22×C3×D8C2212yes—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127—21no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×#121C2214yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C22×C3×C4⋊C4—21no—
26C3×#492—24no—
27C3×#459—24no—
28C3×#449—216no—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2214yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2214yes—
34C2×C3×C4×C8C2211yescharacteristic
35C3×#392—22no—
36C3×#390—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C2×#547C2×C311yescharacteristic, complemented
40C2×#492C2×C311yescharacteristic, complemented
41C2×#459C2×C311yescharacteristic, complemented
42C2×#449C2×C314yescomplemented
43#2541C2×C311yescharacteristic, complemented
44#2000C2×C314yescomplemented
45#1981C2×C311yescharacteristic, complemented
46#1954C2×C311yescharacteristic, complemented
47C4×#102C2×C311yescharacteristic, complemented
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—28no—
51C2×C3×#41—44no—
52C22×C3×Q8C2312yes—
53C22×C3×D4—22no—
54C22×C3×D4—24no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4D411yescharacteristic
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4—21no—
62C23×C3×C4C2312yes—
63C2×C3×SD16—28no—
64C2×C3×SD16—28no—
65C2×C3×SD16—44no—
66C2×C3×D8—28no—
67C2×C3×D8—24no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yescomplemented
72C3×#127—22no—
73C3×#127—22no—
74C3×#127—44no—
75C3×#123—216no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—24no—
78C3×#121—28no—
79C3×#121—28no—
80C3×#121—44no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4D414yescomplemented
85C2×C3×C4⋊C4—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C3×C4⋊C8—22no—
99C3×#103—28no—
100C3×#102—24no—
101C3×#102D418yescomplemented
102C3×C22⋊C8—28no—
103C3×C4×C8—22no—
104C3×#95—21no—
105C22×#41C22×C312yes—
106C22×SD16C22×C314yes—
107C22×D8C22×C312yes—
108#547—24no—
109#547—24no—
110C2×C4⋊Q8C22×C311yescharacteristic
111C2×#127—21no—
112C2×#127C22×C311yescharacteristic
113C2×#123—22no—
114C2×C22⋊Q8C22×C312yes—
115C2×#121—21no—
116C2×#121C22×C314yes—
117C2×C4×D4C22×C311yescharacteristic
118C2×C4×D4C22×C311yescharacteristic
119C22×C4⋊C4—21no—
120#492—24no—
121#459—24no—
122#449—216no—
123C2×C4⋊C8C22×C311yescharacteristic
124C2×#103C22×C314yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×C22⋊C8C22×C314yes—
128C2×C4×C8C22×C311yescharacteristic
129#392—22no—
130#390—21no—
131C4×C4⋊C4C22×C311yescharacteristic
132#340C22×C312yes—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3C2×D412yes—
136C3×#41—416no—
137C3×#41—48no—
138C3×#41—48no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×Q8—24no—
142C2×C3×Q8C2×D414yes—
143C2×C3×D4—28no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—24no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—48no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—82no—
161C2×C3×D4—84no—
162C2×C3×D4C2×D414yes—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—28no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—48no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C3×SD16—416no—
181C3×D8—48no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—24no—
185C2×C3×C8—22no—
186C2×C3×C8C2×D414yes—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—48no—
199C3×C42—42no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—28no—
205C2×#41—44no—
206C22×Q8C23×C312yes—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C3×D411yescharacteristic
212C22×D4C23×C312yes—
213C22×D4C23×C311yescharacteristic
214C23×C4—21no—
215C23×C4—22no—
216C23×C4C23×C312yes—
217C2×SD16—28no—
218C2×SD16—28no—
219C2×SD16—44no—
220C2×D8—28no—
221C2×D8—24no—
222C22×C8C23×C312yes—
223C22×C8C23×C312yes—
224C4⋊Q8—22no—
225C4⋊Q8C3×D414yescomplemented
226#127—22no—
227#127—22no—
228#127—44no—
229#123—216no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232#121—28no—
233#121—28no—
234#121—44no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4C3×D414yescomplemented
239C2×C4⋊C4—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C3×D411yescharacteristic
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C4⋊C4C23×C312yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4C3×D412yes—
250C2×C42—21no—
251C2×C42C23×C311yescharacteristic
252C4⋊C8—22no—
253#103—28no—
254#102—24no—
255#102C3×D418yescomplemented
256C22⋊C8—28no—
257C4×C8—22no—
258#95—21no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—44no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—44no—
274C23×C3C22×D411yescharacteristic
275C3×Q8—28no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×D4—22no—
279C3×D4—24no—
280C3×D4—416no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—44no—
284C3×D4—44no—
285C3×D4—84no—
286C3×D4—816no—
287C3×D4—84no—
288C3×D4#12714yescomplemented
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—416no—
296C2×C3×C4—44no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—44no—
304C2×C3×C4—48no—
305C2×C3×C4#12012yes—
306C2×C3×C4#12012yes—
307C2×C3×C4#13512yes—
308C2×C3×C4AGL(1,ℤ/8)12yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D412yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4H(5,𝔽2)12yes—
313C3×C8—24no—
314C3×C8—42no—
315C24—21no—
316C24—22no—
317C24C2×C3×D412yes—
318#41—416no—
319#41—48no—
320#41—48no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8C2×C3×D414yes—
325C2×D4—28no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—82no—
343C2×D4—84no—
344C2×D4C2×C3×D414yes—
345C2×D4C2×C3×D412yes—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C24×C311yescharacteristic
362SD16—416no—
363D8—48no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—22no—
368C2×C8C2×C3×D414yes—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4C2×C3×D412yes—
376C4⋊C4C2×C3×D414yes—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—48no—
381C42—42no—
382C42C2×C3×D412yes—
383C42C2×C3×D412yes—
384C22×C3—24no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—84no—
398C22×C3—84no—
399C22×C3C2×#12011yescharacteristic
400C22×C3D4212yes—
401C22×C3C2×#13511yescharacteristic
402C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C3#57712yes—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—44no—
407C3×C4—42no—
408C3×C4—44no—
409C3×C4#44914yescomplemented
410C3×C4C2×#12711yescharacteristic
411C3×C4#53511yescharacteristic
412C3×C4D4212yes—
413C23—22no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—44no—
428C23C22×C3×D411yescharacteristic
429Q8—28no—
430Q8—42no—
431Q8—42no—
432D4—22no—
433D4—24no—
434D4—416no—
435D4—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—84no—
440D4—816no—
441D4—84no—
442D4C3×#12714yescomplemented
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—416no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4C3×#12012yes—
460C2×C4C3×#12012yes—
461C2×C4C3×#13512yes—
462C2×C4C3×AGL(1,ℤ/8)12yes—
463C2×C4C22×C3×D411yescharacteristic
464C2×C4C22×C3×D412yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C3×H(5,𝔽2)12yes—
467C8—24no—
468C8—42no—
469C2×C3—44no—
470C2×C3—82no—
471C2×C3—162no—
472C2×C3#243311yescharacteristic
473C2×C3C2×#44911yescharacteristic
474C2×C3#301011yescharacteristic
475C2×C3#327212yes—
476C2×C3#328412yes—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—44no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—84no—
491C22—84no—
492C22C2×C3×#12011yescharacteristic
493C22C3×D4212yes—
494C22C2×C3×#13511yescharacteristic
495C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
496C22C3×#57712yes—
497C4—22no—
498C4—42no—
499C4—44no—
500C4—42no—
501C4—44no—
502C4C3×#44914yescomplemented
503C4C2×C3×#12711yescharacteristic
504C4C3×#53511yescharacteristic
505C4C3×D4212yes—
506C3#2575411yescharacteristic, direct product factor, Sylow
507C2—44no—
508C2—82no—
509C2—162no—
510C2C3×#243311yescharacteristic
511C2C2×C3×#44911yescharacteristic
512C2C3×#301011yescharacteristic
513C2C3×#327212yes—
514C2C3×#328412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C2×C3×#492C2semidirect
3. C2×C3×#459C2semidirect
4. C2×C3×#449C2semidirect
5. C3×#2541C2semidirect
6. C3×#2000C2semidirect
7. C3×#1981C2semidirect
8. C3×#1954C2semidirect
9. C3×C4×#102C2semidirect
10. #25754C3direct (N × H)
39. C2×#547C2×C3semidirect
40. C2×#492C2×C3semidirect
41. C2×#459C2×C3semidirect
42. C2×#449C2×C3semidirect
43. #2541C2×C3semidirect
44. #2000C2×C3semidirect
45. #1981C2×C3semidirect
46. #1954C2×C3semidirect
47. C4×#102C2×C3semidirect
71. C3×C4⋊Q8D4semidirect
84. C3×C4×D4D4semidirect
101. C3×#102D4semidirect
225. C4⋊Q8C3×D4semidirect
238. C4×D4C3×D4semidirect
255. #102C3×D4semidirect
288. C3×D4#127semidirect
409. C3×C4#449semidirect
442. D4C3×#127semidirect
502. C4C3×#449semidirect
506. C3#25754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4922
C2×C3×#4592
C2×C3×#4492
C3×#25412
C3×#20002
C3×#19812
C3×#19542
C3×C4×#1022
#257543