Finite Group Databasebeta

← C3×#29592 #10648199

Subgroups — C3×#29592 #10648199

Order 768.

1864 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic, complemented
4C3×#2847C211yesmaximal, characteristic, complemented
5C3×#2033C214yesmaximal, complemented
6C3×#1999C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1786C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#29592C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C22×C3×D8C2212yescomplemented
15C2×C3×#121—24no—
16C2×C3×#121—22no—
17C2×C3×#121—24no—
18C2×C3×#121C2214yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2212yescomplemented
21C2×C3×#120—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4—21no—
24C3×#480—24no—
25C3×#447—24no—
26C2×C3×#107C2212yescomplemented
27C2×C3×#102C2212yescomplemented
28C2×C3×#102C2212yescomplemented
29C2×C3×#102C2214yescomplemented
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2214yescomplemented
33C3×#386—22no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×#380C2212yescomplemented
37C3×#340C2214yescomplemented
38C22×#121C2×C311yescharacteristic, complemented
39C2×#480C2×C311yescharacteristic, complemented
40C2×#447C2×C311yescharacteristic, complemented
41#2847C2×C311yescharacteristic, complemented
42#2033C2×C314yescomplemented
43#1999C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1786C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C25×C3D411yescharacteristic
49C22×C3×D4—28no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4—22no—
55C22×C3×D4—44no—
56C22×C3×D4—44no—
57C22×C3×D4—44no—
58C22×C3×D4D414yescomplemented
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2312yes—
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C23×C3×C4—22no—
64C23×C3×C4—44no—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D412yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C2×C3×D8—24no—
70C2×C3×D8—44no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×#121—216no—
74C3×#121—28no—
75C3×#121—28no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—44no—
79C3×#120—44no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4D414yescomplemented
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C3×#107—24no—
93C3×#102—24no—
94C3×#102—24no—
95C3×#102—44no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8—44no—
99C3×#95—21no—
100C3×#95—22no—
101C23×D4—21no—
102C23×D4C22×C311yescharacteristic
103C24×C4C22×C311yescharacteristic
104C22×D8C22×C312yescomplemented
105C2×#121—24no—
106C2×#121—22no—
107C2×#121—24no—
108C2×#121C22×C311yescharacteristic
109C2×#121C22×C314yes—
110C2×#121C22×C312yescomplemented
111C2×#120—21no—
112C22×C4⋊C4C22×C311yescharacteristic
113C22×C22⋊C4—21no—
114#480—24no—
115#447—24no—
116C2×#107C22×C312yescomplemented
117C2×#102C22×C312yescomplemented
118C2×#102C22×C312yescomplemented
119C2×#102C22×C314yescomplemented
120C2×C22⋊C8C22×C311yescharacteristic
121C2×C22⋊C8C22×C311yescharacteristic
122C2×C22⋊C8C22×C314yescomplemented
123#386—22no—
124#386—21no—
125#380C22×C311yescharacteristic
126#380C22×C312yescomplemented
127#340C22×C314yescomplemented
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—21no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3—44no—
136C24×C3—44no—
137C24×C3—42no—
138C24×C3—42no—
139C24×C3D814yescomplemented
140C24×C3C2×D412yes—
141C24×C3C2×D411yescharacteristic
142C2×C3×D4—216no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—48no—
151C2×C3×D4—44no—
152C2×C3×D4—48no—
153C2×C3×D4—48no—
154C2×C3×D4—432no—
155C2×C3×D4—48no—
156C2×C3×D4—44no—
157C2×C3×D4—44no—
158C2×C3×D4—42no—
159C2×C3×D4—44no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4C2×D414yes—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—48no—
178C22×C3×C4—48no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2411yescharacteristic
186C3×D8—48no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—24no—
190C2×C3×C8—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4C2×D414yes—
197C3×C22⋊C4—216no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C25—21no—
203C25C3×D411yescharacteristic
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4C3×D414yescomplemented
214C22×D4C23×C311yescharacteristic
215C22×D4C23×C312yes—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4C3×D411yescharacteristic
221C23×C4C3×D412yes—
222C23×C4C23×C312yes—
223C23×C4C23×C311yescharacteristic
224C2×D8—24no—
225C2×D8—44no—
226C22×C8C23×C312yes—
227C22×C8C23×C312yes—
228#121—216no—
229#121—28no—
230#121—28no—
231#121—22no—
232#121—22no—
233#121—44no—
234#120—44no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4C3×D414yescomplemented
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C312yes—
240C2×C4⋊C4C23×C312yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247#107—24no—
248#102—24no—
249#102—24no—
250#102—44no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—44no—
254#95—21no—
255#95—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—28no—
265C23×C3—24no—
266C23×C3—48no—
267C23×C3—44no—
268C23×C3—48no—
269C23×C3—44no—
270C23×C3—48no—
271C23×C3—48no—
272C23×C3—44no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—44no—
276C23×C3—48no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—88no—
283C23×C3—88no—
284C23×C3#12012yes—
285C23×C3C2×D814yes—
286C23×C3C2×D812yes—
287C23×C3AGL(1,ℤ/8)12yes—
288C23×C3C22×D411yescharacteristic
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—48no—
292C3×D4—432no—
293C3×D4—42no—
294C3×D4—44no—
295C3×D4—42no—
296C3×D4—88no—
297C3×D4—84no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—48no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—48no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4#12012yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4H(5,𝔽2)14yes—
322C3×C8—42no—
323C3×C8—42no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—21no—
329C24—42no—
330C24—42no—
331C24—44no—
332C24—44no—
333C24—42no—
334C24—42no—
335C24C3×D814yescomplemented
336C24C2×C3×D412yes—
337C24C2×C3×D411yescharacteristic
338C2×D4—216no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—432no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—42no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4C2×C3×D414yes—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D412yes—
380C22×C4C2×C3×D412yes—
381C22×C4C24×C311yescharacteristic
382D8—48no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—24no—
392C4⋊C4C2×C3×D414yes—
393C22⋊C4—216no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—22no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—416no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—816no—
414C22×C3—84no—
415C22×C3—84no—
416C22×C3—81no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3#44714yes—
422C22×C3C2×#12011yescharacteristic
423C22×C3#53412yes—
424C22×C3C22×D811yescharacteristic
425C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
426C22×C3#57512yes—
427C3×C4—22no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4—82no—
431C3×C4—81no—
432C3×C4#53412yes—
433C3×C4#53512yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—28no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—48no—
445C23—44no—
446C23—48no—
447C23—44no—
448C23—48no—
449C23—48no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—48no—
454C23—44no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—88no—
461C23—88no—
462C23C3×#12012yes—
463C23C2×C3×D814yes—
464C23C2×C3×D812yes—
465C23C3×AGL(1,ℤ/8)12yes—
466C23C22×C3×D411yescharacteristic
467D4—24no—
468D4—24no—
469D4—48no—
470D4—432no—
471D4—42no—
472D4—44no—
473D4—42no—
474D4—88no—
475D4—84no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4C3×#12012yes—
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C22×C3×D411yescharacteristic
499C2×C4C3×H(5,𝔽2)14yes—
500C8—42no—
501C8—42no—
502C2×C3—24no—
503C2×C3—44no—
504C2×C3—84no—
505C2×C3—161no—
506C2×C3#261311yescharacteristic
507C2×C3C2×#44711yescharacteristic
508C2×C3#300511yescharacteristic
509C2×C3#318612yes—
510C2×C3#319412yes—
511C22—24no—
512C22—24no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—416no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—816no—
527C22—84no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22C3×#44714yes—
535C22C2×C3×#12011yescharacteristic
536C22C3×#53412yes—
537C22C22×C3×D811yescharacteristic
538C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
539C22C3×#57512yes—
540C4—22no—
541C4—44no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4C3×#53412yes—
546C4C3×#53512yes—
547C3#2959211yescharacteristic, direct product factor, Sylow
548C2—24no—
549C2—44no—
550C2—84no—
551C2—161no—
552C2C3×#261311yescharacteristic
553C2C2×C3×#44711yescharacteristic
554C2C3×#300511yescharacteristic
555C2C3×#318612yes—
556C2C3×#319412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#121C2semidirect
2. C2×C3×#480C2semidirect
3. C2×C3×#447C2semidirect
4. C3×#2847C2semidirect
5. C3×#2033C2semidirect
6. C3×#1999C2semidirect
7. C3×#1823C2semidirect
8. C3×#1786C2semidirect
9. C3×#1779C2semidirect
10. #29592C3direct (N × H)
14. C22×C3×D8C22semidirect
20. C2×C3×#121C22semidirect
26. C2×C3×#107C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
29. C2×C3×#102C22semidirect
32. C2×C3×C22⋊C8C22semidirect
36. C3×#380C22semidirect
37. C3×#340C22semidirect
38. C22×#121C2×C3semidirect
39. C2×#480C2×C3semidirect
40. C2×#447C2×C3semidirect
41. #2847C2×C3semidirect
42. #2033C2×C3semidirect
43. #1999C2×C3semidirect
44. #1823C2×C3semidirect
45. #1786C2×C3semidirect
46. #1779C2×C3semidirect
58. C22×C3×D4D4semidirect
82. C2×C3×C4⋊C4D4semidirect
104. C22×D8C22×C3semidirect
110. C2×#121C22×C3semidirect
116. C2×#107C22×C3semidirect
117. C2×#102C22×C3semidirect
118. C2×#102C22×C3semidirect
119. C2×#102C22×C3semidirect
122. C2×C22⋊C8C22×C3semidirect
126. #380C22×C3semidirect
127. #340C22×C3semidirect
139. C24×C3D8semidirect
213. C22×D4C3×D4semidirect
237. C2×C4⋊C4C3×D4semidirect
335. C24C3×D8semidirect
547. C3#29592direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#4802
C2×C3×#4472
C3×#28472
C3×#20332
C3×#19992
C3×#18232
C3×#17862
C3×#17792
#295923