Finite Group Databasebeta

← C3×#29714 #10648321

Subgroups — C3×#29714 #10648321

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#450C212yesmaximal
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic
4C3×#2071C212yesmaximal, complemented
5C3×#2055C214yesmaximal, complemented
6C3×#2013C212yesmaximal, complemented
7C3×#1872C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29714C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×SD16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic, complemented
13C2×C3×C22⋊Q8C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4—21no—
16C22×C3×C22⋊C4—21no—
17C22×C3×C42C2211yescharacteristic
18C3×#450—28no—
19C2×C3×C4⋊C8C2212yescomplemented
20C2×C3×#103C2214yescomplemented
21C2×C3×#102C2214yes—
22C2×C3×C22⋊C8C2212yes—
23C3×#391C2212yescomplemented
24C3×#390—21no—
25C3×#390—21no—
26C3×#390C2212yes—
27C3×#390C2212yes—
28C3×#389—21no—
29C3×#389—21no—
30C3×#384C2211yescharacteristic, complemented
31C3×#380—21no—
32C3×#380—21no—
33C3×#340C2214yescomplemented
34C3×#336C2212yescomplemented
35C2×#450C2×C312yes—
36#2857C2×C311yescharacteristic, complemented
37C2×#390C2×C311yescharacteristic
38#2071C2×C312yescomplemented
39#2055C2×C314yescomplemented
40#2013C2×C312yescomplemented
41#1872C2×C312yescomplemented
42#1797C2×C311yescharacteristic, complemented
43C25×C3—21no—
44C22×C3×Q8C2312yes—
45C22×C3×D4—21no—
46C22×C3×D4—22no—
47C22×C3×D4—42no—
48C22×C3×D4—42no—
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×SD16—28no—
56C2×C3×SD16—44no—
57C2×C3×SD16—44no—
58C22×C3×C8C2314yes—
59C3×C4⋊Q8—42no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—24no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—44no—
77C2×C3×C22⋊C4—44no—
78C2×C3×C42—42no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×C4⋊C8—42no—
84C3×#103—44no—
85C3×#102—28no—
86C3×C22⋊C8—24no—
87C3×#95—22no—
88C3×#95—22no—
89C23×D4C22×C311yescharacteristic
90C22×SD16C22×C314yes—
91C2×C4⋊Q8C22×C311yescharacteristic, complemented
92C2×C22⋊Q8C22×C312yes—
93C22×C4⋊C4C22×C311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C42C22×C311yescharacteristic
97#450—28no—
98C2×C4⋊C8C22×C312yescomplemented
99C2×#103C22×C314yescomplemented
100C2×#102C22×C314yes—
101C2×C22⋊C8C22×C312yes—
102#391C22×C312yescomplemented
103#390—21no—
104#390—21no—
105#390C22×C312yes—
106#390C22×C312yes—
107#389—21no—
108#389—21no—
109#384C22×C311yescharacteristic, complemented
110#380—21no—
111#380—21no—
112#340C22×C314yescomplemented
113#336C22×C312yescomplemented
114C24×C3—24no—
115C24×C3—22no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—42no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3C2×D411yescharacteristic
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—24no—
126C2×C3×Q8—24no—
127C2×C3×D4—24no—
128C2×C3×D4—22no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—48no—
132C2×C3×D4—42no—
133C2×C3×D4—42no—
134C2×C3×D4—48no—
135C2×C3×D4C2×D414yes—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4SD1614yescomplemented
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4112yes—
157C22×C3×C4C2411yescharacteristic
158C3×SD16—416no—
159C2×C3×C8—24no—
160C2×C3×C8—24no—
161C2×C3×C8—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C22⋊C4—44no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—41no—
179C3×C42—41no—
180C25—21no—
181C22×Q8C23×C312yes—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C3×D414yes—
187C22×D4C23×C312yes—
188C23×C4—21no—
189C23×C4C3×D411yescharacteristic
190C23×C4C3×D411yescharacteristic
191C23×C4C23×C311yescharacteristic
192C2×SD16—28no—
193C2×SD16—44no—
194C2×SD16—44no—
195C22×C8C23×C314yes—
196C4⋊Q8—42no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C23×C312yes—
207C2×C4⋊C4C23×C312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C42—42no—
216C2×C42C3×D411yescharacteristic
217C2×C42C3×D411yescharacteristic
218C2×C42C23×C311yescharacteristic
219C2×C42C23×C311yescharacteristic
220C4⋊C8—42no—
221#103—44no—
222#102—28no—
223C22⋊C8—24no—
224#95—22no—
225#95—22no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—24no—
231C23×C3—48no—
232C23×C3—48no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—48no—
240C23×C3—84no—
241C23×C3—84no—
242C23×C3—84no—
243C23×C3—84no—
244C23×C3#12112yes—
245C23×C3C2×SD1612yes—
246C23×C3#13712yes—
247C23×C3C22×D411yescharacteristic
248C3×Q8—44no—
249C3×Q8—44no—
250C3×D4—28no—
251C3×D4—48no—
252C3×D4—44no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—84no—
278C2×C3×C4#12112yes—
279C2×C3×C4C2×SD1612yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4AGL(1,ℤ/8)12yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4H(5,𝔽2)12yes—
285C2×C3×C4H(5,𝔽2)12yes—
286C3×C8—44no—
287C24—24no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24C2×C3×D411yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—48no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—48no—
308C2×D4C2×C3×D414yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C3×SD1614yescomplemented
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C3×#4112yes—
330C22×C4C24×C311yescharacteristic
331SD16—416no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C42—22no—
350C42—22no—
351C42—41no—
352C42—41no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—816no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3#46114yes—
368C22×C3C2×#12111yescharacteristic
369C22×C3#53411yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3C22×SD1611yescharacteristic
372C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
373C22×C3C2×#13711yescharacteristic
374C22×C3#57511yescharacteristic
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—41no—
380C3×C4—82no—
381C3×C4—82no—
382C3×C4—82no—
383C3×C4#53412yes—
384C3×C4#53812yes—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—24no—
390C23—48no—
391C23—48no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—48no—
397C23—42no—
398C23—42no—
399C23—84no—
400C23—84no—
401C23—84no—
402C23—84no—
403C23C3×#12112yes—
404C23C2×C3×SD1612yes—
405C23C3×#13712yes—
406C23C22×C3×D411yescharacteristic
407Q8—44no—
408Q8—44no—
409D4—28no—
410D4—48no—
411D4—44no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—84no—
437C2×C4C3×#12112yes—
438C2×C4C2×C3×SD1612yes—
439C2×C4C2×C3×SD1612yes—
440C2×C4C3×AGL(1,ℤ/8)12yes—
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C2×C3×#4111yescharacteristic
443C2×C4C3×H(5,𝔽2)12yes—
444C2×C4C3×H(5,𝔽2)12yes—
445C8—44no—
446C2×C3—24no—
447C2×C3—84no—
448C2×C3#261711yescharacteristic
449C2×C3C2×#46111yescharacteristic
450C2×C3#303611yescharacteristic
451C2×C3#318711yescharacteristic
452C2×C3#319311yescharacteristic
453C2×C3#320811yescharacteristic
454C2×C3#321511yescharacteristic
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—816no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22C3×#46114yes—
470C22C2×C3×#12111yescharacteristic
471C22C3×#53411yescharacteristic
472C22C3×#53511yescharacteristic
473C22C22×C3×SD1611yescharacteristic
474C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
475C22C2×C3×#13711yescharacteristic
476C22C3×#57511yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—41no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4C3×#53412yes—
486C4C3×#53812yes—
487C3#2971411yescharacteristic, direct product factor, Sylow
488C2—24no—
489C2—84no—
490C2C3×#261711yescharacteristic
491C2C2×C3×#46111yescharacteristic
492C2C3×#303611yescharacteristic
493C2C3×#318711yescharacteristic
494C2C3×#319311yescharacteristic
495C2C3×#320811yescharacteristic
496C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
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
2. C3×#2857C2semidirect
4. C3×#2071C2semidirect
5. C3×#2055C2semidirect
6. C3×#2013C2semidirect
7. C3×#1872C2semidirect
8. C3×#1797C2semidirect
9. #29714C3direct (N × H)
12. C2×C3×C4⋊Q8C22semidirect
19. C2×C3×C4⋊C8C22semidirect
20. C2×C3×#103C22semidirect
23. C3×#391C22semidirect
30. C3×#384C22semidirect
33. C3×#340C22semidirect
34. C3×#336C22semidirect
36. #2857C2×C3semidirect
38. #2071C2×C3semidirect
39. #2055C2×C3semidirect
40. #2013C2×C3semidirect
41. #1872C2×C3semidirect
42. #1797C2×C3semidirect
91. C2×C4⋊Q8C22×C3semidirect
98. C2×C4⋊C8C22×C3semidirect
99. C2×#103C22×C3semidirect
102. #391C22×C3semidirect
109. #384C22×C3semidirect
112. #340C22×C3semidirect
113. #336C22×C3semidirect
149. C22×C3×C4SD16semidirect
322. C22×C4C3×SD16semidirect
487. C3#29714direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C3×#28572
C2×C3×#3902
C3×#20712
C3×#20552
C3×#20132
C3×#18722
C3×#17972
#297143