Finite Group Databasebeta

← C3×#29660 #10648267

Subgroups — C3×#29660 #10648267

Order 768.

1552 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#459C212yesmaximal, complemented
3C3×#2856C211yesmaximal, characteristic, complemented
4C3×#2023C212yesmaximal, complemented
5C3×#2011C212yesmaximal, complemented
6C3×#1999C214yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29660C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41—21no—
11C22×C3×D8C2214yes—
12C3×#532—28no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#121—21no—
18C2×C3×#121—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#459—28no—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#102C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C3×#390—21no—
31C3×#390—21no—
32C3×#390C2212yes—
33C3×#384C2211yescharacteristic
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#532C2×C311yescharacteristic, complemented
37C2×#459C2×C312yescomplemented
38#2856C2×C311yescharacteristic, complemented
39#2023C2×C312yescomplemented
40#2011C2×C312yescomplemented
41#1999C2×C314yescomplemented
42#1802C2×C312yescomplemented
43#1797C2×C311yescharacteristic, complemented
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—21no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4C2312yes—
55C22×C3×D4C2312yes—
56C23×C3×C4—22no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×D8—28no—
61C2×C3×D8—44no—
62C2×C3×D8—44no—
63C22×C3×C8C2314yes—
64C3×C4⋊Q8—24no—
65C3×#127—24no—
66C3×#127—24no—
67C3×#124—28no—
68C3×#121—28no—
69C3×#121—24no—
70C3×#121—24no—
71C3×#121—44no—
72C3×#121—44no—
73C3×#121—44no—
74C3×C4×D4—28no—
75C3×C4×D4—24no—
76C3×C4×D4D418yescomplemented
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—24no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×C4⋊C8—24no—
93C3×#102—28no—
94C3×#102—44no—
95C3×C22⋊C8—42no—
96C22×#41—21no—
97C22×D8C22×C314yes—
98#532—28no—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#127C22×C311yescharacteristic
101C2×#127C22×C311yescharacteristic
102C2×#124—21no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C22×C312yes—
107C2×#121C22×C312yes—
108C2×C4×D4—21no—
109C2×C4×D4C22×C312yes—
110C22×C42C22×C311yescharacteristic
111#459—28no—
112C2×C4⋊C8C22×C312yes—
113C2×#102C22×C314yes—
114C2×#102C22×C314yes—
115C2×C22⋊C8C22×C312yes—
116#390—21no—
117#390—21no—
118#390C22×C312yes—
119#384C22×C311yescharacteristic
120#340C22×C314yes—
121#336C22×C312yes—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C2×D411yescharacteristic
125C3×#41—432no—
126C2×C3×Q8—24no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×D4—24no—
130C2×C3×D4—28no—
131C2×C3×D4—24no—
132C2×C3×D4—22no—
133C2×C3×D4—24no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—28no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—41no—
169C22×C3×C4—44no—
170C22×C3×C4—44no—
171C22×C3×C4D814yescomplemented
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C3×D8—416no—
179C2×C3×C8—24no—
180C2×C3×C8—24no—
181C2×C3×C8—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4C2×D414yes—
188C3×C22⋊C4—28no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D414yes—
197C2×#41—28no—
198C2×#41—44no—
199C2×#41—44no—
200C22×Q8—21no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4C23×C312yes—
208C22×D4C23×C312yes—
209C23×C4—22no—
210C23×C4C3×D411yescharacteristic
211C23×C4C3×D411yescharacteristic
212C23×C4C23×C311yescharacteristic
213C2×D8—28no—
214C2×D8—44no—
215C2×D8—44no—
216C22×C8C23×C314yes—
217C4⋊Q8—24no—
218#127—24no—
219#127—24no—
220#124—28no—
221#121—28no—
222#121—24no—
223#121—24no—
224#121—44no—
225#121—44no—
226#121—44no—
227C4×D4—28no—
228C4×D4—24no—
229C4×D4C3×D418yescomplemented
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C312yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C42—24no—
241C2×C42C3×D411yescharacteristic
242C2×C42C3×D411yescharacteristic
243C2×C42C23×C311yescharacteristic
244C2×C42C23×C311yescharacteristic
245C4⋊C8—24no—
246#102—28no—
247#102—44no—
248C22⋊C8—42no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—24no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3C2×D812yes—
270C23×C3C22×D411yescharacteristic
271C3×Q8—48no—
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—48no—
276C3×D4—416no—
277C3×D4—42no—
278C3×D4—44no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—84no—
282C3×D4—88no—
283C3×D4—84no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—28no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—44no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—416no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C2×D812yes—
307C2×C3×C4C2×D812yes—
308C2×C3×C4AGL(1,ℤ/8)12yes—
309C2×C3×C4AGL(1,ℤ/8)12yes—
310C2×C3×C4C22×D412yes—
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C2×C3×C4H(5,𝔽2)12yes—
313C2×C3×C42−1+411yescharacteristic
314C3×C8—44no—
315C24—22no—
316C24—22no—
317C24C2×C3×D411yescharacteristic
318#41—432no—
319C2×Q8—24no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—24no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4C2×C3×D414yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4C3×D814yescomplemented
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C24×C311yescharacteristic
371D8—416no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4C2×C3×D414yes—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42C2×C3×D414yes—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3C2×#12011yescharacteristic
412C22×C3#53511yescharacteristic
413C22×C3#53711yescharacteristic
414C22×C3C22×D811yescharacteristic
415C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C3#57511yescharacteristic
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—44no—
423C3×C4—82no—
424C3×C4#44714yes—
425C3×C4#53512yes—
426C3×C4#53712yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C2×C3×D812yes—
448C23C22×C3×D411yescharacteristic
449Q8—48no—
450D4—24no—
451D4—24no—
452D4—48no—
453D4—48no—
454D4—416no—
455D4—42no—
456D4—44no—
457D4—42no—
458D4—44no—
459D4—84no—
460D4—88no—
461D4—84no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4C3×#12012yes—
483C2×C4C3×#12012yes—
484C2×C4C2×C3×D812yes—
485C2×C4C2×C3×D812yes—
486C2×C4C3×AGL(1,ℤ/8)12yes—
487C2×C4C3×AGL(1,ℤ/8)12yes—
488C2×C4C22×C3×D412yes—
489C2×C4C3×H(5,𝔽2)11yescharacteristic
490C2×C4C3×H(5,𝔽2)12yes—
491C2×C4C3×2−1+411yescharacteristic
492C8—44no—
493C2×C3—42no—
494C2×C3—84no—
495C2×C3—162no—
496C2×C3#261511yescharacteristic
497C2×C3C2×#44711yescharacteristic
498C2×C3#300611yescharacteristic
499C2×C3#320511yescharacteristic
500C2×C3#321411yescharacteristic
501C2×C3#324511yescharacteristic
502C2×C3#325311yescharacteristic
503C22—22no—
504C22—22no—
505C22—22no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22C2×C3×#12011yescharacteristic
525C22C3×#53511yescharacteristic
526C22C3×#53711yescharacteristic
527C22C22×C3×D811yescharacteristic
528C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
529C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
530C22C3×#57511yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—44no—
535C4—41no—
536C4—82no—
537C4C3×#44714yes—
538C4C3×#53512yes—
539C4C3×#53712yes—
540C3#2966011yescharacteristic, direct product factor, Sylow
541C2—42no—
542C2—84no—
543C2—162no—
544C2C3×#261511yescharacteristic
545C2C2×C3×#44711yescharacteristic
546C2C3×#300611yescharacteristic
547C2C3×#320511yescharacteristic
548C2C3×#321411yescharacteristic
549C2C3×#324511yescharacteristic
550C2C3×#325311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#532C2semidirect
2. C2×C3×#459C2semidirect
3. C3×#2856C2semidirect
4. C3×#2023C2semidirect
5. C3×#2011C2semidirect
6. C3×#1999C2semidirect
7. C3×#1802C2semidirect
8. C3×#1797C2semidirect
9. #29660C3direct (N × H)
36. C2×#532C2×C3semidirect
37. C2×#459C2×C3semidirect
38. #2856C2×C3semidirect
39. #2023C2×C3semidirect
40. #2011C2×C3semidirect
41. #1999C2×C3semidirect
42. #1802C2×C3semidirect
43. #1797C2×C3semidirect
76. C3×C4×D4D4semidirect
171. C22×C3×C4D8semidirect
229. C4×D4C3×D4semidirect
364. C22×C4C3×D8semidirect
540. C3#29660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4592
C3×#28562
C3×#20232
C3×#20112
C3×#19992
C3×#18022
C3×#17972
#296603