Finite Group Databasebeta

← C3×#29800 #10648407

Subgroups — C3×#29800 #10648407

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#463C212yesmaximal, complemented
2C3×#2858C211yesmaximal, characteristic, complemented
3C3×#2293C211yesmaximal, characteristic, complemented
4C3×#2043C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#2011C211yesmaximal, characteristic
10C3×#1983C212yesmaximal, complemented
11C3×#1831C211yesmaximal, characteristic, complemented
12#29800C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#117—21no—
21C3×#463—28no—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C3×#390—21no—
30C3×#390—21no—
31C3×#390—21no—
32C3×#390—21no—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×C4×C22⋊C4—21no—
41C3×C43C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44C3×#336C2211yescharacteristic
45C2×#463C2×C312yescomplemented
46#2858C2×C311yescharacteristic, complemented
47#2293C2×C311yescharacteristic, complemented
48#2043C2×C311yescharacteristic, complemented
49#2042C2×C312yescomplemented
50#2041C2×C311yescharacteristic, complemented
51#2013C2×C311yescharacteristic, complemented
52#2012C2×C312yescomplemented
53#2011C2×C311yescharacteristic
54#1983C2×C312yescomplemented
55#1831C2×C311yescharacteristic, complemented
56C22×C3×Q8C2312yes—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×SD16—24no—
63C2×C3×SD16—42no—
64C2×C3×SD16—42no—
65C2×C3×D8—24no—
66C2×C3×D8—42no—
67C2×C3×D8—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—42no—
71C3×#127—42no—
72C3×#124—24no—
73C3×#124—24no—
74C3×#124—42no—
75C3×#124—42no—
76C3×C4×D4—22no—
77C3×C4×D4D414yescomplemented
78C3×#117—44no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C42—21no—
96C2×C3×C42D411yescharacteristic
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C3×C4⋊C8—24no—
104C3×#103—24no—
105C3×#103—42no—
106C3×#103—42no—
107C3×#102—24no—
108C3×#102—42no—
109C3×#102—42no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C22×SD16C22×C312yes—
114C22×D8C22×C312yes—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#127C22×C311yescharacteristic
117C2×#124C22×C312yes—
118C2×#124C22×C312yes—
119C2×C4×D4C22×C311yescharacteristic
120C2×#117—21no—
121#463—28no—
122C2×C4⋊C8C22×C312yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×#103C22×C312yes—
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×#102C22×C312yes—
129#390—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#390C22×C311yescharacteristic
134#390C22×C311yescharacteristic
135#389C22×C311yescharacteristic
136#389C22×C311yescharacteristic
137#388—21no—
138#388—21no—
139C4×C4⋊C4C22×C311yescharacteristic
140C4×C22⋊C4—21no—
141C43C22×C311yescharacteristic
142#336C22×C311yescharacteristic
143#336C22×C312yes—
144#336C22×C311yescharacteristic
145C24×C3—21no—
146C24×C3—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4C2×D412yes—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C3×SD16—48no—
184C3×D8—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—41no—
213C3×C42—41no—
214C3×C42—42no—
215C3×C42C2×D412yes—
216C3×C42#4112yes—
217C22×Q8C23×C312yes—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4C23×C312yes—
221C22×D4C23×C311yescharacteristic
222C23×C4—21no—
223C2×SD16—24no—
224C2×SD16—42no—
225C2×SD16—42no—
226C2×D8—24no—
227C2×D8—42no—
228C2×D8—42no—
229C22×C8C23×C312yes—
230C22×C8C23×C312yes—
231C4⋊Q8—42no—
232#127—42no—
233#124—24no—
234#124—24no—
235#124—42no—
236#124—42no—
237C4×D4—22no—
238C4×D4C3×D414yescomplemented
239#117—44no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42C3×D411yescharacteristic
258C2×C42C3×D411yescharacteristic
259C2×C42C3×D411yescharacteristic
260C2×C42C3×D411yescharacteristic
261C2×C42C23×C311yescharacteristic
262C2×C42C23×C311yescharacteristic
263C2×C42C23×C311yescharacteristic
264C4⋊C8—24no—
265#103—24no—
266#103—42no—
267#103—42no—
268#102—24no—
269#102—42no—
270#102—42no—
271#95—21no—
272#95—21no—
273#95—21no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3C22×D411yescharacteristic
289C3×Q8—42no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×D4—24no—
294C3×D4—42no—
295C3×D4—42no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—84no—
300C3×D4—84no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—48no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4#12112yes—
329C2×C3×C4#12112yes—
330C2×C3×C4#13512yes—
331C2×C3×C4AGL(1,ℤ/8)12yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4H(5,𝔽2)12yes—
335C2×C3×C4H(5,𝔽2)12yes—
336C3×C8—42no—
337C3×C8—42no—
338C24—21no—
339C24—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4C2×C3×D412yes—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C24×C311yescharacteristic
376SD16—48no—
377D8—48no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4C2×C3×D412yes—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42C2×C3×D412yes—
409C42C3×#4112yes—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—81no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3C2×#12111yescharacteristic
425C22×C3#53511yescharacteristic
426C22×C3#53511yescharacteristic
427C22×C3C2×#13511yescharacteristic
428C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
429C22×C3#57511yescharacteristic
430C22×C3#57511yescharacteristic
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—82no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4#53512yes—
442C3×C4#54012yes—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23C22×C3×D411yescharacteristic
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462D4—24no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—84no—
469D4—84no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4C3×#12112yes—
498C2×C4C3×#12112yes—
499C2×C4C3×#13512yes—
500C2×C4C3×AGL(1,ℤ/8)12yes—
501C2×C4C22×C3×D411yescharacteristic
502C2×C4C2×C3×#4111yescharacteristic
503C2×C4C3×H(5,𝔽2)12yes—
504C2×C4C3×H(5,𝔽2)12yes—
505C8—42no—
506C8—42no—
507C2×C3—82no—
508C2×C3—162no—
509C2×C3#261911yescharacteristic
510C2×C3#303511yescharacteristic
511C2×C3#303611yescharacteristic
512C2×C3#320711yescharacteristic
513C2×C3#321111yescharacteristic
514C2×C3#321311yescharacteristic
515C2×C3#321911yescharacteristic
516C22—42no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22C2×C3×#12111yescharacteristic
531C22C3×#53511yescharacteristic
532C22C3×#53511yescharacteristic
533C22C2×C3×#13511yescharacteristic
534C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
535C22C3×#57511yescharacteristic
536C22C3×#57511yescharacteristic
537C4—22no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4C3×#53512yes—
548C4C3×#54012yes—
549C3#2980011yescharacteristic, direct product factor, Sylow
550C2—82no—
551C2—162no—
552C2C3×#261911yescharacteristic
553C2C3×#303511yescharacteristic
554C2C3×#303611yescharacteristic
555C2C3×#320711yescharacteristic
556C2C3×#321111yescharacteristic
557C2C3×#321311yescharacteristic
558C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#463C2semidirect
2. C3×#2858C2semidirect
3. C3×#2293C2semidirect
4. C3×#2043C2semidirect
5. C3×#2042C2semidirect
6. C3×#2041C2semidirect
7. C3×#2013C2semidirect
8. C3×#2012C2semidirect
10. C3×#1983C2semidirect
11. C3×#1831C2semidirect
12. #29800C3direct (N × H)
45. C2×#463C2×C3semidirect
46. #2858C2×C3semidirect
47. #2293C2×C3semidirect
48. #2043C2×C3semidirect
49. #2042C2×C3semidirect
50. #2041C2×C3semidirect
51. #2013C2×C3semidirect
52. #2012C2×C3semidirect
54. #1983C2×C3semidirect
55. #1831C2×C3semidirect
77. C3×C4×D4D4semidirect
238. C4×D4C3×D4semidirect
549. C3#29800direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4632
C3×#28582
C3×#22932
C3×#20432
C3×#20422
C3×#20412
C3×#20132
C3×#20122
C3×#20112
C3×#19832
C3×#18312
#298003