Finite Group Databasebeta

← C3×#29758 #10648365

Subgroups — C3×#29758 #10648365

Order 768.

952 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#482C212yesmaximal, complemented
2C3×#2856C211yesmaximal, characteristic
3C3×#2296C211yesmaximal, characteristic, complemented
4C3×#2084C211yesmaximal, characteristic, complemented
5C3×#2083C211yesmaximal, characteristic, complemented
6C3×#2067C212yesmaximal, complemented
7C3×#2065C212yesmaximal, complemented
8C3×#2041C211yesmaximal, characteristic, complemented
9C3×#2041C211yesmaximal, characteristic, complemented
10C3×#1875C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29758C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#121—21no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#482—28no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#390—21no—
30C3×#390—21no—
31C3×#390C2212yes—
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#385—21no—
37C3×#385—21no—
38C3×#384C2211yescharacteristic
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C2×#482C2×C312yescomplemented
46#2856C2×C311yescharacteristic
47#2296C2×C311yescharacteristic, complemented
48#2084C2×C311yescharacteristic, complemented
49#2083C2×C311yescharacteristic, complemented
50#2067C2×C312yescomplemented
51#2065C2×C312yescomplemented
52#2041C2×C311yescharacteristic, complemented
53#2041C2×C311yescharacteristic, complemented
54#1875C2×C312yescomplemented
55#1797C2×C311yescharacteristic, complemented
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#127—42no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—44no—
69C3×#121—44no—
70C3×#117—22no—
71C3×#117—42no—
72C3×#117D414yescomplemented
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42—42no—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×#107—24no—
97C3×#106—24no—
98C3×C4⋊C8—42no—
99C3×#102—24no—
100C3×#102—24no—
101C3×#102—42no—
102C3×#102—42no—
103C3×C22⋊C8—24no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C2×#127C22×C311yescharacteristic
109C2×#121—21no—
110C2×#121—21no—
111C2×#121C22×C312yes—
112C2×#117C22×C311yescharacteristic
113C2×#117C22×C311yescharacteristic
114C22×C42C22×C311yescharacteristic
115#482—28no—
116C2×#107C22×C312yes—
117C2×#106C22×C312yes—
118C2×C4⋊C8C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122C2×#102C22×C312yes—
123C2×C22⋊C8C22×C312yes—
124#390—21no—
125#390—21no—
126#390C22×C312yes—
127#389C22×C311yescharacteristic
128#389C22×C311yescharacteristic
129#389C22×C311yescharacteristic
130#389C22×C311yescharacteristic
131#385—21no—
132#385—21no—
133#384C22×C311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136#340C22×C312yes—
137#340C22×C312yes—
138#336C22×C311yescharacteristic
139#336C22×C311yescharacteristic
140C24×C3—22no—
141C24×C3C2×D411yescharacteristic
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—44no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4C2×D412yes—
197C3×C4⋊C4C2×D412yes—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—44no—
203C3×C42—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—41no—
207C3×C42—41no—
208C3×C42—42no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C23×C312yes—
213C23×C4C3×D411yescharacteristic
214C23×C4C3×D411yescharacteristic
215C23×C4C23×C311yescharacteristic
216C22×C8C23×C312yes—
217C22×C8C23×C312yes—
218#127—42no—
219#121—24no—
220#121—24no—
221#121—44no—
222#121—44no—
223#117—22no—
224#117—42no—
225#117C3×D414yescomplemented
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—42no—
245C2×C42C3×D411yescharacteristic
246C2×C42C3×D411yescharacteristic
247C2×C42C23×C311yescharacteristic
248C2×C42C23×C311yescharacteristic
249#107—24no—
250#106—24no—
251C4⋊C8—42no—
252#102—24no—
253#102—24no—
254#102—42no—
255#102—42no—
256C22⋊C8—24no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3#12112yes—
275C23×C3C22×D411yescharacteristic
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—84no—
281C3×D4—84no—
282C3×D4—88no—
283C2×C3×C4—21no—
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—21no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—44no—
299C2×C3×C4—44no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4#12112yes—
314C2×C3×C4#13512yes—
315C2×C3×C4AGL(1,ℤ/8)12yes—
316C2×C3×C4AGL(1,ℤ/8)12yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)12yes—
320C2×C3×C42−1+412yes—
321C3×C8—42no—
322C3×C8—42no—
323C24—22no—
324C24C2×C3×D411yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C3×#4112yes—
360C22×C4C24×C311yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4C2×C3×D412yes—
380C4⋊C4C2×C3×D412yes—
381C22⋊C4—24no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C42—24no—
387C42—22no—
388C42—22no—
389C42—41no—
390C42—41no—
391C42—42no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3C2×#12111yescharacteristic
407C22×C3#53411yescharacteristic
408C22×C3#53611yescharacteristic
409C22×C3C2×#13511yescharacteristic
410C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
412C22×C3#57511yescharacteristic
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4—82no—
423C3×C4#53712yes—
424C3×C4#53912yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23C3×#12112yes—
439C23C22×C3×D411yescharacteristic
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—84no—
445D4—84no—
446D4—88no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4C3×#12112yes—
478C2×C4C3×#13512yes—
479C2×C4C3×AGL(1,ℤ/8)12yes—
480C2×C4C3×AGL(1,ℤ/8)12yes—
481C2×C4C22×C3×D411yescharacteristic
482C2×C4C2×C3×#4111yescharacteristic
483C2×C4C3×H(5,𝔽2)12yes—
484C2×C4C3×2−1+412yes—
485C8—42no—
486C8—42no—
487C2×C3—42no—
488C2×C3—162no—
489C2×C3#261811yescharacteristic
490C2×C3#303511yescharacteristic
491C2×C3#303811yescharacteristic
492C2×C3#318811yescharacteristic
493C2×C3#319411yescharacteristic
494C2×C3#322811yescharacteristic
495C2×C3#323511yescharacteristic
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22C2×C3×#12111yescharacteristic
511C22C3×#53411yescharacteristic
512C22C3×#53611yescharacteristic
513C22C2×C3×#13511yescharacteristic
514C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
515C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
516C22C3×#57511yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—82no—
527C4C3×#53712yes—
528C4C3×#53912yes—
529C3#2975811yescharacteristic, direct product factor, Sylow
530C2—42no—
531C2—162no—
532C2C3×#261811yescharacteristic
533C2C3×#303511yescharacteristic
534C2C3×#303811yescharacteristic
535C2C3×#318811yescharacteristic
536C2C3×#319411yescharacteristic
537C2C3×#322811yescharacteristic
538C2C3×#323511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
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×#482C2semidirect
3. C3×#2296C2semidirect
4. C3×#2084C2semidirect
5. C3×#2083C2semidirect
6. C3×#2067C2semidirect
7. C3×#2065C2semidirect
8. C3×#2041C2semidirect
9. C3×#2041C2semidirect
10. C3×#1875C2semidirect
11. C3×#1797C2semidirect
12. #29758C3direct (N × H)
45. C2×#482C2×C3semidirect
47. #2296C2×C3semidirect
48. #2084C2×C3semidirect
49. #2083C2×C3semidirect
50. #2067C2×C3semidirect
51. #2065C2×C3semidirect
52. #2041C2×C3semidirect
53. #2041C2×C3semidirect
54. #1875C2×C3semidirect
55. #1797C2×C3semidirect
72. C3×#117D4semidirect
225. #117C3×D4semidirect
529. C3#29758direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4822
C3×#28562
C3×#22962
C3×#20842
C3×#20832
C3×#20672
C3×#20652
C3×#20412
C3×#18752
C3×#17972
#297583