Finite Group Databasebeta

← C3×#29675 #10648282

Subgroups — C3×#29675 #10648282

Order 768.

1202 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic, complemented
2C2×C3×#463C212yesmaximal, complemented
3C3×#2620C211yesmaximal, characteristic, complemented
4C3×#2026C211yesmaximal, characteristic, complemented
5C3×#2025C211yesmaximal, characteristic, complemented
6C3×#2014C211yesmaximal, characteristic, complemented
7C3×#2011C211yesmaximal, characteristic, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1999C212yesmaximal, complemented
10C3×#1802C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29675C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C3×#540—28no—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—22no—
22C2×C3×#121C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×#117C2211yescharacteristic
26C3×#463—28no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#396—22no—
34C3×#394—22no—
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#384C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C2×#540C2×C311yescharacteristic, complemented
43C2×#463C2×C312yescomplemented
44#2620C2×C311yescharacteristic, complemented
45#2026C2×C311yescharacteristic, complemented
46#2025C2×C311yescharacteristic, complemented
47#2014C2×C311yescharacteristic, complemented
48#2011C2×C311yescharacteristic, complemented
49#2000C2×C312yescomplemented
50#1999C2×C312yescomplemented
51#1802C2×C312yescomplemented
52#1798C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2312yes—
59C23×C3×C4—22no—
60C23×C3×C4C2311yescharacteristic
61C2×C3×SD16—24no—
62C2×C3×SD16—42no—
63C2×C3×SD16—42no—
64C2×C3×D8—24no—
65C2×C3×D8—42no—
66C2×C3×D8—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×#127—24no—
70C3×#125—24no—
71C3×#124—24no—
72C3×#123—216no—
73C3×C22⋊Q8—42no—
74C3×#121—216no—
75C3×#121—24no—
76C3×#121—24no—
77C3×#121—42no—
78C3×C4×D4—24no—
79C3×C4×D4D418yescomplemented
80C3×#117—24no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D412yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—24no—
98C3×#103—24no—
99C3×#102—24no—
100C3×#102—42no—
101C3×#102—42no—
102C3×C22⋊C8—42no—
103C3×#95—22no—
104C22×SD16C22×C312yes—
105C22×D8C22×C312yes—
106#540—28no—
107C2×#127C22×C311yescharacteristic
108C2×#125C22×C311yescharacteristic
109C2×#124C22×C311yescharacteristic
110C2×#123—22no—
111C2×C22⋊Q8C22×C311yescharacteristic
112C2×#121—22no—
113C2×#121C22×C312yes—
114C2×#121C22×C311yescharacteristic
115C2×C4×D4C22×C312yes—
116C2×#117C22×C311yescharacteristic
117#463—28no—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122C2×#102C22×C312yes—
123C2×C22⋊C8C22×C312yes—
124#396—22no—
125#394—22no—
126#391C22×C311yescharacteristic
127#390C22×C311yescharacteristic
128#384C22×C311yescharacteristic
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C311yescharacteristic
132#336C22×C311yescharacteristic
133C24×C3—22no—
134C24×C3—21no—
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×D4—28no—
141C2×C3×D4—28no—
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—21no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—42no—
152C2×C3×D4—82no—
153C2×C3×D4—82no—
154C2×C3×D4—82no—
155C2×C3×D4—82no—
156C2×C3×D4—82no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—28no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—82no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C3×SD16—48no—
174C3×D8—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—82no—
188C3×C4⋊C4C2×D414yes—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—28no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—82no—
196C3×C42—22no—
197C3×C42C2×D414yes—
198C22×Q8C23×C311yescharacteristic
199C22×D4—22no—
200C22×D4—21no—
201C22×D4—22no—
202C22×D4C23×C311yescharacteristic
203C22×D4C23×C312yes—
204C23×C4—22no—
205C23×C4C23×C311yescharacteristic
206C2×SD16—24no—
207C2×SD16—42no—
208C2×SD16—42no—
209C2×D8—24no—
210C2×D8—42no—
211C2×D8—42no—
212C22×C8C23×C312yes—
213C22×C8C23×C312yes—
214#127—24no—
215#125—24no—
216#124—24no—
217#123—216no—
218C22⋊Q8—42no—
219#121—216no—
220#121—24no—
221#121—24no—
222#121—42no—
223C4×D4—24no—
224C4×D4C3×D418yescomplemented
225#117—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C3×D412yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C312yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C3×D412yes—
240C2×C42C23×C311yescharacteristic
241C2×C42C23×C311yescharacteristic
242C4⋊C8—24no—
243#103—24no—
244#102—24no—
245#102—42no—
246#102—42no—
247C22⋊C8—42no—
248#95—22no—
249C23×C3—21no—
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3C22×D411yescharacteristic
267C3×Q8—42no—
268C3×Q8—42no—
269C3×D4—24no—
270C3×D4—24no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—88no—
276C3×D4—88no—
277C3×D4—84no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—88no—
299C2×C3×C4#12014yes—
300C2×C3×C4C22×D412yes—
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C4H(5,𝔽2)12yes—
303C2×C3×C42−1+411yescharacteristic
304C3×C8—42no—
305C3×C8—42no—
306C24—22no—
307C24—21no—
308C24C2×C3×D411yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—42no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4C2×C3×D414yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—28no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D412yes—
345C22×C4C24×C311yescharacteristic
346SD16—48no—
347D8—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—82no—
361C4⋊C4C2×C3×D414yes—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—82no—
369C42—22no—
370C42C2×C3×D414yes—
371C22×C3—41no—
372C22×C3—44no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—42no—
376C22×C3—44no—
377C22×C3—42no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3C2×#12011yescharacteristic
389C22×C3#53511yescharacteristic
390C22×C3#53711yescharacteristic
391C22×C3#57612yes—
392C22×C3#57712yes—
393C3×C4—24no—
394C3×C4—41no—
395C3×C4—42no—
396C3×C4—44no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4#53512yes—
401C3×C4#53712yes—
402C23—22no—
403C23—22no—
404C23—21no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C22×C3×D411yescharacteristic
420Q8—42no—
421Q8—42no—
422D4—24no—
423D4—24no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—88no—
429D4—88no—
430D4—84no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—88no—
452C2×C4C3×#12014yes—
453C2×C4C22×C3×D412yes—
454C2×C4C3×H(5,𝔽2)11yescharacteristic
455C2×C4C3×H(5,𝔽2)12yes—
456C2×C4C3×2−1+411yescharacteristic
457C8—42no—
458C8—42no—
459C2×C3—81no—
460C2×C3—84no—
461C2×C3—161no—
462C2×C3#261511yescharacteristic
463C2×C3#301212yes—
464C2×C3#322011yescharacteristic
465C2×C3#322111yescharacteristic
466C2×C3#325712yes—
467C22—41no—
468C22—41no—
469C22—42no—
470C22—44no—
471C22—42no—
472C22—44no—
473C22—41no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22—82no—
482C22—82no—
483C22—81no—
484C22C2×C3×#12011yescharacteristic
485C22C3×#53511yescharacteristic
486C22C3×#53711yescharacteristic
487C22C3×#57612yes—
488C22C3×#57712yes—
489C4—24no—
490C4—42no—
491C4—44no—
492C4—41no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4C3×#53512yes—
497C4C3×#53712yes—
498C3#2967511yescharacteristic, direct product factor, Sylow
499C2—81no—
500C2—84no—
501C2—161no—
502C2C3×#261511yescharacteristic
503C2C3×#301212yes—
504C2C3×#322011yescharacteristic
505C2C3×#322111yescharacteristic
506C2C3×#325712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#540C2semidirect
2. C2×C3×#463C2semidirect
3. C3×#2620C2semidirect
4. C3×#2026C2semidirect
5. C3×#2025C2semidirect
6. C3×#2014C2semidirect
7. C3×#2011C2semidirect
8. C3×#2000C2semidirect
9. C3×#1999C2semidirect
10. C3×#1802C2semidirect
11. C3×#1798C2semidirect
12. #29675C3direct (N × H)
42. C2×#540C2×C3semidirect
43. C2×#463C2×C3semidirect
44. #2620C2×C3semidirect
45. #2026C2×C3semidirect
46. #2025C2×C3semidirect
47. #2014C2×C3semidirect
48. #2011C2×C3semidirect
49. #2000C2×C3semidirect
50. #1999C2×C3semidirect
51. #1802C2×C3semidirect
52. #1798C2×C3semidirect
79. C3×C4×D4D4semidirect
224. C4×D4C3×D4semidirect
498. C3#29675direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4632
C3×#26202
C3×#20262
C3×#20252
C3×#20142
C3×#20112
C3×#20002
C3×#19992
C3×#18022
C3×#17982
#296753