Finite Group Databasebeta

← C3×#29801 #10648408

Subgroups — C3×#29801 #10648408

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#464C212yesmaximal, complemented
2C3×#2858C211yesmaximal, characteristic
3C3×#2295C211yesmaximal, characteristic, complemented
4C3×#2043C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C211yesmaximal, characteristic
7C3×#2016C211yesmaximal, characteristic, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#2014C211yesmaximal, characteristic
10C3×#1983C212yesmaximal, complemented
11C3×#1831C211yesmaximal, characteristic, complemented
12#29801C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C3×#464—28no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C3×#391C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#390—21no—
31C3×#390—21no—
32C3×#390—21no—
33C3×#390—21no—
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#387—21no—
38C3×C4×C4⋊C4—21no—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#376—21no—
41C3×C43C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44C3×#336C2211yescharacteristic
45C2×#464C2×C312yescomplemented
46#2858C2×C311yescharacteristic
47#2295C2×C311yescharacteristic, complemented
48#2043C2×C311yescharacteristic, complemented
49#2042C2×C312yescomplemented
50#2041C2×C311yescharacteristic
51#2016C2×C311yescharacteristic, complemented
52#2015C2×C312yescomplemented
53#2014C2×C311yescharacteristic
54#1983C2×C312yescomplemented
55#1831C2×C311yescharacteristic, complemented
56C22×C3×Q8C2312yes—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C2×C3×Q16—24no—
62C2×C3×Q16—42no—
63C2×C3×Q16—42no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C2×C3×SD16—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—42no—
70C3×#127—42no—
71C3×#124—24no—
72C3×#124—24no—
73C3×#124—42no—
74C3×#124—42no—
75C3×C4×Q8—22no—
76C3×C4×Q8D414yescomplemented
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
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×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×C4⋊C8—24no—
99C3×#103—24no—
100C3×#103—42no—
101C3×#103—42no—
102C3×#102—24no—
103C3×#102—42no—
104C3×#102—42no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C22×Q16C22×C312yes—
113C22×SD16C22×C312yes—
114C2×C4⋊Q8C22×C311yescharacteristic
115C2×#127C22×C311yescharacteristic
116C2×#124C22×C312yes—
117C2×#124C22×C312yes—
118C2×C4×Q8C22×C311yescharacteristic
119#464—28no—
120C2×C4⋊C8C22×C312yes—
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#102C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127#391C22×C311yescharacteristic
128#391C22×C311yescharacteristic
129#390—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#389C22×C311yescharacteristic
134#389C22×C311yescharacteristic
135#387—21no—
136#387—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C22×C311yescharacteristic
139#376—21no—
140C43C22×C311yescharacteristic
141#336C22×C311yescharacteristic
142#336C22×C312yes—
143#336C22×C311yescharacteristic
144C24×C3—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—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×Q8—22no—
154C2×C3×Q8C2×D412yes—
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—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C3×Q16—48no—
180C3×SD16—48no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—41no—
206C3×C42—41no—
207C3×C42—42no—
208C3×C42C2×D412yes—
209C3×C42#4112yes—
210C22×Q8C23×C312yes—
211C22×Q8C23×C311yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C23×C312yes—
215C2×Q16—24no—
216C2×Q16—42no—
217C2×Q16—42no—
218C2×SD16—24no—
219C2×SD16—42no—
220C2×SD16—42no—
221C22×C8C23×C312yes—
222C22×C8C23×C312yes—
223C4⋊Q8—42no—
224#127—42no—
225#124—24no—
226#124—24no—
227#124—42no—
228#124—42no—
229C4×Q8—22no—
230C4×Q8C3×D414yescomplemented
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C3×D411yescharacteristic
246C2×C42C3×D411yescharacteristic
247C2×C42C3×D411yescharacteristic
248C2×C42C3×D411yescharacteristic
249C2×C42C23×C311yescharacteristic
250C2×C42C23×C311yescharacteristic
251C2×C42C23×C311yescharacteristic
252C4⋊C8—24no—
253#103—24no—
254#103—42no—
255#103—42no—
256#102—24no—
257#102—42no—
258#102—42no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C22×D411yescharacteristic
274C3×Q8—24no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×Q8—42no—
279C3×Q8—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—84no—
285C3×D4—84no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4#12112yes—
316C2×C3×C4#12112yes—
317C2×C3×C4#13512yes—
318C2×C3×C4#13712yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)12yes—
322C2×C3×C4H(5,𝔽2)12yes—
323C3×C8—42no—
324C3×C8—42no—
325C24—22no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×C3×D412yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C24×C311yescharacteristic
360Q16—48no—
361SD16—48no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4C2×C3×D412yes—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—42no—
389C42C2×C3×D412yes—
390C42C3×#4112yes—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3C2×#12111yescharacteristic
399C22×C3#53511yescharacteristic
400C22×C3#53511yescharacteristic
401C22×C3C2×#13511yescharacteristic
402C22×C3C2×#13711yescharacteristic
403C22×C3#57511yescharacteristic
404C22×C3#57511yescharacteristic
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—42no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4#53512yes—
417C3×C4#54012yes—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23C22×C3×D411yescharacteristic
426Q8—24no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—84no—
437D4—84no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4C3×#12112yes—
468C2×C4C3×#12112yes—
469C2×C4C3×#13512yes—
470C2×C4C3×#13712yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C2×C3×#4111yescharacteristic
473C2×C4C3×H(5,𝔽2)12yes—
474C2×C4C3×H(5,𝔽2)12yes—
475C8—42no—
476C8—42no—
477C2×C3—162no—
478C2×C3#261911yescharacteristic
479C2×C3#303511yescharacteristic
480C2×C3#303711yescharacteristic
481C2×C3#321011yescharacteristic
482C2×C3#321111yescharacteristic
483C2×C3#321511yescharacteristic
484C2×C3#321911yescharacteristic
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22C2×C3×#12111yescharacteristic
493C22C3×#53511yescharacteristic
494C22C3×#53511yescharacteristic
495C22C2×C3×#13511yescharacteristic
496C22C2×C3×#13711yescharacteristic
497C22C3×#57511yescharacteristic
498C22C3×#57511yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—41no—
505C4—41no—
506C4—42no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4C3×#53512yes—
511C4C3×#54012yes—
512C3#2980111yescharacteristic, direct product factor, Sylow
513C2—162no—
514C2C3×#261911yescharacteristic
515C2C3×#303511yescharacteristic
516C2C3×#303711yescharacteristic
517C2C3×#321011yescharacteristic
518C2C3×#321111yescharacteristic
519C2C3×#321511yescharacteristic
520C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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×#464C2semidirect
3. C3×#2295C2semidirect
4. C3×#2043C2semidirect
5. C3×#2042C2semidirect
7. C3×#2016C2semidirect
8. C3×#2015C2semidirect
10. C3×#1983C2semidirect
11. C3×#1831C2semidirect
12. #29801C3direct (N × H)
45. C2×#464C2×C3semidirect
47. #2295C2×C3semidirect
48. #2043C2×C3semidirect
49. #2042C2×C3semidirect
51. #2016C2×C3semidirect
52. #2015C2×C3semidirect
54. #1983C2×C3semidirect
55. #1831C2×C3semidirect
76. C3×C4×Q8D4semidirect
230. C4×Q8C3×D4semidirect
512. C3#29801direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4642
C3×#28582
C3×#22952
C3×#20432
C3×#20422
C3×#20412
C3×#20162
C3×#20152
C3×#20142
C3×#19832
C3×#18312
#298013