Finite Group Databasebeta

← C3×#29741 #10648348

Subgroups — C3×#29741 #10648348

Order 768.

1024 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic, complemented
2C2×C3×#408C211yesmaximal, characteristic, complemented
3C3×#2616C211yesmaximal, characteristic, complemented
4C3×#2615C211yesmaximal, characteristic
5C3×#2077C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2034C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1895C211yesmaximal, characteristic, complemented
10C3×#1894C211yesmaximal, characteristic, complemented
11C3×#1807C211yesmaximal, characteristic, complemented
12#29741C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2212yes—
21C3×#465—24no—
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2212yes—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#408—24no—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#395—21no—
32C3×#394—22no—
33C3×#394—21no—
34C3×#393—21no—
35C3×#392—22no—
36C3×#392—21no—
37C3×#380C2212yes—
38C3×#380C2211yescharacteristic
39C3×#380C2212yes—
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#465C2×C311yescharacteristic, complemented
45C2×#408C2×C311yescharacteristic, complemented
46#2616C2×C311yescharacteristic, complemented
47#2615C2×C311yescharacteristic
48#2077C2×C312yescomplemented
49#2035C2×C312yescomplemented
50#2034C2×C312yescomplemented
51#2000C2×C312yescomplemented
52#1895C2×C311yescharacteristic, complemented
53#1894C2×C311yescharacteristic, complemented
54#1807C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—42no—
58C2×C3×#41—42no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4D412yes—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C2×C3×SD16—22no—
69C2×C3×SD16—42no—
70C2×C3×M16—22no—
71C2×C3×M16—42no—
72C22×C3×C8—42no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2312yes—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—42no—
80C3×C22⋊Q8—42no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—42no—
84C3×#121—42no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C3×#106—22no—
100C3×#106—42no—
101C3×#103—22no—
102C3×#103—42no—
103C3×#102—22no—
104C3×#102—42no—
105C3×C22⋊C8—24no—
106C3×C22⋊C8—24no—
107C3×#95—22no—
108C3×#95—21no—
109C3×#95—22no—
110C3×#95—21no—
111C22×#41C22×C311yescharacteristic
112C22×SD16C22×C311yescharacteristic
113C22×M16C22×C311yescharacteristic
114C23×C8C22×C311yescharacteristic
115C2×C22⋊Q8C22×C311yescharacteristic
116C2×C22⋊Q8C22×C312yes—
117C2×#121C22×C311yescharacteristic
118C2×#121C22×C312yes—
119#465—24no—
120C2×#106C22×C311yescharacteristic
121C2×#106C22×C312yes—
122C2×#103C22×C311yescharacteristic
123C2×#103C22×C312yes—
124C2×#102C22×C311yescharacteristic
125C2×#102C22×C312yes—
126#408—24no—
127C2×C22⋊C8C22×C312yes—
128C2×C22⋊C8C22×C312yes—
129#395—21no—
130#394—22no—
131#394—21no—
132#393—21no—
133#392—22no—
134#392—21no—
135#380C22×C311yescharacteristic
136#380C22×C311yescharacteristic
137#380C22×C312yes—
138#380C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142C24×C3—21no—
143C24×C3C2×D412yes—
144C24×C3C2×D411yescharacteristic
145C3×#41—44no—
146C3×#41—44no—
147C3×#41—48no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—82no—
166C2×C3×D4—82no—
167C2×C3×D4—82no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4#4112yes—
185C22×C3×C4C2411yescharacteristic
186C3×SD16—44no—
187C3×M16—44no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—44no—
194C2×C3×C8C2×D412yes—
195C2×C3×C8C2×D412yes—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—42no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—44no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—81no—
211C3×C22⋊C4—82no—
212C3×C22⋊C4—81no—
213C3×C22⋊C4—82no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C311yescharacteristic
220C22×D4—22no—
221C22×D4—21no—
222C22×D4C3×D412yes—
223C22×D4C3×D411yescharacteristic
224C22×D4C23×C311yescharacteristic
225C23×C4C23×C312yes—
226C23×C4C23×C311yescharacteristic
227C2×SD16—22no—
228C2×SD16—42no—
229C2×M16—22no—
230C2×M16—42no—
231C22×C8—42no—
232C22×C8C3×D414yescomplemented
233C22×C8C23×C312yes—
234C22×C8C23×C311yescharacteristic
235C22×C8C23×C311yescharacteristic
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—42no—
239C22⋊Q8—42no—
240#121—22no—
241#121—22no—
242#121—42no—
243#121—42no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C312yes—
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C312yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258#106—22no—
259#106—42no—
260#103—22no—
261#103—42no—
262#102—22no—
263#102—42no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95—21no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3#12112yes—
287C23×C3C22×D411yescharacteristic
288C3×Q8—44no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—42no—
292C3×D4—44no—
293C3×D4—48no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—42no—
297C3×D4—84no—
298C3×D4—88no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4#12112yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)12yes—
327C2×C3×C4H(5,𝔽2)12yes—
328C3×C8—22no—
329C3×C8—22no—
330C3×C8—42no—
331C24—21no—
332C24C2×C3×D412yes—
333C24C2×C3×D411yescharacteristic
334#41—44no—
335#41—44no—
336#41—48no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D412yes—
373C22×C4C3×#4112yes—
374C22×C4C24×C311yescharacteristic
375SD16—44no—
376M16—44no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—44no—
383C2×C8C2×C3×D412yes—
384C2×C8C2×C3×D412yes—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—81no—
400C22⋊C4—82no—
401C22⋊C4—81no—
402C22⋊C4—82no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—81no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3C2×#12111yescharacteristic
421C22×C3#53412yes—
422C22×C3#57711yescharacteristic
423C22×C3#57712yes—
424C22×C3#57711yescharacteristic
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—81no—
428C3×C4—82no—
429C3×C4—82no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4#53411yescharacteristic
433C3×C4#53511yescharacteristic
434C3×C4#53812yes—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23C3×#12112yes—
452C23C22×C3×D411yescharacteristic
453Q8—44no—
454Q8—41no—
455Q8—41no—
456Q8—42no—
457D4—44no—
458D4—48no—
459D4—41no—
460D4—41no—
461D4—42no—
462D4—84no—
463D4—88no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4C3×#12112yes—
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C2×C3×#4111yescharacteristic
491C2×C4C3×H(5,𝔽2)12yes—
492C2×C4C3×H(5,𝔽2)12yes—
493C8—22no—
494C8—22no—
495C8—42no—
496C2×C3—42no—
497C2×C3—82no—
498C2×C3—161no—
499C2×C3#261711yescharacteristic
500C2×C3#305711yescharacteristic
501C2×C3#306011yescharacteristic
502C2×C3#320012yes—
503C2×C3#320412yes—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22C2×C3×#12111yescharacteristic
522C22C3×#53412yes—
523C22C3×#57711yescharacteristic
524C22C3×#57712yes—
525C22C3×#57711yescharacteristic
526C4—22no—
527C4—42no—
528C4—81no—
529C4—82no—
530C4—82no—
531C4—81no—
532C4—81no—
533C4C3×#53411yescharacteristic
534C4C3×#53511yescharacteristic
535C4C3×#53812yes—
536C3#2974111yescharacteristic, direct product factor, Sylow
537C2—42no—
538C2—82no—
539C2—161no—
540C2C3×#261711yescharacteristic
541C2C3×#305711yescharacteristic
542C2C3×#306011yescharacteristic
543C2C3×#320012yes—
544C2C3×#320412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#465C2semidirect
2. C2×C3×#408C2semidirect
3. C3×#2616C2semidirect
5. C3×#2077C2semidirect
6. C3×#2035C2semidirect
7. C3×#2034C2semidirect
8. C3×#2000C2semidirect
9. C3×#1895C2semidirect
10. C3×#1894C2semidirect
11. C3×#1807C2semidirect
12. #29741C3direct (N × H)
44. C2×#465C2×C3semidirect
45. C2×#408C2×C3semidirect
46. #2616C2×C3semidirect
48. #2077C2×C3semidirect
49. #2035C2×C3semidirect
50. #2034C2×C3semidirect
51. #2000C2×C3semidirect
52. #1895C2×C3semidirect
53. #1894C2×C3semidirect
54. #1807C2×C3semidirect
73. C22×C3×C8D4semidirect
232. C22×C8C3×D4semidirect
536. C3#29741direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4652
C2×C3×#4082
C3×#26162
C3×#26152
C3×#20772
C3×#20352
C3×#20342
C3×#20002
C3×#18952
C3×#18942
C3×#18072
#297413