Finite Group Databasebeta

← C3×#29727 #10648334

Subgroups — C3×#29727 #10648334

Order 768.

1124 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#450C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic
3C3×#2620C211yesmaximal, characteristic, complemented
4C3×#2309C211yesmaximal, characteristic, complemented
5C3×#2071C212yesmaximal, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic, complemented
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29727C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#124C2211yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#450—24no—
22C3×#447—24no—
23C2×C3×C4⋊C8C2212yescomplemented
24C2×C3×#103C2212yescomplemented
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#396—22no—
31C3×#394—22no—
32C3×#391C2211yescharacteristic, complemented
33C3×#390—22no—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#390C2211yescharacteristic, complemented
37C3×#384C2211yescharacteristic, complemented
38C3×#379—22no—
39C3×#340C2212yescomplemented
40C3×#340C2212yescomplemented
41C3×#336C2212yescomplemented
42C2×#450C2×C311yescharacteristic, complemented
43C2×#447C2×C311yescharacteristic
44#2620C2×C311yescharacteristic, complemented
45#2309C2×C311yescharacteristic, complemented
46#2071C2×C312yescomplemented
47#2055C2×C312yescomplemented
48#2054C2×C312yescomplemented
49#2012C2×C312yescomplemented
50#1875C2×C311yescharacteristic, complemented
51#1873C2×C311yescharacteristic, complemented
52#1798C2×C311yescharacteristic, complemented
53C25×C3—21no—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—42no—
59C22×C3×D4—42no—
60C22×C3×D4D414yes—
61C22×C3×D4C2311yescharacteristic
62C22×C3×D4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C2×C3×SD16—42no—
67C2×C3×D8—24no—
68C2×C3×D8—42no—
69C2×C3×D8—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×#124—42no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#117—42no—
78C3×#117—42no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×C4⋊C8—42no—
99C3×#103—42no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#102—42no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105C3×#95—22no—
106C23×D4C22×C311yescharacteristic
107C22×SD16C22×C312yes—
108C22×D8C22×C312yes—
109C2×#124C22×C311yescharacteristic, complemented
110C2×C22⋊Q8C22×C311yescharacteristic
111C2×#121C22×C311yescharacteristic
112C2×#117C22×C311yescharacteristic
113C2×#117C22×C311yescharacteristic
114#450—24no—
115#447—24no—
116C2×C4⋊C8C22×C312yescomplemented
117C2×#103C22×C312yescomplemented
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yescomplemented
121C2×C22⋊C8C22×C311yescharacteristic
122C2×C22⋊C8C22×C311yescharacteristic
123#396—22no—
124#394—22no—
125#391C22×C311yescharacteristic, complemented
126#390—22no—
127#390C22×C312yes—
128#390C22×C312yes—
129#390C22×C311yescharacteristic, complemented
130#384C22×C311yescharacteristic, complemented
131#379—22no—
132#340C22×C312yescomplemented
133#340C22×C312yescomplemented
134#336C22×C312yescomplemented
135C24×C3—22no—
136C24×C3—24no—
137C24×C3—21no—
138C24×C3—42no—
139C24×C3—42no—
140C24×C3—82no—
141C24×C3—82no—
142C24×C3C2×D411yescharacteristic
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×D4—24no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—48no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—48no—
161C2×C3×D4—42no—
162C2×C3×D4—82no—
163C2×C3×D4C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C3×SD16—48no—
177C3×D8—48no—
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—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—82no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—82no—
199C3×C22⋊C4—82no—
200C3×C22⋊C4—82no—
201C3×C22⋊C4—82no—
202C3×C22⋊C4—82no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—42no—
206C25—21no—
207C22×Q8C23×C311yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4C3×D414yes—
214C22×D4C23×C311yescharacteristic
215C22×D4C23×C312yes—
216C23×C4C23×C311yescharacteristic
217C2×SD16—24no—
218C2×SD16—42no—
219C2×SD16—42no—
220C2×D8—24no—
221C2×D8—42no—
222C2×D8—42no—
223C22×C8C23×C312yes—
224C22×C8C23×C312yes—
225#124—42no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228#121—22no—
229#121—22no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C3×D412yes—
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C3×D412yes—
248C2×C42—21no—
249C2×C42C23×C311yescharacteristic
250C2×C42C23×C311yescharacteristic
251C4⋊C8—42no—
252#103—42no—
253#102—24no—
254#102—24no—
255#102—42no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258#95—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—24no—
264C23×C3—48no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—84no—
277C23×C3—84no—
278C23×C3—88no—
279C23×C3—88no—
280C23×C3#12112yes—
281C23×C3C22×D411yescharacteristic
282C3×Q8—42no—
283C3×Q8—42no—
284C3×D4—24no—
285C3×D4—24no—
286C3×D4—48no—
287C3×D4—42no—
288C3×D4—42no—
289C3×D4—42no—
290C3×D4—42no—
291C3×D4—84no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4#12112yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)12yes—
316C2×C3×C4H(5,𝔽2)12yes—
317C3×C8—42no—
318C3×C8—42no—
319C24—22no—
320C24—24no—
321C24—21no—
322C24—42no—
323C24—42no—
324C24—82no—
325C24—82no—
326C24C2×C3×D411yescharacteristic
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×D4—24no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—48no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—42no—
346C2×D4—82no—
347C2×D4C2×C3×D414yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4C2×C3×D412yes—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D412yes—
358C22×C4C3×#4112yes—
359C22×C4C24×C311yescharacteristic
360SD16—48no—
361D8—48no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—22no—
388C42—22no—
389C42—42no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—816no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—81no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3C2×#12111yescharacteristic
409C22×C3#53411yescharacteristic
410C22×C3#53511yescharacteristic
411C22×C3#57612yes—
412C22×C3#57712yes—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—81no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4#53412yes—
421C3×C4#53812yes—
422C23—22no—
423C23—22no—
424C23—21no—
425C23—21no—
426C23—24no—
427C23—48no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—84no—
440C23—84no—
441C23—88no—
442C23—88no—
443C23C3×#12112yes—
444C23C22×C3×D411yescharacteristic
445Q8—42no—
446Q8—42no—
447D4—24no—
448D4—24no—
449D4—48no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—84no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4C3×#12112yes—
476C2×C4C22×C3×D411yescharacteristic
477C2×C4C2×C3×#4111yescharacteristic
478C2×C4C3×H(5,𝔽2)12yes—
479C2×C4C3×H(5,𝔽2)12yes—
480C8—42no—
481C8—42no—
482C2×C3—42no—
483C2×C3—84no—
484C2×C3—161no—
485C2×C3#261711yescharacteristic
486C2×C3#304111yescharacteristic
487C2×C3#304211yescharacteristic
488C2×C3#319812yes—
489C2×C3#322412yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—816no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22—81no—
508C22C2×C3×#12111yescharacteristic
509C22C3×#53411yescharacteristic
510C22C3×#53511yescharacteristic
511C22C3×#57612yes—
512C22C3×#57712yes—
513C4—22no—
514C4—42no—
515C4—42no—
516C4—81no—
517C4—82no—
518C4—81no—
519C4—81no—
520C4C3×#53412yes—
521C4C3×#53812yes—
522C3#2972711yescharacteristic, direct product factor, Sylow
523C2—42no—
524C2—84no—
525C2—161no—
526C2C3×#261711yescharacteristic
527C2C3×#304111yescharacteristic
528C2C3×#304211yescharacteristic
529C2C3×#319812yes—
530C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
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×#450C2semidirect
3. C3×#2620C2semidirect
4. C3×#2309C2semidirect
5. C3×#2071C2semidirect
6. C3×#2055C2semidirect
7. C3×#2054C2semidirect
8. C3×#2012C2semidirect
9. C3×#1875C2semidirect
10. C3×#1873C2semidirect
11. C3×#1798C2semidirect
12. #29727C3direct (N × H)
16. C2×C3×#124C22semidirect
23. C2×C3×C4⋊C8C22semidirect
24. C2×C3×#103C22semidirect
27. C2×C3×#102C22semidirect
32. C3×#391C22semidirect
36. C3×#390C22semidirect
37. C3×#384C22semidirect
39. C3×#340C22semidirect
40. C3×#340C22semidirect
41. C3×#336C22semidirect
42. C2×#450C2×C3semidirect
44. #2620C2×C3semidirect
45. #2309C2×C3semidirect
46. #2071C2×C3semidirect
47. #2055C2×C3semidirect
48. #2054C2×C3semidirect
49. #2012C2×C3semidirect
50. #1875C2×C3semidirect
51. #1873C2×C3semidirect
52. #1798C2×C3semidirect
109. C2×#124C22×C3semidirect
116. C2×C4⋊C8C22×C3semidirect
117. C2×#103C22×C3semidirect
120. C2×#102C22×C3semidirect
125. #391C22×C3semidirect
129. #390C22×C3semidirect
130. #384C22×C3semidirect
132. #340C22×C3semidirect
133. #340C22×C3semidirect
134. #336C22×C3semidirect
522. C3#29727direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C2×C3×#4472
C3×#26202
C3×#23092
C3×#20712
C3×#20552
C3×#20542
C3×#20122
C3×#18752
C3×#18732
C3×#17982
#297273