Finite Group Databasebeta

← C3×#24629 #10643236

Subgroups — C3×#24629 #10643236

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#462C211yesmaximal, characteristic, complemented
2C2×C3×C4×SD16C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic
4C3×#2355C211yesmaximal, characteristic, complemented
5C3×#2056C212yesmaximal, complemented
6C3×#2015C212yesmaximal
7C3×#1983C212yesmaximal
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1927C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic
12#24629C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#462—24no—
20C3×C4×SD16—24no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#397—22no—
32C3×#391C2212yes—
33C3×#391C2212yes—
34C3×#387—22no—
35C3×#384C2211yescharacteristic
36C3×#382—21no—
37C3×#380—21no—
38C3×#380—21no—
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#462C2×C311yescharacteristic, complemented
45C2×C4×SD16C2×C311yescharacteristic
46#2547C2×C311yescharacteristic
47#2355C2×C311yescharacteristic, complemented
48#2056C2×C312yescomplemented
49#2015C2×C312yes—
50#1983C2×C312yes—
51#1956C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1927C2×C312yescomplemented
54#1802C2×C311yescharacteristic
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×Q16—24no—
60C2×C3×Q16—42no—
61C2×C3×Q16—42no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×#124—24no—
71C3×#124—42no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42—22no—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C3×#106—22no—
94C3×C4⋊C8—22no—
95C3×C4⋊C8—42no—
96C3×#103—22no—
97C3×#103—24no—
98C3×#103—24no—
99C3×#103—42no—
100C3×#102—22no—
101C3×#102—42no—
102C3×C4×C8—22no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95D412yes—
109C22×Q16C22×C312yes—
110C22×SD16C22×C311yescharacteristic
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×D4C22×C311yescharacteristic
115#462—24no—
116C4×SD16—24no—
117C2×#106C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C311yescharacteristic
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#102C22×C311yescharacteristic
125C2×#102C22×C312yes—
126C2×C4×C8C22×C311yescharacteristic
127#397—22no—
128#391C22×C312yes—
129#391C22×C312yes—
130#387—22no—
131#384C22×C311yescharacteristic
132#382—21no—
133#380—21no—
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C22×C312yes—
137C4×C4⋊C4C22×C311yescharacteristic
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4#4112yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C3×Q16—48no—
173C3×SD16—28no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—22no—
182C2×C3×C8#4112yes—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—81no—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—82no—
203C3×C42C2×D412yes—
204C3×C42#4112yes—
205C22×Q8C23×C312yes—
206C22×Q8C23×C311yescharacteristic
207C22×D4C23×C311yescharacteristic
208C23×C4—21no—
209C2×Q16—24no—
210C2×Q16—42no—
211C2×Q16—42no—
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16—22no—
215C22×C8C23×C312yes—
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C311yescharacteristic
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220#124—24no—
221#124—42no—
222C4×Q8—22no—
223C4×Q8C3×D414yes—
224C4×D4—22no—
225C4×D4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C3×D411yescharacteristic
232C2×C4⋊C4C3×D411yescharacteristic
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42C23×C312yes—
242C2×C42C23×C311yescharacteristic
243#106—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246#103—22no—
247#103—24no—
248#103—24no—
249#103—42no—
250#102—22no—
251#102—42no—
252C4×C8—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95C3×D412yes—
259C23×C3—22no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3C22×D411yescharacteristic
267C3×Q8—22no—
268C3×Q8—22no—
269C3×Q8—41no—
270C3×Q8—41no—
271C3×Q8—42no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—41no—
278C3×D4—41no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—84no—
303C2×C3×C4#12112yes—
304C2×C3×C4#12112yes—
305C2×C3×C4#13512yes—
306C2×C3×C4#13712yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C3×C8—22no—
314C3×C8—41no—
315C3×C8—42no—
316C24—21no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8C2×C3×D412yes—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4C3×#4112yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C24×C311yescharacteristic
348Q16—48no—
349SD16—28no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8C3×#4112yes—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—81no—
371C4⋊C4C2×C3×D412yes—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C42—24no—
376C42—22no—
377C42—22no—
378C42—82no—
379C42C2×C3×D412yes—
380C42C3×#4112yes—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3C2×#12111yescharacteristic
389C22×C3#54611yescharacteristic
390C22×C3#55011yescharacteristic
391C22×C3C2×#13511yescharacteristic
392C22×C3C2×#13711yescharacteristic
393C22×C3#57711yescharacteristic
394C22×C3#57811yescharacteristic
395C3×C4—22no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—82no—
401C3×C4—82no—
402C3×C4—81no—
403C3×C4#46414yes—
404C3×C4#52511yescharacteristic
405C3×C4#53511yescharacteristic
406C3×C4#55311yescharacteristic
407C3×C4#55511yescharacteristic
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23C22×C3×D411yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424D4—22no—
425D4—22no—
426D4—41no—
427D4—41no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4C3×#12112yes—
453C2×C4C3×#12112yes—
454C2×C4C3×#13512yes—
455C2×C4C3×#13712yes—
456C2×C4C22×C3×D411yescharacteristic
457C2×C4C2×C3×#4111yescharacteristic
458C2×C4C2×C3×#4111yescharacteristic
459C2×C4C2×C3×#4111yescharacteristic
460C2×C4C3×H(5,𝔽2)11yescharacteristic
461C2×C4C3×H(5,𝔽2)11yescharacteristic
462C8—22no—
463C8—42no—
464C8—41no—
465C2×C3—82no—
466C2×C3#241411yescharacteristic
467C2×C3C2×#46411yescharacteristic
468C2×C3#304611yescharacteristic
469C2×C3#330211yescharacteristic
470C2×C3#333111yescharacteristic
471C2×C3#339411yescharacteristic
472C2×C3#339811yescharacteristic
473C22—42no—
474C22—42no—
475C22—42no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22C2×C3×#12111yescharacteristic
481C22C3×#54611yescharacteristic
482C22C3×#55011yescharacteristic
483C22C2×C3×#13511yescharacteristic
484C22C2×C3×#13711yescharacteristic
485C22C3×#57711yescharacteristic
486C22C3×#57811yescharacteristic
487C4—22no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—82no—
493C4—82no—
494C4—81no—
495C4C3×#46414yes—
496C4C3×#52511yescharacteristic
497C4C3×#53511yescharacteristic
498C4C3×#55311yescharacteristic
499C4C3×#55511yescharacteristic
500C3#2462911yescharacteristic, direct product factor, Sylow
501C2—82no—
502C2C3×#241411yescharacteristic
503C2C2×C3×#46411yescharacteristic
504C2C3×#304611yescharacteristic
505C2C3×#330211yescharacteristic
506C2C3×#333111yescharacteristic
507C2C3×#339411yescharacteristic
508C2C3×#339811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#462C2semidirect
4. C3×#2355C2semidirect
5. C3×#2056C2semidirect
8. C3×#1956C2semidirect
9. C3×#1939C2semidirect
10. C3×#1927C2semidirect
12. #24629C3direct (N × H)
44. C2×#462C2×C3semidirect
47. #2355C2×C3semidirect
48. #2056C2×C3semidirect
51. #1956C2×C3semidirect
52. #1939C2×C3semidirect
53. #1927C2×C3semidirect
500. C3#24629direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×C3×C4×SD162
C3×#25472
C3×#23552
C3×#20562
C3×#20152
C3×#19832
C3×#19562
C3×#19392
C3×#19272
C3×#18022
#246293