Finite Group Databasebeta

← C3×#25662 #10644269

Subgroups — C3×#25662 #10644269

Order 768.

1028 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic
9C3×#1865C212yesmaximal, complemented
10C3×#1864C212yesmaximal
11C3×#1763C211yesmaximal, characteristic, complemented
12#25662C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#126—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C22⋊Q8—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C3×#489—24no—
28C3×#475—24no—
29C2×C3×#106C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#395—21no—
39C3×#395—21no—
40C3×#391—21no—
41C3×#389C2212yes—
42C3×#389C2212yes—
43C3×#384—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#556C2×C311yescharacteristic
49C2×#489C2×C311yescharacteristic
50C2×#475C2×C311yescharacteristic, complemented
51#2567C2×C311yescharacteristic, complemented
52#2043C2×C312yescomplemented
53#2042C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic
56#1865C2×C312yescomplemented
57#1864C2×C312yes—
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#126—28no—
72C3×#126—28no—
73C3×#124—22no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#124D414yes—
77C3×C22⋊Q8—28no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—28no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#106—24no—
112C3×C4⋊C8—22no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#103—24no—
116C3×#103—24no—
117C3×#102—24no—
118C3×#102—24no—
119C3×#97—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#556—24no—
124#556—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#126—21no—
129C2×#124C22×C311yescharacteristic
130C2×#124C22×C311yescharacteristic
131C2×C22⋊Q8—21no—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117—21no—
137#489—24no—
138#475—24no—
139C2×#106C22×C312yes—
140C2×C4⋊C8C22×C311yescharacteristic
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×#97C22×C311yescharacteristic
148#395—21no—
149#395—21no—
150#391—21no—
151#389C22×C312yes—
152#389C22×C312yes—
153#384—21no—
154C4×C4⋊C4C22×C311yescharacteristic
155#340C22×C312yes—
156#336C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—41no—
168C2×C3×Q8—42no—
169C2×C3×Q8—82no—
170C2×C3×Q8C2×D412yes—
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4C2×D412yes—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—82no—
210C3×C4⋊C4—82no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4#4114yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C42—24no—
219C3×C42—24no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—42no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227C22×Q8—21no—
228C22×Q8C3×D411yescharacteristic
229C22×Q8C23×C311yescharacteristic
230C22×Q8C23×C311yescharacteristic
231C22×D4C23×C311yescharacteristic
232C23×C4—21no—
233C22×C8C23×C312yes—
234C22×C8C23×C312yes—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239#126—28no—
240#126—28no—
241#124—22no—
242#124—22no—
243#124—22no—
244#124C3×D414yes—
245C22⋊Q8—28no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8C3×D414yes—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C3×D411yescharacteristic
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C23×C311yescharacteristic
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C311yescharacteristic
279#106—24no—
280C4⋊C8—22no—
281#103—24no—
282#103—24no—
283#103—24no—
284#103—24no—
285#102—24no—
286#102—24no—
287#97—22no—
288#95—21no—
289#95—22no—
290#95—22no—
291C23×C3—22no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3C22×D411yescharacteristic
297C3×Q8—22no—
298C3×Q8—22no—
299C3×Q8—24no—
300C3×Q8—44no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—42no—
304C3×Q8—84no—
305C3×D4—24no—
306C3×D4—41no—
307C3×D4—41no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—48no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4#12012yes—
335C2×C3×C4#12012yes—
336C2×C3×C4#13712yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C42−1+412yes—
341C3×C8—42no—
342C3×C8—42no—
343C24—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—42no—
354C2×Q8—82no—
355C2×Q8C2×C3×D412yes—
356C2×Q8C2×C3×D412yes—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4C2×C3×D412yes—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C24×C311yescharacteristic
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4C2×C3×D412yes—
397C4⋊C4C3×#4114yes—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C42—24no—
404C42—24no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—42no—
410C42C2×C3×D412yes—
411C42C2×C3×D412yes—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3C2×#12011yescharacteristic
418C22×C3#54812yes—
419C22×C3C2×#13711yescharacteristic
420C22×C3#57511yescharacteristic
421C22×C3#57712yes—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—44no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—41no—
429C3×C4—42no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—82no—
433C3×C4#53211yescharacteristic
434C3×C4#53611yescharacteristic
435C3×C4#54712yes—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23C22×C3×D411yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—24no—
445Q8—44no—
446Q8—41no—
447Q8—41no—
448Q8—42no—
449Q8—84no—
450D4—24no—
451D4—41no—
452D4—41no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4C3×#12012yes—
480C2×C4C3×#12012yes—
481C2×C4C3×#13712yes—
482C2×C4C22×C3×D411yescharacteristic
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C2×C3×#4112yes—
485C2×C4C3×2−1+412yes—
486C8—42no—
487C8—42no—
488C2×C3—82no—
489C2×C3#243211yescharacteristic
490C2×C3#300711yescharacteristic
491C2×C3#301011yescharacteristic
492C2×C3#336912yes—
493C2×C3#337612yes—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—82no—
499C22C2×C3×#12011yescharacteristic
500C22C3×#54812yes—
501C22C2×C3×#13711yescharacteristic
502C22C3×#57511yescharacteristic
503C22C3×#57712yes—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—41no—
510C4—42no—
511C4—41no—
512C4—41no—
513C4—44no—
514C4—82no—
515C4C3×#53211yescharacteristic
516C4C3×#53611yescharacteristic
517C4C3×#54712yes—
518C3#2566211yescharacteristic, direct product factor, Sylow
519C2—82no—
520C2C3×#243211yescharacteristic
521C2C3×#300711yescharacteristic
522C2C3×#301011yescharacteristic
523C2C3×#336912yes—
524C2C3×#337612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#475C2semidirect
4. C3×#2567C2semidirect
5. C3×#2043C2semidirect
7. C3×#1985C2semidirect
9. C3×#1865C2semidirect
11. C3×#1763C2semidirect
12. #25662C3direct (N × H)
50. C2×#475C2×C3semidirect
51. #2567C2×C3semidirect
52. #2043C2×C3semidirect
54. #1985C2×C3semidirect
56. #1865C2×C3semidirect
58. #1763C2×C3semidirect
518. C3#25662direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4892
C2×C3×#4752
C3×#25672
C3×#20432
C3×#20422
C3×#19852
C3×#19832
C3×#18652
C3×#18642
C3×#17632
#256623