Finite Group Databasebeta

← C3×#27652 #10646259

Subgroups — C3×#27652 #10646259

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#485C212yesmaximal
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C3×#2285C211yesmaximal, characteristic
4C3×#2084C211yesmaximal, characteristic, complemented
5C3×#2083C211yesmaximal, characteristic, complemented
6C3×#2072C211yesmaximal, characteristic, complemented
7C3×#2072C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1825C212yesmaximal
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic
12#27652C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×#117C2212yes—
16C22×C3×C42C2211yescharacteristic
17C3×#485—28no—
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×C22⋊C8C2212yes—
24C3×#391C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#387—21no—
27C3×#385—21no—
28C3×#384C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C3×#382C2212yes—
31C3×#380—21no—
32C3×#380—21no—
33C3×#380C2211yescharacteristic
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C2×#485C2×C312yes—
46C4×C22⋊Q8C2×C311yescharacteristic
47#2285C2×C311yescharacteristic
48#2084C2×C311yescharacteristic, complemented
49#2083C2×C311yescharacteristic, complemented
50#2072C2×C311yescharacteristic, complemented
51#2072C2×C311yescharacteristic, complemented
52#1949C2×C312yescomplemented
53#1825C2×C312yes—
54#1805C2×C312yescomplemented
55#1797C2×C311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8D414yes—
64C3×C4×Q8—24no—
65C3×#117—24no—
66C3×#117—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—24no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42—22no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C3×#107—24no—
91C3×#106—24no—
92C3×C4⋊C8—24no—
93C3×#103—24no—
94C3×#103—24no—
95C3×C22⋊C8—24no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C2×C22⋊Q8C22×C311yescharacteristic
104C2×C4×Q8C22×C311yescharacteristic
105C2×#117C22×C312yes—
106C22×C42C22×C311yescharacteristic
107#485—28no—
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×C22⋊C8C22×C312yes—
114#391C22×C311yescharacteristic
115#389C22×C311yescharacteristic
116#387—21no—
117#385—21no—
118#384C22×C311yescharacteristic
119#384C22×C311yescharacteristic
120#382C22×C312yes—
121#380—21no—
122#380—21no—
123#380C22×C311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—21no—
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C311yescharacteristic
134#336C22×C311yescharacteristic
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8C2×D412yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4#4112yes—
161C22×C3×C4C2411yescharacteristic
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—44no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—42no—
190C3×C42—44no—
191C3×C42—42no—
192C3×C42#4114yes—
193C22×Q8C23×C311yescharacteristic
194C23×C4C3×D411yescharacteristic
195C23×C4C3×D411yescharacteristic
196C23×C4C23×C311yescharacteristic
197C22×C8C23×C312yes—
198C22×C8C23×C312yes—
199C22⋊Q8—22no—
200C22⋊Q8C3×D414yes—
201C4×Q8—24no—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C3×D411yescharacteristic
224C2×C42C3×D411yescharacteristic
225C2×C42C23×C311yescharacteristic
226C2×C42C23×C311yescharacteristic
227#107—24no—
228#106—24no—
229C4⋊C8—24no—
230#103—24no—
231#103—24no—
232C22⋊C8—24no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3#12112yes—
247C23×C3C22×D411yescharacteristic
248C3×Q8—24no—
249C3×Q8—42no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4#12112yes—
283C2×C3×C4#12312yes—
284C2×C3×C4#12312yes—
285C2×C3×C4#13512yes—
286C2×C3×C4#13712yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+412yes—
291C3×C8—42no—
292C3×C8—42no—
293C24C2×C3×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8C2×C3×D412yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C3×#4112yes—
319C22×C4C24×C311yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C4⋊C4—24no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4C2×C3×D412yes—
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—44no—
349C42—42no—
350C42C3×#4114yes—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3C2×#12111yescharacteristic
359C22×C3C2×#12311yescharacteristic
360C22×C3#53611yescharacteristic
361C22×C3C2×#13511yescharacteristic
362C22×C3C2×#13711yescharacteristic
363C22×C3#57511yescharacteristic
364C22×C3#57511yescharacteristic
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—82no—
375C3×C4—82no—
376C3×C4#54712yes—
377C3×C4#55212yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23C3×#12112yes—
385C23C22×C3×D411yescharacteristic
386Q8—24no—
387Q8—42no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4C3×#12112yes—
421C2×C4C3×#12312yes—
422C2×C4C3×#12312yes—
423C2×C4C3×#13512yes—
424C2×C4C3×#13712yes—
425C2×C4C22×C3×D411yescharacteristic
426C2×C4C2×C3×#4112yes—
427C2×C4C2×C3×#4111yescharacteristic
428C2×C4C3×2−1+412yes—
429C8—42no—
430C8—42no—
431C2×C3—42no—
432C2×C3#248611yescharacteristic
433C2×C3#303511yescharacteristic
434C2×C3#303711yescharacteristic
435C2×C3#309111yescharacteristic
436C2×C3#309511yescharacteristic
437C2×C3#323111yescharacteristic
438C2×C3#324011yescharacteristic
439C22—22no—
440C22—22no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22C2×C3×#12111yescharacteristic
447C22C2×C3×#12311yescharacteristic
448C22C3×#53611yescharacteristic
449C22C2×C3×#13511yescharacteristic
450C22C2×C3×#13711yescharacteristic
451C22C3×#57511yescharacteristic
452C22C3×#57511yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—82no—
462C4—82no—
463C4—81no—
464C4C3×#54712yes—
465C4C3×#55212yes—
466C3#2765211yescharacteristic, direct product factor, Sylow
467C2—42no—
468C2C3×#248611yescharacteristic
469C2C3×#303511yescharacteristic
470C2C3×#303711yescharacteristic
471C2C3×#309111yescharacteristic
472C2C3×#309511yescharacteristic
473C2C3×#323111yescharacteristic
474C2C3×#324011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2084C2semidirect
5. C3×#2083C2semidirect
6. C3×#2072C2semidirect
7. C3×#2072C2semidirect
8. C3×#1949C2semidirect
10. C3×#1805C2semidirect
12. #27652C3direct (N × H)
48. #2084C2×C3semidirect
49. #2083C2×C3semidirect
50. #2072C2×C3semidirect
51. #2072C2×C3semidirect
52. #1949C2×C3semidirect
54. #1805C2×C3semidirect
466. C3#27652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4852
C3×C4×C22⋊Q82
C3×#22852
C3×#20842
C3×#20832
C3×#20722
C3×#19492
C3×#18252
C3×#18052
C3×#17972
#276523