Finite Group Databasebeta

← C3×#27649 #10646256

Subgroups — C3×#27649 #10646256

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 514 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×#2085C212yesmaximal, complemented
5C3×#2074C212yesmaximal, complemented
6C3×#1948C211yesmaximal, characteristic, complemented
7C3×#1947C211yesmaximal, characteristic, complemented
8C3×#1825C212yesmaximal
9C3×#1803C211yesmaximal, characteristic, complemented
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic
12#27649C311yesmaximal, 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⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C22⋊C8C2212yes—
25C3×#391C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#385—21no—
29C3×#384C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C3×#382C2212yes—
32C3×#380—21no—
33C3×#380—21no—
34C3×#380C2211yescharacteristic
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4—21no—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C22⋊C4—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C2×#485C2×C312yes—
46C4×C22⋊Q8C2×C311yescharacteristic
47#2285C2×C311yescharacteristic
48#2085C2×C312yescomplemented
49#2074C2×C312yescomplemented
50#1948C2×C311yescharacteristic, complemented
51#1947C2×C311yescharacteristic, complemented
52#1825C2×C312yes—
53#1803C2×C311yescharacteristic, complemented
54#1803C2×C311yescharacteristic, complemented
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—22no—
65C3×C4×Q8—22no—
66C3×#117—24no—
67C3×#117—24no—
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⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42—22no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#106—24no—
94C3×C4⋊C8—22no—
95C3×C4⋊C8—22no—
96C3×#103—24no—
97C3×#103—24no—
98C3×C22⋊C8—24no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C2×C22⋊Q8C22×C311yescharacteristic
107C2×C4×Q8C22×C311yescharacteristic
108C2×#117C22×C312yes—
109C22×C42C22×C311yescharacteristic
110#485—28no—
111C2×#107C22×C312yes—
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#103C22×C312yes—
116C2×#103C22×C312yes—
117C2×C22⋊C8C22×C312yes—
118#391C22×C311yescharacteristic
119#389C22×C311yescharacteristic
120#387—21no—
121#385—21no—
122#384C22×C311yescharacteristic
123#384C22×C311yescharacteristic
124#382C22×C312yes—
125#380—21no—
126#380—21no—
127#380C22×C311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C22×C311yescharacteristic
132C4×C22⋊C4—21no—
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137#336C22×C312yes—
138C24×C3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8C2×D412yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—44no—
200C3×C42—42no—
201C3×C42—42no—
202C3×C42#4112yes—
203C3×C42#4112yes—
204C22×Q8C23×C311yescharacteristic
205C23×C4C3×D411yescharacteristic
206C23×C4C3×D411yescharacteristic
207C23×C4C23×C311yescharacteristic
208C22×C8C23×C312yes—
209C22×C8C23×C312yes—
210C22⋊Q8—22no—
211C22⋊Q8C3×D414yes—
212C4×Q8—22no—
213C4×Q8—22no—
214#117—24no—
215#117—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C23×C312yes—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C3×D411yescharacteristic
237C2×C42C3×D411yescharacteristic
238C2×C42C23×C311yescharacteristic
239C2×C42C23×C311yescharacteristic
240#107—24no—
241#106—24no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244#103—24no—
245#103—24no—
246C22⋊C8—24no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3#12112yes—
261C23×C3C22×D411yescharacteristic
262C3×Q8—22no—
263C3×Q8—22no—
264C3×Q8—41no—
265C3×Q8—41no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—44no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4#12112yes—
301C2×C3×C4#12312yes—
302C2×C3×C4#12312yes—
303C2×C3×C4#13512yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C3×C8—42no—
312C3×C8—42no—
313C24C2×C3×D411yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8C2×C3×D412yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C3×#4112yes—
343C22×C4C24×C311yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4C2×C3×D412yes—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C42—22no—
371C42—22no—
372C42—21no—
373C42—21no—
374C42—44no—
375C42—42no—
376C42—42no—
377C42C3×#4112yes—
378C42C3×#4112yes—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3C2×#12111yescharacteristic
387C22×C3C2×#12311yescharacteristic
388C22×C3#53611yescharacteristic
389C22×C3C2×#13511yescharacteristic
390C22×C3C2×#13711yescharacteristic
391C22×C3#57511yescharacteristic
392C22×C3#57511yescharacteristic
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4—82no—
402C3×C4—82no—
403C3×C4#48514yescomplemented
404C3×C4#54711yescharacteristic
405C3×C4#54711yescharacteristic
406C3×C4#55211yescharacteristic
407C3×C4#55211yescharacteristic
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23C3×#12112yes—
415C23C22×C3×D411yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4C3×#12112yes—
455C2×C4C3×#12312yes—
456C2×C4C3×#12312yes—
457C2×C4C3×#13512yes—
458C2×C4C3×#13712yes—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C2×C3×#4111yescharacteristic
461C2×C4C2×C3×#4111yescharacteristic
462C2×C4C2×C3×#4111yescharacteristic
463C2×C4C3×2−1+411yescharacteristic
464C2×C4C3×2−1+411yescharacteristic
465C8—42no—
466C8—42no—
467C2×C3—42no—
468C2×C3#248611yescharacteristic
469C2×C3#303511yescharacteristic
470C2×C3#303711yescharacteristic
471C2×C3C2×#48511yescharacteristic
472C2×C3#309311yescharacteristic
473C2×C3#323311yescharacteristic
474C2×C3#323711yescharacteristic
475C22—22no—
476C22—22no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22C2×C3×#12111yescharacteristic
483C22C2×C3×#12311yescharacteristic
484C22C3×#53611yescharacteristic
485C22C2×C3×#13511yescharacteristic
486C22C2×C3×#13711yescharacteristic
487C22C3×#57511yescharacteristic
488C22C3×#57511yescharacteristic
489C4—22no—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—81no—
496C4—82no—
497C4—82no—
498C4—81no—
499C4C3×#48514yescomplemented
500C4C3×#54711yescharacteristic
501C4C3×#54711yescharacteristic
502C4C3×#55211yescharacteristic
503C4C3×#55211yescharacteristic
504C3#2764911yescharacteristic, direct product factor, Sylow
505C2—42no—
506C2C3×#248611yescharacteristic
507C2C3×#303511yescharacteristic
508C2C3×#303711yescharacteristic
509C2C2×C3×#48511yescharacteristic
510C2C3×#309311yescharacteristic
511C2C3×#323311yescharacteristic
512C2C3×#323711yescharacteristic
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×#2085C2semidirect
5. C3×#2074C2semidirect
6. C3×#1948C2semidirect
7. C3×#1947C2semidirect
9. C3×#1803C2semidirect
10. C3×#1803C2semidirect
12. #27649C3direct (N × H)
48. #2085C2×C3semidirect
49. #2074C2×C3semidirect
50. #1948C2×C3semidirect
51. #1947C2×C3semidirect
53. #1803C2×C3semidirect
54. #1803C2×C3semidirect
403. C3×C4#485semidirect
499. C4C3×#485semidirect
504. C3#27649direct (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×#20852
C3×#20742
C3×#19482
C3×#19472
C3×#18252
C3×#18032
C3×#17972
#276493