Finite Group Databasebeta

← C3×#27650 #10646257

Subgroups — C3×#27650 #10646257

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#482C212yesmaximal
2C3×C4×#121C211yesmaximal, characteristic
3C3×#2285C211yesmaximal, characteristic, complemented
4C3×#2084C211yesmaximal, characteristic, complemented
5C3×#2083C211yesmaximal, characteristic, complemented
6C3×#2071C211yesmaximal, characteristic, complemented
7C3×#2071C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1825C212yesmaximal, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27650C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#482—28no—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C22⋊C8C2212yes—
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#388—21no—
28C3×#386—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⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C2×#482C2×C312yes—
46C4×#121C2×C311yescharacteristic
47#2285C2×C311yescharacteristic, complemented
48#2084C2×C311yescharacteristic, complemented
49#2083C2×C311yescharacteristic, complemented
50#2071C2×C311yescharacteristic, complemented
51#2071C2×C311yescharacteristic, complemented
52#1949C2×C312yescomplemented
53#1825C2×C312yescomplemented
54#1804C2×C312yescomplemented
55#1797C2×C311yescharacteristic, complemented
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×#121—22no—
65C3×#121D414yes—
66C3×C4×D4—24no—
67C3×C4×D4—44no—
68C3×#117—24no—
69C3×#117—24no—
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—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—24no—
87C2×C3×C42—21no—
88C2×C3×C42—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×#107—24no—
94C3×#106—24no—
95C3×C4⋊C8—24no—
96C3×#102—24no—
97C3×#102—24no—
98C3×C22⋊C8—24no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—21no—
104C2×#121C22×C311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C22×C311yescharacteristic
107C2×#117C22×C312yes—
108C22×C42C22×C311yescharacteristic
109#482—28no—
110C2×#107C22×C312yes—
111C2×#106C22×C312yes—
112C2×C4⋊C8C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C312yes—
116#390C22×C311yescharacteristic
117#389C22×C311yescharacteristic
118#388—21no—
119#386—21no—
120#384C22×C311yescharacteristic
121#384C22×C311yescharacteristic
122#382C22×C312yes—
123#380—21no—
124#380—21no—
125#380C22×C311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C22⋊C4—21no—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C311yescharacteristic
135#336C22×C311yescharacteristic
136C24×C3—21no—
137C24×C3C2×D411yescharacteristic
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4C2×D412yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
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—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C3×C42—44no—
199C3×C42#4114yes—
200C22×D4—21no—
201C22×D4C23×C311yescharacteristic
202C23×C4—21no—
203C23×C4C3×D411yescharacteristic
204C23×C4C3×D411yescharacteristic
205C23×C4C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208#121—22no—
209#121C3×D414yes—
210C4×D4—24no—
211C4×D4—44no—
212#117—24no—
213#117—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D411yescharacteristic
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C311yescharacteristic
237#107—24no—
238#106—24no—
239C4⋊C8—24no—
240#102—24no—
241#102—24no—
242C22⋊C8—24no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3#12112yes—
262C23×C3C22×D411yescharacteristic
263C3×D4—24no—
264C3×D4—48no—
265C3×D4—42no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—84no—
296C2×C3×C4#12112yes—
297C2×C3×C4#12312yes—
298C2×C3×C4#12312yes—
299C2×C3×C4#13512yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4112yes—
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C42−1+412yes—
305C3×C8—42no—
306C3×C8—42no—
307C24—21no—
308C24C2×C3×D411yescharacteristic
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4C2×C3×D412yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C3×#4112yes—
339C22×C4C24×C311yescharacteristic
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4C2×C3×D412yes—
359C22⋊C4—24no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—42no—
369C42—44no—
370C42C3×#4114yes—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3C2×#12111yescharacteristic
386C22×C3C2×#12311yescharacteristic
387C22×C3#53611yescharacteristic
388C22×C3C2×#13511yescharacteristic
389C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C3#57511yescharacteristic
391C22×C3#57511yescharacteristic
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—82no—
401C3×C4—82no—
402C3×C4#54712yes—
403C3×C4#55212yes—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C3×#12112yes—
418C23C22×C3×D411yescharacteristic
419D4—24no—
420D4—48no—
421D4—42no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—84no—
452C2×C4C3×#12112yes—
453C2×C4C3×#12312yes—
454C2×C4C3×#12312yes—
455C2×C4C3×#13512yes—
456C2×C4C3×AGL(1,ℤ/8)12yes—
457C2×C4C22×C3×D411yescharacteristic
458C2×C4C2×C3×#4112yes—
459C2×C4C2×C3×#4111yescharacteristic
460C2×C4C3×2−1+412yes—
461C8—42no—
462C8—42no—
463C2×C3—42no—
464C2×C3—82no—
465C2×C3#248611yescharacteristic
466C2×C3#303511yescharacteristic
467C2×C3#303611yescharacteristic
468C2×C3#309111yescharacteristic
469C2×C3#309411yescharacteristic
470C2×C3#322811yescharacteristic
471C2×C3#324011yescharacteristic
472C22—22no—
473C22—22no—
474C22—22no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22C2×C3×#12111yescharacteristic
487C22C2×C3×#12311yescharacteristic
488C22C3×#53611yescharacteristic
489C22C2×C3×#13511yescharacteristic
490C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
491C22C3×#57511yescharacteristic
492C22C3×#57511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—81no—
500C4—82no—
501C4—82no—
502C4—81no—
503C4C3×#54712yes—
504C4C3×#55212yes—
505C3#2765011yescharacteristic, direct product factor, Sylow
506C2—42no—
507C2—82no—
508C2C3×#248611yescharacteristic
509C2C3×#303511yescharacteristic
510C2C3×#303611yescharacteristic
511C2C3×#309111yescharacteristic
512C2C3×#309411yescharacteristic
513C2C3×#322811yescharacteristic
514C2C3×#324011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
3. C3×#2285C2semidirect
4. C3×#2084C2semidirect
5. C3×#2083C2semidirect
6. C3×#2071C2semidirect
7. C3×#2071C2semidirect
8. C3×#1949C2semidirect
9. C3×#1825C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #27650C3direct (N × H)
47. #2285C2×C3semidirect
48. #2084C2×C3semidirect
49. #2083C2×C3semidirect
50. #2071C2×C3semidirect
51. #2071C2×C3semidirect
52. #1949C2×C3semidirect
53. #1825C2×C3semidirect
54. #1804C2×C3semidirect
55. #1797C2×C3semidirect
505. C3#27650direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4822
C3×C4×#1212
C3×#22852
C3×#20842
C3×#20832
C3×#20712
C3×#19492
C3×#18252
C3×#18042
C3×#17972
#276503