Finite Group Databasebeta

← C3×#27651 #10646258

Subgroups — C3×#27651 #10646258

Order 768.

922 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#481C211yesmaximal, characteristic
2C2×C3×#480C211yesmaximal, characteristic
3C3×C4×#121C211yesmaximal, characteristic
4C2×C3×#382C211yesmaximal, characteristic, complemented
5C3×#2085C212yesmaximal, complemented
6C3×#2073C212yesmaximal, complemented
7C3×#1949C212yesmaximal, complemented
8C3×#1824C211yesmaximal, characteristic, complemented
9C3×#1823C211yesmaximal, characteristic, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27651C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#481—24no—
20C3×#480—24no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#388—21no—
31C3×#386—21no—
32C3×#382—21no—
33C3×#382—21no—
34C3×#382C2212yes—
35C3×#382C2212yes—
36C3×#380C2211yescharacteristic
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C2×C3×#95—21no—
40C2×C3×#95—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#336C2212yes—
46C2×#481C2×C311yescharacteristic
47C2×#480C2×C311yescharacteristic
48C4×#121C2×C311yescharacteristic
49C2×#382C2×C311yescharacteristic, complemented
50#2085C2×C312yescomplemented
51#2073C2×C312yescomplemented
52#1949C2×C312yescomplemented
53#1824C2×C311yescharacteristic, complemented
54#1823C2×C311yescharacteristic, complemented
55#1804C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4—21no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×#121—22no—
68C3×#121D414yes—
69C3×C4×D4—24no—
70C3×C4×D4—44no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—24no—
89C2×C3×C42—21no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#107—24no—
95C3×#106—24no—
96C3×C4⋊C8—24no—
97C3×#102—24no—
98C3×#102—24no—
99C3×C22⋊C8—22no—
100C3×C22⋊C8—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C2×#121C22×C311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C4⋊C4C22×C311yescharacteristic
113C22×C42C22×C311yescharacteristic
114#481—24no—
115#480—24no—
116C2×#107C22×C312yes—
117C2×#106C22×C312yes—
118C2×C4⋊C8C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×C22⋊C8C22×C311yescharacteristic
122C2×C22⋊C8C22×C311yescharacteristic
123#390C22×C311yescharacteristic
124#389C22×C311yescharacteristic
125#388—21no—
126#386—21no—
127#382—21no—
128#382—21no—
129#382C22×C312yes—
130#382C22×C312yes—
131#380C22×C311yescharacteristic
132C4×C4⋊C4C22×C311yescharacteristic
133C4×C22⋊C4—21no—
134C2×#95—21no—
135C2×#95—21no—
136#340C22×C312yes—
137#340C22×C312yes—
138#340C22×C312yes—
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C24×C3C2×D411yescharacteristic
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4C2×D412yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4—41no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—42no—
206C3×C42#4114yes—
207C22×D4—21no—
208C22×D4C23×C311yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C3×D411yescharacteristic
213C23×C4C3×D411yescharacteristic
214C23×C4C23×C311yescharacteristic
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217#121—22no—
218#121C3×D414yes—
219C4×D4—24no—
220C4×D4—44no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—24no—
239C2×C42—21no—
240C2×C42C3×D411yescharacteristic
241C2×C42C3×D411yescharacteristic
242C2×C42C23×C311yescharacteristic
243C2×C42C23×C311yescharacteristic
244#107—24no—
245#106—24no—
246C4⋊C8—24no—
247#102—24no—
248#102—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3#12112yes—
270C23×C3#13512yes—
271C23×C3AGL(1,ℤ/8)12yes—
272C23×C3C22×D411yescharacteristic
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—42no—
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—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—84no—
308C2×C3×C4—84no—
309C2×C3×C4—84no—
310C2×C3×C4#12112yes—
311C2×C3×C4#12312yes—
312C2×C3×C4#12312yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C42−1+412yes—
317C3×C8—42no—
318C3×C8—42no—
319C24—21no—
320C24C2×C3×D411yescharacteristic
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4C2×C3×D412yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D412yes—
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C3×#4112yes—
357C22×C4C24×C311yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4C2×C3×D412yes—
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—42no—
384C42C3×#4114yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3#46314yes—
399C22×C3C2×#12111yescharacteristic
400C22×C3C2×#12311yescharacteristic
401C22×C3#53611yescharacteristic
402C22×C3C2×#13511yescharacteristic
403C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C3#57511yescharacteristic
405C22×C3#57511yescharacteristic
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—42no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4—82no—
415C3×C4—82no—
416C3×C4#54712yes—
417C3×C4#55212yes—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C3×#12112yes—
430C23C3×#13512yes—
431C23C3×AGL(1,ℤ/8)12yes—
432C23C22×C3×D411yescharacteristic
433D4—24no—
434D4—48no—
435D4—42no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4C3×#12112yes—
471C2×C4C3×#12312yes—
472C2×C4C3×#12312yes—
473C2×C4C22×C3×D411yescharacteristic
474C2×C4C2×C3×#4112yes—
475C2×C4C2×C3×#4111yescharacteristic
476C2×C4C3×2−1+412yes—
477C8—42no—
478C8—42no—
479C2×C3—24no—
480C2×C3—82no—
481C2×C3#248611yescharacteristic
482C2×C3C2×#46311yescharacteristic
483C2×C3#304011yescharacteristic
484C2×C3#309111yescharacteristic
485C2×C3#309411yescharacteristic
486C2×C3#323211yescharacteristic
487C2×C3#323411yescharacteristic
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22C3×#46314yes—
502C22C2×C3×#12111yescharacteristic
503C22C2×C3×#12311yescharacteristic
504C22C3×#53611yescharacteristic
505C22C2×C3×#13511yescharacteristic
506C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
507C22C3×#57511yescharacteristic
508C22C3×#57511yescharacteristic
509C4—22no—
510C4—22no—
511C4—22no—
512C4—42no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—82no—
517C4—82no—
518C4—81no—
519C4C3×#54712yes—
520C4C3×#55212yes—
521C3#2765111yescharacteristic, direct product factor, Sylow
522C2—24no—
523C2—82no—
524C2C3×#248611yescharacteristic
525C2C2×C3×#46311yescharacteristic
526C2C3×#304011yescharacteristic
527C2C3×#309111yescharacteristic
528C2C3×#309411yescharacteristic
529C2C3×#323211yescharacteristic
530C2C3×#323411yescharacteristic
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
4. C2×C3×#382C2semidirect
5. C3×#2085C2semidirect
6. C3×#2073C2semidirect
7. C3×#1949C2semidirect
8. C3×#1824C2semidirect
9. C3×#1823C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #27651C3direct (N × H)
49. C2×#382C2×C3semidirect
50. #2085C2×C3semidirect
51. #2073C2×C3semidirect
52. #1949C2×C3semidirect
53. #1824C2×C3semidirect
54. #1823C2×C3semidirect
55. #1804C2×C3semidirect
56. #1797C2×C3semidirect
521. C3#27651direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4812
C2×C3×#4802
C3×C4×#1212
C2×C3×#3822
C3×#20852
C3×#20732
C3×#19492
C3×#18242
C3×#18232
C3×#18042
C3×#17972
#276513