Finite Group Databasebeta

← C3×#27775 #10646382

Subgroups — C3×#27775 #10646382

Order 768.

962 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×#490C212yesmaximal, complemented
2C3×#2552C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic
4C3×#2041C212yesmaximal, complemented
5C3×#2025C212yesmaximal, complemented
6C3×#1982C211yesmaximal, characteristic, complemented
7C3×#1957C211yesmaximal, characteristic, complemented
8C3×#1936C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic, complemented
10C3×#1770C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic, complemented
12#27775C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#127C2212yes—
15C2×C3×#125C2212yes—
16C2×C3×#121—21no—
17C2×C3×#121—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C3×#490—28no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#398—22no—
31C3×#392—22no—
32C3×#390C2212yes—
33C3×#389—21no—
34C3×#389C2212yes—
35C3×#384—21no—
36C3×#384C2211yescharacteristic
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41C3×#340C2212yes—
42C3×#336C2212yes—
43C2×#490C2×C312yescomplemented
44#2552C2×C311yescharacteristic, complemented
45#2541C2×C311yescharacteristic
46#2041C2×C312yescomplemented
47#2025C2×C312yescomplemented
48#1982C2×C311yescharacteristic, complemented
49#1957C2×C311yescharacteristic, complemented
50#1936C2×C312yescomplemented
51#1802C2×C311yescharacteristic, complemented
52#1770C2×C311yescharacteristic, complemented
53C4×#102C2×C311yescharacteristic, complemented
54C22×C3×D4—21no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C3×#127—24no—
65C3×#127—22no—
66C3×#127—22no—
67C3×#127—42no—
68C3×#125—24no—
69C3×#125—42no—
70C3×#121—44no—
71C3×#121—44no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×C3×C42C2312yes—
90C2×C3×C42C2311yescharacteristic
91C3×C4⋊C8—22no—
92C3×#102—24no—
93C3×#102—24no—
94C3×#102—24no—
95C3×#102—22no—
96C3×#102—24no—
97C3×#102—42no—
98C3×#102D414yescomplemented
99C3×#97—24no—
100C3×C4×C8—22no—
101C3×#95—21no—
102C3×#95—22no—
103C3×#95D412yes—
104C2×#127C22×C311yescharacteristic
105C2×#127C22×C312yes—
106C2×#125C22×C312yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×C4×D4C22×C311yescharacteristic
110#490—28no—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
116C2×#102C22×C311yescharacteristic
117C2×#102C22×C312yes—
118C2×#102C22×C311yescharacteristic
119C2×#97C22×C312yes—
120C2×C4×C8C22×C311yescharacteristic
121#398—22no—
122#392—22no—
123#390C22×C312yes—
124#389—21no—
125#389C22×C312yes—
126#384—21no—
127#384C22×C311yescharacteristic
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C4⋊C4C22×C311yescharacteristic
130#340C22×C312yes—
131#340C22×C311yescharacteristic
132#340C22×C311yescharacteristic
133#336C22×C312yes—
134C24×C3—21no—
135C24×C3—22no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—81no—
156C2×C3×D4—81no—
157C2×C3×D4—82no—
158C2×C3×D4—82no—
159C2×C3×D4C2×D412yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8C2×D412yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—81no—
193C3×C4⋊C4—82no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4#4112yes—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—42no—
202C3×C42#4112yes—
203C3×C42#4112yes—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4C23×C312yes—
209C22×D4C23×C311yescharacteristic
210C23×C4—21no—
211C22×C8C23×C311yescharacteristic
212C22×C8C23×C312yes—
213C22×C8C23×C311yescharacteristic
214#127—24no—
215#127—22no—
216#127—22no—
217#127—42no—
218#125—24no—
219#125—42no—
220#121—44no—
221#121—44no—
222C4×D4—22no—
223C4×D4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C3×D411yescharacteristic
230C2×C4⋊C4C3×D411yescharacteristic
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C42—21no—
239C2×C42C23×C312yes—
240C2×C42C23×C311yescharacteristic
241C4⋊C8—22no—
242#102—24no—
243#102—24no—
244#102—24no—
245#102—22no—
246#102—24no—
247#102—42no—
248#102C3×D414yescomplemented
249#97—24no—
250C4×C8—22no—
251#95—21no—
252#95—22no—
253#95C3×D412yes—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3C22×D411yescharacteristic
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—42no—
272C3×D4—41no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—41no—
276C3×D4—42no—
277C3×D4—84no—
278C3×D4—88no—
279C3×D4—88no—
280C3×D4—84no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—84no—
302C2×C3×C4#12412yes—
303C2×C3×C4#12412yes—
304C2×C3×C4AGL(1,ℤ/8)12yes—
305C2×C3×C4AGL(1,ℤ/8)12yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C3×C8—22no—
313C3×C8—42no—
314C3×C8—41no—
315C24—21no—
316C24—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4C2×C3×D412yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C24×C311yescharacteristic
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8C2×C3×D412yes—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4C2×C3×D412yes—
376C4⋊C4C3×#4112yes—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42C3×#4112yes—
384C42C3×#4112yes—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3C2×#12411yescharacteristic
400C22×C3D4211yescharacteristic
401C22×C3#54811yescharacteristic
402C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C3#57611yescharacteristic
405C22×C3#57711yescharacteristic
406C3×C4—22no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—81no—
412C3×C4—82no—
413C3×C4#49014yescomplemented
414C3×C4#53211yescharacteristic
415C3×C4#54211yescharacteristic
416C3×C4#54711yescharacteristic
417C3×C4#55511yescharacteristic
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C22×C3×D411yescharacteristic
433D4—22no—
434D4—22no—
435D4—42no—
436D4—41no—
437D4—42no—
438D4—42no—
439D4—41no—
440D4—42no—
441D4—84no—
442D4—88no—
443D4—88no—
444D4—84no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4C3×#12412yes—
467C2×C4C3×#12412yes—
468C2×C4C3×AGL(1,ℤ/8)12yes—
469C2×C4C3×AGL(1,ℤ/8)12yes—
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C2×C3×#4111yescharacteristic
472C2×C4C2×C3×#4111yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C3×H(5,𝔽2)11yescharacteristic
475C2×C4C3×2−1+411yescharacteristic
476C8—22no—
477C8—42no—
478C8—41no—
479C2×C3—82no—
480C2×C3—162no—
481C2×C3#249111yescharacteristic
482C2×C3C2×#49011yescharacteristic
483C2×C3#313611yescharacteristic
484C2×C3#327211yescharacteristic
485C2×C3#327411yescharacteristic
486C2×C3#336611yescharacteristic
487C2×C3#336811yescharacteristic
488C22—42no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22C2×C3×#12411yescharacteristic
503C22C3×D4211yescharacteristic
504C22C3×#54811yescharacteristic
505C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
506C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
507C22C3×#57611yescharacteristic
508C22C3×#57711yescharacteristic
509C4—22no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—81no—
515C4—82no—
516C4C3×#49014yescomplemented
517C4C3×#53211yescharacteristic
518C4C3×#54211yescharacteristic
519C4C3×#54711yescharacteristic
520C4C3×#55511yescharacteristic
521C3#2777511yescharacteristic, direct product factor, Sylow
522C2—82no—
523C2—162no—
524C2C3×#249111yescharacteristic
525C2C2×C3×#49011yescharacteristic
526C2C3×#313611yescharacteristic
527C2C3×#327211yescharacteristic
528C2C3×#327411yescharacteristic
529C2C3×#336611yescharacteristic
530C2C3×#336811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#490C2semidirect
2. C3×#2552C2semidirect
4. C3×#2041C2semidirect
5. C3×#2025C2semidirect
6. C3×#1982C2semidirect
7. C3×#1957C2semidirect
8. C3×#1936C2semidirect
9. C3×#1802C2semidirect
10. C3×#1770C2semidirect
11. C3×C4×#102C2semidirect
12. #27775C3direct (N × H)
43. C2×#490C2×C3semidirect
44. #2552C2×C3semidirect
46. #2041C2×C3semidirect
47. #2025C2×C3semidirect
48. #1982C2×C3semidirect
49. #1957C2×C3semidirect
50. #1936C2×C3semidirect
51. #1802C2×C3semidirect
52. #1770C2×C3semidirect
53. C4×#102C2×C3semidirect
98. C3×#102D4semidirect
248. #102C3×D4semidirect
413. C3×C4#490semidirect
516. C4C3×#490semidirect
521. C3#27775direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4902
C3×#25522
C3×#25412
C3×#20412
C3×#20252
C3×#19822
C3×#19572
C3×#19362
C3×#18022
C3×#17702
C3×C4×#1022
#277753