Finite Group Databasebeta

← C3×#29499 #10648106

Subgroups — C3×#29499 #10648106

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic, complemented
2C3×C4×#127C211yesmaximal, characteristic, complemented
3C3×#2084C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6C3×#2011C211yesmaximal, characteristic
7C3×#1972C212yesmaximal, complemented
8C3×#1936C212yesmaximal, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1745C211yesmaximal, characteristic, complemented
12#29499C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#126—21no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#97C2212yes—
25C3×#390—21no—
26C3×#390—21no—
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#390C2212yes—
30C3×#390C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#388—21no—
35C3×#388—21no—
36C3×#388—21no—
37C3×#384C2211yescharacteristic
38C3×C43C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2860C2×C311yescharacteristic, complemented
45C4×#127C2×C311yescharacteristic, complemented
46#2084C2×C311yescharacteristic, complemented
47#2071C2×C311yescharacteristic, complemented
48#2041C2×C311yescharacteristic, complemented
49#2011C2×C311yescharacteristic
50#1972C2×C312yescomplemented
51#1936C2×C312yescomplemented
52#1879C2×C312yescomplemented
53#1864C2×C312yescomplemented
54#1745C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2312yes—
59C23×C3×C4—22no—
60C2×C3×D8—24no—
61C2×C3×D8—24no—
62C2×C3×D8—42no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#127—24no—
66C3×#126—44no—
67C3×C4×D4—42no—
68C3×C4×D4—42no—
69C3×C4×D4—44no—
70C3×C4×D4—44no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×#106—42no—
97C3×#102—24no—
98C3×#102—24no—
99C3×#102—42no—
100C3×#102—42no—
101C3×#97—24no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C22×D8C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#126—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C22×C312yes—
111C2×#106C22×C312yes—
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
116C2×#97C22×C312yes—
117#390—21no—
118#390—21no—
119#390C22×C312yes—
120#390C22×C311yescharacteristic
121#390C22×C311yescharacteristic
122#390C22×C311yescharacteristic
123#389C22×C311yescharacteristic
124#389C22×C311yescharacteristic
125#389C22×C311yescharacteristic
126#388—21no—
127#388—21no—
128#388—21no—
129#384C22×C311yescharacteristic
130C43C22×C311yescharacteristic
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C311yescharacteristic
134#336C22×C311yescharacteristic
135#336C22×C312yes—
136C24×C3—22no—
137C24×C3—21no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4#4114yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—44no—
159C22×C3×C4—44no—
160C22×C3×C4—44no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C3×D8—48no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8#4114yes—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42—41no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4C23×C311yescharacteristic
206C22×D4C23×C312yes—
207C23×C4—22no—
208C2×D8—24no—
209C2×D8—24no—
210C2×D8—42no—
211C22×C8C23×C312yes—
212C22×C8C23×C312yes—
213#127—24no—
214#126—44no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42C3×D411yescharacteristic
238C2×C42C3×D411yescharacteristic
239C2×C42C3×D411yescharacteristic
240C2×C42C3×D411yescharacteristic
241C2×C42C23×C311yescharacteristic
242C2×C42C23×C311yescharacteristic
243C2×C42C23×C311yescharacteristic
244#106—42no—
245#102—24no—
246#102—24no—
247#102—42no—
248#102—42no—
249#97—24no—
250#95—21no—
251#95—21no—
252#95—21no—
253C23×C3—24no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3C22×D411yescharacteristic
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—48no—
271C3×D4—48no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
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—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
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—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—88no—
304C2×C3×C4AGL(1,ℤ/8)12yes—
305C2×C3×C4AGL(1,ℤ/8)12yes—
306C2×C3×C4AGL(1,ℤ/8)12yes—
307C2×C3×C4C2×#4112yes—
308C2×C3×C4C2×#4112yes—
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C3×C8—24no—
312C3×C8—42no—
313C24—22no—
314C24—21no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4C3×#4114yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345D8—48no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8C3×#4114yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—21no—
378C42—21no—
379C42—41no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3#52511yescharacteristic
395C22×C3#53211yescharacteristic
396C22×C3#53511yescharacteristic
397C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
399C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C3#57511yescharacteristic
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—41no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4—82no—
411C3×C4#55112yes—
412C3×C4#55312yes—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C22×C3×D411yescharacteristic
428D4—24no—
429D4—24no—
430D4—48no—
431D4—48no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—88no—
464C2×C4C3×AGL(1,ℤ/8)12yes—
465C2×C4C3×AGL(1,ℤ/8)12yes—
466C2×C4C3×AGL(1,ℤ/8)12yes—
467C2×C4C2×C3×#4112yes—
468C2×C4C2×C3×#4112yes—
469C2×C4C3×H(5,𝔽2)11yescharacteristic
470C2×C4C3×H(5,𝔽2)11yescharacteristic
471C8—24no—
472C8—42no—
473C2×C3—84no—
474C2×C3—161no—
475C2×C3#260311yescharacteristic
476C2×C3#310911yescharacteristic
477C2×C3#311411yescharacteristic
478C2×C3#317111yescharacteristic
479C2×C3#318411yescharacteristic
480C2×C3#321411yescharacteristic
481C2×C3#321711yescharacteristic
482C22—44no—
483C22—44no—
484C22—44no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22C3×#52511yescharacteristic
497C22C3×#53211yescharacteristic
498C22C3×#53511yescharacteristic
499C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
500C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
501C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
502C22C3×#57511yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—81no—
510C4—81no—
511C4—82no—
512C4—81no—
513C4C3×#55112yes—
514C4C3×#55312yes—
515C3#2949911yescharacteristic, direct product factor, Sylow
516C2—84no—
517C2—161no—
518C2C3×#260311yescharacteristic
519C2C3×#310911yescharacteristic
520C2C3×#311411yescharacteristic
521C2C3×#317111yescharacteristic
522C2C3×#318411yescharacteristic
523C2C3×#321411yescharacteristic
524C2C3×#321711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2860C2semidirect
2. C3×C4×#127C2semidirect
3. C3×#2084C2semidirect
4. C3×#2071C2semidirect
5. C3×#2041C2semidirect
7. C3×#1972C2semidirect
8. C3×#1936C2semidirect
9. C3×#1879C2semidirect
10. C3×#1864C2semidirect
11. C3×#1745C2semidirect
12. #29499C3direct (N × H)
44. #2860C2×C3semidirect
45. C4×#127C2×C3semidirect
46. #2084C2×C3semidirect
47. #2071C2×C3semidirect
48. #2041C2×C3semidirect
50. #1972C2×C3semidirect
51. #1936C2×C3semidirect
52. #1879C2×C3semidirect
53. #1864C2×C3semidirect
54. #1745C2×C3semidirect
515. C3#29499direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×C4×#1272
C3×#20842
C3×#20712
C3×#20412
C3×#20112
C3×#19722
C3×#19362
C3×#18792
C3×#18642
C3×#17452
#294993