Finite Group Databasebeta

← C3×#29558 #10648165

Subgroups — C3×#29558 #10648165

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2858C211yesmaximal, characteristic, complemented
2C3×#2293C211yesmaximal, characteristic, complemented
3C3×#2058C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic, complemented
6C3×#2011C211yesmaximal, characteristic
7C3×#1972C212yesmaximal, complemented
8C3×#1936C212yesmaximal, complemented
9C3×#1926C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1745C211yesmaximal, characteristic, complemented
12#29558C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117—21no—
20C2×C3×#106C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#97C2212yes—
26C3×#390—21no—
27C3×#390—21no—
28C3×#390C2211yescharacteristic
29C3×#390C2212yes—
30C3×#390C2211yescharacteristic
31C3×#390C2212yes—
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#388—21no—
35C3×#388—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C3×C43C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2858C2×C311yescharacteristic, complemented
45#2293C2×C311yescharacteristic, complemented
46#2058C2×C311yescharacteristic, complemented
47#2041C2×C311yescharacteristic, complemented
48#2023C2×C311yescharacteristic, complemented
49#2011C2×C311yescharacteristic
50#1972C2×C312yescomplemented
51#1936C2×C312yescomplemented
52#1926C2×C312yescomplemented
53#1879C2×C312yescomplemented
54#1745C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C2×C3×D8—24no—
62C2×C3×D8—24no—
63C2×C3×D8—42no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—42no—
67C3×#127—42no—
68C3×#124—44no—
69C3×#124—44no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×#117—44no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
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—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—21no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×#106—42no—
98C3×#102—24no—
99C3×#102—24no—
100C3×#102—42no—
101C3×#102—42no—
102C3×#97—24no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C22×D8C22×C312yes—
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#127C22×C311yescharacteristic
109C2×#124—21no—
110C2×#124—21no—
111C2×C4×D4C22×C311yescharacteristic
112C2×#117—21no—
113C2×#106C22×C312yes—
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×#102C22×C312yes—
118C2×#97C22×C312yes—
119#390—21no—
120#390—21no—
121#390C22×C312yes—
122#390C22×C312yes—
123#390C22×C311yescharacteristic
124#390C22×C311yescharacteristic
125#389C22×C311yescharacteristic
126#389C22×C311yescharacteristic
127#388—21no—
128#388—21no—
129C4×C4⋊C4C22×C311yescharacteristic
130C4×C22⋊C4—21no—
131C43C22×C311yescharacteristic
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C311yescharacteristic
135#336C22×C312yes—
136#336C22×C311yescharacteristic
137C24×C3—21no—
138C24×C3—22no—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4#4112yes—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C3×D8—48no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8#4114yes—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4#4112yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—44no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—22no—
204C3×C42—41no—
205C3×C42—42no—
206C22×Q8—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C23×C312yes—
210C22×D4C23×C311yescharacteristic
211C23×C4—21no—
212C2×D8—24no—
213C2×D8—24no—
214C2×D8—42no—
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217C4⋊Q8—42no—
218#127—42no—
219#124—44no—
220#124—44no—
221C4×D4—22no—
222C4×D4—22no—
223#117—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C23×C312yes—
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42C3×D411yescharacteristic
242C2×C42C3×D411yescharacteristic
243C2×C42C3×D411yescharacteristic
244C2×C42C3×D411yescharacteristic
245C2×C42C23×C311yescharacteristic
246C2×C42C23×C311yescharacteristic
247C2×C42C23×C311yescharacteristic
248#106—42no—
249#102—24no—
250#102—24no—
251#102—42no—
252#102—42no—
253#97—24no—
254#95—21no—
255#95—21no—
256#95—21no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C3×Q8—84no—
273C3×D4—24no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—84no—
280C3×D4—84no—
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—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—48no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4AGL(1,ℤ/8)12yes—
310C2×C3×C4AGL(1,ℤ/8)12yes—
311C2×C3×C4AGL(1,ℤ/8)12yes—
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)12yes—
316C3×C8—24no—
317C3×C8—42no—
318C24—21no—
319C24—22no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4C3×#4112yes—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C24×C311yescharacteristic
353D8—48no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8C3×#4114yes—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4C3×#4112yes—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—41no—
386C42—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3#52511yescharacteristic
402C22×C3#53211yescharacteristic
403C22×C3#53511yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
407C22×C3#57511yescharacteristic
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—82no—
418C3×C4#55112yes—
419C3×C4#55512yes—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23C22×C3×D411yescharacteristic
435Q8—84no—
436D4—24no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—84no—
443D4—84no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—48no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4C3×AGL(1,ℤ/8)12yes—
473C2×C4C3×AGL(1,ℤ/8)12yes—
474C2×C4C3×AGL(1,ℤ/8)12yes—
475C2×C4C2×C3×#4111yescharacteristic
476C2×C4C2×C3×#4112yes—
477C2×C4C2×C3×#4111yescharacteristic
478C2×C4C3×H(5,𝔽2)12yes—
479C8—24no—
480C8—42no—
481C2×C3—82no—
482C2×C3—162no—
483C2×C3#261011yescharacteristic
484C2×C3#310911yescharacteristic
485C2×C3#311511yescharacteristic
486C2×C3#317111yescharacteristic
487C2×C3#318411yescharacteristic
488C2×C3#321311yescharacteristic
489C2×C3#321711yescharacteristic
490C22—42no—
491C22—42no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22C3×#52511yescharacteristic
505C22C3×#53211yescharacteristic
506C22C3×#53511yescharacteristic
507C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
508C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
509C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
510C22C3×#57511yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—41no—
518C4—81no—
519C4—81no—
520C4—82no—
521C4C3×#55112yes—
522C4C3×#55512yes—
523C3#2955811yescharacteristic, direct product factor, Sylow
524C2—82no—
525C2—162no—
526C2C3×#261011yescharacteristic
527C2C3×#310911yescharacteristic
528C2C3×#311511yescharacteristic
529C2C3×#317111yescharacteristic
530C2C3×#318411yescharacteristic
531C2C3×#321311yescharacteristic
532C2C3×#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×#2858C2semidirect
2. C3×#2293C2semidirect
3. C3×#2058C2semidirect
4. C3×#2041C2semidirect
5. C3×#2023C2semidirect
7. C3×#1972C2semidirect
8. C3×#1936C2semidirect
9. C3×#1926C2semidirect
10. C3×#1879C2semidirect
11. C3×#1745C2semidirect
12. #29558C3direct (N × H)
44. #2858C2×C3semidirect
45. #2293C2×C3semidirect
46. #2058C2×C3semidirect
47. #2041C2×C3semidirect
48. #2023C2×C3semidirect
50. #1972C2×C3semidirect
51. #1936C2×C3semidirect
52. #1926C2×C3semidirect
53. #1879C2×C3semidirect
54. #1745C2×C3semidirect
523. C3#29558direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28582
C3×#22932
C3×#20582
C3×#20412
C3×#20232
C3×#20112
C3×#19722
C3×#19362
C3×#19262
C3×#18792
C3×#17452
#295583