Finite Group Databasebeta

← C3×#29162 #10647769

Subgroups — C3×#29162 #10647769

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic, complemented
2C3×C4×#121C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2094C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6C3×#2023C211yesmaximal, characteristic, complemented
7C3×#1919C212yesmaximal, complemented
8C3×#1895C212yesmaximal
9C3×#1862C212yesmaximal, complemented
10C3×#1809C212yesmaximal, complemented
11C3×#1728C211yesmaximal, characteristic, complemented
12#29162C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×#121C2211yescharacteristic
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×C3×#107C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C3×#391—21no—
25C3×#390C2211yescharacteristic
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#389C2212yes—
29C3×#388—21no—
30C3×#386—21no—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4—21no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2857C2×C311yescharacteristic, complemented
45C4×#121C2×C311yescharacteristic
46#2100C2×C311yescharacteristic, complemented
47#2094C2×C311yescharacteristic, complemented
48#2057C2×C311yescharacteristic, complemented
49#2023C2×C311yescharacteristic, complemented
50#1919C2×C312yescomplemented
51#1895C2×C312yes—
52#1862C2×C312yescomplemented
53#1809C2×C312yescomplemented
54#1728C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×M16—24no—
63C2×C3×M16—24no—
64C2×C3×M16—42no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—42no—
68C3×C22⋊Q8—44no—
69C3×#121—22no—
70C3×#121Q814yescomplemented
71C3×C4×D4—42no—
72C3×C4×D4—44no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42—42no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×#107—42no—
96C3×#106—24no—
97C3×#102—24no—
98C3×#102—42no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—22no—
103C3×#95—22no—
104C22×M16C22×C312yes—
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C22⋊Q8—21no—
107C2×#121C22×C311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4C22×C311yescharacteristic
110C22×C42C22×C311yescharacteristic
111C2×#107C22×C312yes—
112C2×#106C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115#391—21no—
116#390C22×C311yescharacteristic
117#389—21no—
118#389C22×C311yescharacteristic
119#389C22×C312yes—
120#388—21no—
121#386—21no—
122#384C22×C311yescharacteristic
123#380—21no—
124#380C22×C311yescharacteristic
125#380C22×C312yes—
126C4×C4⋊C4C22×C311yescharacteristic
127C4×C22⋊C4—21no—
128#340C22×C312yes—
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C312yes—
132#336C22×C311yescharacteristic
133#336C22×C311yescharacteristic
134#336C22×C312yes—
135C24×C3—21no—
136C24×C3C2×D411yescharacteristic
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×D4—24no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4C2×Q812yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×Q812yes—
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C3×M16—48no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8#4114yes—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4C2×Q812yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—41no—
200C3×C42—41no—
201C3×C42—42no—
202C22×Q8—21no—
203C22×D4—21no—
204C22×D4C23×C311yescharacteristic
205C23×C4—21no—
206C23×C4C3×D411yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C23×C311yescharacteristic
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—42no—
212C22×C8C23×C312yes—
213C22×C8C23×C312yes—
214C4⋊Q8—42no—
215C22⋊Q8—44no—
216#121—22no—
217#121C3×Q814yescomplemented
218C4×D4—42no—
219C4×D4—44no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C312yes—
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42—42no—
238C2×C42C3×D411yescharacteristic
239C2×C42C3×D411yescharacteristic
240C2×C42C23×C311yescharacteristic
241C2×C42C23×C311yescharacteristic
242#107—42no—
243#106—24no—
244#102—24no—
245#102—42no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3C22⋊Q812yes—
265C23×C3C22×D411yescharacteristic
266C3×Q8—84no—
267C3×D4—24no—
268C3×D4—48no—
269C3×D4—42no—
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—21no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
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—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—84no—
301C2×C3×C4C22⋊Q812yes—
302C2×C3×C4AGL(1,ℤ/8)12yes—
303C2×C3×C4AGL(1,ℤ/8)12yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×Q811yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4H(5,𝔽2)12yes—
309C3×C8—24no—
310C3×C8—42no—
311C24—21no—
312C24C2×C3×D411yescharacteristic
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4C2×C3×Q812yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×Q812yes—
345C22×C4C3×#4112yes—
346C22×C4C24×C311yescharacteristic
347M16—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8C3×#4114yes—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4C2×C3×Q812yes—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C42—22no—
374C42—22no—
375C42—41no—
376C42—41no—
377C42—42no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3C2×C22⋊Q811yescharacteristic
393C22×C3#53211yescharacteristic
394C22×C3#53411yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
397C22×C3C2×#13711yescharacteristic
398C22×C3#57511yescharacteristic
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—41no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4—82no—
409C3×C4#55112yes—
410C3×C4#55412yes—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C3×C22⋊Q812yes—
425C23C22×C3×D411yescharacteristic
426Q8—84no—
427D4—24no—
428D4—48no—
429D4—42no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4C3×C22⋊Q812yes—
462C2×C4C3×AGL(1,ℤ/8)12yes—
463C2×C4C3×AGL(1,ℤ/8)12yes—
464C2×C4C3×#13712yes—
465C2×C4C22×C3×Q811yescharacteristic
466C2×C4C2×C3×#4111yescharacteristic
467C2×C4C2×C3×#4112yes—
468C2×C4C3×H(5,𝔽2)12yes—
469C8—24no—
470C8—42no—
471C2×C3—42no—
472C2×C3—82no—
473C2×C3#256411yescharacteristic
474C2×C3#307711yescharacteristic
475C2×C3#307911yescharacteristic
476C2×C3#317111yescharacteristic
477C2×C3#318211yescharacteristic
478C2×C3#319211yescharacteristic
479C2×C3#319611yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22C2×C3×C22⋊Q811yescharacteristic
495C22C3×#53211yescharacteristic
496C22C3×#53411yescharacteristic
497C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
498C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
499C22C2×C3×#13711yescharacteristic
500C22C3×#57511yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—42no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4—82no—
511C4C3×#55112yes—
512C4C3×#55412yes—
513C3#2916211yescharacteristic, direct product factor, Sylow
514C2—42no—
515C2—82no—
516C2C3×#256411yescharacteristic
517C2C3×#307711yescharacteristic
518C2C3×#307911yescharacteristic
519C2C3×#317111yescharacteristic
520C2C3×#318211yescharacteristic
521C2C3×#319211yescharacteristic
522C2C3×#319611yescharacteristic
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
1. C3×#2857C2semidirect
3. C3×#2100C2semidirect
4. C3×#2094C2semidirect
5. C3×#2057C2semidirect
6. C3×#2023C2semidirect
7. C3×#1919C2semidirect
9. C3×#1862C2semidirect
10. C3×#1809C2semidirect
11. C3×#1728C2semidirect
12. #29162C3direct (N × H)
44. #2857C2×C3semidirect
46. #2100C2×C3semidirect
47. #2094C2×C3semidirect
48. #2057C2×C3semidirect
49. #2023C2×C3semidirect
50. #1919C2×C3semidirect
52. #1862C2×C3semidirect
53. #1809C2×C3semidirect
54. #1728C2×C3semidirect
70. C3×#121Q8semidirect
217. #121C3×Q8semidirect
513. C3#29162direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28572
C3×C4×#1212
C3×#21002
C3×#20942
C3×#20572
C3×#20232
C3×#19192
C3×#18952
C3×#18622
C3×#18092
C3×#17282
#291623