Finite Group Databasebeta

← C3×#29330 #10647937

Subgroups — C3×#29330 #10647937

Order 768.

880 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2850C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2042C212yesmaximal, complemented
5C3×#1934C212yesmaximal, complemented
6C3×#1933C212yesmaximal, complemented
7C3×#1874C211yesmaximal, characteristic
8C3×#1874C211yesmaximal, characteristic
9C3×#1872C211yesmaximal, characteristic, complemented
10C3×#1872C211yesmaximal, characteristic, complemented
11C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
12#29330C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2212yes—
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389—21no—
30C3×#389—21no—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C3×#388—21no—
34C3×#388—21no—
35C3×#386—21no—
36C3×#386—21no—
37C3×#382C2211yescharacteristic
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44#2850C2×C311yescharacteristic
45C2×#389C2×C311yescharacteristic, complemented
46#2101C2×C312yescomplemented
47#2042C2×C312yescomplemented
48#1934C2×C312yescomplemented
49#1933C2×C312yescomplemented
50#1874C2×C311yescharacteristic
51#1874C2×C311yescharacteristic
52#1872C2×C311yescharacteristic, complemented
53#1872C2×C311yescharacteristic, complemented
54C4×C22⋊C8C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#124—42no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×#121—22no—
69C3×#121—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4—44no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
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—42no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×#103—24no—
94C3×#103—42no—
95C3×#102—24no—
96C3×#102—42no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8—22no—
99C3×C4×C8—42no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C2×#124C22×C311yescharacteristic
107C2×C22⋊Q8C22×C311yescharacteristic
108C2×#121C22×C311yescharacteristic
109C22×C4⋊C4—21no—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C42C22×C311yescharacteristic
113C2×#103C22×C312yes—
114C2×#103C22×C312yes—
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×C22⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C4×C8C22×C312yes—
120#391C22×C311yescharacteristic
121#390C22×C311yescharacteristic
122#389—21no—
123#389—21no—
124#389C22×C312yes—
125#389C22×C312yes—
126#388—21no—
127#388—21no—
128#386—21no—
129#386—21no—
130#382C22×C311yescharacteristic
131C2×#95—21no—
132#340C22×C312yes—
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#336C22×C312yes—
137C24×C3—21no—
138C24×C3C2×D411yescharacteristic
139C2×C3×Q8—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—44no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—44no—
195C3×C4⋊C4#4114yes—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—41no—
205C3×C42—41no—
206C22×Q8C23×C311yescharacteristic
207C22×D4—21no—
208C22×D4C23×C311yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C3×D411yescharacteristic
212C23×C4C3×D411yescharacteristic
213C23×C4C23×C311yescharacteristic
214C22×C8C23×C312yes—
215C22×C8C23×C312yes—
216#124—42no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219#121—22no—
220#121—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C312yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42—42no—
240C2×C42C3×D411yescharacteristic
241C2×C42C3×D411yescharacteristic
242C2×C42C23×C311yescharacteristic
243C2×C42C23×C311yescharacteristic
244#103—24no—
245#103—42no—
246#102—24no—
247#102—42no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C4×C8—42no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3#12412yes—
269C23×C3#13512yes—
270C23×C3#13512yes—
271C23×C3C22×D411yescharacteristic
272C3×Q8—42no—
273C3×Q8—42no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—84no—
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—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—48no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—84no—
310C2×C3×C4#12412yes—
311C2×C3×C4AGL(1,ℤ/8)12yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)12yes—
317C3×C8—42no—
318C3×C8—42no—
319C24—21no—
320C24C2×C3×D411yescharacteristic
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—22no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C3×#4112yes—
357C22×C4C3×#4112yes—
358C22×C4C24×C311yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4C3×#4114yes—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3#48814yes—
402C22×C3#52511yescharacteristic
403C22×C3C2×#12411yescharacteristic
404C22×C3#53411yescharacteristic
405C22×C3C2×#13511yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—44no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4—82no—
419C3×C4#55312yes—
420C3×C4#55312yes—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23C3×#12412yes—
433C23C3×#13512yes—
434C23C3×#13512yes—
435C23C22×C3×D411yescharacteristic
436Q8—42no—
437Q8—42no—
438D4—42no—
439D4—42no—
440D4—84no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4C3×#12412yes—
475C2×C4C3×AGL(1,ℤ/8)12yes—
476C2×C4C3×#13712yes—
477C2×C4C2×C3×#4112yes—
478C2×C4C2×C3×#4111yescharacteristic
479C2×C4C2×C3×#4111yescharacteristic
480C2×C4C3×H(5,𝔽2)12yes—
481C8—42no—
482C8—42no—
483C2×C3—24no—
484C2×C3—161no—
485C2×C3#259111yescharacteristic
486C2×C3#310511yescharacteristic
487C2×C3#310811yescharacteristic
488C2×C3C2×#48811yescharacteristic
489C2×C3#314111yescharacteristic
490C2×C3#318811yescharacteristic
491C2×C3#319011yescharacteristic
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22C3×#48814yes—
506C22C3×#52511yescharacteristic
507C22C2×C3×#12411yescharacteristic
508C22C3×#53411yescharacteristic
509C22C2×C3×#13511yescharacteristic
510C22C2×C3×#13511yescharacteristic
511C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
512C22C2×C3×#13711yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—44no—
519C4—81no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4C3×#55312yes—
524C4C3×#55312yes—
525C3#2933011yescharacteristic, direct product factor, Sylow
526C2—24no—
527C2—161no—
528C2C3×#259111yescharacteristic
529C2C3×#310511yescharacteristic
530C2C3×#310811yescharacteristic
531C2C2×C3×#48811yescharacteristic
532C2C3×#314111yescharacteristic
533C2C3×#318811yescharacteristic
534C2C3×#319011yescharacteristic
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
2. C2×C3×#389C2semidirect
3. C3×#2101C2semidirect
4. C3×#2042C2semidirect
5. C3×#1934C2semidirect
6. C3×#1933C2semidirect
9. C3×#1872C2semidirect
10. C3×#1872C2semidirect
11. C3×C4×C22⋊C8C2semidirect
12. #29330C3direct (N × H)
45. C2×#389C2×C3semidirect
46. #2101C2×C3semidirect
47. #2042C2×C3semidirect
48. #1934C2×C3semidirect
49. #1933C2×C3semidirect
52. #1872C2×C3semidirect
53. #1872C2×C3semidirect
54. C4×C22⋊C8C2×C3semidirect
525. C3#29330direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28502
C2×C3×#3892
C3×#21012
C3×#20422
C3×#19342
C3×#19332
C3×#18742
C3×#18722
C3×C4×C22⋊C82
#293303