Finite Group Databasebeta

← C3×#24339 #10642946

Subgroups — C3×#24339 #10642946

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3C2×C3×#442C211yesmaximal, characteristic
4C3×#2580C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2025C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal
10C3×#1862C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24339C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C3×#554—24no—
15C3×#554—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C3×#474—24no—
26C3×#442—24no—
27C2×C3×#107C2212yes—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#399—21no—
36C3×#393—21no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#385—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#554C2×C311yescharacteristic, complemented
47C2×#474C2×C311yescharacteristic, complemented
48C2×#442C2×C311yescharacteristic
49#2580C2×C311yescharacteristic, complemented
50#2059C2×C312yescomplemented
51#2025C2×C312yescomplemented
52#1993C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1879C2×C312yes—
55#1862C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C23×C3×C4—21no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8Q814yescomplemented
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yes—
72C3×#125—24no—
73C3×#125—24no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—216no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4D414yes—
84C2×C3×C4⋊C4—28no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C42C2312yes—
101C2×C3×C42C2311yescharacteristic
102C3×#107—24no—
103C3×#106—22no—
104C3×C4⋊C8—22no—
105C3×#102—22no—
106C3×#102—24no—
107C3×#102—22no—
108C3×#102—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—22no—
113C22×D8C22×C311yescharacteristic
114#554—24no—
115#554—24no—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#125C22×C312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×D4C22×C311yescharacteristic
122C2×C4×D4C22×C312yes—
123C2×C4×D4C22×C311yescharacteristic
124C22×C4⋊C4—21no—
125#474—24no—
126#442—24no—
127C2×#107C22×C312yes—
128C2×#106C22×C311yescharacteristic
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#102C22×C312yes—
131C2×#102C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#97C22×C311yescharacteristic
135#399—21no—
136#393—21no—
137#390C22×C312yes—
138#389C22×C312yes—
139#385—22no—
140#382C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C24×C3—21no—
147C24×C3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—42no—
151C2×C3×Q8—82no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4C2×D412yes—
161C2×C3×D4C2×Q812yes—
162C2×C3×D4C2×Q812yes—
163C2×C3×D4#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—28no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—44no—
171C22×C3×C4—42no—
172C22×C3×C4—82no—
173C22×C3×C4—82no—
174C22×C3×C4—82no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C3×D8—28no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
189C2×C3×C8C2×Q812yes—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—82no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D414yes—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—82no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C22×Q8—21no—
214C22×Q8C3×D411yescharacteristic
215C22×D4C23×C311yescharacteristic
216C22×D4C23×C311yescharacteristic
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C2×D8—22no—
221C2×D8—22no—
222C2×D8C3×Q814yescomplemented
223C22×C8C23×C311yescharacteristic
224C22×C8C23×C312yes—
225C22×C8C23×C311yescharacteristic
226C4⋊Q8—22no—
227C4⋊Q8C3×D414yes—
228#125—24no—
229#125—24no—
230C22⋊Q8—28no—
231C22⋊Q8—216no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4C3×D414yes—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C3×D411yescharacteristic
246C2×C4⋊C4C3×D412yes—
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C4⋊C4C23×C312yes—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42C23×C312yes—
257C2×C42C23×C311yescharacteristic
258#107—24no—
259#106—22no—
260C4⋊C8—22no—
261#102—22no—
262#102—24no—
263#102—22no—
264#102—24no—
265#97—22no—
266#95—21no—
267#95—21no—
268#95—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C3×Q8—44no—
281C3×Q8—84no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4C22⋊Q814yes—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—48no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—84no—
310C2×C3×C4—88no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12012yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×Q811yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319C3×C8—22no—
320C3×C8—41no—
321C3×C8—42no—
322C24—21no—
323C24—21no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—42no—
327C2×Q8—82no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4C2×C3×D412yes—
337C2×D4C2×C3×Q812yes—
338C2×D4C2×C3×Q812yes—
339C2×D4C3×#4112yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C24×C311yescharacteristic
356D8—28no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8C2×C3×Q812yes—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—82no—
378C4⋊C4C2×C3×D412yes—
379C4⋊C4C2×C3×D414yes—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42C2×C3×D412yes—
388C42C2×C3×D412yes—
389C22×C3—24no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3D4×Q811yescharacteristic
402C22×C3#57512yes—
403C22×C3#57612yes—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4—82no—
411C3×C4—81no—
412C3×C4C2×C22⋊Q811yescharacteristic
413C3×C4#53611yescharacteristic
414C3×C4#54811yescharacteristic
415C3×C4D4×Q811yescharacteristic
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C22×C3×D411yescharacteristic
427Q8—44no—
428Q8—84no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4C3×C22⋊Q814yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4—88no—
458C2×C4C3×#12012yes—
459C2×C4C3×#12012yes—
460C2×C4C22×C3×D411yescharacteristic
461C2×C4C22×C3×D411yescharacteristic
462C2×C4C22×C3×Q811yescharacteristic
463C2×C4C2×C3×#4111yescharacteristic
464C2×C4C3×2−1+411yescharacteristic
465C2×C4C3×2−1+411yescharacteristic
466C8—22no—
467C8—41no—
468C8—42no—
469C2×C3—44no—
470C2×C3—82no—
471C2×C3#240911yescharacteristic
472C2×C3#300511yescharacteristic
473C2×C3#300811yescharacteristic
474C2×C3#337012yes—
475C2×C3#338912yes—
476C22—24no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22C2×C3×#12011yescharacteristic
487C22C3×#54811yescharacteristic
488C22C3×D4×Q811yescharacteristic
489C22C3×#57512yes—
490C22C3×#57612yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—81no—
498C4—82no—
499C4C2×C3×C22⋊Q811yescharacteristic
500C4C3×#53611yescharacteristic
501C4C3×#54811yescharacteristic
502C4C3×D4×Q811yescharacteristic
503C3#2433911yescharacteristic, direct product factor, Sylow
504C2—44no—
505C2—82no—
506C2C3×#240911yescharacteristic
507C2C3×#300511yescharacteristic
508C2C3×#300811yescharacteristic
509C2C3×#337012yes—
510C2C3×#338912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#554C2semidirect
2. C2×C3×#474C2semidirect
4. C3×#2580C2semidirect
5. C3×#2059C2semidirect
6. C3×#2025C2semidirect
7. C3×#1993C2semidirect
8. C3×#1985C2semidirect
10. C3×#1862C2semidirect
11. C3×#1804C2semidirect
12. #24339C3direct (N × H)
46. C2×#554C2×C3semidirect
47. C2×#474C2×C3semidirect
49. #2580C2×C3semidirect
50. #2059C2×C3semidirect
51. #2025C2×C3semidirect
52. #1993C2×C3semidirect
53. #1985C2×C3semidirect
55. #1862C2×C3semidirect
56. #1804C2×C3semidirect
66. C2×C3×D8Q8semidirect
222. C2×D8C3×Q8semidirect
503. C3#24339direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#4742
C2×C3×#4422
C3×#25802
C3×#20592
C3×#20252
C3×#19932
C3×#19852
C3×#18792
C3×#18622
C3×#18042
#243393