Finite Group Databasebeta

← C3×#25599 #10644206

Subgroups — C3×#25599 #10644206

Order 768.

1578 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#493C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#2856C211yesmaximal, characteristic, complemented
5C3×#1999C214yesmaximal, complemented
6C3×#1990C211yesmaximal, characteristic, complemented
7C3×#1957C211yesmaximal, characteristic, complemented
8C3×#1879C214yesmaximal, complemented
9C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
10#25599C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C22×C3×D8C2214yes—
13C3×#532—24no—
14C3×#532—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2212yes—
26C22×C3×C42C2211yescharacteristic
27C3×#493—24no—
28C3×#486—24no—
29C2×C3×#102C2214yes—
30C2×C3×#102C2214yes—
31C2×C3×C22⋊C8C2212yes—
32C2×C3×C4×C8C2211yescharacteristic
33C2×C3×C4×C8C2211yescharacteristic
34C3×#390—21no—
35C3×#390—21no—
36C3×#390C2212yes—
37C3×#384C2211yescharacteristic
38C3×#340C2214yes—
39C3×#336C2212yes—
40C2×#532C2×C311yescharacteristic, complemented
41C2×#493C2×C311yescharacteristic, complemented
42C2×#486C2×C311yescharacteristic, complemented
43#2856C2×C311yescharacteristic, complemented
44#1999C2×C314yescomplemented
45#1990C2×C311yescharacteristic, complemented
46#1957C2×C311yescharacteristic, complemented
47#1879C2×C314yescomplemented
48C4×C22⋊C8C2×C311yescharacteristic, complemented
49C2×C3×#41—28no—
50C2×C3×#41—44no—
51C2×C3×#41—44no—
52C22×C3×Q8—21no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2312yes—
61C23×C3×C4—22no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×D8—28no—
66C2×C3×D8—28no—
67C2×C3×D8—44no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yescomplemented
72C3×#127—22no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#127D414yescomplemented
76C3×#124—28no—
77C3×#121—28no—
78C3×#121—24no—
79C3×#121—24no—
80C3×#121—44no—
81C3×#121—44no—
82C3×#121—44no—
83C3×C4×D4—28no—
84C3×C4×D4—24no—
85C3×C4×D4—24no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C42—22no—
97C2×C3×C42—22no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C3×#102—28no—
103C3×#102—44no—
104C3×C22⋊C8—24no—
105C3×C4×C8—22no—
106C3×C4×C8—22no—
107C22×#41—21no—
108C22×D8C22×C314yes—
109#532—24no—
110#532—24no—
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×#127C22×C311yescharacteristic
113C2×#127C22×C311yescharacteristic
114C2×#124—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C22×C312yes—
119C2×#121C22×C312yes—
120C2×C4×D4—21no—
121C2×C4×D4C22×C312yes—
122C22×C42C22×C311yescharacteristic
123#493—24no—
124#486—24no—
125C2×#102C22×C314yes—
126C2×#102C22×C314yes—
127C2×C22⋊C8C22×C312yes—
128C2×C4×C8C22×C311yescharacteristic
129C2×C4×C8C22×C311yescharacteristic
130#390—21no—
131#390—21no—
132#390C22×C312yes—
133#384C22×C311yescharacteristic
134#340C22×C314yes—
135#336C22×C312yes—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3C2×D411yescharacteristic
139C3×#41—432no—
140C2×C3×Q8—24no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×D4—28no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—44no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—28no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4D814yescomplemented
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×D8—416no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8#4114yescomplemented
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4C2×D414yes—
203C3×C22⋊C4—28no—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—44no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214C2×#41—28no—
215C2×#41—44no—
216C2×#41—44no—
217C22×Q8—21no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C23×C312yes—
225C22×D4C23×C312yes—
226C23×C4—22no—
227C23×C4C3×D411yescharacteristic
228C23×C4C3×D411yescharacteristic
229C23×C4C23×C311yescharacteristic
230C2×D8—28no—
231C2×D8—28no—
232C2×D8—44no—
233C22×C8C23×C312yes—
234C22×C8C23×C312yes—
235C4⋊Q8—22no—
236C4⋊Q8C3×D414yescomplemented
237#127—22no—
238#127—22no—
239#127—22no—
240#127C3×D414yescomplemented
241#124—28no—
242#121—28no—
243#121—24no—
244#121—24no—
245#121—44no—
246#121—44no—
247#121—44no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C23×C312yes—
255C2×C4⋊C4C23×C312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42C3×D411yescharacteristic
264C2×C42C3×D411yescharacteristic
265C2×C42C23×C311yescharacteristic
266C2×C42C23×C311yescharacteristic
267#102—28no—
268#102—44no—
269C22⋊C8—24no—
270C4×C8—22no—
271C4×C8—22no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3C2×D812yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—48no—
295C3×D4—28no—
296C3×D4—416no—
297C3×D4—44no—
298C3×D4—44no—
299C3×D4—44no—
300C3×D4—48no—
301C3×D4—48no—
302C3×D4—88no—
303C3×D4—84no—
304C3×D4—84no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—28no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—416no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—44no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12012yes—
326C2×C3×C4C2×D812yes—
327C2×C3×C4C2×D812yes—
328C2×C3×C4#13512yes—
329C2×C3×C4AGL(1,ℤ/8)12yes—
330C2×C3×C4AGL(1,ℤ/8)12yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C4H(5,𝔽2)12yes—
335C3×C8—24no—
336C3×C8—42no—
337C24—22no—
338C24—22no—
339C24C2×C3×D411yescharacteristic
340#41—432no—
341C2×Q8—24no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4C2×C3×D414yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4C3×D814yescomplemented
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C24×C311yescharacteristic
391D8—416no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8C3×#4114yescomplemented
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4C2×C3×D414yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—22no—
410C42—22no—
411C42—21no—
412C42—21no—
413C42C2×C3×D412yes—
414C42C2×C3×D412yes—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#53211yescharacteristic
438C22×C3#53511yescharacteristic
439C22×C3C22×D811yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—44no—
446C3×C4—41no—
447C3×C4—82no—
448C3×C4#44714yes—
449C3×C4#44914yes—
450C3×C4#54612yes—
451C3×C4#55012yes—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C2×C3×D812yes—
473C23C22×C3×D411yescharacteristic
474Q8—48no—
475D4—28no—
476D4—416no—
477D4—48no—
478D4—48no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—88no—
483D4—84no—
484D4—84no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4C3×#12012yes—
505C2×C4C3×#12012yes—
506C2×C4C2×C3×D812yes—
507C2×C4C2×C3×D812yes—
508C2×C4C3×#13512yes—
509C2×C4C3×AGL(1,ℤ/8)12yes—
510C2×C4C3×AGL(1,ℤ/8)12yes—
511C2×C4C22×C3×D411yescharacteristic
512C2×C4C22×C3×D411yescharacteristic
513C2×C4C2×C3×#4112yes—
514C2×C4C3×H(5,𝔽2)12yes—
515C8—24no—
516C8—42no—
517C2×C3—42no—
518C2×C3—84no—
519C2×C3—162no—
520C2×C3#243111yescharacteristic
521C2×C3C2×#44711yescharacteristic
522C2×C3C2×#44911yescharacteristic
523C2×C3#316911yescharacteristic
524C2×C3#318411yescharacteristic
525C2×C3#320511yescharacteristic
526C2×C3#320711yescharacteristic
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22C2×C3×#12011yescharacteristic
549C22C3×#53211yescharacteristic
550C22C3×#53511yescharacteristic
551C22C22×C3×D811yescharacteristic
552C22C2×C3×#13511yescharacteristic
553C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
554C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
555C4—22no—
556C4—41no—
557C4—44no—
558C4—41no—
559C4—82no—
560C4C3×#44714yes—
561C4C3×#44914yes—
562C4C3×#54612yes—
563C4C3×#55012yes—
564C3#2559911yescharacteristic, direct product factor, Sylow
565C2—42no—
566C2—84no—
567C2—162no—
568C2C3×#243111yescharacteristic
569C2C2×C3×#44711yescharacteristic
570C2C2×C3×#44911yescharacteristic
571C2C3×#316911yescharacteristic
572C2C3×#318411yescharacteristic
573C2C3×#320511yescharacteristic
574C2C3×#320711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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. C2×C3×#532C2semidirect
2. C2×C3×#493C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#2856C2semidirect
5. C3×#1999C2semidirect
6. C3×#1990C2semidirect
7. C3×#1957C2semidirect
8. C3×#1879C2semidirect
9. C3×C4×C22⋊C8C2semidirect
10. #25599C3direct (N × H)
40. C2×#532C2×C3semidirect
41. C2×#493C2×C3semidirect
42. C2×#486C2×C3semidirect
43. #2856C2×C3semidirect
44. #1999C2×C3semidirect
45. #1990C2×C3semidirect
46. #1957C2×C3semidirect
47. #1879C2×C3semidirect
48. C4×C22⋊C8C2×C3semidirect
71. C3×C4⋊Q8D4semidirect
75. C3×#127D4semidirect
183. C22×C3×C4D8semidirect
196. C2×C3×C8#41semidirect
236. C4⋊Q8C3×D4semidirect
240. #127C3×D4semidirect
384. C22×C4C3×D8semidirect
397. C2×C8C3×#41semidirect
564. C3#25599direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4932
C2×C3×#4862
C3×#28562
C3×#19992
C3×#19902
C3×#19572
C3×#18792
C3×C4×C22⋊C82
#255993