Finite Group Databasebeta

← C3×#25598 #10644205

Subgroups — C3×#25598 #10644205

Order 768.

1298 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
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#2856C211yesmaximal, characteristic
5C3×#2000C214yesmaximal, complemented
6C3×#1990C211yesmaximal, characteristic, complemented
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1880C214yesmaximal, complemented
9C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
10#25598C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2214yes—
12C3×#533—24no—
13C3×#533—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×C4×Q8—21no—
24C2×C3×C4×Q8C2212yes—
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#492—24no—
28C3×#486—24no—
29C2×C3×#103C2214yes—
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×#533C2×C311yescharacteristic, complemented
41C2×#492C2×C311yescharacteristic, complemented
42C2×#486C2×C311yescharacteristic, complemented
43#2856C2×C311yescharacteristic
44#2000C2×C314yescomplemented
45#1990C2×C311yescharacteristic, complemented
46#1956C2×C311yescharacteristic, complemented
47#1880C2×C314yescomplemented
48C4×C22⋊C8C2×C311yescharacteristic, complemented
49C22×C3×Q8—21no—
50C22×C3×Q8C2312yes—
51C22×C3×D4—22no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4C2312yes—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×SD16—28no—
59C2×C3×SD16—28no—
60C2×C3×SD16—44no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yescomplemented
66C3×C4⋊Q8D414yescomplemented
67C3×#127—22no—
68C3×#127—22no—
69C3×#125—28no—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×#121—44no—
74C3×#121—44no—
75C3×#121—44no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×#117—28no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C42—22no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42D411yescharacteristic
97C2×C3×C42D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C3×#103—44no—
101C3×#102—28no—
102C3×C22⋊C8—24no—
103C3×C4×C8—22no—
104C3×C4×C8—22no—
105C22×SD16C22×C314yes—
106#533—24no—
107#533—24no—
108C2×C4⋊Q8C22×C311yescharacteristic
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×#127C22×C311yescharacteristic
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C22×C312yes—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C22×C312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C22×C312yes—
119C2×#117—21no—
120C22×C42C22×C311yescharacteristic
121#492—24no—
122#486—24no—
123C2×#103C22×C314yes—
124C2×#102C22×C314yes—
125C2×C22⋊C8C22×C312yes—
126C2×C4×C8C22×C311yescharacteristic
127C2×C4×C8C22×C311yescharacteristic
128#390—21no—
129#390—21no—
130#390C22×C312yes—
131#384C22×C311yescharacteristic
132#340C22×C314yes—
133#336C22×C312yes—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×Q8C2×D414yes—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4SD1614yescomplemented
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×SD16—416no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8#4114yes—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D414yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C42—28no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203C22×Q8—21no—
204C22×Q8C23×C312yes—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C23×C312yes—
209C23×C4C3×D411yescharacteristic
210C23×C4C3×D411yescharacteristic
211C23×C4C23×C311yescharacteristic
212C2×SD16—28no—
213C2×SD16—28no—
214C2×SD16—44no—
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8C3×D414yescomplemented
220C4⋊Q8C3×D414yescomplemented
221#127—22no—
222#127—22no—
223#125—28no—
224C22⋊Q8—28no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227#121—44no—
228#121—44no—
229#121—44no—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233#117—28no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C23×C312yes—
242C2×C4⋊C4C23×C312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C3×D411yescharacteristic
251C2×C42C3×D411yescharacteristic
252C2×C42C23×C311yescharacteristic
253C2×C42C23×C311yescharacteristic
254#103—44no—
255#102—28no—
256C22⋊C8—24no—
257C4×C8—22no—
258C4×C8—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3C2×SD1612yes—
273C23×C3C22×D411yescharacteristic
274C3×Q8—28no—
275C3×Q8—44no—
276C3×Q8—48no—
277C3×D4—44no—
278C3×D4—44no—
279C3×D4—88no—
280C3×D4—84no—
281C3×D4—84no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—28no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4#12012yes—
304C2×C3×C4#12012yes—
305C2×C3×C4C2×SD1612yes—
306C2×C3×C4C2×SD1612yes—
307C2×C3×C4#13512yes—
308C2×C3×C4AGL(1,ℤ/8)12yes—
309C2×C3×C4#13712yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×C3×C4H(5,𝔽2)12yes—
314C3×C8—24no—
315C3×C8—42no—
316C24—22no—
317C24C2×C3×D411yescharacteristic
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8C2×C3×D414yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4C3×SD1614yescomplemented
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C24×C311yescharacteristic
357SD16—416no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8C3×#4114yes—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4C2×C3×D414yes—
374C22⋊C4—28no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C42—28no—
379C42—22no—
380C42—22no—
381C42—21no—
382C42—21no—
383C42C2×C3×D412yes—
384C42C2×C3×D412yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53211yescharacteristic
401C22×C3#53511yescharacteristic
402C22×C3C22×SD1611yescharacteristic
403C22×C3C2×#13511yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C3×C4—22no—
407C3×C4—41no—
408C3×C4—44no—
409C3×C4—41no—
410C3×C4—44no—
411C3×C4—82no—
412C3×C4#44814yes—
413C3×C4#44914yes—
414C3×C4#54612yes—
415C3×C4#55012yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×C3×SD1612yes—
430C23C22×C3×D411yescharacteristic
431Q8—28no—
432Q8—48no—
433Q8—44no—
434D4—44no—
435D4—44no—
436D4—88no—
437D4—84no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4C3×#12012yes—
461C2×C4C3×#12012yes—
462C2×C4C2×C3×SD1612yes—
463C2×C4C2×C3×SD1612yes—
464C2×C4C3×#13512yes—
465C2×C4C3×AGL(1,ℤ/8)12yes—
466C2×C4C3×#13712yes—
467C2×C4C22×C3×D411yescharacteristic
468C2×C4C22×C3×D411yescharacteristic
469C2×C4C2×C3×#4112yes—
470C2×C4C3×H(5,𝔽2)12yes—
471C8—24no—
472C8—42no—
473C2×C3—42no—
474C2×C3—162no—
475C2×C3#243111yescharacteristic
476C2×C3C2×#44811yescharacteristic
477C2×C3C2×#44911yescharacteristic
478C2×C3#316711yescharacteristic
479C2×C3#318211yescharacteristic
480C2×C3#320811yescharacteristic
481C2×C3#321011yescharacteristic
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22C2×C3×#12011yescharacteristic
497C22C3×#53211yescharacteristic
498C22C3×#53511yescharacteristic
499C22C22×C3×SD1611yescharacteristic
500C22C2×C3×#13511yescharacteristic
501C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
502C22C2×C3×#13711yescharacteristic
503C4—22no—
504C4—41no—
505C4—44no—
506C4—41no—
507C4—44no—
508C4—82no—
509C4C3×#44814yes—
510C4C3×#44914yes—
511C4C3×#54612yes—
512C4C3×#55012yes—
513C3#2559811yescharacteristic, direct product factor, Sylow
514C2—42no—
515C2—162no—
516C2C3×#243111yescharacteristic
517C2C2×C3×#44811yescharacteristic
518C2C2×C3×#44911yescharacteristic
519C2C3×#316711yescharacteristic
520C2C3×#318211yescharacteristic
521C2C3×#320811yescharacteristic
522C2C3×#321011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
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×#533C2semidirect
2. C2×C3×#492C2semidirect
3. C2×C3×#486C2semidirect
5. C3×#2000C2semidirect
6. C3×#1990C2semidirect
7. C3×#1956C2semidirect
8. C3×#1880C2semidirect
9. C3×C4×C22⋊C8C2semidirect
10. #25598C3direct (N × H)
40. C2×#533C2×C3semidirect
41. C2×#492C2×C3semidirect
42. C2×#486C2×C3semidirect
44. #2000C2×C3semidirect
45. #1990C2×C3semidirect
46. #1956C2×C3semidirect
47. #1880C2×C3semidirect
48. C4×C22⋊C8C2×C3semidirect
65. C3×C4⋊Q8D4semidirect
66. C3×C4⋊Q8D4semidirect
168. C22×C3×C4SD16semidirect
219. C4⋊Q8C3×D4semidirect
220. C4⋊Q8C3×D4semidirect
350. C22×C4C3×SD16semidirect
513. C3#25598direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4922
C2×C3×#4862
C3×#28562
C3×#20002
C3×#19902
C3×#19562
C3×#18802
C3×C4×C22⋊C82
#255983