Finite Group Databasebeta

← C3×#25567 #10644174

Subgroups — C3×#25567 #10644174

Order 768.

1522 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×C3×#486C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C3×#2541C211yesmaximal, characteristic, complemented
5C3×#1999C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1874C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25567C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C22×C3×D8C2212yes—
13C3×#547—24no—
14C3×#547—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127—21no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×#121C2212yes—
21C2×C3×#121C2214yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4C2212yes—
25C3×#486—24no—
26C3×#459—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2214yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C22⋊C8C2214yes—
32C2×C3×C4×C8C2211yescharacteristic
33C3×#392—22no—
34C3×#390—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2214yes—
37C3×#340C2212yes—
38C2×#547C2×C311yescharacteristic, complemented
39C2×#486C2×C311yescharacteristic, complemented
40C2×#459C2×C311yescharacteristic, complemented
41#2541C2×C311yescharacteristic, complemented
42#1999C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1874C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C2×C3×#41—28no—
48C2×C3×#41—44no—
49C2×C3×#41—44no—
50C22×C3×Q8—21no—
51C22×C3×D4—22no—
52C22×C3×D4—24no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4D411yescharacteristic
56C22×C3×D4C2312yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C23×C3×C4C2312yes—
61C2×C3×D8—24no—
62C2×C3×D8—44no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yescomplemented
67C3×#127—22no—
68C3×#127—22no—
69C3×#127—44no—
70C3×#123—216no—
71C3×C22⋊Q8—28no—
72C3×#121—24no—
73C3×#121—24no—
74C3×#121—28no—
75C3×#121—44no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4D414yescomplemented
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C3×C4⋊C8—22no—
95C3×#102—24no—
96C3×#102—28no—
97C3×#102—24no—
98C3×C22⋊C8—44no—
99C3×C4×C8—22no—
100C3×#95—21no—
101C22×#41—21no—
102C22×D8C22×C312yes—
103#547—24no—
104#547—24no—
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×#127—21no—
107C2×#127C22×C311yescharacteristic
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×#121C22×C312yes—
111C2×#121C22×C314yes—
112C2×C4×D4C22×C311yescharacteristic
113C2×C4×D4C22×C311yescharacteristic
114C22×C4⋊C4C22×C312yes—
115#486—24no—
116#459—24no—
117C2×C4⋊C8C22×C311yescharacteristic
118C2×#102C22×C314yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×C22⋊C8C22×C314yes—
122C2×C4×C8C22×C311yescharacteristic
123#392—22no—
124#390—21no—
125C4×C4⋊C4C22×C311yescharacteristic
126#340C22×C312yes—
127#340C22×C314yes—
128C24×C3—21no—
129C24×C3—22no—
130C24×C3C2×D412yes—
131C3×#41—432no—
132C2×C3×Q8—24no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×D4—28no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—48no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—48no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—82no—
154C2×C3×D4—84no—
155C2×C3×D4D814yescomplemented
156C2×C3×D4C2×D412yes—
157C22×C3×C4—28no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—44no—
167C22×C3×C4—44no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C3×D8—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—24no—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D414yes—
188C3×C4⋊C4#4114yescomplemented
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—48no—
193C3×C42—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196C2×#41—28no—
197C2×#41—44no—
198C2×#41—44no—
199C22×Q8—21no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C3×D411yescharacteristic
205C22×D4C23×C312yes—
206C22×D4C23×C311yescharacteristic
207C23×C4—22no—
208C23×C4—21no—
209C23×C4C23×C312yes—
210C2×D8—24no—
211C2×D8—44no—
212C22×C8C23×C312yes—
213C22×C8C23×C312yes—
214C4⋊Q8—22no—
215C4⋊Q8C3×D414yescomplemented
216#127—22no—
217#127—22no—
218#127—44no—
219#123—216no—
220C22⋊Q8—28no—
221#121—24no—
222#121—24no—
223#121—28no—
224#121—44no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4C3×D414yescomplemented
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4C3×D411yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C312yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4C3×D412yes—
241C2×C42—21no—
242C2×C42C23×C311yescharacteristic
243C4⋊C8—22no—
244#102—24no—
245#102—28no—
246#102—24no—
247C22⋊C8—44no—
248C4×C8—22no—
249#95—21no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—44no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3C2×D814yes—
265C23×C3C22×D411yescharacteristic
266C3×Q8—48no—
267C3×D4—22no—
268C3×D4—22no—
269C3×D4—28no—
270C3×D4—416no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—44no—
275C3×D4—84no—
276C3×D4—816no—
277C3×D4—84no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—416no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—48no—
295C2×C3×C4#12012yes—
296C2×C3×C4#12012yes—
297C2×C3×C4C2×D812yes—
298C2×C3×C4AGL(1,ℤ/8)12yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C4H(5,𝔽2)12yes—
303C3×C8—42no—
304C3×C8—42no—
305C24—21no—
306C24—22no—
307C24C2×C3×D412yes—
308#41—432no—
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×D4—28no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—48no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—82no—
331C2×D4—84no—
332C2×D4C3×D814yescomplemented
333C2×D4C2×C3×D412yes—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C24×C311yescharacteristic
351D8—48no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4C2×C3×D412yes—
364C4⋊C4C2×C3×D414yes—
365C4⋊C4C3×#4114yescomplemented
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—48no—
370C42—42no—
371C42C2×C3×D412yes—
372C42C2×C3×D412yes—
373C22×C3—24no—
374C22×C3—28no—
375C22×C3—44no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—84no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—84no—
388C22×C3C2×#12011yescharacteristic
389C22×C3#54612yes—
390C22×C3C22×D811yescharacteristic
391C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
392C22×C3#57712yes—
393C3×C4—22no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—44no—
397C3×C4—44no—
398C3×C4#44714yes—
399C3×C4#53211yescharacteristic
400C3×C4#53511yescharacteristic
401C3×C4#55012yes—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23C2×C3×D814yes—
417C23C22×C3×D411yescharacteristic
418Q8—48no—
419D4—28no—
420D4—22no—
421D4—22no—
422D4—416no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—44no—
427D4—84no—
428D4—816no—
429D4—84no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—416no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4C3×#12012yes—
448C2×C4C3×#12012yes—
449C2×C4C2×C3×D812yes—
450C2×C4C3×AGL(1,ℤ/8)12yes—
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C22×C3×D411yescharacteristic
453C2×C4C2×C3×#4112yes—
454C2×C4C3×H(5,𝔽2)12yes—
455C8—42no—
456C8—42no—
457C2×C3—44no—
458C2×C3—82no—
459C2×C3—162no—
460C2×C3#243111yescharacteristic
461C2×C3C2×#44711yescharacteristic
462C2×C3#301011yescharacteristic
463C2×C3#329412yes—
464C2×C3#333212yes—
465C22—24no—
466C22—28no—
467C22—42no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—84no—
479C22—84no—
480C22C2×C3×#12011yescharacteristic
481C22C3×#54612yes—
482C22C22×C3×D811yescharacteristic
483C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
484C22C3×#57712yes—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—44no—
489C4—44no—
490C4C3×#44714yes—
491C4C3×#53211yescharacteristic
492C4C3×#53511yescharacteristic
493C4C3×#55012yes—
494C3#2556711yescharacteristic, direct product factor, Sylow
495C2—44no—
496C2—82no—
497C2—162no—
498C2C3×#243111yescharacteristic
499C2C2×C3×#44711yescharacteristic
500C2C3×#301011yescharacteristic
501C2C3×#329412yes—
502C2C3×#333212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C2×C3×#486C2semidirect
3. C2×C3×#459C2semidirect
4. C3×#2541C2semidirect
5. C3×#1999C2semidirect
6. C3×#1988C2semidirect
7. C3×#1954C2semidirect
8. C3×#1874C2semidirect
9. C3×C4×#102C2semidirect
10. #25567C3direct (N × H)
38. C2×#547C2×C3semidirect
39. C2×#486C2×C3semidirect
40. C2×#459C2×C3semidirect
41. #2541C2×C3semidirect
42. #1999C2×C3semidirect
43. #1988C2×C3semidirect
44. #1954C2×C3semidirect
45. #1874C2×C3semidirect
46. C4×#102C2×C3semidirect
66. C3×C4⋊Q8D4semidirect
79. C3×C4×D4D4semidirect
155. C2×C3×D4D8semidirect
188. C3×C4⋊C4#41semidirect
215. C4⋊Q8C3×D4semidirect
228. C4×D4C3×D4semidirect
332. C2×D4C3×D8semidirect
365. C4⋊C4C3×#41semidirect
494. C3#25567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4862
C2×C3×#4592
C3×#25412
C3×#19992
C3×#19882
C3×#19542
C3×#18742
C3×C4×#1022
#255673