Finite Group Databasebeta

← C3×#25772 #10644379

Subgroups — C3×#25772 #10644379

Order 768.

1618 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×#3496C211yesmaximal, characteristic
2C2×C3×#458C214yesmaximal, complemented
3C3×#2501C212yesmaximal, complemented
4C3×#2486C211yesmaximal, characteristic, complemented
5C3×#2022C214yesmaximal, complemented
6C3×#1964C212yesmaximal, complemented
7C3×C4×#99C211yesmaximal, characteristic, complemented
8#25772C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×H(5,𝔽2)C2212yes—
10C3×#547—22no—
11C3×#546—24no—
12C3×D42—22no—
13C3×#537—21no—
14C3×#534—21no—
15C3×#534C2212yes—
16C3×#532—21no—
17C2×C3×#123C2212yes—
18C2×C3×#123C2218yescomplemented
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic, complemented
23C3×#458—216no—
24C2×C3×#99C2212yes—
25C2×C3×#99C2218yescomplemented
26C3×#399—24no—
27C3×#398—21no—
28C3×#397—21no—
29C3×#390—22no—
30C3×#389—21no—
31C3×#384—22no—
32C3×#381C2212yescomplemented
33C3×#380—21no—
34C3×#380C2211yescharacteristic, complemented
35C3×C4×C22⋊C4C2212yescomplemented
36C3×#342C2212yescomplemented
37#3496C2×C311yescharacteristic
38C2×#458C2×C314yescomplemented
39#2501C2×C312yescomplemented
40#2486C2×C311yescharacteristic, complemented
41#2022C2×C314yescomplemented
42#1964C2×C312yescomplemented
43C4×#99C2×C311yescharacteristic, complemented
44C3×H(5,𝔽2)—28no—
45C3×H(5,𝔽2)—44no—
46C2×C3×#41—22no—
47C2×C3×#41—24no—
48C22×C3×D4—22no—
49C22×C3×D4—24no—
50C22×C3×D4—21no—
51C22×C3×D4D412yes—
52C22×C3×D4C2314yes—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C3×C4⋊Q8—21no—
58C3×#127—21no—
59C3×#124—22no—
60C3×#123—24no—
61C3×#123—24no—
62C3×#123—28no—
63C3×#123—216no—
64C3×#123—48no—
65C3×C22⋊Q8—24no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#121—24no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#121—22no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#120—24no—
76C3×#120—24no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—28no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—28no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D414yes—
96C2×C3×C22⋊C4C2312yes—
97C2×C3×C22⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C42—22no—
100C2×C3×C42D411yescharacteristic
101C3×#99—216no—
102C3×#99D418yescomplemented
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—22no—
107C2×H(5,𝔽2)C22×C312yes—
108#547—22no—
109#546—24no—
110D42—22no—
111#537—21no—
112#534—21no—
113#534C22×C312yes—
114#532—21no—
115C2×#123C22×C312yes—
116C2×#123C22×C318yescomplemented
117C2×#121—21no—
118C2×#121C22×C312yes—
119C2×C4×D4C22×C311yescharacteristic
120C22×C4⋊C4C22×C311yescharacteristic, complemented
121#458—216no—
122C2×#99C22×C312yes—
123C2×#99C22×C318yescomplemented
124#399—24no—
125#398—21no—
126#397—21no—
127#390—22no—
128#389—21no—
129#384—22no—
130#381C22×C312yescomplemented
131#380—21no—
132#380C22×C311yescharacteristic, complemented
133C4×C22⋊C4C22×C312yescomplemented
134#342C22×C312yescomplemented
135C24×C3—22no—
136C24×C3C2×D414yes—
137C24×C3C2×D411yescharacteristic
138C24×C3C2411yescharacteristic
139C3×#41—416no—
140C3×#41—44no—
141C3×#41—44no—
142C2×C3×Q8—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—28no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—44no—
156C2×C3×D4—416no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4—82no—
160C2×C3×D4C2×D414yes—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—416no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—41no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4—44no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—416no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4C2×D418yescomplemented
203C3×C42—21no—
204C3×C42—21no—
205C3×C42—44no—
206H(5,𝔽2)—28no—
207H(5,𝔽2)—44no—
208C2×#41—22no—
209C2×#41—24no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4C3×D412yes—
214C22×D4C23×C314yes—
215C22×D4C23×C311yescharacteristic
216C23×C4C3×D411yescharacteristic
217C23×C4C23×C312yes—
218C23×C4C23×C311yescharacteristic
219C4⋊Q8—21no—
220#127—21no—
221#124—22no—
222#123—24no—
223#123—24no—
224#123—28no—
225#123—216no—
226#123—48no—
227C22⋊Q8—24no—
228#121—22no—
229#121—22no—
230#121—24no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#120—24no—
238#120—24no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C3×D411yescharacteristic
252C2×C4⋊C4C3×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4C3×D411yescharacteristic
257C2×C22⋊C4C3×D414yes—
258C2×C22⋊C4C23×C312yes—
259C2×C22⋊C4C23×C311yescharacteristic
260C2×C22⋊C4C23×C314yes—
261C2×C42—22no—
262C2×C42C3×D411yescharacteristic
263#99—216no—
264#99C3×D418yescomplemented
265#95—22no—
266#95—22no—
267#95—21no—
268#95—22no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—24no—
272C23×C3—28no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—82no—
281C23×C3—82no—
282C23×C3#12712yes—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D412yes—
285C23×C3C22×D412yes—
286C23×C3H(5,𝔽2)12yes—
287C3×Q8—44no—
288C3×D4—28no—
289C3×D4—44no—
290C3×D4—48no—
291C3×D4—44no—
292C3×D4—44no—
293C3×D4—48no—
294C3×D4—44no—
295C3×D4—82no—
296C3×D4—82no—
297C3×D4—82no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—24no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—48no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—48no—
323C2×C3×C4—81no—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12012yes—
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12712yes—
328C24—22no—
329C24C2×C3×D414yes—
330C24C2×C3×D411yescharacteristic
331C24C24×C311yescharacteristic
332#41—416no—
333#41—44no—
334#41—44no—
335C2×Q8—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—28no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—416no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—82no—
353C2×D4C2×C3×D414yes—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C2×C3×D414yes—
367C22×C4C2×C3×D412yes—
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—416no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—44no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—416no—
394C22⋊C4—44no—
395C22⋊C4C2×C3×D418yescomplemented
396C42—21no—
397C42—21no—
398C42—44no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—81no—
413C22×C3—88no—
414C22×C3—164no—
415C22×C3C2×#12011yescharacteristic
416C22×C3C2×#12711yescharacteristic
417C22×C3#53511yescharacteristic
418C22×C3D4214yes—
419C3×C4—22no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—48no—
423C3×C4—44no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—82no—
427C3×C4—82no—
428C3×C4#45812yescomplemented
429C23—21no—
430C23—21no—
431C23—24no—
432C23—28no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—82no—
441C23—82no—
442C23C3×#12712yes—
443C23C22×C3×D411yescharacteristic
444C23C22×C3×D412yes—
445C23C22×C3×D412yes—
446C23C3×H(5,𝔽2)12yes—
447Q8—44no—
448D4—28no—
449D4—44no—
450D4—48no—
451D4—44no—
452D4—44no—
453D4—48no—
454D4—44no—
455D4—82no—
456D4—82no—
457D4—82no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—81no—
484C2×C4C3×#12011yescharacteristic
485C2×C4C3×#12012yes—
486C2×C4C3×#12011yescharacteristic
487C2×C4C3×#12712yes—
488C2×C3—24no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—42no—
492C2×C3—84no—
493C2×C3#243311yescharacteristic
494C2×C3C2×#45811yescharacteristic
495C2×C3#302811yescharacteristic
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—81no—
510C22—88no—
511C22—164no—
512C22C2×C3×#12011yescharacteristic
513C22C2×C3×#12711yescharacteristic
514C22C3×#53511yescharacteristic
515C22C3×D4214yes—
516C4—22no—
517C4—21no—
518C4—22no—
519C4—48no—
520C4—44no—
521C4—41no—
522C4—41no—
523C4—82no—
524C4—82no—
525C4C3×#45812yescomplemented
526C3#2577211yescharacteristic, direct product factor, Sylow
527C2—24no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2C3×#243311yescharacteristic
533C2C2×C3×#45811yescharacteristic
534C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#458C2semidirect
3. C3×#2501C2semidirect
4. C3×#2486C2semidirect
5. C3×#2022C2semidirect
6. C3×#1964C2semidirect
7. C3×C4×#99C2semidirect
8. #25772C3direct (N × H)
18. C2×C3×#123C22semidirect
22. C22×C3×C4⋊C4C22semidirect
25. C2×C3×#99C22semidirect
32. C3×#381C22semidirect
34. C3×#380C22semidirect
35. C3×C4×C22⋊C4C22semidirect
36. C3×#342C22semidirect
38. C2×#458C2×C3semidirect
39. #2501C2×C3semidirect
40. #2486C2×C3semidirect
41. #2022C2×C3semidirect
42. #1964C2×C3semidirect
43. C4×#99C2×C3semidirect
102. C3×#99D4semidirect
116. C2×#123C22×C3semidirect
120. C22×C4⋊C4C22×C3semidirect
123. C2×#99C22×C3semidirect
130. #381C22×C3semidirect
132. #380C22×C3semidirect
133. C4×C22⋊C4C22×C3semidirect
134. #342C22×C3semidirect
202. C3×C22⋊C4C2×D4semidirect
264. #99C3×D4semidirect
395. C22⋊C4C2×C3×D4semidirect
428. C3×C4#458semidirect
525. C4C3×#458semidirect
526. C3#25772direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34962
C2×C3×#4582
C3×#25012
C3×#24862
C3×#20222
C3×#19642
C3×C4×#992
#257723