Finite Group Databasebeta

← C3×#29634 #10648241

Subgroups — C3×#29634 #10648241

Order 768.

1908 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic, complemented
2C3×#2405C211yesmaximal, characteristic, complemented
3C3×#2403C212yesmaximal, complemented
4C3×#2044C214yesmaximal, complemented
5C3×#2021C214yesmaximal, complemented
6C3×#1819C211yesmaximal, characteristic, complemented
7C3×#1781C212yesmaximal, complemented
8#29634C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2212yes—
10C23×C3×D4C2211yescharacteristic
11C3×#534—24no—
12C3×#534—22no—
13C3×#534C2214yes—
14C2×C3×#123—22no—
15C2×C3×#123—21no—
16C2×C3×#121—21no—
17C2×C3×#121—21no—
18C2×C3×#120—22no—
19C2×C3×#120—22no—
20C2×C3×#120C2214yescomplemented
21C22×C3×C22⋊C4C2211yescharacteristic
22C2×C3×#99C2218yescomplemented
23C3×#394—22no—
24C3×#392—24no—
25C3×#386—22no—
26C3×#386—21no—
27C3×#386C2214yescomplemented
28C3×#385C2212yescomplemented
29C3×#379C2212yes—
30C3×#379C2211yescharacteristic
31C3×#379C2212yescomplemented
32C3×#342C2214yescomplemented
33C2×#534C2×C311yescharacteristic, complemented
34#2405C2×C311yescharacteristic, complemented
35#2403C2×C312yescomplemented
36#2044C2×C314yescomplemented
37#2021C2×C314yescomplemented
38#1819C2×C311yescharacteristic, complemented
39#1781C2×C312yescomplemented
40C25×C3D412yescomplemented
41C25×C3C2311yescharacteristic
42C25×C3C2311yescharacteristic
43C22×C3×D4—22no—
44C22×C3×D4—21no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4—21no—
53C22×C3×D4—24no—
54C22×C3×D4—44no—
55C22×C3×D4—48no—
56C22×C3×D4D414yescomplemented
57C22×C3×D4C2314yes—
58C23×C3×C4D412yescomplemented
59C23×C3×C4D411yescharacteristic, complemented
60C23×C3×C4D411yescharacteristic, complemented
61C23×C3×C4C2311yescharacteristic
62C3×#123—28no—
63C3×#123—28no—
64C3×#123—44no—
65C3×#121—24no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—24no—
69C3×#120—28no—
70C3×#120—28no—
71C3×#120—48no—
72C3×#120—48no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yescomplemented
89C2×C3×C22⋊C4C2312yes—
90C2×C3×C22⋊C4C2314yes—
91C2×C3×C22⋊C4C2312yes—
92C3×#99—48no—
93C3×#95—22no—
94C3×#95—24no—
95C23×D4C22×C312yes—
96C23×D4C22×C311yescharacteristic
97#534—24no—
98#534—22no—
99#534C22×C314yes—
100C2×#123—22no—
101C2×#123—21no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#120—22no—
105C2×#120—22no—
106C2×#120C22×C314yescomplemented
107C22×C22⋊C4C22×C311yescharacteristic
108C2×#99C22×C318yescomplemented
109#394—22no—
110#392—24no—
111#386—22no—
112#386—21no—
113#386C22×C314yescomplemented
114#385C22×C312yescomplemented
115#379C22×C312yes—
116#379C22×C311yescharacteristic
117#379C22×C312yescomplemented
118#342C22×C314yescomplemented
119C24×C3—24no—
120C24×C3—22no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—24no—
125C24×C3—44no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3—44no—
129C24×C3—81no—
130C24×C3—81no—
131C24×C3—82no—
132C24×C3C2×D412yes—
133C24×C3C2×D411yescharacteristic
134C24×C3C2×D414yes—
135C24×C3C2×D412yes—
136C24×C3C2×D411yescharacteristic
137C24×C3C2411yescharacteristic
138C2×C3×D4—28no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—28no—
145C2×C3×D4—28no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—44no—
154C2×C3×D4—48no—
155C2×C3×D4—48no—
156C2×C3×D4—48no—
157C2×C3×D4—48no—
158C2×C3×D4—42no—
159C2×C3×D4—44no—
160C2×C3×D4—42no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4—48no—
164C2×C3×D4—48no—
165C2×C3×D4—82no—
166C2×C3×D4—82no—
167C2×C3×D4—84no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—42no—
179C22×C3×C4—44no—
180C22×C3×C4—42no—
181C22×C3×C4—81no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—44no—
188C3×C22⋊C4—28no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—48no—
195C3×C22⋊C4—48no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—82no—
200C3×C22⋊C4—81no—
201C3×C22⋊C4—82no—
202C25C3×D412yescomplemented
203C25C23×C311yescharacteristic
204C25C23×C311yescharacteristic
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—24no—
216C22×D4—44no—
217C22×D4—48no—
218C22×D4C3×D414yescomplemented
219C22×D4C23×C314yes—
220C23×C4C3×D411yescharacteristic, complemented
221C23×C4C3×D412yescomplemented
222C23×C4C3×D411yescharacteristic, complemented
223C23×C4C23×C311yescharacteristic
224#123—28no—
225#123—28no—
226#123—44no—
227#121—24no—
228#121—24no—
229#121—24no—
230#121—24no—
231#120—28no—
232#120—28no—
233#120—48no—
234#120—48no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4C3×D412yescomplemented
251C2×C22⋊C4C23×C314yes—
252C2×C22⋊C4C23×C312yes—
253C2×C22⋊C4C23×C312yes—
254#99—48no—
255#95—22no—
256#95—24no—
257C23×C3—24no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—44no—
283C23×C3—44no—
284C23×C3—44no—
285C23×C3—41no—
286C23×C3—44no—
287C23×C3—42no—
288C23×C3—44no—
289C23×C3—82no—
290C23×C3—82no—
291C23×C3—88no—
292C23×C3—88no—
293C23×C3—816no—
294C23×C3#12012yes—
295C23×C3#12011yescharacteristic
296C23×C3#12011yescharacteristic
297C23×C3C22×D412yes—
298C23×C3C22×D411yescharacteristic
299C23×C3H(5,𝔽2)14yes—
300C3×D4—48no—
301C3×D4—48no—
302C3×D4—44no—
303C3×D4—48no—
304C3×D4—48no—
305C3×D4—48no—
306C3×D4—44no—
307C3×D4—84no—
308C3×D4—84no—
309C3×D4—816no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—44no—
328C2×C3×C4—42no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—81no—
333C2×C3×C4—81no—
334C24—24no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—44no—
341C24—42no—
342C24—42no—
343C24—44no—
344C24—81no—
345C24—81no—
346C24—82no—
347C24C2×C3×D412yes—
348C24C2×C3×D411yescharacteristic
349C24C2×C3×D414yes—
350C24C2×C3×D412yes—
351C24C2×C3×D411yescharacteristic
352C24C24×C311yescharacteristic
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—42no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—84no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—81no—
397C22×C4C2×C3×D414yes—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D412yes—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—44no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—28no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—82no—
415C22⋊C4—81no—
416C22⋊C4—82no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—48no—
432C22×C3—48no—
433C22×C3—48no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—84no—
440C22×C3—82no—
441C22×C3—84no—
442C22×C3—84no—
443C22×C3—82no—
444C22×C3—88no—
445C22×C3—88no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3B(4,𝔽2)12yes—
449C22×C3C2×#12011yescharacteristic
450C22×C3#53414yes—
451C22×C3#53512yes—
452C3×C4—44no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—82no—
457C3×C4—84no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—41no—
488C23—44no—
489C23—42no—
490C23—82no—
491C23—82no—
492C23—88no—
493C23—88no—
494C23—816no—
495C23C3×#12012yes—
496C23C3×#12011yescharacteristic
497C23C3×#12011yescharacteristic
498C23C22×C3×D412yes—
499C23C22×C3×D411yescharacteristic
500C23C3×H(5,𝔽2)14yes—
501D4—44no—
502D4—48no—
503D4—48no—
504D4—48no—
505D4—44no—
506D4—48no—
507D4—48no—
508D4—84no—
509D4—84no—
510D4—816no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C3—24no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—41no—
539C2×C3—42no—
540C2×C3—82no—
541C2×C3—84no—
542C2×C3#261311yescharacteristic
543C2×C3C2×B(4,𝔽2)11yescharacteristic
544C2×C3#302611yescharacteristic
545C22—21no—
546C22—22no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—21no—
557C22—44no—
558C22—48no—
559C22—44no—
560C22—48no—
561C22—48no—
562C22—44no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—88no—
573C22—88no—
574C22—81no—
575C22—81no—
576C22C3×B(4,𝔽2)12yes—
577C22C2×C3×#12011yescharacteristic
578C22C3×#53414yes—
579C22C3×#53512yes—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—44no—
584C4—84no—
585C4—82no—
586C3#2963411yescharacteristic, direct product factor, Sylow
587C2—24no—
588C2—22no—
589C2—22no—
590C2—42no—
591C2—41no—
592C2—82no—
593C2—84no—
594C2C3×#261311yescharacteristic
595C2C2×C3×B(4,𝔽2)11yescharacteristic
596C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#534C2semidirect
2. C3×#2405C2semidirect
3. C3×#2403C2semidirect
4. C3×#2044C2semidirect
5. C3×#2021C2semidirect
6. C3×#1819C2semidirect
7. C3×#1781C2semidirect
8. #29634C3direct (N × H)
20. C2×C3×#120C22semidirect
22. C2×C3×#99C22semidirect
27. C3×#386C22semidirect
28. C3×#385C22semidirect
31. C3×#379C22semidirect
32. C3×#342C22semidirect
33. C2×#534C2×C3semidirect
34. #2405C2×C3semidirect
35. #2403C2×C3semidirect
36. #2044C2×C3semidirect
37. #2021C2×C3semidirect
38. #1819C2×C3semidirect
39. #1781C2×C3semidirect
40. C25×C3D4semidirect
56. C22×C3×D4D4semidirect
58. C23×C3×C4D4semidirect
59. C23×C3×C4D4semidirect
60. C23×C3×C4D4semidirect
88. C2×C3×C22⋊C4D4semidirect
106. C2×#120C22×C3semidirect
108. C2×#99C22×C3semidirect
113. #386C22×C3semidirect
114. #385C22×C3semidirect
117. #379C22×C3semidirect
118. #342C22×C3semidirect
202. C25C3×D4semidirect
218. C22×D4C3×D4semidirect
220. C23×C4C3×D4semidirect
221. C23×C4C3×D4semidirect
222. C23×C4C3×D4semidirect
250. C2×C22⋊C4C3×D4semidirect
586. C3#29634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C3×#24052
C3×#24032
C3×#20442
C3×#20212
C3×#18192
C3×#17812
#296343