Finite Group Databasebeta

← C3×#25771 #10644378

Subgroups — C3×#25771 #10644378

Order 768.

2090 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C2×C3×B(4,𝔽2)C214yesmaximal, complemented
3C3×#2433C212yesmaximal, complemented
4C3×#2408C211yesmaximal, characteristic, complemented
5C3×#2021C214yesmaximal, complemented
6C3×#1964C212yesmaximal, complemented
7C3×C4×#99C211yesmaximal, characteristic, complemented
8#25771C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×H(5,𝔽2)C2212yes—
10C23×C3×D4C2211yescharacteristic, complemented
11C3×#547—22no—
12C3×#546—24no—
13C3×D42—22no—
14C3×#537—21no—
15C3×#534—21no—
16C3×#534C2212yes—
17C3×#532—21no—
18C2×C3×#127—22no—
19C2×C3×#121—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×#120C2212yes—
23C2×C3×#120C2218yescomplemented
24C2×C3×C4×D4C2211yescharacteristic
25C3×B(4,𝔽2)—216no—
26C2×C3×#99C2212yes—
27C2×C3×#99C2218yescomplemented
28C3×#394—21no—
29C3×#392—21no—
30C3×#392—24no—
31C3×#390—22no—
32C3×#390—21no—
33C3×#380C2211yescharacteristic, complemented
34C3×#379C2212yescomplemented
35C3×C4×C22⋊C4C2212yescomplemented
36C3×#342C2212yescomplemented
37#3496C2×C311yescharacteristic, complemented
38C2×B(4,𝔽2)C2×C314yescomplemented
39#2433C2×C312yescomplemented
40#2408C2×C311yescharacteristic, complemented
41#2021C2×C314yescomplemented
42#1964C2×C312yescomplemented
43C4×#99C2×C311yescharacteristic, complemented
44C25×C3C2312yes—
45C3×H(5,𝔽2)—28no—
46C3×H(5,𝔽2)—44no—
47C2×C3×#41—22no—
48C2×C3×#41—24no—
49C22×C3×D4—22no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—28no—
54C22×C3×D4—21no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—44no—
59C22×C3×D4D412yescomplemented
60C22×C3×D4C2314yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D411yescharacteristic, complemented
63C23×C3×C4C2311yescharacteristic
64C3×C4⋊Q8—21no—
65C3×#127—21no—
66C3×#127—48no—
67C3×#124—22no—
68C3×#123—24no—
69C3×#123—24no—
70C3×C22⋊Q8—24no—
71C3×#121—22no—
72C3×#121—22no—
73C3×#121—24no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—44no—
81C3×#120—24no—
82C3×#120—24no—
83C3×#120—28no—
84C3×#120—216no—
85C3×#120—48no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D412yescomplemented
93C2×C3×C4⋊C4D411yescharacteristic, complemented
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—28no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4D411yescharacteristic, complemented
100C2×C3×C22⋊C4D414yescomplemented
101C2×C3×C22⋊C4C2312yes—
102C2×C3×C22⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C42—22no—
105C2×C3×C42D411yescharacteristic, complemented
106C3×#99—216no—
107C3×#99D418yescomplemented
108C3×#95—22no—
109C3×#95—21no—
110C2×H(5,𝔽2)C22×C312yes—
111C23×D4C22×C311yescharacteristic, complemented
112#547—22no—
113#546—24no—
114D42—22no—
115#537—21no—
116#534—21no—
117#534C22×C312yes—
118#532—21no—
119C2×#127—22no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C22×C312yes—
123C2×#120C22×C312yes—
124C2×#120C22×C318yescomplemented
125C2×C4×D4C22×C311yescharacteristic
126B(4,𝔽2)—216no—
127C2×#99C22×C312yes—
128C2×#99C22×C318yescomplemented
129#394—21no—
130#392—21no—
131#392—24no—
132#390—22no—
133#390—21no—
134#380C22×C311yescharacteristic, complemented
135#379C22×C312yescomplemented
136C4×C22⋊C4C22×C312yescomplemented
137#342C22×C312yescomplemented
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—24no—
141C24×C3—24no—
142C24×C3—24no—
143C24×C3—44no—
144C24×C3—82no—
145C24×C3C2×D414yes—
146C24×C3C2×D411yescharacteristic
147C24×C3C2411yescharacteristic
148C3×#41—416no—
149C3×#41—44no—
150C3×#41—44no—
151C2×C3×Q8—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—24no—
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—24no—
161C2×C3×D4—28no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—44no—
167C2×C3×D4—42no—
168C2×C3×D4—44no—
169C2×C3×D4—416no—
170C2×C3×D4—416no—
171C2×C3×D4—44no—
172C2×C3×D4—48no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4—416no—
177C2×C3×D4—44no—
178C2×C3×D4—41no—
179C2×C3×D4—42no—
180C2×C3×D4—41no—
181C2×C3×D4—82no—
182C2×C3×D4—88no—
183C2×C3×D4—82no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—416no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—82no—
219C3×C22⋊C4C2×D418yescomplemented
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—44no—
223C25C23×C312yes—
224H(5,𝔽2)—28no—
225H(5,𝔽2)—44no—
226C2×#41—22no—
227C2×#41—24no—
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—28no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—44no—
238C22×D4C3×D412yescomplemented
239C22×D4C23×C314yes—
240C22×D4C23×C311yescharacteristic
241C23×C4C3×D411yescharacteristic, complemented
242C23×C4C23×C311yescharacteristic
243C4⋊Q8—21no—
244#127—21no—
245#127—48no—
246#124—22no—
247#123—24no—
248#123—24no—
249C22⋊Q8—24no—
250#121—22no—
251#121—22no—
252#121—24no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—44no—
260#120—24no—
261#120—24no—
262#120—28no—
263#120—216no—
264#120—48no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C3×D411yescharacteristic, complemented
272C2×C4⋊C4C3×D412yescomplemented
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C3×D411yescharacteristic, complemented
279C2×C22⋊C4C3×D414yescomplemented
280C2×C22⋊C4C23×C312yes—
281C2×C22⋊C4C23×C311yescharacteristic
282C2×C22⋊C4C23×C314yes—
283C2×C42—22no—
284C2×C42C3×D411yescharacteristic, complemented
285#99—216no—
286#99C3×D418yescomplemented
287#95—22no—
288#95—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—28no—
293C23×C3—24no—
294C23×C3—28no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3—48no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—44no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—48no—
311C23×C3—88no—
312C23×C3—88no—
313C23×C3—82no—
314C23×C3—82no—
315C23×C3#12712yescomplemented
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D412yes—
318C23×C3C22×D412yes—
319C23×C3H(5,𝔽2)12yes—
320C3×Q8—44no—
321C3×D4—28no—
322C3×D4—44no—
323C3×D4—48no—
324C3×D4—44no—
325C3×D4—48no—
326C3×D4—44no—
327C3×D4—44no—
328C3×D4—48no—
329C3×D4—44no—
330C3×D4—44no—
331C3×D4—82no—
332C3×D4—82no—
333C3×D4—82no—
334C3×D4—832no—
335C3×D4—88no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—24no—
343C2×C3×C4—28no—
344C2×C3×C4—28no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—48no—
348C2×C3×C4—44no—
349C2×C3×C4—44no—
350C2×C3×C4—44no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—81no—
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12012yes—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12712yescomplemented
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—44no—
368C24—82no—
369C24C2×C3×D414yes—
370C24C2×C3×D411yescharacteristic
371C24C24×C311yescharacteristic
372#41—416no—
373#41—44no—
374#41—44no—
375C2×Q8—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—416no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—82no—
406C2×D4—88no—
407C2×D4—82no—
408C2×D4C2×C3×D414yes—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C2×C3×D414yes—
418C22×C4C2×C3×D412yes—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D412yes—
424C22×C4C2×C3×D411yescharacteristic
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—416no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4C2×C3×D418yescomplemented
444C42—21no—
445C42—21no—
446C42—44no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—416no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—81no—
465C22×C3—88no—
466C22×C3—88no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—164no—
472C22×C3C2×#12011yescharacteristic
473C22×C3C2×#12711yescharacteristic
474C22×C3#53511yescharacteristic
475C22×C3D4214yes—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—48no—
480C3×C4—44no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—82no—
484C3×C4B(4,𝔽2)12yescomplemented
485C23—21no—
486C23—21no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—48no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—48no—
507C23—88no—
508C23—88no—
509C23—82no—
510C23—82no—
511C23C3×#12712yescomplemented
512C23C22×C3×D411yescharacteristic
513C23C22×C3×D412yes—
514C23C22×C3×D412yes—
515C23C3×H(5,𝔽2)12yes—
516Q8—44no—
517D4—28no—
518D4—44no—
519D4—48no—
520D4—44no—
521D4—48no—
522D4—44no—
523D4—44no—
524D4—48no—
525D4—44no—
526D4—44no—
527D4—82no—
528D4—82no—
529D4—82no—
530D4—832no—
531D4—88no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4C3×#12011yescharacteristic
555C2×C4C3×#12012yes—
556C2×C4C3×#12011yescharacteristic
557C2×C4C3×#12712yescomplemented
558C2×C3—24no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—42no—
562C2×C3—84no—
563C2×C3—84no—
564C2×C3#243311yescharacteristic
565C2×C3C2×B(4,𝔽2)11yescharacteristic
566C2×C3#302711yescharacteristic
567C22—21no—
568C22—22no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—416no—
576C22—44no—
577C22—44no—
578C22—41no—
579C22—41no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—81no—
585C22—88no—
586C22—88no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—164no—
592C22C2×C3×#12011yescharacteristic
593C22C2×C3×#12711yescharacteristic
594C22C3×#53511yescharacteristic
595C22C3×D4214yes—
596C4—22no—
597C4—21no—
598C4—22no—
599C4—48no—
600C4—44no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4C3×B(4,𝔽2)12yescomplemented
605C3#2577111yescharacteristic, direct product factor, Sylow
606C2—24no—
607C2—21no—
608C2—21no—
609C2—42no—
610C2—84no—
611C2—84no—
612C2C3×#243311yescharacteristic
613C2C2×C3×B(4,𝔽2)11yescharacteristic
614C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
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
1. C3×#3496C2semidirect
2. C2×C3×B(4,𝔽2)C2semidirect
3. C3×#2433C2semidirect
4. C3×#2408C2semidirect
5. C3×#2021C2semidirect
6. C3×#1964C2semidirect
7. C3×C4×#99C2semidirect
8. #25771C3direct (N × H)
10. C23×C3×D4C22semidirect
23. C2×C3×#120C22semidirect
27. C2×C3×#99C22semidirect
33. C3×#380C22semidirect
34. C3×#379C22semidirect
35. C3×C4×C22⋊C4C22semidirect
36. C3×#342C22semidirect
37. #3496C2×C3semidirect
38. C2×B(4,𝔽2)C2×C3semidirect
39. #2433C2×C3semidirect
40. #2408C2×C3semidirect
41. #2021C2×C3semidirect
42. #1964C2×C3semidirect
43. C4×#99C2×C3semidirect
59. C22×C3×D4D4semidirect
62. C23×C3×C4D4semidirect
92. C2×C3×C4⋊C4D4semidirect
93. C2×C3×C4⋊C4D4semidirect
99. C2×C3×C22⋊C4D4semidirect
100. C2×C3×C22⋊C4D4semidirect
105. C2×C3×C42D4semidirect
107. C3×#99D4semidirect
111. C23×D4C22×C3semidirect
124. C2×#120C22×C3semidirect
128. C2×#99C22×C3semidirect
134. #380C22×C3semidirect
135. #379C22×C3semidirect
136. C4×C22⋊C4C22×C3semidirect
137. #342C22×C3semidirect
219. C3×C22⋊C4C2×D4semidirect
238. C22×D4C3×D4semidirect
241. C23×C4C3×D4semidirect
271. C2×C4⋊C4C3×D4semidirect
272. C2×C4⋊C4C3×D4semidirect
278. C2×C22⋊C4C3×D4semidirect
279. C2×C22⋊C4C3×D4semidirect
284. C2×C42C3×D4semidirect
286. #99C3×D4semidirect
315. C23×C3#127semidirect
361. C2×C3×C4#127semidirect
443. C22⋊C4C2×C3×D4semidirect
484. C3×C4B(4,𝔽2)semidirect
511. C23C3×#127semidirect
557. C2×C4C3×#127semidirect
604. C4C3×B(4,𝔽2)semidirect
605. C3#25771direct (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×B(4,𝔽2)2
C3×#24332
C3×#24082
C3×#20212
C3×#19642
C3×C4×#992
#257713