Finite Group Databasebeta

← #11635874

Subgroups — #11635874

Order 768.

5070 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#127C211yesmaximal, characteristic, complemented
2C2×D4×#67C214yesmaximal, complemented
3C2×#5717C211yesmaximal, characteristic, complemented
4#84053C214yesmaximal, complemented
5#84052C214yesmaximal, complemented
6#83190C214yesmaximal, complemented
7#83188C212yesmaximal, complemented
8#83186C214yesmaximal, complemented
9#83175C214yesmaximal, complemented
10#83173C211yesmaximal, characteristic, complemented
11C4×#930C212yesmaximal, complemented
12#59126—31nomaximal, Sylow
13C23×C3×D4C2214yescomplemented
14C23×C3×D4C2212yescomplemented
15C23×#67C2214yescomplemented
16C22×D3×D4C2214yescomplemented
17C2×C3×#127—28no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#127C2214yescomplemented
20C2×C3×#127C2211yescharacteristic, complemented
21C2×C3×#127C2212yescomplemented
22C22×C3×C42C2211yescharacteristic, complemented
23C2×#953C2218yescomplemented
24C2×#940C2214yescomplemented
25C2×#939C2218yescomplemented
26D4×#67—232no—
27C2×#938C2214yescomplemented
28C2×#938C2214yescomplemented
29C2×#938C2214yescomplemented
30C2×#937C2214yescomplemented
31C2×#937C2218yescomplemented
32C2×#937C2214yescomplemented
33C2×D4×Dic3C2214yescomplemented
34C2×#930C2212yes—
35C2×C4×#67C2214yescomplemented
36#5717—28no—
37C2×C4×D12C2212yescomplemented
38#5247C2214yescomplemented
39#5223C2218yescomplemented
40#5223C2218yescomplemented
41#5223C2218yescomplemented
42#5222C2218yescomplemented
43#5221C2214yescomplemented
44#5221C2214yescomplemented
45#5220C2214yescomplemented
46#5220C2214yescomplemented
47#5213C2214yescomplemented
48#5212C2212yescomplemented
49#4940C2214yescomplemented
50#4939C2212yescomplemented
51C4×(C2×D3)⋊C4C2214yescomplemented
52C4×C4⋊Dic3C2212yescomplemented
53#4928C2212yescomplemented
54C2×D42—34no—
55C2×D42—31no—
56C22×#127D311yescharacteristic, complemented
57#2669—34no—
58#2541—32no—
59#2433—34no—
60#2428—34no—
61#2408—34no—
62#2403—34no—
63#2366—31no—
64C4×#121—32no—
65C25×C3C2314yes—
66C24×D3C2314yes—
67C22×C3×D4—216no—
68C22×C3×D4—28no—
69C22×C3×D4—216no—
70C22×C3×D4—48no—
71C22×C3×D4—44no—
72C22×C3×D4—44no—
73C22×C3×D4C2314yes—
74C22×C3×D4C2314yes—
75C22×C3×D4C2318yes—
76C22×C3×D4C2314yes—
77C22×C3×D4C2312yes—
78C22×C3×D4C2312yes—
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2312yes—
81C22×#67—232no—
82C22×#67—48no—
83C22×#67C2318yes—
84C22×#67C2318yes—
85C22×#67C2314yes—
86C22×#67C2318yes—
87C23×Dic3C2314yes—
88C2×D3×D4—216no—
89C2×D3×D4—216no—
90C2×D3×D4—48no—
91C22×D12C2312yes—
92C22×C4×D3C2314yes—
93C3×#127—216no—
94C3×#127—28no—
95C3×#127—24no—
96C3×#127D418yescomplemented
97C2×C3×C42—24no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101#953—232no—
102C2×#217C2314yes—
103C2×#217C2318yes—
104C2×#217C2318yes—
105C2×#217C2314yes—
106C2×#217C2314yes—
107C2×#217C2318yes—
108#940—216no—
109#939—232no—
110#938—216no—
111#938—216no—
112#938—48no—
113#937—216no—
114#937—232no—
115#937—48no—
116D4×Dic3—216no—
117#930D4116yescomplemented
118C4×#67—216no—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×Dic3⋊C4C2314yes—
127C2×C4×Dic3C2314yes—
128C4×D12—28no—
129C3⋊#95C2314yes—
130C3⋊#95C2314yes—
131C3⋊#95C2314yes—
132C23×D4—34no—
133C23×D4—34no—
134C23×D4C2×D314yescomplemented
135C23×D4C2×D312yescomplemented
136D42—632no—
137D42—68no—
138C2×#127—28no—
139C2×#127—34no—
140C2×#127C2×D311yescharacteristic
141C2×#127C2×D312yescomplemented
142C2×#127C2×D314yescomplemented
143C2×#127C2×D311yescharacteristic, complemented
144C2×#121—32no—
145C2×#121—34no—
146C2×#121—38no—
147C2×#121—34no—
148C2×#121—34no—
149C2×#120—38no—
150C2×#120—34no—
151C2×#120—38no—
152C2×#120—34no—
153C2×C4×D4—34no—
154C2×C4×D4—34no—
155C2×C4×D4—32no—
156C22×C42C2×D311yescharacteristic, complemented
157#394—38no—
158#392—38no—
159#392—38no—
160#392—38no—
161#390—34no—
162#390—34no—
163#390—32no—
164#389—32no—
165#388—34no—
166#386—34no—
167#386—34no—
168#386—34no—
169#380—32no—
170#379—34no—
171C4×C4⋊C4—32no—
172C4×C22⋊C4—34no—
173C24×C3—28no—
174C24×C3—216no—
175C24×C3—28no—
176C24×C3—44no—
177C24×C3—48no—
178C24×C3C2414yes—
179C24×C3C2414yes—
180C24×C3C2411yescharacteristic
181C23×D3—28no—
182C23×D3—216no—
183C23×D3—48no—
184C23×D3—44no—
185C23×D3—44no—
186C23×D3C2414yes—
187C2×C3×D4—28no—
188C2×C3×D4—216no—
189C2×C3×D4—28no—
190C2×C3×D4—216no—
191C2×C3×D4—28no—
192C2×C3×D4—24no—
193C2×C3×D4—28no—
194C2×C3×D4—28no—
195C2×C3×D4—28no—
196C2×C3×D4—28no—
197C2×C3×D4—464no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—44no—
201C2×C3×D4—48no—
202C2×C3×D4—42no—
203C2×C3×D4—42no—
204C2×C3×D4—432no—
205C2×C3×D4C2×D4116yescomplemented
206C2×C3×D4C2×D418yescomplemented
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—24no—
212C22×C3×C4C2×D418yescomplemented
213C22×C3×C4C2411yescharacteristic
214C22×C3×C4C2412yes—
215C22×C3×C4C2411yescharacteristic
216C22×C3×C4C2412yes—
217C2×#67—28no—
218C2×#67—28no—
219C2×#67—216no—
220C2×#67—216no—
221C2×#67—216no—
222C2×#67—216no—
223C2×#67—216no—
224C2×#67—48no—
225C2×#67—48no—
226C2×#67—48no—
227C2×#67—48no—
228C2×#67—464no—
229C2×#67C2×D4116yescomplemented
230C22×Dic3—216no—
231C22×Dic3—44no—
232C22×Dic3C2414yes—
233C22×Dic3C2414yes—
234C22×Dic3C2414yes—
235D3×D4—464no—
236C2×D12—24no—
237C2×D12—24no—
238C2×D12C2×D418yescomplemented
239C2×C4×D3—28no—
240C2×C4×D3—28no—
241C2×C4×D3—28no—
242C3×C42—22no—
243C3×C42—24no—
244C3×C42C2×D414yescomplemented
245#217—28no—
246#217—28no—
247#217—216no—
248#217—216no—
249#217—44no—
250#217—48no—
251(C2×D3)⋊C4—28no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4C2×D4116yescomplemented
255C4⋊Dic3—24no—
256C4⋊Dic3—24no—
257C4⋊Dic3C2×D418yescomplemented
258Dic3⋊C4—28no—
259C4×Dic3—28no—
260C25—34no—
261C25C22×D314yes—
262C22×D4—216no—
263C22×D4—28no—
264C22×D4—216no—
265C22×D4—34no—
266C22×D4—38no—
267C22×D4—38no—
268C22×D4—38no—
269C22×D4—32no—
270C22×D4—48no—
271C22×D4—44no—
272C22×D4—44no—
273C22×D4—616no—
274C22×D4—632no—
275C22×D4—616no—
276C22×D4—128no—
277C22×D4—128no—
278C22×D4C22×D314yes—
279C22×D4C22×D318yes—
280C22×D4C22×D314yes—
281C22×D4C22×D314yes—
282C22×D4C22×D312yes—
283C22×D4C22×D312yes—
284C23×C4—34no—
285C23×C4—34no—
286C23×C4C22×D311yescharacteristic
287C23×C4C22×D312yes—
288#127—216no—
289#127—28no—
290#127—24no—
291#127—616no—
292#127#6718yescomplemented
293#121—316no—
294#121—632no—
295#121—616no—
296#121—616no—
297#121—128no—
298#120—632no—
299#120—632no—
300#120—616no—
301#120—128no—
302C4×D4—616no—
303C4×D4—616no—
304C4×D4—68no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—32no—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—38no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—38no—
314C2×C22⋊C4—34no—
315C2×C22⋊C4—34no—
316C2×C22⋊C4—34no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—38no—
319C2×C42—24no—
320C2×C42—34no—
321C2×C42C22×D312yes—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324#95—34no—
325#95—34no—
326#95—34no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—28no—
331C23×C3—24no—
332C23×C3—24no—
333C23×C3—28no—
334C23×C3—28no—
335C23×C3—24no—
336C23×C3—28no—
337C23×C3—24no—
338C23×C3—24no—
339C23×C3—432no—
340C23×C3—416no—
341C23×C3—416no—
342C23×C3—816no—
343C23×C3—88no—
344C23×C3C22×D414yes—
345C23×C3C22×D418yes—
346C23×C3H(5,𝔽2)12yes—
347C23×C3C2511yescharacteristic
348C22×D3—24no—
349C22×D3—24no—
350C22×D3—24no—
351C22×D3—24no—
352C22×D3—24no—
353C22×D3—24no—
354C22×D3—416no—
355C22×D3—432no—
356C22×D3—816no—
357C22×D3—88no—
358C22×D3—88no—
359C22×D3C22×D418yes—
360C3×D4—232no—
361C3×D4—48no—
362C3×D4—432no—
363C3×D4—416no—
364C3×D4—48no—
365C3×D4—416no—
366C2×C3×C4—28no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—216no—
374C2×C3×C4C22×D414yes—
375C2×C3×C4C22×D414yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D414yes—
378C2×C3×C4H(5,𝔽2)12yes—
379C2×C3×C4H(5,𝔽2)14yes—
380#67—232no—
381#67—432no—
382#67—432no—
383#67—816no—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—416no—
393C2×Dic3C22×D418yes—
394D12—216no—
395C4×D3—416no—
396C24—28no—
397C24—216no—
398C24—28no—
399C24—34no—
400C24—44no—
401C24—48no—
402C24—616no—
403C24—68no—
404C24—124no—
405C24—128no—
406C24—124no—
407C24C23×D314yes—
408C24C23×D314yes—
409C24C23×D311yescharacteristic
410C2×D4—216no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—216no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—316no—
421C2×D4—38no—
422C2×D4—464no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—432no—
430C2×D4—616no—
431C2×D4—68no—
432C2×D4—68no—
433C2×D4—616no—
434C2×D4—616no—
435C2×D4—616no—
436C2×D4—616no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—128no—
440C2×D4—128no—
441C2×D4—128no—
442C2×D4—128no—
443C2×D4—1264no—
444C2×D4—1264no—
445C2×D4C2×#67116yescomplemented
446C2×D4C2×#6718yescomplemented
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—34no—
455C22×C4—616no—
456C22×C4—68no—
457C22×C4—68no—
458C22×C4—68no—
459C22×C4—124no—
460C22×C4D3×D418yescomplemented
461C22×C4C23×D312yes—
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D311yescharacteristic
464C22×C4C23×D312yes—
465C4⋊C4—38no—
466C4⋊C4—68no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C22⋊C4—316no—
470C22⋊C4—616no—
471C22⋊C4—616no—
472C22⋊C4—68no—
473C22⋊C4—68no—
474C22⋊C4—68no—
475C22⋊C4—68no—
476C22⋊C4—68no—
477C22⋊C4—124no—
478C22⋊C4—128no—
479C42—22no—
480C42—24no—
481C42—68no—
482C42C2×#6714yescomplemented
483C22×C3—24no—
484C22×C3—22no—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—216no—
488C22×C3—28no—
489C22×C3—44no—
490C22×C3—48no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—48no—
494C22×C3—44no—
495C22×C3—48no—
496C22×C3—44no—
497C22×C3—832no—
498C22×C3D4214yescomplemented
499C22×C3C23×D412yes—
500C22×C3C23×D411yescharacteristic
501C22×C3C2×H(5,𝔽2)11yescharacteristic
502C22×C3C2×H(5,𝔽2)12yes—
503C22×C3C2×H(5,𝔽2)11yescharacteristic
504C2×D3—28no—
505C2×D3—28no—
506C2×D3—28no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—832no—
512C3×C4—24no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4D4218yescomplemented
516Dic3—28no—
517Dic3—44no—
518Dic3—44no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—38no—
532C23—432no—
533C23—416no—
534C23—416no—
535C23—64no—
536C23—64no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23—64no—
541C23—816no—
542C23—88no—
543C23—1232no—
544C23—1216no—
545C23—248no—
546C23—2416no—
547C23—248no—
548C23C2×D3×D414yes—
549C23#100412yes—
550C23C22×#6718yes—
551C23C24×D311yescharacteristic
552D4—232no—
553D4—48no—
554D4—432no—
555D4—416no—
556D4—48no—
557D4—416no—
558D4—632no—
559D4—616no—
560D4—1232no—
561D4—1232no—
562D4—2416no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—216no—
571C2×C4—38no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—1216no—
581C2×C4—1216no—
582C2×C4C2×D3×D414yes—
583C2×C4C2×D3×D414yes—
584C2×C4#100412yes—
585C2×C4#100414yes—
586C2×C4C22×#6712yes—
587C2×C4C22×#6714yes—
588C2×C3—24no—
589C2×C3—48no—
590C2×C3—84no—
591C2×C3C2×D4211yescharacteristic
592C2×C3C2×D4212yes—
593C2×C3#348411yescharacteristic
594C2×C3#349812yes—
595C2×C3#354311yescharacteristic
596D3—48no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—22no—
601C22—216no—
602C22—28no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—68no—
612C22—68no—
613C22—68no—
614C22—832no—
615C22—124no—
616C22—124no—
617C22—124no—
618C22—124no—
619C22—2432no—
620C22#571714yescomplemented
621C22C22×D3×D412yes—
622C22C2×#100411yescharacteristic
623C22C2×#100411yescharacteristic
624C22C2×#100412yes—
625C22C23×#6711yescharacteristic
626C4—24no—
627C4—22no—
628C4—22no—
629C4—68no—
630C4—124no—
631C4—124no—
632C4D4×#6718yescomplemented
633C3#5912611yescharacteristic, complemented, Sylow
634C2—24no—
635C2—48no—
636C2—84no—
637C2—128no—
638C2C2×#571711yescharacteristic
639C2#8619512yes—
640C2#8619711yescharacteristic
641C2C2×D4×#6712yes—
642C2#8723911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC22×C3×#127C2
hypercenterC22×#127D3
hypocenterC3#59126
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#127

Complements

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

NH ≅ G/Ntype
1. C22×C3×#127C2semidirect
2. C2×D4×#67C2semidirect
3. C2×#5717C2semidirect
4. #84053C2semidirect
5. #84052C2semidirect
6. #83190C2semidirect
7. #83188C2semidirect
8. #83186C2semidirect
9. #83175C2semidirect
10. #83173C2semidirect
11. C4×#930C2semidirect
13. C23×C3×D4C22semidirect
14. C23×C3×D4C22semidirect
15. C23×#67C22semidirect
16. C22×D3×D4C22semidirect
19. C2×C3×#127C22semidirect
20. C2×C3×#127C22semidirect
21. C2×C3×#127C22semidirect
22. C22×C3×C42C22semidirect
23. C2×#953C22semidirect
24. C2×#940C22semidirect
25. C2×#939C22semidirect
27. C2×#938C22semidirect
28. C2×#938C22semidirect
29. C2×#938C22semidirect
30. C2×#937C22semidirect
31. C2×#937C22semidirect
32. C2×#937C22semidirect
33. C2×D4×Dic3C22semidirect
35. C2×C4×#67C22semidirect
37. C2×C4×D12C22semidirect
38. #5247C22semidirect
39. #5223C22semidirect
40. #5223C22semidirect
41. #5223C22semidirect
42. #5222C22semidirect
43. #5221C22semidirect
44. #5221C22semidirect
45. #5220C22semidirect
46. #5220C22semidirect
47. #5213C22semidirect
48. #5212C22semidirect
49. #4940C22semidirect
50. #4939C22semidirect
51. C4×(C2×D3)⋊C4C22semidirect
52. C4×C4⋊Dic3C22semidirect
53. #4928C22semidirect
56. C22×#127D3semidirect
96. C3×#127D4semidirect
117. #930D4semidirect
134. C23×D4C2×D3semidirect
135. C23×D4C2×D3semidirect
141. C2×#127C2×D3semidirect
142. C2×#127C2×D3semidirect
143. C2×#127C2×D3semidirect
156. C22×C42C2×D3semidirect
205. C2×C3×D4C2×D4semidirect
206. C2×C3×D4C2×D4semidirect
212. C22×C3×C4C2×D4semidirect
229. C2×#67C2×D4semidirect
238. C2×D12C2×D4semidirect
244. C3×C42C2×D4semidirect
254. (C2×D3)⋊C4C2×D4semidirect
257. C4⋊Dic3C2×D4semidirect
292. #127#67semidirect
445. C2×D4C2×#67semidirect
446. C2×D4C2×#67semidirect
460. C22×C4D3×D4semidirect
482. C42C2×#67semidirect
498. C22×C3D42semidirect
515. C3×C4D42semidirect
620. C22#5717semidirect
632. C4D4×#67semidirect
633. C3#59126semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1272
C2×D4×#672
C2×#57172
#840532
#840522
#831902
#831882
#831862
#831752
#831732
C4×#9302
#591263