Finite Group Databasebeta

← #10811317

Subgroups — #10811317

Order 768.

1696 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5550C211yesmaximal, characteristic
2#74007C212yesmaximal, complemented
3C3×#1799C211yesmaximal, characteristic, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#70036C214yesmaximal, complemented
6#69529C212yesmaximal, complemented
7#68969C212yesmaximal, complemented
8D3×#342C212yesmaximal, complemented
9#22476—31nomaximal, Sylow
10C22×#235C411yescharacteristic, complemented
11C23×D12C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×#937C2212yes—
14C2×#930C411yescharacteristic, complemented
15C2×#924C2211yescharacteristic
16#5550—22no—
17#5550—24no—
18#5550—22no—
19C2×#894C412yescomplemented
20C2×#893—22no—
21C2×#884C412yescomplemented
22C2×D3×C22⋊C4C2212yes—
23C2×D3×C22⋊C4C2214yes—
24C2×#875C411yescharacteristic, complemented
25C2×C4×D12C411yescharacteristic, complemented
26C2×C3×#99C2212yes—
27C3×#380C2212yes—
28C2×#834C2212yes—
29C4×#217C2212yes—
30#4988—22no—
31#4955C2214yes—
32#4665—24no—
33C3×#342C2212yes—
34#4527C2214yes—
35#4527C2214yes—
36#4526C2211yescharacteristic
37#4526C2211yescharacteristic
38C2×#546—31no—
39D4×C22⋊C4—32no—
40#1996—32no—
41#1898—34no—
42#1819—32no—
43#1799D311yescharacteristic, complemented
44C4×#99—31no—
45C2×#342—32no—
46C24×D3C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4C2×C411yescharacteristic, complemented
49C23×C3×C4C2311yescharacteristic
50C22×#67C2×C412yescomplemented
51C22×#67C2×C412yescomplemented
52C2×#235—22no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—44no—
56C22×D12—44no—
57C22×D12C2×C411yescharacteristic
58C22×D12C2×C411yescharacteristic
59C22×D12C2×C412yes—
60C22×D12D412yes—
61C22×D12D414yes—
62C22×D12D412yes—
63C22×C4×D3—24no—
64C22×C4×D3C2×C412yescomplemented
65C22×C4×D3C2×C412yescomplemented
66C22×Dic6C2×C411yescharacteristic, complemented
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
71C2×C3×C22⋊C4C2×C411yescharacteristic, complemented
72C2×C3×C22⋊C4C2314yes—
73C2×C3×C22⋊C4C2311yescharacteristic
74C2×C3×C42C2×C411yescharacteristic, complemented
75C2×#217C2312yes—
76C2×#217C2312yes—
77C2×#217C2312yes—
78#937—24no—
79#937—24no—
80#930—22no—
81#930—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4C2×C412yescomplemented
86C2×(C2×D3)⋊C4C2×C412yescomplemented
87C2×(C2×D3)⋊C4C2×C412yescomplemented
88C2×(C2×D3)⋊C4C2×C412yescomplemented
89C2×C4⋊Dic3C2×C411yescharacteristic, complemented
90#924—22no—
91#924—22no—
92C2×Dic3⋊C4C2×C412yescomplemented
93C2×C4×Dic3—22no—
94#894—24no—
95#894—24no—
96#893—48no—
97#884—24no—
98#884—24no—
99D3×C22⋊C4—24no—
100D3×C22⋊C4—24no—
101D3×C22⋊C4—28no—
102D3×C22⋊C4—44no—
103#875—22no—
104#875—22no—
105C4×D12—22no—
106C4×D12—21no—
107C4×D12—21no—
108C3×#99—42no—
109C3×#95—22no—
110#834—24no—
111C3⋊#95—22no—
112C22×#41—31no—
113C23×D4—31no—
114#546—62no—
115#546—64no—
116#546—62no—
117C2×#124—31no—
118C2×#123—32no—
119C2×C22⋊Q8—31no—
120C2×#121—31no—
121C2×#121—32no—
122C2×#120—32no—
123C2×C4×D4—31no—
124C2×C4×D4—62no—
125C2×C4×D4C2×D311yescharacteristic
126C22×C22⋊C4—32no—
127C22×C22⋊C4—34no—
128C2×#99—32no—
129C2×#99C2×D312yes—
130#390—62no—
131#386—64no—
132#380C2×D312yes—
133#379—34no—
134C4×C22⋊C4—32no—
135#342—31no—
136#342—34no—
137#342—34no—
138#342—31no—
139#342C2×D312yes—
140C24×C3C22×C411yescharacteristic
141C24×C3C2411yescharacteristic
142C23×D3—24no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—44no—
146C23×D3—82no—
147C23×D3C22×C412yes—
148C23×D3C22×C412yes—
149C23×D3C2×D414yes—
150C23×D3C2×D414yes—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—24no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C2×#67—22no—
168C2×#67—22no—
169C2×#67—24no—
170C2×#67—44no—
171C2×#67—82no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3C22×C412yes—
175C22×Dic3C22×C412yes—
176#235—48no—
177#235—44no—
178#235—44no—
179C2×D12—28no—
180C2×D12—24no—
181C2×D12—42no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—48no—
185C2×D12C22⋊C412yes—
186C2×D12C22⋊C411yescharacteristic
187C2×D12C22⋊C412yes—
188C2×D12C22⋊C414yes—
189C2×D12C22⋊C412yes—
190C2×D12C22⋊C414yes—
191C2×D12C22⋊C411yescharacteristic
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—44no—
197C2×C4×D3—48no—
198C2×C4×D3—48no—
199C2×C4×D3—84no—
200C2×C4×D3—82no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—42no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—44no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C42—21no—
211C3×C42—21no—
212#217—24no—
213#217—22no—
214#217—22no—
215#217—22no—
216#217#4114yescomplemented
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—48no—
223(C2×D3)⋊C4—48no—
224(C2×D3)⋊C4—84no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227Dic3⋊C4—24no—
228C4×Dic3—44no—
229C25—32no—
230C2×#41—61no—
231C2×#41—62no—
232C2×#41—61no—
233C2×#41—124no—
234C22×Q8—31no—
235C22×D4—32no—
236C22×D4—32no—
237C22×D4—31no—
238C22×D4—32no—
239C22×D4—31no—
240C22×D4—32no—
241C22×D4—34no—
242C22×D4—32no—
243C22×D4—124no—
244C22×D4C22×D311yescharacteristic
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—64no—
248C23×C4C4×D311yescharacteristic, complemented
249C23×C4C22×D311yescharacteristic
250#124—62no—
251#124—62no—
252#123—64no—
253#123—64no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256#121—64no—
257#121—64no—
258#121—62no—
259#121—62no—
260#120—64no—
261#120—64no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—61no—
265C4×D4—62no—
266C4×D4—61no—
267C4×D4—128no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4C4×D311yescharacteristic, complemented
272C2×C22⋊C4—32no—
273C2×C22⋊C4—32no—
274C2×C22⋊C4—32no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—68no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—124no—
286C2×C22⋊C4C4×D311yescharacteristic, complemented
287C2×C22⋊C4C22×D314yes—
288C2×C22⋊C4C22×D311yescharacteristic
289C2×C42—62no—
290C2×C42C4×D311yescharacteristic, complemented
291#99—42no—
292#99—64no—
293#95—22no—
294#95—62no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3C23×C411yescharacteristic
303C23×C3C22×D412yes—
304C23×C3C2×#4112yes—
305C23×C3H(5,𝔽2)11yescharacteristic
306C23×C32−1+411yescharacteristic
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—28no—
316C22×D3—24no—
317C22×D3—42no—
318C22×D3—44no—
319C22×D3—44no—
320C22×D3—42no—
321C22×D3—44no—
322C22×D3—42no—
323C22×D3—84no—
324C22×D3—88no—
325C22×D3—84no—
326C22×D3C2×C22⋊C414yes—
327C22×D3C2×C22⋊C412yes—
328C22×D3C2×C22⋊C412yes—
329C3×D4—42no—
330C3×D4—42no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—44no—
347C2×C3×C4C2×C22⋊C412yes—
348C2×C3×C4C2×C22⋊C411yescharacteristic
349C2×C3×C4C2×C22⋊C411yescharacteristic
350#67—44no—
351#67—44no—
352#67—84no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363D12—28no—
364D12—24no—
365D12—24no—
366D12—22no—
367D12—42no—
368D12—44no—
369D12#9914yescomplemented
370C4×D3—44no—
371C4×D3—44no—
372C4×D3—82no—
373C4×D3—82no—
374C4×D3—816no—
375Dic6—42no—
376Dic6—42no—
377C24—32no—
378C24—34no—
379C24—32no—
380C24—34no—
381C24—62no—
382C24—62no—
383C24—64no—
384C24—124no—
385C24—242no—
386C24C2×C4×D311yescharacteristic
387C24C23×D311yescharacteristic
388#41—124no—
389#41—128no—
390#41—124no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—122no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—31no—
399C2×D4—32no—
400C2×D4—31no—
401C2×D4—34no—
402C2×D4—32no—
403C2×D4—34no—
404C2×D4—32no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—64no—
410C2×D4—124no—
411C2×D4—122no—
412C2×D4—124no—
413C2×D4—128no—
414C2×D4—124no—
415C2×D4—242no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—128no—
430C22×C4—124no—
431C22×C4—128no—
432C22×C4—244no—
433C22×C4—242no—
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×C4×D311yescharacteristic
436C22×C4C2×C4×D311yescharacteristic
437C22×C4C2×C4×D311yescharacteristic
438C22×C4C2×C4×D311yescharacteristic
439C22×C4D3×D412yes—
440C22×C4D3×D412yes—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—64no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—34no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—128no—
461C22⋊C4—128no—
462C22⋊C4—244no—
463C42—21no—
464C42—21no—
465C42—124no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—42no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C22×C22⋊C411yescharacteristic
479C22×C3C4×#4111yescharacteristic
480C22×C3#52011yescharacteristic
481C22×C3#54612yes—
482C22×C3#54712yes—
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—24no—
487C2×D3—22no—
488C2×D3—24no—
489C2×D3—22no—
490C2×D3—24no—
491C2×D3—48no—
492C2×D3—48no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—84no—
496C2×D3—84no—
497C2×D3—82no—
498C2×D3—82no—
499C2×D3C2×#9914yes—
500C3×C4—21no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—84no—
506C3×C4C2×#9912yes—
507Dic3—44no—
508Dic3—42no—
509Dic3—42no—
510Dic3—82no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—32no—
515C23—34no—
516C23—32no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—68no—
522C23—64no—
523C23—62no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—62no—
528C23—62no—
529C23—64no—
530C23—64no—
531C23—124no—
532C23—122no—
533C23—124no—
534C23—124no—
535C23—122no—
536C23—122no—
537C23—244no—
538C23—244no—
539C23—248no—
540C23C22×C4×D311yescharacteristic
541C23C2×D3×D412yes—
542C23#100411yescharacteristic
543C23#100711yescharacteristic
544C23D3×#4112yes—
545Q8—122no—
546Q8—122no—
547D4—34no—
548D4—42no—
549D4—42no—
550D4—64no—
551D4—68no—
552D4—62no—
553D4—64no—
554D4—122no—
555D4—124no—
556D4—124no—
557D4—124no—
558D4—244no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—124no—
579C2×C4—124no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—242no—
588C2×C4—242no—
589C2×C4—2416no—
590C2×C4D3×C22⋊C411yescharacteristic
591C2×C4D3×C22⋊C412yes—
592C2×C4D3×C22⋊C411yescharacteristic
593C2×C3—22no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—81no—
598C2×C3#233911yescharacteristic
599C2×C3C22×#9911yescharacteristic
600C2×C3#288311yescharacteristic
601D3—24no—
602D3—42no—
603D3—44no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—34no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—62no—
616C22—62no—
617C22—64no—
618C22—62no—
619C22—62no—
620C22—64no—
621C22—62no—
622C22—64no—
623C22—81no—
624C22—81no—
625C22—128no—
626C22—128no—
627C22—124no—
628C22—124no—
629C22—244no—
630C22—244no—
631C22—242no—
632C22—242no—
633C22C2×D3×C22⋊C411yescharacteristic
634C22#550411yescharacteristic
635C22#552211yescharacteristic
636C22#561212yes—
637C22#563112yes—
638C4—21no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—84no—
644C4—124no—
645C4—122no—
646C4—122no—
647C4—242no—
648C4D3×#9912yes—
649C3#2247611yescharacteristic, complemented, Sylow
650C2—22no—
651C2—22no—
652C2—21no—
653C2—21no—
654C2—64no—
655C2—81no—
656C2—122no—
657C2—124no—
658C2#7400811yescharacteristic
659C2C2×D3×#9911yescharacteristic
660C2#7869411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1799C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5550C2
cosocleC24×C3C24
hypercenter#1799D3
hypocenterC3#22476
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #74007C2semidirect
3. C3×#1799C2semidirect
4. C4×#834C2semidirect
5. #70036C2semidirect
6. #69529C2semidirect
7. #68969C2semidirect
8. D3×#342C2semidirect
10. C22×#235C4semidirect
14. C2×#930C4semidirect
19. C2×#894C4semidirect
21. C2×#884C4semidirect
24. C2×#875C4semidirect
25. C2×C4×D12C4semidirect
43. #1799D3semidirect
48. C23×C3×C4C2×C4semidirect
50. C22×#67C2×C4semidirect
51. C22×#67C2×C4semidirect
64. C22×C4×D3C2×C4semidirect
65. C22×C4×D3C2×C4semidirect
66. C22×Dic6C2×C4semidirect
70. C2×C3×C4⋊C4C2×C4semidirect
71. C2×C3×C22⋊C4C2×C4semidirect
74. C2×C3×C42C2×C4semidirect
85. C2×(C2×D3)⋊C4C2×C4semidirect
86. C2×(C2×D3)⋊C4C2×C4semidirect
87. C2×(C2×D3)⋊C4C2×C4semidirect
88. C2×(C2×D3)⋊C4C2×C4semidirect
89. C2×C4⋊Dic3C2×C4semidirect
92. C2×Dic3⋊C4C2×C4semidirect
216. #217#41semidirect
248. C23×C4C4×D3semidirect
271. C2×C4⋊C4C4×D3semidirect
286. C2×C22⋊C4C4×D3semidirect
290. C2×C42C4×D3semidirect
369. D12#99semidirect
649. C3#22476semidirect

Maximal subgroups

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

subgroupindex
C2×#55502
#740072
C3×#17992
C4×#8342
#700362
#695292
#689692
D3×#3422
#224763