Finite Group Databasebeta

← #10826316

Subgroups — #10826316

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×#5817C211yesmaximal, characteristic
3#84068C211yesmaximal, characteristic, complemented
4C2×#5227C211yesmaximal, characteristic, complemented
5C2×#5187C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#72072C212yesmaximal, complemented
8#72071C212yesmaximal, complemented
9#72065C212yesmaximal, complemented
10#71975C211yesmaximal, characteristic, complemented
11#71973C211yesmaximal, characteristic, complemented
12#23859—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×#239C2211yescharacteristic
16C22×C3×Q16—24no—
17C22×C3×Q16—28no—
18C22×C3×Q16C2212yes—
19C22×C3×Q16C2212yes—
20C23×C3×C8C2211yescharacteristic
21#5817—24no—
22#5817—24no—
23C2×#947—21no—
24C2×#947C2211yescharacteristic
25C2×#946—21no—
26C2×#946C2211yescharacteristic
27C2×#946C2212yes—
28C2×Q8×Dic3C2211yescharacteristic
29C22×#215C2212yes—
30C2×#930—21no—
31C2×#930C2211yescharacteristic
32C2×C4×#67—21no—
33C2×C4×#67C2211yescharacteristic
34#5231—21no—
35#5230—21no—
36#5229C2212yes—
37#5227—24no—
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41#5213—21no—
42#5212C2211yescharacteristic
43#5210—21no—
44C2×#830C2211yescharacteristic
45#5187—28no—
46C2×#821C2212yes—
47C2×(C2×D3)⋊C8C2212yes—
48#4551C2212yes—
49#4550C2212yes—
50C23×Q16D311yescharacteristic, complemented
51C2×#548—31no—
52C2×#452—32no—
53C2×#448—31no—
54#2409—31no—
55#2015—32no—
56#2001—32no—
57#1893—31no—
58#1878—32no—
59#1869—31no—
60#1787—31no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—21no—
64C22×C3×Q8—24no—
65C22×C3×Q8—24no—
66C22×C3×Q8—22no—
67C22×C3×Q8—22no—
68C22×C3×Q8—42no—
69C22×C3×Q8—42no—
70C22×C3×Q8D414yes—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×Q8C2312yes—
73C22×C3×Q8C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×#67D411yescharacteristic
79C2×#239—22no—
80C2×#239—22no—
81C2×#239—24no—
82C2×#239—42no—
83C22×D12—21no—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C4×D3C2311yescharacteristic
87C2×C3×Q16—232no—
88C2×C3×Q16—24no—
89C2×C3×Q16—24no—
90C2×C3×Q16—24no—
91C2×C3×Q16—24no—
92C2×C3×Q16—24no—
93C2×C3×Q16D418yescomplemented
94C22×C3×C8—22no—
95C22×C3×C8—24no—
96C22×C3×C8C2312yes—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99#947—28no—
100#947—22no—
101#947—22no—
102#946—28no—
103#946—22no—
104#946—22no—
105#946—24no—
106#946—42no—
107Q8×Dic3—22no—
108Q8×Dic3—22no—
109C2×#215—24no—
110C2×#215—24no—
111C2×#215—42no—
112#930—28no—
113#930—22no—
114#930D414yes—
115C4×#67—28no—
116C4×#67—22no—
117C4×#67—22no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—21no—
120C2×(C2×D3)⋊C4—21no—
121C2×(C2×D3)⋊C4—21no—
122C2×(C2×D3)⋊C4—22no—
123C2×(C2×D3)⋊C4D411yescharacteristic
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—22no—
129C2×Dic3⋊C4D411yescharacteristic
130C2×C4×Dic3—21no—
131C2×C4×Dic3C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133#835—24no—
134#835—24no—
135#835—42no—
136C3⋊#95—21no—
137C3⋊#95—22no—
138C3⋊#95—22no—
139#830—22no—
140#821—24no—
141(C2×D3)⋊C8—24no—
142C22×#41—31no—
143C23×Q8C2×D311yescharacteristic
144C23×Q8C2×D311yescharacteristic
145C22×Q16—24no—
146C22×Q16—28no—
147C22×Q16C2×D312yes—
148C22×Q16C2×D312yes—
149C22×SD16—32no—
150C23×C8C2×D311yescharacteristic
151#548—64no—
152#548—64no—
153C2×#124—31no—
154C2×#124—61no—
155C2×C22⋊Q8—31no—
156C2×C22⋊Q8—32no—
157C2×C22⋊Q8—61no—
158C2×#121—31no—
159C2×#121—61no—
160C2×C4×Q8—31no—
161C2×C4×D4—31no—
162C2×C4×D4—61no—
163#452—68no—
164#448—64no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#102—32no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—32no—
171#395—61no—
172#393—61no—
173#391—32no—
174#386—61no—
175#385—61no—
176#380—31no—
177#340—32no—
178#336—32no—
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C23×D3C2×D411yescharacteristic
182C2×C3×Q8—24no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—28no—
188C2×C3×Q8—28no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—44no—
196C2×C3×Q8—44no—
197C2×C3×Q8—44no—
198C2×C3×Q8—48no—
199C2×C3×Q8—48no—
200C2×C3×Q8—44no—
201C2×C3×Q8—42no—
202C2×C3×Q8C2×D414yes—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8C2×D414yes—
205C2×C3×Q8C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—44no—
213C22×C3×C4—44no—
214C22×C3×C4—42no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2411yescharacteristic
219C2×#67—22no—
220C2×#67—24no—
221C2×#67—22no—
222C2×#67—22no—
223C2×#67—42no—
224C2×#67—42no—
225C22×Dic3—21no—
226C22×Dic3—22no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230#239—48no—
231#239—44no—
232#239—44no—
233C2×D12—22no—
234C2×D12—22no—
235C2×D12—24no—
236C2×D12—21no—
237C2×D12—44no—
238C2×D12C2×D412yes—
239C2×C4×D3—24no—
240C2×C4×D3—22no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3—22no—
244C2×C4×D3—44no—
245C3×Q16—216no—
246C3×Q16—216no—
247C2×C3×C8—28no—
248C2×C3×C8—22no—
249C2×C3×C8—22no—
250C2×C3×C8C2×D414yes—
251#217—22no—
252#217—42no—
253#215—48no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—44no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—24no—
266Dic3⋊C4—22no—
267Dic3⋊C4—44no—
268C4×Dic3—24no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×C3⋊C8—22no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—64no—
277C2×#41—122no—
278C22×Q8—21no—
279C22×Q8—22no—
280C22×Q8—21no—
281C22×Q8—24no—
282C22×Q8—24no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—42no—
286C22×Q8—42no—
287C22×Q8#6714yes—
288C22×Q8C22×D311yescharacteristic
289C22×Q8C22×D312yes—
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C22×D4—31no—
293C22×D4—61no—
294C22×D4—61no—
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—31no—
298C23×C4—61no—
299C23×C4C22×D311yescharacteristic
300C2×Q16—232no—
301C2×Q16—24no—
302C2×Q16—24no—
303C2×Q16—24no—
304C2×Q16—24no—
305C2×Q16—24no—
306C2×Q16#6718yescomplemented
307C2×SD16—64no—
308C2×SD16—64no—
309C2×SD16—122no—
310C22×C8—22no—
311C22×C8—24no—
312C22×C8—32no—
313C22×C8C22×D312yes—
314#124—62no—
315#124—68no—
316#124—62no—
317C22⋊Q8—62no—
318C22⋊Q8—68no—
319C22⋊Q8—62no—
320C22⋊Q8—64no—
321C22⋊Q8—122no—
322#121—34no—
323#121—68no—
324#121—62no—
325C4×Q8—62no—
326C4×Q8—62no—
327C4×D4—68no—
328C4×D4—62no—
329C4×D4—62no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—62no—
336C2×C22⋊C4—31no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—62no—
343C2×C22⋊C4—61no—
344C2×C42—31no—
345C2×C42—61no—
346#103—64no—
347#103—64no—
348#103—122no—
349#102—64no—
350C22⋊C8—62no—
351C22⋊C8—64no—
352#95—62no—
353#95—62no—
354#95—61no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3#12012yes—
358C23×C3#13512yes—
359C23×C3#13712yes—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—22no—
363C22×D3—22no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—22no—
367C22×D3—42no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—42no—
371C3×Q8—22no—
372C3×Q8—24no—
373C3×Q8—22no—
374C3×Q8—24no—
375C3×Q8—22no—
376C3×Q8—24no—
377C3×Q8—48no—
378C3×Q8—48no—
379C3×Q8—42no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×Q8#12014yes—
383C2×C3×C4—22no—
384C2×C3×C4—24no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—22no—
390C2×C3×C4—24no—
391C2×C3×C4—48no—
392C2×C3×C4—48no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4#12012yes—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D412yes—
400C2×C3×C4H(5,𝔽2)12yes—
401#67—44no—
402#67—48no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—42no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416D12—24no—
417D12—48no—
418D12—41no—
419D12—41no—
420C4×D3—48no—
421C4×D3—41no—
422C4×D3—41no—
423C4×D3—42no—
424C3×C8—24no—
425C3⋊C8—42no—
426C24—31no—
427C24—61no—
428C24D3×D411yescharacteristic
429#41—128no—
430#41—124no—
431#41—124no—
432C2×Q8—24no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—28no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8C2×#6712yes—
453C2×Q8C2×#6714yes—
454C2×Q8C2×#6714yes—
455C2×Q8C2×#6712yes—
456C2×D4—32no—
457C2×D4—64no—
458C2×D4—62no—
459C2×D4—62no—
460C2×D4—64no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—61no—
465C2×D4—124no—
466C2×D4—122no—
467C2×D4—122no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—61no—
486C22×C4—62no—
487C22×C4—124no—
488C22×C4D3×D411yescharacteristic
489C22×C4D3×D412yes—
490C22×C4C2×#6712yes—
491C22×C4C23×D311yescharacteristic
492Q16—216no—
493Q16—216no—
494SD16—128no—
495C2×C8—28no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—62no—
499C2×C8—62no—
500C2×C8—62no—
501C2×C8C2×#6714yes—
502C4⋊C4—32no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—62no—
506C4⋊C4—61no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—124no—
510C22⋊C4—64no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—64no—
516C22⋊C4—124no—
517C22⋊C4—122no—
518C42—64no—
519C42—61no—
520C42—61no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—24no—
524C22×C3—24no—
525C22×C3#45214yescomplemented
526C22×C3C2×#12011yescharacteristic
527C22×C3D4212yes—
528C22×C3C2×#13511yescharacteristic
529C22×C3C2×#13711yescharacteristic
530C22×C3#57812yes—
531C2×D3—42no—
532C2×D3—41no—
533C2×D3—41no—
534C2×D3—42no—
535C2×D3—41no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—82no—
540C2×D3—82no—
541C3×C4—24no—
542C3×C4—22no—
543C3×C4—42no—
544C3×C4—44no—
545C3×C4C2×#12011yescharacteristic
546C3×C4#53411yescharacteristic
547C3×C4D4212yes—
548Dic3—41no—
549Dic3—41no—
550Dic3—42no—
551Dic3—41no—
552Dic3—82no—
553C23—22no—
554C23—22no—
555C23—61no—
556C23—61no—
557C23—62no—
558C23—62no—
559C23—61no—
560C23—62no—
561C23—121no—
562C23—121no—
563C23—122no—
564C23—122no—
565C23#91812yes—
566C23#91912yes—
567C23#93712yes—
568C23C2×D3×D411yescharacteristic
569Q8—24no—
570Q8—22no—
571Q8—24no—
572Q8—22no—
573Q8—22no—
574Q8—24no—
575Q8—48no—
576Q8—48no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8#95314yes—
581D4—64no—
582D4—128no—
583D4—124no—
584D4—128no—
585D4—121no—
586D4—121no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—62no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—128no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4#93712yes—
618C2×C4C2×D3×D411yescharacteristic
619C2×C4#100412yes—
620C2×C4C22×#6712yes—
621C2×C4C22×#6711yescharacteristic
622C8—24no—
623C8—122no—
624C2×C3—24no—
625C2×C3#240311yescharacteristic
626C2×C3C2×#45211yescharacteristic
627C2×C3#301111yescharacteristic
628C2×C3#327012yes—
629C2×C3#328512yes—
630D3—81no—
631D3—82no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—122no—
637C22—121no—
638C22—121no—
639C22—122no—
640C22—121no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—242no—
645C22—242no—
646C22#518714yescomplemented
647C22C2×#91811yescharacteristic
648C22C2×#91911yescharacteristic
649C22#576612yes—
650C22C2×#93711yescharacteristic
651C22D4×#6712yes—
652C4—22no—
653C4—24no—
654C4—42no—
655C4—44no—
656C4—121no—
657C4—121no—
658C4—122no—
659C4—121no—
660C4—242no—
661C4D4×#6712yes—
662C4#580411yescharacteristic
663C4C2×#95311yescharacteristic
664C3#2385911yescharacteristic, complemented, Sylow
665C2—24no—
666C2—241no—
667C2—242no—
668C2C2×#518711yescharacteristic
669C2#8388411yescharacteristic
670C2#8388812yes—
671C2#8388912yes—
672C2#8405211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×Q16C2
socleC23×C3C22×D4
prime radicalC2×#5817C2
hypercenterC23×Q16D3
hypocenterC3#23859
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
1. C23×C3×Q16C2semidirect
3. #84068C2semidirect
4. C2×#5227C2semidirect
5. C2×#5187C2semidirect
6. #72132C2semidirect
7. #72072C2semidirect
8. #72071C2semidirect
9. #72065C2semidirect
10. #71975C2semidirect
11. #71973C2semidirect
50. C23×Q16D3semidirect
93. C2×C3×Q16D4semidirect
306. C2×Q16#67semidirect
525. C22×C3#452semidirect
646. C22#5187semidirect
664. C3#23859semidirect

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×#58172
#840682
C2×#52272
C2×#51872
#721322
#720722
#720712
#720652
#719752
#719732
#238593