Finite Group Databasebeta

← #10826279

Subgroups — #10826279

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×Q8×#67C211yesmaximal, characteristic
3#84068C211yesmaximal, characteristic
4C2×#5228C211yesmaximal, characteristic, complemented
5C2×#5186C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#72073C212yesmaximal, complemented
8#72071C212yesmaximal, complemented
9#72065C212yesmaximal, complemented
10#71975C211yesmaximal, characteristic
11#71972C211yesmaximal, characteristic, complemented
12#23856—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×D3×Q8C2211yescharacteristic
16C22×C3×Q16—24no—
17C22×C3×Q16—28no—
18C22×C3×Q16C2212yes—
19C22×C3×Q16C2212yes—
20C23×C3×C8C2211yescharacteristic
21Q8×#67—24no—
22Q8×#67—24no—
23C2×#946—21no—
24C2×#946C2211yescharacteristic
25C2×#946C2212yes—
26C2×Q8×Dic3C2211yescharacteristic
27C2×#944—21no—
28C2×#944C2211yescharacteristic
29C22×#216C2212yes—
30C2×C4×#67—21no—
31C2×C4×#67C2211yescharacteristic
32C2×#924—21no—
33C2×#924C2211yescharacteristic
34#5231—21no—
35#5230—21no—
36#5229C2212yes—
37#5228—24no—
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41#5213—21no—
42#5212C2211yescharacteristic
43#5210—21no—
44C2×#830C2211yescharacteristic
45#5186—28no—
46C2×(C2×D3)⋊C8C2212yes—
47C2×#816C2212yes—
48#4551C2212yes—
49#4550C2212yes—
50C23×Q16D311yescharacteristic, complemented
51C2×D4×Q8—31no—
52C2×#451—31no—
53C2×#451—32no—
54#2409—31no—
55#2016—32no—
56#2001—32no—
57#1892—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×#67D411yescharacteristic
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—24no—
81C2×D3×Q8—42no—
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×Dic6C2311yescharacteristic
86C2×C3×Q16—232no—
87C2×C3×Q16—24no—
88C2×C3×Q16—24no—
89C2×C3×Q16—24no—
90C2×C3×Q16—24no—
91C2×C3×Q16—24no—
92C2×C3×Q16D418yescomplemented
93C22×C3×C8—22no—
94C22×C3×C8—24no—
95C22×C3×C8C2312yes—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99#946—28no—
100#946—22no—
101#946—22no—
102#946—24no—
103#946—42no—
104Q8×Dic3—22no—
105Q8×Dic3—22no—
106#944—28no—
107#944—22no—
108#944—22no—
109C2×#216—24no—
110C2×#216—24no—
111C2×#216—42no—
112C4×#67—28no—
113C4×#67—22no—
114C4×#67—22no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4D411yescharacteristic
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2311yescharacteristic
121#924—28no—
122#924—22no—
123#924D414yes—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
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(C2×D3)⋊C8—24no—
141#816—24no—
142C23×Q8—31no—
143C23×Q8C2×D311yescharacteristic
144C23×Q8C2×D311yescharacteristic
145C22×Q16—24no—
146C22×Q16—28no—
147C22×Q16—32no—
148C22×Q16C2×D312yes—
149C22×Q16C2×D312yes—
150C23×C8C2×D311yescharacteristic
151D4×Q8—64no—
152D4×Q8—64no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—61no—
155C2×C22⋊Q8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—32no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×C4×Q8—31no—
161C2×C4×D4—31no—
162C2×C4×D4—61no—
163#451—64no—
164#451—68no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#103—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×D3C2×D411yescharacteristic
181C2×C3×Q8—24no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—28no—
187C2×C3×Q8—28no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—44no—
195C2×C3×Q8—44no—
196C2×C3×Q8—44no—
197C2×C3×Q8—48no—
198C2×C3×Q8—48no—
199C2×C3×Q8—44no—
200C2×C3×Q8—42no—
201C2×C3×Q8C2×D414yes—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8C2×D414yes—
204C2×C3×Q8C2×D412yes—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—24no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—44no—
212C22×C3×C4—44no—
213C22×C3×C4—42no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2411yescharacteristic
218C2×#67—22no—
219C2×#67—22no—
220C2×#67Q1614yescomplemented
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—22no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227D3×Q8—48no—
228D3×Q8—44no—
229D3×Q8—44no—
230C2×C4×D3—24no—
231C2×C4×D3—22no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—22no—
235C2×C4×D3—44no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—24no—
239C2×Dic6—21no—
240C2×Dic6—44no—
241C2×Dic6C2×D412yes—
242C3×Q16—216no—
243C3×Q16—216no—
244C2×C3×C8—28no—
245C2×C3×C8—22no—
246C2×C3×C8—22no—
247C2×C3×C8C2×D414yes—
248#217—22no—
249#217—24no—
250#217—42no—
251#216—48no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—44no—
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—22no—
258C4⋊Dic3—22no—
259C4⋊Dic3C2×D412yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264Dic3⋊C4—22no—
265Dic3⋊C4—44no—
266C4×Dic3—24no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—21no—
275C22×Q8—24no—
276C22×Q8—24no—
277C22×Q8—22no—
278C22×Q8—22no—
279C22×Q8—31no—
280C22×Q8—42no—
281C22×Q8—42no—
282C22×Q8—61no—
283C22×Q8—62no—
284C22×Q8—62no—
285C22×Q8—64no—
286C22×Q8—122no—
287C22×Q8#6714yes—
288C22×Q8C22×D311yescharacteristic
289C22×Q8C22×D312yes—
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C23×C4—21no—
293C23×C4—21no—
294C23×C4—31no—
295C23×C4—61no—
296C23×C4C22×D311yescharacteristic
297C2×Q16—232no—
298C2×Q16—24no—
299C2×Q16—24no—
300C2×Q16—24no—
301C2×Q16—24no—
302C2×Q16—24no—
303C2×Q16—64no—
304C2×Q16—64no—
305C2×Q16—122no—
306C2×Q16#6718yescomplemented
307C22×C8—22no—
308C22×C8—24no—
309C22×C8—32no—
310C22×C8C22×D312yes—
311C4⋊Q8—62no—
312C4⋊Q8—68no—
313C4⋊Q8—62no—
314C22⋊Q8—34no—
315C22⋊Q8—62no—
316C22⋊Q8—68no—
317C22⋊Q8—68no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C22⋊Q8—64no—
321C22⋊Q8—122no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×D4—68no—
325C4×D4—62no—
326C4×D4—62no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
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—62no—
341C2×C22⋊C4—61no—
342C2×C42—31no—
343C2×C42—61no—
344#103—64no—
345#103—64no—
346#103—64no—
347#103—122no—
348C22⋊C8—62no—
349C22⋊C8—64no—
350#95—62no—
351#95—62no—
352#95—61no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3#12012yes—
356C23×C3C2×Q1612yes—
357C23×C3#13712yes—
358C23×C3C22×D411yescharacteristic
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—21no—
362C22×D3—22no—
363C22×D3C2×Q1612yescomplemented
364C3×Q8—22no—
365C3×Q8—24no—
366C3×Q8—22no—
367C3×Q8—24no—
368C3×Q8—22no—
369C3×Q8—24no—
370C3×Q8—48no—
371C3×Q8—48no—
372C3×Q8—42no—
373C3×Q8—42no—
374C3×Q8—42no—
375C3×Q8#12014yes—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—48no—
385C2×C3×C4—48no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4#12012yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D412yes—
393C2×C3×C4H(5,𝔽2)12yes—
394#67—28no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3C2×Q1612yescomplemented
409C4×D3—48no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—42no—
413Dic6—24no—
414Dic6—48no—
415Dic6—41no—
416Dic6—41no—
417C3×C8—24no—
418C3⋊C8—42no—
419C24—31no—
420C24D3×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—32no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—48no—
439C2×Q8—48no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—64no—
443C2×Q8—61no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—124no—
447C2×Q8—128no—
448C2×Q8—124no—
449C2×Q8—124no—
450C2×Q8C2×#6712yes—
451C2×Q8C2×#6714yes—
452C2×Q8C2×#6714yes—
453C2×Q8C2×#6712yes—
454C2×D4—34no—
455C2×D4—62no—
456C2×D4—62no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—61no—
476C22×C4—62no—
477C22×C4—124no—
478C22×C4D3×D411yescharacteristic
479C22×C4D3×D412yes—
480C22×C4C2×#6712yes—
481C22×C4C23×D311yescharacteristic
482Q16—216no—
483Q16—216no—
484Q16—128no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8C2×#6714yes—
492C4⋊C4—32no—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—61no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—124no—
503C22⋊C4—64no—
504C22⋊C4—62no—
505C22⋊C4—64no—
506C22⋊C4—62no—
507C22⋊C4—124no—
508C22⋊C4—122no—
509C42—64no—
510C42—61no—
511C42—61no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3#45114yescomplemented
517C22×C3C2×#12011yescharacteristic
518C22×C3D4212yes—
519C22×C3C22×Q1611yescharacteristic
520C22×C3C2×#13711yescharacteristic
521C22×C3#57812yes—
522C2×D3—22no—
523C2×D3—24no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—42no—
527C3×C4—24no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—44no—
531C3×C4C2×#12011yescharacteristic
532C3×C4#53411yescharacteristic
533C3×C4D4212yes—
534Dic3—22no—
535Dic3—42no—
536Dic3—41no—
537Dic3—41no—
538Dic3—42no—
539Dic3—82no—
540C23—22no—
541C23—22no—
542C23—32no—
543C23—61no—
544C23—62no—
545C23—61no—
546C23—62no—
547C23D3×Q1612yes—
548C23#91812yes—
549C23#93712yes—
550C23C2×D3×D411yescharacteristic
551Q8—24no—
552Q8—22no—
553Q8—24no—
554Q8—22no—
555Q8—22no—
556Q8—24no—
557Q8—48no—
558Q8—48no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—64no—
563Q8—128no—
564Q8—121no—
565Q8—121no—
566Q8#95314yes—
567D4—68no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—32no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—128no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4#93712yes—
600C2×C4C2×D3×D411yescharacteristic
601C2×C4#100412yes—
602C2×C4C22×#6712yes—
603C2×C4C22×#6711yescharacteristic
604C8—24no—
605C8—122no—
606C2×C3—24no—
607C2×C3#240311yescharacteristic
608C2×C3C2×#45111yescharacteristic
609C2×C3#301111yescharacteristic
610C2×C3#327012yes—
611C2×C3D4×Q1612yes—
612D3—42no—
613C22—22no—
614C22—22no—
615C22—24no—
616C22—24no—
617C22—62no—
618C22—64no—
619C22—121no—
620C22—121no—
621C22—122no—
622C22#518614yescomplemented
623C22C2×D3×Q1611yescharacteristic
624C22C2×#91811yescharacteristic
625C22#576612yes—
626C22C2×#93711yescharacteristic
627C22D4×#6712yes—
628C4—22no—
629C4—24no—
630C4—42no—
631C4—44no—
632C4—62no—
633C4—121no—
634C4—122no—
635C4—121no—
636C4—122no—
637C4—242no—
638C4D4×#6712yes—
639C4#580411yescharacteristic
640C4C2×#95311yescharacteristic
641C3#2385611yescharacteristic, complemented, Sylow
642C2—24no—
643C2—122no—
644C2C2×#518611yescharacteristic
645C2#8388411yescharacteristic
646C2#8388812yes—
647C2Q16×#6712yes—
648C2#8405211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×Q16C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC23×Q16D3
hypocenterC3#23856
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
4. C2×#5228C2semidirect
5. C2×#5186C2semidirect
6. #72132C2semidirect
7. #72073C2semidirect
8. #72071C2semidirect
9. #72065C2semidirect
11. #71972C2semidirect
50. C23×Q16D3semidirect
92. C2×C3×Q16D4semidirect
220. C2×#67Q16semidirect
306. C2×Q16#67semidirect
363. C22×D3C2×Q16semidirect
408. C2×Dic3C2×Q16semidirect
516. C22×C3#451semidirect
622. C22#5186semidirect
641. C3#23856semidirect

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×Q8×#672
#840682
C2×#52282
C2×#51862
#721322
#720732
#720712
#720652
#719752
#719722
#238563