Finite Group Databasebeta

← #10727099

Subgroups — #10727099

Order 768.

2242 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic, complemented
3C2×#5210C211yesmaximal, characteristic
4C22×C3⋊#95C211yesmaximal, characteristic
5C2×#4942C212yesmaximal
6#72134C211yesmaximal, characteristic, complemented
7#71568C212yesmaximal, complemented
8#71568C211yesmaximal, characteristic, complemented
9#71566C212yesmaximal, complemented
10#71566C211yesmaximal, characteristic, complemented
11#71544C212yesmaximal
12#14758—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C24×Dic3C2212yes—
16C2×C3×C22⋊Q8—24no—
17C2×C3×C22⋊Q8—28no—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2211yescharacteristic
22C22×#217C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27#5230—24no—
28#5210—24no—
29C2×C3⋊#95—24no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41#4942—28no—
42C22×C22⋊Q8D311yescharacteristic
43C2×#395—31no—
44C2×#385—31no—
45C2×#380—32no—
46C22×#95—31no—
47#1446—31no—
48#1444—32no—
49#1444—31no—
50#1441—31no—
51#1441—32no—
52C22⋊#95—32no—
53C25×C3C2311yescharacteristic
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4—24no—
60C23×C3×C4—22no—
61C23×C3×C4—22no—
62C23×C3×C4—42no—
63C23×C3×C4D414yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—24no—
69C23×Dic3—24no—
70C23×Dic3—24no—
71C23×Dic3—24no—
72C23×Dic3—42no—
73C23×Dic3—42no—
74C23×Dic3Q818yescomplemented
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2312yes—
81C3×C22⋊Q8—232no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4D418yes—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217—24no—
93C2×#217—22no—
94C2×#217Q814yescomplemented
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—42no—
104C2×Dic3⋊C4—42no—
105C2×Dic3⋊C4—42no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C3⋊#95—42no—
116C3⋊#95D414yes—
117C23×Q8C2×D311yescharacteristic
118C24×C4—32no—
119C24×C4C2×D311yescharacteristic
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—28no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D312yes—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4C2×D312yes—
131#395—64no—
132#385—64no—
133#380—68no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—31no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—31no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—64no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C24×C3C2×D414yes—
154C24×C3C2×Q812yes—
155C24×C3C2×Q814yes—
156C24×C3#4112yes—
157C24×C3C2411yescharacteristic
158C2×C3×Q8—28no—
159C2×C3×Q8—48no—
160C2×C3×Q8—48no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—28no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—28no—
183C22×C3×C4—44no—
184C22×C3×C4—48no—
185C22×C3×C4—48no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4#4112yes—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—216no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—21no—
213C22×Dic3—216no—
214C22×Dic3—21no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—22no—
221C22×Dic3—22no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—48no—
225C22×Dic3—48no—
226C22×Dic3—48no—
227C22×Dic3—48no—
228C22×Dic3—48no—
229C22×Dic3C2×D414yes—
230C22×Dic3C2×Q812yes—
231C22×Dic3C2×Q812yes—
232C22×Dic3C2×Q814yes—
233C22×Dic3C2×Q814yes—
234C22×Dic3#4114yes—
235C3×C4⋊C4—28no—
236C3×C4⋊C4—216no—
237C3×C22⋊C4—216no—
238#217—28no—
239#217—44no—
240Dic3⋊C4—48no—
241Dic3⋊C4—48no—
242Dic3⋊C4—44no—
243Dic3⋊C4—44no—
244C25C22×D311yescharacteristic
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—42no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—32no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—32no—
257C23×C4—38no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—42no—
261C23×C4—64no—
262C23×C4—64no—
263C23×C4—64no—
264C23×C4—64no—
265C23×C4—122no—
266C23×C4—122no—
267C23×C4#6714yes—
268C23×C4C22×D312yes—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D311yescharacteristic
272C22⋊Q8—232no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—122no—
287C2×C4⋊C4—122no—
288C2×C4⋊C4—122no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—34no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4#6718yes—
297#95—34no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—64no—
303#95—64no—
304#95—62no—
305#95—62no—
306#95—122no—
307#95—122no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—24no—
326C23×C3#12014yes—
327C23×C3C22⋊Q814yes—
328C23×C3C22⋊Q818yes—
329C23×C3C4⋊Q814yes—
330C23×C3C4⋊Q818yes—
331C23×C3C22×D411yescharacteristic
332C23×C3C22×D411yescharacteristic
333C23×C3C22×D412yes—
334C23×C3C22×Q812yes—
335C23×C3C22×Q811yescharacteristic
336C23×C3C2×#4111yescharacteristic
337C23×C3C2×#4111yescharacteristic
338C23×C3C2×#4112yes—
339C23×C3H(5,𝔽2)12yes—
340C23×C3H(5,𝔽2)11yescharacteristic
341C23×C32−1+411yescharacteristic
342C3×Q8—416no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—416no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4#12114yes—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—432no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3C22⋊Q818yescomplemented
405C24—21no—
406C24—21no—
407C24—22no—
408C24—22no—
409C24—21no—
410C24—21no—
411C24D3×Q812yes—
412C24D3×Q814yes—
413C24#23912yes—
414C24C2×#6712yes—
415C24C2×#6714yes—
416C24C23×D311yescharacteristic
417C2×Q8—28no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—44no—
421C2×Q8—44no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—34no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—34no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—616no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—616no—
457C22×C4—61no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—62no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—128no—
484C22×C4—128no—
485C22×C4—128no—
486C22×C4—128no—
487C22×C4—128no—
488C22×C4D3×D412yes—
489C22×C4#23712yes—
490C22×C4C2×#6714yes—
491C22×C4C2×#6714yes—
492C22×C4C2×#6712yes—
493C22×C4C2×#6712yes—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—128no—
497C4⋊C4—128no—
498C4⋊C4—124no—
499C4⋊C4—124no—
500C22⋊C4—216no—
501C22⋊C4—68no—
502C22⋊C4—124no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—28no—
510C22×C3—28no—
511C22×C3—28no—
512C22×C3—28no—
513C22×C3#39518yescomplemented
514C22×C3C2×#12011yescharacteristic
515C22×C3C2×#12111yescharacteristic
516C22×C3C2×C22⋊Q811yescharacteristic
517C22×C3C2×C22⋊Q812yes—
518C22×C3C2×C22⋊Q812yes—
519C22×C3C2×C4⋊Q811yescharacteristic
520C22×C3C2×C4⋊Q812yes—
521C22×C3#53211yescharacteristic
522C22×C3#53312yes—
523C22×C3#53411yescharacteristic
524C22×C3#53711yescharacteristic
525C22×C3#54312yes—
526C22×C3#54412yes—
527C22×C3D4212yes—
528C22×C3#54612yes—
529C22×C3#54612yes—
530C22×C3#54611yescharacteristic
531C22×C3#54712yes—
532C22×C3D4×Q811yescharacteristic
533C22×C3#55112yes—
534C22×C3#55211yescharacteristic
535C22×C3#55411yescharacteristic
536C22×C3#55412yes—
537C3×C4—24no—
538C3×C4—42no—
539C3×C4—44no—
540C3×C4—44no—
541Dic3—28no—
542Dic3—42no—
543Dic3—42no—
544Dic3—44no—
545Dic3—81no—
546Dic3—81no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—22no—
565C23#94418yes—
566C23#94414yes—
567C23#94614yes—
568C23#94618yes—
569C23#95314yes—
570C23C2×D3×D411yescharacteristic
571C23C2×#23711yescharacteristic
572C23#100412yes—
573C23C2×D3×Q811yescharacteristic
574C23C2×D3×Q812yes—
575C23C2×#23911yescharacteristic
576C23D3×#4112yes—
577C23#100911yescharacteristic
578C23#101011yescharacteristic
579C23C22×#6712yes—
580C23C22×#6711yescharacteristic
581Q8—416no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—38no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—64no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—64no—
621C2×C4—1232no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—121no—
625C2×C4—122no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—122no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—121no—
633C2×C4—122no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4#93914yes—
644C2×C3—28no—
645C2×C3C2×#39511yescharacteristic
646C2×C3#241011yescharacteristic
647C2×C3#241711yescharacteristic
648C2×C3#243011yescharacteristic
649C2×C3#243712yes—
650C2×C3#243812yes—
651C2×C3#244012yes—
652C2×C3#254912yes—
653C2×C3#256411yescharacteristic
654C2×C3#265911yescharacteristic
655C2×C3#285711yescharacteristic
656C22—28no—
657C22—28no—
658C22—24no—
659C22—28no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22#523018yescomplemented
667C22#562212yes—
668C22#562312yes—
669C22D3×C22⋊Q812yes—
670C22#563011yescharacteristic
671C22#563211yescharacteristic
672C22#563412yes—
673C22#563511yescharacteristic
674C22#563612yes—
675C22#563712yes—
676C22#564211yescharacteristic
677C22D4×#6712yes—
678C22C2×#93911yescharacteristic
679C22#580411yescharacteristic
680C22#580612yes—
681C22C2×#94412yes—
682C22C2×#94411yescharacteristic
683C22C2×#94612yes—
684C22C2×#94611yescharacteristic
685C22#582412yes—
686C22#582912yes—
687C22#583211yescharacteristic
688C22#583311yescharacteristic
689C22C2×#95311yescharacteristic
690C4—24no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—68no—
695C4—122no—
696C4—122no—
697C4—124no—
698C4—241no—
699C4—241no—
700C3#1475811yescharacteristic, complemented, Sylow
701C2—28no—
702C2#8276212yes—
703C2#8276312yes—
704C2#8278112yes—
705C2#8278411yescharacteristic
706C2#8278911yescharacteristic
707C2#8281111yescharacteristic
708C2#8281511yescharacteristic
709C2#8281612yes—
710C2#8404811yescharacteristic
711C2C2×#523011yescharacteristic
712C2#8409111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C22⋊Q8C2
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#14758
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5230C2semidirect
6. #72134C2semidirect
7. #71568C2semidirect
8. #71568C2semidirect
9. #71566C2semidirect
10. #71566C2semidirect
74. C23×Dic3Q8semidirect
94. C2×#217Q8semidirect
404. C2×Dic3C22⋊Q8semidirect
513. C22×C3#395semidirect
666. C22#5230semidirect
700. C3#14758semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×#52302
C2×#52102
C22×C3⋊#952
C2×#49422
#721342
#715682
#715662
#715442
#147583