Finite Group Databasebeta

← #10732607

Subgroups — #10732607

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#875C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C2×#4973C211yesmaximal, characteristic, complemented
4C2×#4973C211yesmaximal, characteristic, complemented
5C2×#4988C211yesmaximal, characteristic
6C2×#4988C211yesmaximal, characteristic
7#71570C214yesmaximal, complemented
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9#69429C214yesmaximal, complemented
10#15226—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2214yes—
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×#875—24no—
24C2×#875—24no—
25C2×#875—24no—
26C3×#389—24no—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2212yes—
32#4988—24no—
33#4988—24no—
34#4973—24no—
35#4973—24no—
36C22×#124—31no—
37C2×#391—31no—
38C2×#391—31no—
39C2×#390—31no—
40C2×#390—31no—
41C2×#389D311yescharacteristic, complemented
42#1440—34no—
43#1439—34no—
44C4×#95—31no—
45C24×D3C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—22no—
57C22×D12—22no—
58C22×D12—22no—
59C22×D12—22no—
60C22×D12D414yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6D414yes—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—44no—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42D414yes—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—28no—
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—44no—
90C2×(C2×D3)⋊C4—44no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97#875—232no—
98C3×#95—24no—
99C3⋊#95—28no—
100C3⋊#95—28no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C23×Q8—31no—
104C23×D4—31no—
105C2×#124—64no—
106C2×#124—64no—
107C2×#124—64no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C4⋊C4C2×D312yes—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—32no—
113C22×C22⋊C4—34no—
114C22×C22⋊C4—34no—
115C22×C42—31no—
116C22×C42—31no—
117C22×C42C2×D311yescharacteristic
118#391—64no—
119#391—64no—
120#390—64no—
121#390—64no—
122#389—24no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—34no—
126C2×#95—34no—
127C2×#95C2×D312yes—
128C24×C3C2411yescharacteristic
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—24no—
133C23×D3—24no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—24no—
138C23×D3—24no—
139C23×D3C2×D414yes—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4C2×D414yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3C2×D414yes—
186C22×Dic3#4114yes—
187C22×Dic3#4114yes—
188C2×D12—28no—
189C2×D12—28no—
190C2×D12—28no—
191C2×D12—44no—
192C2×D12—44no—
193C2×D12—44no—
194C2×D12—44no—
195C2×Dic6—28no—
196C2×Dic6—28no—
197C2×Dic6—28no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6—44no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—48no—
204C3×C4⋊C4—44no—
205C3×C42—28no—
206(C2×D3)⋊C4—216no—
207(C2×D3)⋊C4—216no—
208(C2×D3)⋊C4—416no—
209(C2×D3)⋊C4—416no—
210C4×Dic3—44no—
211C4×Dic3—44no—
212C25—32no—
213C22×Q8—34no—
214C22×Q8—62no—
215C22×Q8—62no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—62no—
220C22×D4—34no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4—62no—
225C22×D4—62no—
226C22×D4—62no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4C22×D312yes—
231C23×C4C22×D312yes—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235#124—632no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4D1214yes—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—68no—
251C2×C22⋊C4—68no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—124no—
255C2×C22⋊C4—124no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—62no—
259C2×C42—62no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42—62no—
264C2×C42#6714yes—
265#95—24no—
266#95—64no—
267#95—64no—
268#95—68no—
269#95—68no—
270C23×C3C22×D411yescharacteristic
271C23×C3C22×D411yescharacteristic
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C2×#4111yescharacteristic
275C23×C3C2×#4111yescharacteristic
276C23×C3C2×#4111yescharacteristic
277C23×C3C2×#4112yes—
278C23×C3C2×#4112yes—
279C23×C3H(5,𝔽2)12yes—
280C23×C32−1+412yes—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—28no—
285C22×D3—28no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—44no—
291C22×D3—44no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—44no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3—44no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—28no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4#12014yes—
325C2×C3×C4#12114yes—
326C2×C3×C4#12114yes—
327C2×C3×C4#12114yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12314yes—
330C2×C3×C4#12414yes—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352D12—416no—
353Dic6—416no—
354C24—34no—
355C24—62no—
356C24—62no—
357C24—62no—
358C24—64no—
359C24—62no—
360C24—62no—
361C24—64no—
362C24—64no—
363C24—64no—
364C24—62no—
365C24C23×D311yescharacteristic
366C2×Q8—68no—
367C2×Q8—68no—
368C2×Q8—68no—
369C2×Q8—124no—
370C2×Q8—124no—
371C2×Q8—124no—
372C2×Q8—124no—
373C2×D4—68no—
374C2×D4—68no—
375C2×D4—68no—
376C2×D4—124no—
377C2×D4—124no—
378C2×D4—124no—
379C2×D4—124no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4C2×D1214yes—
419C22×C4C2×D1212yes—
420C22×C4#23512yes—
421C22×C4D3×D412yes—
422C22×C4D3×D412yes—
423C22×C4#23912yes—
424C22×C4#23912yes—
425C22×C4C2×#6712yes—
426C22×C4C2×#6712yes—
427C22×C4C2×#6712yes—
428C4⋊C4—28no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C22⋊C4—616no—
432C22⋊C4—616no—
433C22⋊C4—1216no—
434C22⋊C4—1216no—
435C42—28no—
436C42—124no—
437C42—124no—
438C22×C3C2×#12011yescharacteristic
439C22×C3C2×#12111yescharacteristic
440C22×C3C2×#12111yescharacteristic
441C22×C3C2×#12111yescharacteristic
442C22×C3C2×#12311yescharacteristic
443C22×C3C2×#12311yescharacteristic
444C22×C3C2×#12411yescharacteristic
445C22×C3#52912yes—
446C22×C3#52912yes—
447C22×C3#53211yescharacteristic
448C22×C3#53211yescharacteristic
449C22×C3#53511yescharacteristic
450C22×C3#53611yescharacteristic
451C22×C3#54212yes—
452C22×C3#54212yes—
453C22×C3#54612yes—
454C22×C3#54612yes—
455C22×C3#54712yes—
456C22×C3#54812yes—
457C22×C3#54812yes—
458C22×C3#55012yes—
459C22×C3#55612yes—
460C22×C3#55912yes—
461C2×D3—44no—
462C2×D3—48no—
463C2×D3—44no—
464C2×D3—44no—
465C2×D3—48no—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—84no—
469C2×D3—84no—
470C2×D3—82no—
471C2×D3—82no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—44no—
475C3×C4—82no—
476C3×C4#39418yes—
477Dic3—44no—
478Dic3—44no—
479Dic3—44no—
480C23—64no—
481C23—64no—
482C23—64no—
483C23—68no—
484C23—68no—
485C23—124no—
486C23—124no—
487C23—124no—
488C23—124no—
489C23—124no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—124no—
495C23—122no—
496C23—124no—
497C23—122no—
498C23—124no—
499C23—124no—
500C23—122no—
501C23—122no—
502C23—124no—
503C23C22×D1211yescharacteristic
504C23C2×#23511yescharacteristic
505C23C2×D3×D411yescharacteristic
506C23C2×D3×D411yescharacteristic
507C23#100412yes—
508C23C2×#23911yescharacteristic
509C23C2×#23911yescharacteristic
510C23#100712yes—
511C23D3×#4112yes—
512C23D3×#4112yes—
513C23C22×#6711yescharacteristic
514Q8—1216no—
515D4—1216no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4#89414yes—
558C2×C4#89414yes—
559C2×C4#89514yes—
560C2×C4#89514yes—
561C2×C4#93014yes—
562C2×C4#93714yes—
563C2×C4#94714yes—
564C2×C3C2×#39411yescharacteristic
565C2×C3#242612yes—
566C2×C3#242712yes—
567C2×C3#243111yescharacteristic
568C2×C3#243211yescharacteristic
569C2×C3#244712yes—
570C2×C3#244812yes—
571C2×C3#254411yescharacteristic
572C2×C3#254511yescharacteristic
573C2×C3#264011yescharacteristic
574C2×C3#286111yescharacteristic
575D3—84no—
576C22—124no—
577C22—124no—
578C22—128no—
579C22—124no—
580C22—128no—
581C22—242no—
582C22—242no—
583C22—242no—
584C22—244no—
585C22—242no—
586C22—244no—
587C22C2×#89411yescharacteristic
588C22C2×#89411yescharacteristic
589C22C2×#89511yescharacteristic
590C22C2×#89511yescharacteristic
591C22#555412yes—
592C22#555912yes—
593C22#556812yes—
594C22#557612yes—
595C22#559912yes—
596C22#561212yes—
597C22#563112yes—
598C22#563312yes—
599C22#569112yes—
600C22#569112yes—
601C22#573312yes—
602C22#573312yes—
603C22C2×#93011yescharacteristic
604C22C2×#93711yescharacteristic
605C22#580511yescharacteristic
606C22C2×#94711yescharacteristic
607C22#581611yescharacteristic
608C22#583211yescharacteristic
609C22#583211yescharacteristic
610C4—42no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—124no—
615C4—124no—
616C4—124no—
617C4#499118yes—
618C3#1522611yescharacteristic, complemented, Sylow
619C2—244no—
620C2#7600612yes—
621C2#7600712yes—
622C2#7621812yes—
623C2#7621912yes—
624C2C2×#499111yescharacteristic
625C2#8238111yescharacteristic
626C2#8247411yescharacteristic
627C2#8265211yescharacteristic
628C2#8275911yescharacteristic
629C2#8305711yescharacteristic
630C2#8326311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#389C2
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3#15226
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
3. C2×#4973C2semidirect
4. C2×#4973C2semidirect
7. #71570C2semidirect
8. C4×C3⋊#95C2semidirect
9. #69429C2semidirect
41. C2×#389D3semidirect
618. C3#15226semidirect

Maximal subgroups

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

subgroupindex
C22×#8752
C2×C3×#3892
C2×#49732
C2×#49882
#715702
C4×C3⋊#952
#694292
#152263