Finite Group Databasebeta

← #10848769

Subgroups — #10848769

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic, complemented
2#82081C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4C2×#4713C211yesmaximal, characteristic, complemented
5C3×#1749C211yesmaximal, characteristic, complemented
6#69962C212yesmaximal, complemented
7#69961C212yesmaximal, complemented
8#69827C211yesmaximal, characteristic, complemented
9#69821C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68673C212yesmaximal, complemented
12#25605—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930—21no—
15C2×#930—21no—
16C2×#930C2211yescharacteristic
17C2×#930C2212yes—
18C2×#929—21no—
19C2×C4×#67—21no—
20C2×#924C2211yescharacteristic
21C22×D24C2212yes—
22C22×#204C2212yes—
23#5479—24no—
24#5479—24no—
25C2×#876—21no—
26C2×#875C2211yescharacteristic
27C2×#874C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#872—21no—
30C2×#870C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×C3×#97C2211yescharacteristic
34C2×C3×#97C2211yescharacteristic
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×#821C2212yes—
38C2×#816C2212yes—
39#4939—21no—
40#4939—21no—
41#4939C2212yes—
42#4935C2211yescharacteristic
43#4713—24no—
44#4712—24no—
45C3×#336C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#529—31no—
49C2×#496—31no—
50C2×#490—31no—
51#2856—31no—
52#2000—32no—
53#1999—32no—
54#1992—31no—
55#1959—31no—
56#1880—32no—
57#1879—32no—
58#1749D311yescharacteristic, complemented
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—22no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42D411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—21no—
79#930—22no—
80#930—22no—
81#930—42no—
82#930—42no—
83#930—44no—
84#930—44no—
85#929—28no—
86C4×#67—28no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99#924—22no—
100#924—22no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×D24—24no—
107C2×D24—24no—
108C2×D24—42no—
109C2×#204—24no—
110C2×#204—24no—
111C2×#204—42no—
112#876—28no—
113#875—22no—
114#875D414yescomplemented
115#874—22no—
116#874—22no—
117C4×D12—22no—
118C4×D12—22no—
119#872—28no—
120#870—22no—
121#870D414yescomplemented
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×C22⋊C8—24no—
125C3×#97—22no—
126C3×#97—22no—
127#821—24no—
128#821—24no—
129#821—42no—
130#816—42no—
131C22×SD16—32no—
132C22×D8—32no—
133C2×#127—31no—
134#529—64no—
135#529—64no—
136C2×#126—61no—
137C2×#125—31no—
138C2×#124—31no—
139C2×#123—61no—
140C2×C22⋊Q8—31no—
141C2×#121—31no—
142C2×#121—32no—
143C2×#121—61no—
144C2×#121—61no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C2×#117—61no—
149C22×C42C2×D311yescharacteristic
150#496—64no—
151#490—64no—
152C2×#103—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—32no—
156C2×C22⋊C8C2×D312yes—
157C2×#97C2×D311yescharacteristic
158C2×#97C2×D311yescharacteristic
159#390—32no—
160#390—61no—
161#390—61no—
162#384—31no—
163#340—32no—
164#340—32no—
165#336C2×D312yes—
166C24×C3C2×D411yescharacteristic
167C23×D3—21no—
168C23×D3—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C2×#67—44no—
189C2×#67—44no—
190C2×#67—44no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—21no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—42no—
205C2×D12—42no—
206C2×D12—42no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—42no—
210C2×D12C2×D412yes—
211C2×C4×D3—24no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6C2×D412yes—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8#4114yes—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42C2×D412yes—
229C3×C42C2×D412yes—
230#217—24no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—44no—
236(C2×D3)⋊C4—44no—
237(C2×D3)⋊C4—44no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—42no—
243C4⋊Dic3—42no—
244C4⋊Dic3C2×D412yes—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248Dic3⋊C4—24no—
249Dic3⋊C4—24no—
250C4×Dic3—24no—
251D24—48no—
252#204—48no—
253C22×Q8—31no—
254C22×D4—31no—
255C22×D4—32no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4—61no—
259C22×D4—61no—
260C23×C4—61no—
261C23×C4D1211yescharacteristic
262C23×C4D1211yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×D8—64no—
268C2×D8—64no—
269C2×D8—122no—
270C22×C8C22×D312yes—
271C22×C8C22×D312yes—
272#127—62no—
273#127—62no—
274#126—68no—
275#125—34no—
276#125—62no—
277#124—34no—
278#124—62no—
279#123—68no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282#121—62no—
283#121—62no—
284#121—122no—
285#121—122no—
286#121—124no—
287#121—124no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—68no—
293#117—68no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—61no—
314C2×C42D1211yescharacteristic
315C2×C42D1211yescharacteristic
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318#103—122no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—24no—
323#97—22no—
324#97—22no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3AGL(1,ℤ/8)12yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—22no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—44no—
360C2×C3×C4—44no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12012yes—
365C2×C3×C4AGL(1,ℤ/8)12yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4112yes—
369C2×C3×C4H(5,𝔽2)12yes—
370#67—48no—
371#67—88no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384D12—24no—
385D12—42no—
386D12—42no—
387D12—42no—
388D12—42no—
389D12—42no—
390D12—84no—
391D12—84no—
392C4×D3—48no—
393Dic6—24no—
394Dic6—42no—
395C3×C8—24no—
396C3×C8—42no—
397C24—61no—
398C24—62no—
399C24C2×D1211yescharacteristic
400C2×Q8—32no—
401C2×Q8—61no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×D4—32no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—61no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—122no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4—124no—
422C2×D4—124no—
423C2×D4—124no—
424C2×D4—122no—
425C2×D4—122no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—64no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C23×D311yescharacteristic
449SD16—128no—
450D8—128no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8#23514yes—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C22⋊C4—64no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—64no—
482C42D3×D412yes—
483C42D3×D412yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#53211yescharacteristic
493C22×C3#53511yescharacteristic
494C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57511yescharacteristic
497C22×C3#57511yescharacteristic
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C2×D3—82no—
509C2×D3—82no—
510C2×D3—82no—
511C2×D3—82no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4#54612yes—
518C3×C4#55012yes—
519Dic3—42no—
520Dic3—42no—
521Dic3—42no—
522Dic3—82no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23#90812yes—
544C23C22×D1211yescharacteristic
545Q8—64no—
546Q8—122no—
547D4—64no—
548D4—128no—
549D4—122no—
550D4—122no—
551D4—122no—
552D4—122no—
553D4—122no—
554D4—244no—
555D4—244no—
556D4—248no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—128no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4#88212yes—
587C2×C4#88212yes—
588C2×C4#90812yes—
589C2×C4C2×#23512yes—
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×D3×D411yescharacteristic
592C2×C4#100912yes—
593C8—24no—
594C8—42no—
595C2×C3—42no—
596C2×C3#243111yescharacteristic
597C2×C3#300611yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#317111yescharacteristic
600C2×C3#318011yescharacteristic
601C2×C3#321711yescharacteristic
602C2×C3#321911yescharacteristic
603D3—82no—
604D3—162no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—242no—
620C22—242no—
621C22—242no—
622C22—242no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22#547711yescharacteristic
627C22C2×#88211yescharacteristic
628C22#552511yescharacteristic
629C22C2×#90811yescharacteristic
630C22C2×#90811yescharacteristic
631C22#574811yescharacteristic
632C22#574811yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—122no—
639C4—122no—
640C4—122no—
641C4—242no—
642C4#555012yes—
643C4#555312yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—42no—
646C2—242no—
647C2—482no—
648C2#7358611yescharacteristic
649C2#7850011yescharacteristic
650C2#7853211yescharacteristic
651C2#7863911yescharacteristic
652C2#7863911yescharacteristic
653C2#7935311yescharacteristic
654C2#7935411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
hypercenter#1749D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C2×#5479C2semidirect
2. #82081C2semidirect
3. C2×#4712C2semidirect
4. C2×#4713C2semidirect
5. C3×#1749C2semidirect
6. #69962C2semidirect
7. #69961C2semidirect
8. #69827C2semidirect
9. #69821C2semidirect
10. #68686C2semidirect
11. #68673C2semidirect
58. #1749D3semidirect
114. #875D4semidirect
121. #870D4semidirect
644. C3#25605semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
#820812
C2×#47122
C2×#47132
C3×#17492
#699622
#699612
#698272
#698212
#686862
#686732
#256053