Finite Group Databasebeta

← #10827354

Subgroups — #10827354

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic
2C22×#816C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic, complemented
4C2×#4738C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69930C212yesmaximal, complemented
9#69924C211yesmaximal, characteristic, complemented
10#68689C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#23961—31nomaximal, Sylow
13C22×#237—21no—
14C23×Dic6C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#924C2212yes—
22C22×#204C2212yes—
23#5555—24no—
24#5555—24no—
25C2×#897—21no—
26C2×#886—21no—
27C2×#886—21no—
28C2×C4×D12C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2212yes—
32#5220—21no—
33#5212C2211yescharacteristic
34C2×#821C2212yes—
35C2×#816—24no—
36C2×#816C2212yes—
37C2×#816C2212yes—
38C2×#816C2212yes—
39#4979—21no—
40#4947—21no—
41#4946—21no—
42#4929C2212yes—
43#4738—24no—
44C3×#336C2212yes—
45#4551C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#547—31no—
49C2×#448—31no—
50C22×#103—31no—
51#2409—31no—
52#2024—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1872—32no—
57#1869D311yescharacteristic, complemented
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C2×#237—28no—
68C2×#237—44no—
69C2×#237—44no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—22no—
73C22×Dic6—24no—
74C22×Dic6—21no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—42no—
78C22×Dic6C2312yes—
79C22×C3×C8—22no—
80C22×C3×C8—42no—
81C22×C3×C8C2312yes—
82C22×C3×C8C2312yes—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—22no—
91#930—22no—
92#930—22no—
93#930—22no—
94#930D414yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3D414yes—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109#924—24no—
110#924—42no—
111C2×Dic3⋊C4—22no—
112C2×#204—24no—
113C2×#204—24no—
114C2×#204—42no—
115#897—28no—
116#886—28no—
117#886—28no—
118C4×D12—22no—
119C4×D12—22no—
120#869—22no—
121#869—22no—
122C3×C22⋊C8—22no—
123C3×C22⋊C8—42no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126C3⋊#95—21no—
127#821—42no—
128#816—24no—
129#816—24no—
130#816—24no—
131C22×#41—61no—
132C23×Q8—31no—
133C22×SD16—32no—
134C23×C8C2×D311yescharacteristic
135#547—64no—
136#547—64no—
137C2×C4⋊Q8—31no—
138C2×#123—61no—
139C2×#123—61no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4C2×D311yescharacteristic
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148#448—64no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×C22⋊C8C2×D311yescharacteristic
155C2×C22⋊C8C2×D312yes—
156#395—61no—
157#393—61no—
158#391—32no—
159#386—61no—
160#385—61no—
161#380—31no—
162#340—32no—
163#340—32no—
164#340—32no—
165#336C2×D312yes—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D411yescharacteristic
168C23×D3—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4SD1614yescomplemented
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—24no—
187C2×#67—42no—
188C2×#67—42no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—44no—
199C22×Dic3—42no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204#237—432no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—21no—
208C2×D12C2×D412yes—
209C2×C4×D3—24no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—48no—
220C2×Dic6—44no—
221C2×Dic6—44no—
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—42no—
225C2×Dic6#4114yescomplemented
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—44no—
233C3×C4⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C42—21no—
237C3×C42—21no—
238#217—44no—
239#217—42no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—24no—
243C4⋊Dic3—24no—
244C4⋊Dic3—24no—
245C4⋊Dic3—24no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—22no—
250C4⋊Dic3—22no—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3C2×D412yes—
253C4⋊Dic3C2×D412yes—
254Dic3⋊C4—44no—
255#204—48no—
256C2×#41—68no—
257C2×#41—124no—
258C2×#41—124no—
259C22×Q8—32no—
260C22×Q8—64no—
261C22×Q8—61no—
262C22×Q8—61no—
263C22×Q8—62no—
264C22×Q8—62no—
265C22×Q8—122no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4—61no—
269C22×D4D1211yescharacteristic
270C23×C4—61no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—122no—
279C22×C8—22no—
280C22×C8—42no—
281C22×C8C22×D312yes—
282C22×C8C22×D312yes—
283C4⋊Q8—62no—
284C4⋊Q8—62no—
285#123—68no—
286#123—68no—
287C22⋊Q8—68no—
288C22⋊Q8—64no—
289C22⋊Q8—122no—
290#121—34no—
291#121—62no—
292#121—62no—
293#121—62no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—62no—
297C4×D4—62no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4D1211yescharacteristic
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4D1211yescharacteristic
317C2×C22⋊C4D1211yescharacteristic
318C2×C42C22×D311yescharacteristic
319#103—64no—
320#103—64no—
321#103—64no—
322#102—122no—
323C22⋊C8—22no—
324C22⋊C8—42no—
325#95—61no—
326#95—62no—
327#95—62no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—22no—
334C23×C3#12012yes—
335C23×C3C2×SD1612yes—
336C23×C3C2×SD1612yes—
337C23×C3AGL(1,ℤ/8)12yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—42no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—42no—
344C3×D4—28no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4#12012yes—
359C2×C3×C4C2×SD1612yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4H(5,𝔽2)12yes—
364#67—48no—
365#67—48no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—48no—
371C2×Dic3—48no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—48no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384D12—24no—
385D12—41no—
386D12—41no—
387C4×D3—48no—
388Dic6—24no—
389Dic6—24no—
390Dic6—48no—
391Dic6—48no—
392Dic6—42no—
393Dic6—42no—
394C3×C8—42no—
395C3×C8—42no—
396C24—61no—
397C24C2×D1211yescharacteristic
398C24C2×D1211yescharacteristic
399#41—1232no—
400C2×Q8—34no—
401C2×Q8—64no—
402C2×Q8—68no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—124no—
409C2×Q8—128no—
410C2×Q8—124no—
411C2×Q8—122no—
412C2×Q8—122no—
413C2×Q8—122no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—32no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4#20414yescomplemented
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—64no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—122no—
442C22×C4—124no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4C2×D1211yescharacteristic
453C22×C4D3×D412yes—
454C22×C4D3×D412yes—
455C22×C4C23×D311yescharacteristic
456SD16—128no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—22no—
465C4⋊C4—32no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C4⋊C4—124no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—124no—
483C22⋊C4—122no—
484C42—21no—
485C42—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3#45014yes—
496C22×C3C2×#12011yescharacteristic
497C22×C3#54612yes—
498C22×C3C22×SD1611yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57812yes—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—82no—
505C2×D3—82no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4#52511yescharacteristic
511C3×C4#53511yescharacteristic
512C3×C4#54612yes—
513Dic3—44no—
514Dic3—42no—
515Dic3—42no—
516Dic3—42no—
517Dic3—82no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—62no—
525C23—121no—
526C23—121no—
527C23—122no—
528C23—122no—
529C23#88212yes—
530C23C2×#20412yes—
531C23C2×#20412yes—
532C23#90812yes—
533C23C22×D1211yescharacteristic
534Q8—64no—
535Q8—64no—
536Q8—128no—
537Q8—128no—
538Q8—122no—
539Q8—122no—
540D4—28no—
541D4—64no—
542D4—128no—
543D4—128no—
544D4—121no—
545D4—121no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—128no—
564C2×C4—128no—
565C2×C4—128no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—122no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—128no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4#88212yes—
579C2×C4C2×#20412yes—
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4#100412yes—
583C2×C4D3×#4112yes—
584C8—42no—
585C8—42no—
586C2×C3—24no—
587C2×C3—42no—
588C2×C3#240411yescharacteristic
589C2×C3C2×#45011yescharacteristic
590C2×C3#301111yescharacteristic
591C2×C3#329812yes—
592C2×C3#330812yes—
593D3—82no—
594C22—22no—
595C22—24no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—242no—
607C22—242no—
608C22#473114yes—
609C22C2×#88211yescharacteristic
610C22#555412yes—
611C22C22×#20411yescharacteristic
612C22C2×#90811yescharacteristic
613C22#575312yes—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—122no—
619C4—122no—
620C4—122no—
621C4—124no—
622C4—242no—
623C4#559211yescharacteristic
624C4#561012yes—
625C4#565611yescharacteristic
626C3#2396111yescharacteristic, complemented, Sylow
627C2—24no—
628C2—42no—
629C2—242no—
630C2#7488511yescharacteristic
631C2C2×#473111yescharacteristic
632C2#7867911yescharacteristic
633C2#7944712yes—
634C2#7950912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#1869D3
hypocenterC3#23961
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#816C2semidirect
3. #82349C2semidirect
4. C2×#4738C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69962C2semidirect
8. #69930C2semidirect
9. #69924C2semidirect
10. #68689C2semidirect
11. #68672C2semidirect
57. #1869D3semidirect
171. C2×C3×D4SD16semidirect
225. C2×Dic6#41semidirect
426. C2×D4#204semidirect
626. C3#23961semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C22×#8162
#823492
C2×#47382
C3×#18692
#719732
#699622
#699302
#699242
#686892
#686722
#239613