Finite Group Databasebeta

← #10972907

Subgroups — #10972907

Order 768.

2654 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5596C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C3×#2501C211yesmaximal, characteristic, complemented
4#82891C214yesmaximal, complemented
5#82841C214yesmaximal, complemented
6#82654C211yesmaximal, characteristic, complemented
7#82177C211yesmaximal, characteristic, complemented
8#78050C214yesmaximal, complemented
9#78041C214yesmaximal, complemented
10#76352C212yesmaximal, complemented
11#76297C212yesmaximal, complemented
12#74340C212yesmaximal, complemented
13#74321C212yesmaximal, complemented
14#73491C211yesmaximal, characteristic, complemented
15#73437C211yesmaximal, characteristic, complemented
16#35249—31nomaximal, Sylow
17C22×#237C2212yes—
18C23×Dic6C2211yescharacteristic
19C2×C3×#123C2214yes—
20C2×C3×#121C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×#940C2211yescharacteristic
23C2×#939C2212yes—
24C2×#936C2212yes—
25C2×D4×Dic3C2211yescharacteristic
26C2×#930C2211yescharacteristic
27C2×#929C2214yes—
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31#5596—28no—
32C2×#893C2211yescharacteristic
33C2×#888C2211yescharacteristic
34C2×#887—28no—
35C2×#887C2212yes—
36C2×#887C2212yes—
37C2×#887C2212yes—
38C2×#887C2212yes—
39C2×#885C2212yes—
40C2×#67⋊C4C2212yes—
41C2×#875C2212yes—
42C3×#399C2214yes—
43C3×#390C2212yes—
44C3×#384C2212yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46#5220C2214yes—
47C4×#217C2211yescharacteristic
48#4993C2214yes—
49#4990C2218yes—
50#4989C2218yes—
51#4988C2212yes—
52#4979C2218yes—
53#4976C2218yes—
54#4975C2212yes—
55#4973C2212yes—
56#4959C2218yes—
57#4958C2214yes—
58#4957C2218yes—
59#4947C2214yes—
60#4946C2218yes—
61#4945C2218yes—
62#4943C2212yes—
63#4942C2212yes—
64#4932C2212yes—
65C4×Dic3⋊C4C2212yes—
66#4929C2212yes—
67#4673C2214yes—
68#4651C2212yes—
69#4650C2212yes—
70C2×#532—31no—
71C22×C4×Q8—31no—
72#2854—34no—
73#2632—34no—
74#2580—34no—
75#2567—32no—
76#2548—32no—
77#2501—31no—
78#2501D311yescharacteristic, complemented
79#2436—31no—
80#2436—31no—
81#2406—34no—
82#2326—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×D4C2312yes—
86C22×C3×D4C2311yescharacteristic
87C23×C3×C4C2312yes—
88C23×C3×C4C2311yescharacteristic
89C22×#67C2312yes—
90C22×#67C2312yes—
91C22×#67C2312yes—
92C23×Dic3C2312yes—
93C23×Dic3C2311yescharacteristic
94C23×Dic3C2311yescharacteristic
95C23×Dic3C2312yes—
96C2×#237—28no—
97C2×#237—48no—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×Dic6—24no—
101C22×Dic6—28no—
102C22×Dic6C2312yes—
103C22×Dic6C2312yes—
104C22×Dic6C2312yes—
105C3×#123—48no—
106C3×#121—24no—
107C2×C3×C4⋊C4—24no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2318yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2318yes—
115C2×C3×C42C2312yes—
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2318yes—
119#940—24no—
120#939—28no—
121#936—28no—
122D4×Dic3—24no—
123#930—24no—
124#929—48no—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2318yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2318yes—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×Dic3⋊C4—28no—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2318yes—
136C2×C4×Dic3—28no—
137C2×C4×Dic3D418yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2314yes—
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145#893—24no—
146#888—24no—
147#887—28no—
148#887—28no—
149#887—28no—
150#887—28no—
151#885—28no—
152#67⋊C4—28no—
153#875—28no—
154C3×#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2318yes—
160C3⋊#95C2318yes—
161C22×#41—32no—
162C23×Q8—31no—
163#532—68no—
164C2×C4⋊Q8—31no—
165C2×#127—31no—
166C2×#124—32no—
167C2×#124—32no—
168C2×#124—32no—
169C2×#123—34no—
170C2×#123C2×D314yes—
171C2×#121—32no—
172C2×#121—31no—
173C2×#121C2×D311yescharacteristic
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×Q8—32no—
177C2×C4×Q8—32no—
178C2×C4×Q8—68no—
179C2×C4×D4—31no—
180C2×C4×D4—31no—
181C2×C4×D4—32no—
182C22×C4⋊C4—32no—
183C22×C4⋊C4C2×D311yescharacteristic
184C22×C42—31no—
185C22×C42—32no—
186#399—34no—
187#399C2×D314yes—
188#395—38no—
189#394—38no—
190#393—34no—
191#393—34no—
192#391—32no—
193#391—32no—
194#390—32no—
195#390—38no—
196#390C2×D312yes—
197#389—38no—
198#388—38no—
199#388—38no—
200#386—34no—
201#386—34no—
202#385—38no—
203#385—38no—
204#384—32no—
205#384C2×D312yes—
206#383—32no—
207#382—32no—
208#381—32no—
209#380—32no—
210C4×C4⋊C4—32no—
211C4×C22⋊C4—31no—
212C4×C22⋊C4C2×D311yescharacteristic
213#376—32no—
214C24×C3C2412yes—
215C24×C3C2411yescharacteristic
216C23×D3C2412yes—
217C2×C3×D4—24no—
218C2×C3×D4—28no—
219C2×C3×D4—22no—
220C2×C3×D4—24no—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—28no—
224C22×C3×C4C2×D414yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C2×#67—24no—
231C2×#67—28no—
232C2×#67—24no—
233C2×#67—24no—
234C2×#67—28no—
235C2×#67—44no—
236C22×Dic3—24no—
237C22×Dic3—24no—
238C22×Dic3—28no—
239C22×Dic3—24no—
240C22×Dic3—24no—
241C22×Dic3—44no—
242C22×Dic3C2×D414yes—
243C22×Dic3C2×D414yes—
244C22×Dic3C2414yes—
245C22×Dic3C2412yes—
246C22×Dic3C2411yescharacteristic
247C22×Dic3C2411yescharacteristic
248C22×Dic3C2411yescharacteristic
249C22×Dic3C2414yes—
250C22×Dic3C2411yescharacteristic
251#237—432no—
252C2×D12—22no—
253C2×D12—24no—
254C2×C4×D3—24no—
255C2×C4×D3—44no—
256C2×Dic6—232no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C2×Dic6—28no—
261C2×Dic6—24no—
262C2×Dic6—44no—
263C2×Dic6#4118yescomplemented
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—48no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—24no—
270C3×C22⋊C4—48no—
271C3×C42—24no—
272#217—24no—
273#217—24no—
274#217—216no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—216no—
278(C2×D3)⋊C4—48no—
279C4⋊Dic3—22no—
280Dic3⋊C4—24no—
281Dic3⋊C4—28no—
282Dic3⋊C4—28no—
283Dic3⋊C4—28no—
284Dic3⋊C4—48no—
285C4×Dic3—24no—
286C4×Dic3—28no—
287C4×Dic3—28no—
288C4×Dic3C2×D414yes—
289C4×Dic3C2×D414yes—
290C4×Dic3C2×D414yes—
291C4×Dic3C2×D414yes—
292C2×#41—68no—
293C2×#41—128no—
294C22×Q8—32no—
295C22×Q8—32no—
296C22×Q8—32no—
297C22×Q8—68no—
298C22×Q8—64no—
299C22×D4—32no—
300C22×D4—32no—
301C22×D4—32no—
302C22×D4—31no—
303C22×D4C22×D311yescharacteristic
304C22×D4C22×D312yes—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—32no—
309C23×C4—32no—
310C23×C4C22×D312yes—
311C23×C4C22×D311yescharacteristic
312C4⋊Q8—64no—
313#127—64no—
314#124—68no—
315#124—68no—
316#124—68no—
317#123—48no—
318#123—128no—
319#121—24no—
320#121—68no—
321#121—64no—
322C4×Q8—68no—
323C4×Q8—68no—
324C4×Q8—68no—
325C4×Q8—68no—
326C4×D4—68no—
327C4×D4—64no—
328C4×D4—64no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—38no—
336C2×C4⋊C4—68no—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D318yes—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—38no—
347C2×C22⋊C4—38no—
348C2×C22⋊C4C22×D312yes—
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D318yes—
351C2×C42—38no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42—34no—
358C2×C42—32no—
359C2×C42—68no—
360C2×C42C22×D312yes—
361#95—34no—
362#95—34no—
363#95—32no—
364#95—34no—
365#95—38no—
366#95—38no—
367#95C22×D312yes—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—24no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3C22×D412yes—
375C23×C3H(5,𝔽2)14yes—
376C23×C3C2511yescharacteristic
377C22×D3—22no—
378C22×D3—24no—
379C22×D3—24no—
380C22×D3—24no—
381C3×D4—48no—
382C3×D4—44no—
383C2×C3×C4—22no—
384C2×C3×C4—28no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—24no—
388C2×C3×C4—28no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—24no—
392C2×C3×C4—48no—
393C2×C3×C4C22×D412yes—
394C2×C3×C4C22×D412yes—
395C2×C3×C4C2×#4114yes—
396#67—48no—
397#67—48no—
398#67—48no—
399C2×Dic3—216no—
400C2×Dic3—28no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—28no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—24no—
412C2×Dic3—48no—
413C2×Dic3C22×D412yes—
414C2×Dic3C22×D412yes—
415C2×Dic3C22×D412yes—
416C2×Dic3C22×D412yes—
417C2×Dic3C2×#4118yes—
418C2×Dic32−1+414yes—
419D12—44no—
420C4×D3—48no—
421Dic6—216no—
422Dic6—216no—
423Dic6—48no—
424C24—32no—
425C24C23×D312yes—
426C24C23×D311yescharacteristic
427#41—1232no—
428C2×Q8—38no—
429C2×Q8—68no—
430C2×Q8—64no—
431C2×Q8—632no—
432C2×Q8—64no—
433C2×Q8—64no—
434C2×Q8—64no—
435C2×Q8—124no—
436C2×D4—24no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—68no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4—124no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—28no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—31no—
455C22×C4—34no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—34no—
459C22×C4—32no—
460C22×C4—64no—
461C22×C4—68no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4D3×D414yes—
469C22×C4C23×D314yes—
470C22×C4C23×D312yes—
471C22×C4C23×D314yes—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D311yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—68no—
479C4⋊C4—68no—
480C4⋊C4—64no—
481C4⋊C4—68no—
482C4⋊C4—62no—
483C4⋊C4—128no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—48no—
487C22⋊C4—64no—
488C22⋊C4—616no—
489C22⋊C4—64no—
490C22⋊C4—616no—
491C22⋊C4—64no—
492C22⋊C4—64no—
493C22⋊C4—128no—
494C42—24no—
495C42—34no—
496C42—34no—
497C42—34no—
498C42—34no—
499C42—68no—
500C42—68no—
501C42—64no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—28no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—48no—
509C22×C3#53514yes—
510C22×C3C23×D411yescharacteristic
511C22×C3C22×#4112yes—
512C22×C3C2×H(5,𝔽2)12yes—
513C22×C3C2×2−1+412yes—
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—48no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—44no—
521C3×C4—44no—
522C3×C4#52514yes—
523Dic3—28no—
524Dic3—22no—
525Dic3—22no—
526Dic3—22no—
527Dic3—42no—
528Dic3—44no—
529Dic3#53614yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—62no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23C2×D3×D412yes—
541C23#100414yes—
542C23C24×D311yescharacteristic
543Q8—616no—
544Q8—616no—
545Q8—128no—
546D4—48no—
547D4—44no—
548D4—128no—
549D4—128no—
550D4—128no—
551D4—124no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—32no—
562C2×C4—34no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—38no—
567C2×C4—48no—
568C2×C4—68no—
569C2×C4—616no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—68no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—128no—
582C2×C4—128no—
583C2×C4C2×D3×D412yes—
584C2×C4C2×D3×D412yes—
585C2×C4D3×#4114yes—
586C2×C3—24no—
587C2×C3—82no—
588C2×C3C2×#52511yescharacteristic
589C2×C3C2×#53511yescharacteristic
590C2×C3C2×#53611yescharacteristic
591C2×C3#352214yes—
592D3—82no—
593C22—22no—
594C22—28no—
595C22—22no—
596C22—42no—
597C22—48no—
598C22—42no—
599C22—42no—
600C22—122no—
601C22—128no—
602C22—122no—
603C22—122no—
604C22#571514yes—
605C22C22×D3×D411yescharacteristic
606C22C2×#100412yes—
607C22C2×D3×#4112yes—
608C22D3×2−1+412yes—
609C4—22no—
610C4—34no—
611C4—44no—
612C4—44no—
613C4—42no—
614C4—62no—
615C4—62no—
616C4—62no—
617C4—68no—
618C4—124no—
619C4—122no—
620C4#565614yes—
621C3#3524911yescharacteristic, complemented, Sylow
622C2—24no—
623C2—82no—
624C2—242no—
625C2C2×#565611yescharacteristic
626C2C2×#571511yescharacteristic
627C2D3×#53611yescharacteristic
628C2#8668814yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2501C2
prime radicalC2×#5596C2
hypercenter#2501D3
hypocenterC3#35249
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C3×#2501C2semidirect
4. #82891C2semidirect
5. #82841C2semidirect
6. #82654C2semidirect
7. #82177C2semidirect
8. #78050C2semidirect
9. #78041C2semidirect
10. #76352C2semidirect
11. #76297C2semidirect
12. #74340C2semidirect
13. #74321C2semidirect
14. #73491C2semidirect
15. #73437C2semidirect
78. #2501D3semidirect
263. C2×Dic6#41semidirect
621. C3#35249semidirect

Maximal subgroups

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

subgroupindex
C2×#55962
C22×#8872
C3×#25012
#828912
#828412
#826542
#821772
#780502
#780412
#763522
#762972
#743402
#743212
#734912
#734372
#352493