Finite Group Databasebeta

← #11002141

Subgroups — #11002141

Order 768.

2406 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5617C211yesmaximal, characteristic
2C22×#897C211yesmaximal, characteristic, complemented
3C3×#2591C211yesmaximal, characteristic, complemented
4#82665C211yesmaximal, characteristic, complemented
5#82181C211yesmaximal, characteristic, complemented
6#78369C214yesmaximal, complemented
7#77139C214yesmaximal, complemented
8#75925C214yesmaximal, complemented
9#75668C214yesmaximal, complemented
10#74936C212yesmaximal, complemented
11#74906C212yesmaximal, complemented
12#74769C212yesmaximal, complemented
13#74689C212yesmaximal, complemented
14#73567C211yesmaximal, characteristic, complemented
15#73517C211yesmaximal, characteristic, complemented
16#36716—31nomaximal, Sylow
17C23×C4×D3C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#939C2212yes—
22C2×#937C2212yes—
23C2×D4×Dic3C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×(C2×D3)⋊C4C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C2×#924C2211yescharacteristic
29#5617—28no—
30C2×#898C2212yes—
31C2×#897—28no—
32C2×#897C2214yes—
33C2×#897C2214yes—
34C2×#886C2212yes—
35C2×#885C2212yes—
36C2×#883C2214yes—
37C2×#878C2214yescomplemented
38C3×#400C2212yescomplemented
39C3×#394C2212yescomplemented
40C3×#388C2214yescomplemented
41C3×#386C2212yes—
42C3×#386C2212yes—
43C3×C4×C22⋊C4C2211yescharacteristic
44#5221C2212yescomplemented
45#5220C2212yes—
46#5220C2212yes—
47#5210C2212yes—
48#5210C2212yes—
49C4×#217C2211yescharacteristic
50#4981C2212yescomplemented
51#4979C2212yescomplemented
52#4977C2212yescomplemented
53#4976C2214yescomplemented
54#4961C2214yes—
55#4959C2214yes—
56#4956C2214yes—
57#4955C2212yes—
58#4953C2214yescomplemented
59#4950C2212yes—
60#4950C2212yes—
61#4950C2214yescomplemented
62#4948C2214yescomplemented
63#4947C2214yescomplemented
64#4942C2212yes—
65#4940C2214yes—
66#4935C2212yescomplemented
67#4929C2212yescomplemented
68#4675C2214yes—
69#4674C2214yes—
70#4669C2214yes—
71#4668C2214yes—
72#4664C2214yes—
73#4661C2214yescomplemented
74#4661C2214yescomplemented
75#4656C2214yescomplemented
76#4655C2214yescomplemented
77#4646C2214yescomplemented
78C2×#538—31no—
79C22×C22⋊Q8—31no—
80#2849—32no—
81#2700—34no—
82#2698—34no—
83#2681—34no—
84#2680—34no—
85#2622—32no—
86#2591D311yescharacteristic, complemented
87#2583—31no—
88#2579—32no—
89#2504—31no—
90#2502—31no—
91#2409—32no—
92#2312—31no—
93C24×D3C2311yescharacteristic
94C22×C3×D4C2312yes—
95C22×C3×D4C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C23×C3×C4C2311yescharacteristic
99C22×#67C2312yes—
100C22×#67C2312yes—
101C23×Dic3C2311yescharacteristic
102C23×Dic3C2311yescharacteristic
103C23×Dic3C2312yes—
104C23×Dic3C2311yescharacteristic
105C22×C4×D3—24no—
106C22×C4×D3—44no—
107C22×C4×D3C2314yes—
108C22×Dic6—24no—
109C22×Dic6—44no—
110C22×Dic6C2314yes—
111C22×Dic6C2311yescharacteristic
112C3×#121—24no—
113C2×C3×C4⋊C4—24no—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C42C2312yes—
122C2×#217C2312yes—
123C2×#217C2312yes—
124C2×#217C2312yes—
125C2×#217C2314yes—
126C2×#217C2314yes—
127#939—28no—
128#937—28no—
129D4×Dic3—28no—
130C2×(C2×D3)⋊C4—24no—
131C2×(C2×D3)⋊C4D418yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2312yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×C4⋊Dic3—24no—
140C2×C4⋊Dic3—24no—
141C2×C4⋊Dic3C2314yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2312yes—
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2311yescharacteristic
146C2×C4⋊Dic3C2314yes—
147C2×C4⋊Dic3C2314yes—
148#924—24no—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2314yes—
152C2×Dic3⋊C4C2314yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2312yes—
155#898—28no—
156#897—216no—
157#897—216no—
158#886—28no—
159#885—28no—
160#883—48no—
161#878—48no—
162C3×#95C2314yes—
163C3×#95C2314yes—
164C3×#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C23×Q8—31no—
174C24×C4—31no—
175#538—68no—
176C2×#126—32no—
177C2×#124—32no—
178C2×#123—32no—
179C2×#123—34no—
180C2×C22⋊Q8—31no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—34no—
183C2×C22⋊Q8—34no—
184C2×C22⋊Q8—68no—
185C2×#121—32no—
186C2×#121C2×D311yescharacteristic
187C2×#120—32no—
188C2×C4×D4—32no—
189C22×C4⋊C4—31no—
190C22×C4⋊C4—31no—
191C22×C4⋊C4C2×D311yescharacteristic
192C22×C22⋊C4—31no—
193C22×C22⋊C4—31no—
194#400C2×D312yescomplemented
195#398—32no—
196#398—34no—
197#398—34no—
198#397—34no—
199#397—34no—
200#397—34no—
201#396—34no—
202#395—32no—
203#395—34no—
204#394—34no—
205#394—34no—
206#394C2×D312yescomplemented
207#393—34no—
208#391—32no—
209#391—34no—
210#390—32no—
211#389—34no—
212#388—34no—
213#388—34no—
214#388C2×D314yescomplemented
215#387—34no—
216#386—32no—
217#386—32no—
218#386—34no—
219#386C2×D312yes—
220#386C2×D312yes—
221#385—32no—
222#385—32no—
223#385—32no—
224#385—34no—
225#385—32no—
226#385—34no—
227#384—32no—
228#382—32no—
229#381—34no—
230#380—32no—
231#379—32no—
232C4×C22⋊C4—31no—
233C4×C22⋊C4C2×D311yescharacteristic
234C24×C3C2412yes—
235C24×C3C2411yescharacteristic
236C23×D3—24no—
237C23×D3—42no—
238C23×D3—42no—
239C23×D3C2×D414yes—
240C23×D3C2412yes—
241C2×C3×D4—24no—
242C2×C3×D4—28no—
243C2×C3×D4—22no—
244C2×C3×D4—42no—
245C22×C3×C4—22no—
246C22×C3×C4—24no—
247C22×C3×C4—22no—
248C22×C3×C4—42no—
249C22×C3×C4—42no—
250C22×C3×C4C2×D414yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2414yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2411yescharacteristic
256C22×C3×C4C2412yes—
257C2×#67—24no—
258C2×#67—24no—
259C2×#67—44no—
260C2×#67—44no—
261C22×Dic3—22no—
262C22×Dic3—22no—
263C22×Dic3—24no—
264C22×Dic3—42no—
265C22×Dic3—42no—
266C22×Dic3—42no—
267C22×Dic3—44no—
268C22×Dic3C2×D414yes—
269C22×Dic3C2414yes—
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2412yes—
273C22×Dic3C2412yes—
274C22×Dic3C2412yes—
275C2×C4×D3—28no—
276C2×C4×D3—28no—
277C2×C4×D3—44no—
278C2×C4×D3—416no—
279C2×Dic6—28no—
280C2×Dic6—28no—
281C2×Dic6—22no—
282C2×Dic6—44no—
283C2×Dic6—416no—
284C2×Dic6—42no—
285C3×C4⋊C4—24no—
286C3×C4⋊C4—24no—
287C3×C4⋊C4—22no—
288C3×C22⋊C4—24no—
289C3×C22⋊C4—24no—
290C3×C22⋊C4—28no—
291C3×C22⋊C4—44no—
292C3×C42—42no—
293#217—24no—
294#217—24no—
295#217—24no—
296#217—44no—
297#217—44no—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—24no—
300(C2×D3)⋊C4—28no—
301(C2×D3)⋊C4—44no—
302(C2×D3)⋊C4—44no—
303(C2×D3)⋊C4C2×D418yes—
304(C2×D3)⋊C4C2×D418yes—
305C4⋊Dic3—22no—
306C4⋊Dic3—24no—
307C4⋊Dic3—24no—
308C4⋊Dic3—24no—
309C4⋊Dic3—28no—
310C4⋊Dic3—44no—
311C4⋊Dic3—44no—
312Dic3⋊C4—24no—
313Dic3⋊C4—28no—
314Dic3⋊C4—44no—
315Dic3⋊C4—44no—
316C4×Dic3—24no—
317C4×Dic3—42no—
318C25—31no—
319C22×Q8—34no—
320C22×Q8—31no—
321C22×Q8—64no—
322C22×Q8—124no—
323C22×D4—32no—
324C22×D4—32no—
325C22×D4C22×D312yes—
326C22×D4C22×D311yescharacteristic
327C23×C4—31no—
328C23×C4—31no—
329C23×C4—31no—
330C23×C4—32no—
331C23×C4—34no—
332C23×C4—64no—
333C23×C4—124no—
334C23×C4C22×D311yescharacteristic
335C23×C4C22×D311yescharacteristic
336C23×C4C22×D311yescharacteristic
337#126—68no—
338#124—68no—
339#123—68no—
340#123—128no—
341C22⋊Q8—616no—
342C22⋊Q8—616no—
343C22⋊Q8—64no—
344C22⋊Q8—128no—
345#121—24no—
346#121—68no—
347#120—68no—
348C4×D4—68no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—34no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—34no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—64no—
362C2×C4⋊C4—64no—
363C2×C4⋊C4C22×D312yes—
364C2×C4⋊C4C22×D312yes—
365C2×C4⋊C4C22×D311yescharacteristic
366C2×C22⋊C4—38no—
367C2×C22⋊C4—32no—
368C2×C22⋊C4—32no—
369C2×C22⋊C4—32no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—32no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—32no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—34no—
379C2×C22⋊C4—64no—
380C2×C22⋊C4C22×D312yes—
381C2×C22⋊C4C22×D314yes—
382C2×C22⋊C4C22×D312yes—
383C2×C22⋊C4C22×D314yes—
384C2×C42—32no—
385C2×C42—32no—
386C2×C42C22×D312yes—
387#95—32no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—34no—
395#95C22×D312yes—
396#95C22×D314yes—
397#95C22×D314yes—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—24no—
402C23×C3—24no—
403C23×C3—24no—
404C23×C3C22×D412yes—
405C23×C32−1+414yes—
406C23×C3C2511yescharacteristic
407C22×D3—28no—
408C22×D3—24no—
409C22×D3—24no—
410C22×D3—24no—
411C22×D3—44no—
412C22×D3—44no—
413C22×D3—416no—
414C22×D3C22×D414yes—
415C3×D4—48no—
416C3×D4—82no—
417C2×C3×C4—28no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—24no—
427C2×C3×C4—24no—
428C2×C3×C4—24no—
429C2×C3×C4—44no—
430C2×C3×C4—44no—
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4H(5,𝔽2)14yes—
434#67—84no—
435#67—84no—
436C2×Dic3—24no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—24no—
442C2×Dic3—24no—
443C2×Dic3—24no—
444C2×Dic3—24no—
445C2×Dic3—24no—
446C2×Dic3—24no—
447C2×Dic3—24no—
448C2×Dic3—44no—
449C2×Dic3—44no—
450C2×Dic3—44no—
451C2×Dic3—84no—
452C2×Dic3C22×D414yes—
453C4×D3—416no—
454Dic6—416no—
455Dic6—82no—
456C24—32no—
457C24—34no—
458C24—64no—
459C24—122no—
460C24—122no—
461C24C23×D312yes—
462C24C23×D311yescharacteristic
463C2×Q8—68no—
464C2×Q8—62no—
465C2×Q8—68no—
466C2×Q8—124no—
467C2×Q8—122no—
468C2×Q8—1216no—
469C2×D4—24no—
470C2×D4—28no—
471C2×D4—22no—
472C2×D4—42no—
473C2×D4—64no—
474C2×D4—64no—
475C2×D4—124no—
476C2×D4—124no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—32no—
481C22×C4—34no—
482C22×C4—34no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—68no—
490C22×C4—64no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—68no—
494C22×C4—124no—
495C22×C4—1216no—
496C22×C4—122no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4—124no—
500C22×C4D3×D414yes—
501C22×C4C23×D312yes—
502C22×C4C23×D314yes—
503C22×C4C23×D312yes—
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D311yescharacteristic
506C22×C4C23×D312yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—68no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—62no—
516C4⋊C4—68no—
517C4⋊C4—124no—
518C4⋊C4—124no—
519C4⋊C4—124no—
520C4⋊C4—124no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—28no—
524C22⋊C4—38no—
525C22⋊C4—38no—
526C22⋊C4—44no—
527C22⋊C4—68no—
528C22⋊C4—64no—
529C22⋊C4—64no—
530C22⋊C4—64no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—124no—
534C22⋊C4—124no—
535C22⋊C4—124no—
536C22⋊C4—124no—
537C42—42no—
538C42—64no—
539C42—122no—
540C22×C3—22no—
541C22×C3—22no—
542C22×C3—28no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—48no—
547C22×C3#53614yes—
548C22×C3C23×D411yescharacteristic
549C22×C3C2×H(5,𝔽2)12yes—
550C22×C3C2×2−1+412yes—
551C22×C3#58512yes—
552C2×D3—24no—
553C2×D3—24no—
554C2×D3—24no—
555C2×D3—416no—
556C2×D3—48no—
557C3×C4—24no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—42no—
561C3×C4—42no—
562C3×C4—44no—
563Dic3—24no—
564Dic3—42no—
565Dic3—42no—
566Dic3—42no—
567Dic3—42no—
568Dic3—44no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—34no—
576C23—68no—
577C23—64no—
578C23—64no—
579C23—64no—
580C23—124no—
581C23—124no—
582C23—1216no—
583C23C2×D3×D412yes—
584C23#101014yes—
585C23C24×D311yescharacteristic
586Q8—1216no—
587Q8—242no—
588D4—48no—
589D4—82no—
590D4—244no—
591D4—244no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—34no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—1216no—
620C2×C4—124no—
621C2×C4—124no—
622C2×C4—124no—
623C2×C4—244no—
624C2×C4C2×D3×D412yes—
625C2×C4C2×D3×D412yes—
626C2×C4#100914yes—
627C2×C3—24no—
628C2×C3—82no—
629C2×C3C2×#53611yescharacteristic
630C2×C3#348411yescharacteristic
631C2×C3#348511yescharacteristic
632C2×C3#354114yes—
633D3—44no—
634C22—22no—
635C22—22no—
636C22—28no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—48no—
641C22—64no—
642C22—64no—
643C22—64no—
644C22—1216no—
645C22—128no—
646C22#567514yes—
647C22C22×D3×D411yescharacteristic
648C22C2×#100912yes—
649C22C2×#101012yes—
650C22#596612yes—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—64no—
658C4—122no—
659C4—122no—
660C4—122no—
661C4—122no—
662C4—124no—
663C3#3671611yescharacteristic, complemented, Sylow
664C2—24no—
665C2—82no—
666C2—124no—
667C2#8603911yescharacteristic
668C2C2×#567511yescharacteristic
669C2#8629611yescharacteristic
670C2#8689314yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2591C2
prime radicalC2×#5617C2
hypercenter#2591D3
hypocenterC3#36716
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. C22×#897C2semidirect
3. C3×#2591C2semidirect
4. #82665C2semidirect
5. #82181C2semidirect
6. #78369C2semidirect
7. #77139C2semidirect
8. #75925C2semidirect
9. #75668C2semidirect
10. #74936C2semidirect
11. #74906C2semidirect
12. #74769C2semidirect
13. #74689C2semidirect
14. #73567C2semidirect
15. #73517C2semidirect
37. C2×#878C22semidirect
38. C3×#400C22semidirect
39. C3×#394C22semidirect
40. C3×#388C22semidirect
44. #5221C22semidirect
50. #4981C22semidirect
51. #4979C22semidirect
52. #4977C22semidirect
53. #4976C22semidirect
58. #4953C22semidirect
61. #4950C22semidirect
62. #4948C22semidirect
63. #4947C22semidirect
66. #4935C22semidirect
67. #4929C22semidirect
73. #4661C22semidirect
74. #4661C22semidirect
75. #4656C22semidirect
76. #4655C22semidirect
77. #4646C22semidirect
86. #2591D3semidirect
194. #400C2×D3semidirect
206. #394C2×D3semidirect
214. #388C2×D3semidirect
663. C3#36716semidirect

Maximal subgroups

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

subgroupindex
C2×#56172
C22×#8972
C3×#25912
#826652
#821812
#783692
#771392
#759252
#756682
#749362
#749062
#747692
#746892
#735672
#735172
#367163