Finite Group Databasebeta

← #11166153

Subgroups — #11166153

Order 768.

2390 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5530C212yesmaximal, complemented
2C3×#2317C211yesmaximal, characteristic, complemented
3#82286C211yesmaximal, characteristic, complemented
4#82273C211yesmaximal, characteristic, complemented
5#82193C212yesmaximal, complemented
6#76220C212yesmaximal, complemented
7#76211C212yesmaximal, complemented
8#76016C212yesmaximal, complemented
9#76003C212yesmaximal, complemented
10#74630C212yesmaximal, complemented
11#74628C214yesmaximal, complemented
12#74517C212yesmaximal, complemented
13#74374C212yesmaximal, complemented
14#74368C212yesmaximal, complemented
15#74245C214yesmaximal, complemented
16#42952—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2212yes—
21C2×#930C2214yes—
22C2×C4×#67C2212yes—
23#5530—216no—
24C2×#895C2212yes—
25C2×#895C2212yes—
26C2×#895C2214yes—
27C2×#895C2214yes—
28C2×#893C2212yes—
29C2×D3×C4⋊C4C2212yes—
30C2×#890C2212yes—
31C2×#884C2214yes—
32C2×#883C2214yes—
33C2×#876C2214yes—
34C2×#874C2211yescharacteristic
35C2×C4×D12C2214yes—
36C2×C4×D12C2211yescharacteristic
37C3×#389C2212yes—
38C3×#389C2212yes—
39C3×#385C2212yes—
40C3×#385C2212yes—
41C3×#384C2212yes—
42C3×#382C2212yes—
43#5213C2214yes—
44#4994C2212yes—
45#4994C2214yes—
46#4994C2214yes—
47#4994C2214yes—
48#4991C2212yes—
49#4991C2214yes—
50#4991C2211yescharacteristic
51#4991C2212yes—
52#4991C2214yes—
53#4988C2212yes—
54#4987C2212yes—
55#4987C2214yes—
56#4987C2214yes—
57#4984C2212yes—
58#4981C2211yescharacteristic
59#4981C2214yes—
60#4961C2212yes—
61#4960C2212yes—
62#4960C2211yescharacteristic
63#4960C2212yes—
64#4953C2211yescharacteristic
65#4939C2212yes—
66C4×(C2×D3)⋊C4C2212yes—
67#4936C2211yescharacteristic
68C4×C4⋊Dic3C2211yescharacteristic
69#4677C2214yes—
70#4672C2214yes—
71#4671C2214yes—
72#4668C2214yes—
73#4668C2214yes—
74#4665C2214yes—
75#4665C2214yes—
76#4659C2214yes—
77C2×#553—32no—
78#2823—32no—
79#2693—32no—
80#2675—32no—
81#2671—34no—
82#2636—34no—
83#2619—31no—
84#2537—32no—
85#2454—32no—
86#2442—32no—
87#2428—32no—
88#2416—32no—
89#2416—32no—
90#2392—31no—
91#2317D311yescharacteristic, complemented
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2312yes—
94C22×#67C2312yes—
95C22×#67C2314yes—
96C22×D12C2312yes—
97C22×D12C2314yes—
98C22×D12C2311yescharacteristic
99C22×D12C2314yes—
100C22×D12C2312yes—
101C22×D12C2312yes—
102C22×C4×D3C2312yes—
103C22×C4×D3C2314yes—
104C22×C4×D3C2314yes—
105C3×#117—42no—
106C3×#117D418yescomplemented
107C2×C3×C4⋊C4—42no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C22⋊C4C2311yescharacteristic
116C2×C3×C22⋊C4C2311yescharacteristic
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2312yes—
121C2×#217C2312yes—
122#930—28no—
123#930—48no—
124C4×#67—28no—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2312yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2314yes—
142C2×C4⋊Dic3C2311yescharacteristic
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×C4×Dic3C2312yes—
147#895—28no—
148#895—48no—
149#895—48no—
150#895—44no—
151#893—28no—
152D3×C4⋊C4—28no—
153#890—28no—
154#884—216no—
155#883—216no—
156#876—216no—
157#874—42no—
158C4×D12—216no—
159C4×D12—42no—
160C3×#95C2314yes—
161C3×#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2311yescharacteristic
167C3⋊#95C2311yescharacteristic
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170#553—616no—
171C2×#127—31no—
172C2×#126—34no—
173C2×#125—32no—
174C2×#123—34no—
175C2×#121—32no—
176C2×#121—34no—
177C2×#121—32no—
178C2×#121—34no—
179C2×#121—32no—
180C2×#121—34no—
181C2×#121—34no—
182C2×C4×D4—32no—
183C2×C4×D4—32no—
184C2×C4×D4—34no—
185C2×C4×D4—31no—
186C2×#117C2×D311yescharacteristic
187C2×#117C2×D311yescharacteristic
188C22×C4⋊C4—32no—
189C22×C4⋊C4C2×D311yescharacteristic
190#400—34no—
191#399—34no—
192#398—31no—
193#398—34no—
194#397—31no—
195#396—34no—
196#396—34no—
197#396—34no—
198#396—32no—
199#394—32no—
200#394—34no—
201#394—32no—
202#394—34no—
203#394—31no—
204#394—32no—
205#394—34no—
206#392—32no—
207#392—32no—
208#392—31no—
209#392—34no—
210#390—32no—
211#390—32no—
212#389—32no—
213#389C2×D312yes—
214#389C2×D312yes—
215#386—34no—
216#386—34no—
217#386—34no—
218#385—34no—
219#385—34no—
220#385C2×D312yes—
221#385C2×D312yes—
222#384C2×D312yes—
223#382—31no—
224#382C2×D312yes—
225#380—34no—
226#380—32no—
227#380—34no—
228C4×C4⋊C4—31no—
229C4×C22⋊C4—32no—
230C24×C3C2411yescharacteristic
231C23×D3C2414yes—
232C23×D3C2412yes—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—44no—
236C22×C3×C4—42no—
237C22×C3×C4C2×D414yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2414yes—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2414yes—
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C2×#67—24no—
247C2×#67—24no—
248C2×#67—28no—
249C2×#67—42no—
250C2×#67—44no—
251C2×#67—44no—
252C22×Dic3C2412yes—
253C22×Dic3C2412yes—
254C22×Dic3C2414yes—
255C22×Dic3C2412yes—
256C2×D12—28no—
257C2×D12—24no—
258C2×D12—28no—
259C2×D12—24no—
260C2×D12—24no—
261C2×D12—24no—
262C2×D12—28no—
263C2×D12—41no—
264C2×D12—44no—
265C2×D12—42no—
266C2×D12—42no—
267C2×D12—44no—
268C2×D12—44no—
269C2×D12—44no—
270C2×D12—44no—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—24no—
274C2×C4×D3—28no—
275C2×C4×D3—28no—
276C2×C4×D3—44no—
277C2×C4×D3—44no—
278C2×C4×D3—44no—
279C2×C4×D3—44no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—44no—
282C3×C4⋊C4—44no—
283C3×C4⋊C4—42no—
284C3×C4⋊C4—41no—
285C3×C4⋊C4—41no—
286C3×C4⋊C4C2×D418yes—
287C3×C22⋊C4—41no—
288C3×C22⋊C4—41no—
289C3×C22⋊C4C2×D418yes—
290C3×C42—22no—
291C3×C42—22no—
292C3×C42C2×D418yes—
293#217—24no—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—28no—
296(C2×D3)⋊C4—24no—
297(C2×D3)⋊C4—28no—
298(C2×D3)⋊C4—28no—
299(C2×D3)⋊C4—28no—
300(C2×D3)⋊C4—42no—
301(C2×D3)⋊C4—44no—
302(C2×D3)⋊C4—44no—
303(C2×D3)⋊C4—44no—
304(C2×D3)⋊C4—42no—
305(C2×D3)⋊C4—44no—
306(C2×D3)⋊C4—44no—
307C4⋊Dic3—28no—
308C4⋊Dic3—24no—
309C4⋊Dic3—41no—
310C4⋊Dic3—42no—
311C4⋊Dic3—44no—
312Dic3⋊C4—24no—
313Dic3⋊C4—28no—
314Dic3⋊C4—28no—
315C4×Dic3—24no—
316C22×D4—32no—
317C22×D4—34no—
318C22×D4—32no—
319C22×D4—31no—
320C22×D4—34no—
321C22×D4—34no—
322C22×D4—32no—
323C22×D4—32no—
324C23×C4—32no—
325C23×C4—34no—
326C23×C4—34no—
327C23×C4C22×D311yescharacteristic
328C23×C4C22×D312yes—
329#127—122no—
330#126—616no—
331#125—68no—
332#123—616no—
333#121—68no—
334#121—68no—
335#121—616no—
336#121—128no—
337#121—128no—
338#121—128no—
339#121—124no—
340C4×D4—68no—
341C4×D4—68no—
342C4×D4—616no—
343C4×D4—122no—
344#117—42no—
345#117D1218yescomplemented
346C2×C4⋊C4—32no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—68no—
356C2×C4⋊C4C22×D312yes—
357C2×C4⋊C4C22×D312yes—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C4⋊C4C22×D314yes—
361C2×C4⋊C4C22×D314yes—
362C2×C4⋊C4C22×D312yes—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—32no—
368C2×C22⋊C4—34no—
369C2×C22⋊C4—34no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—34no—
374C2×C22⋊C4—34no—
375C2×C22⋊C4—32no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4C22×D311yescharacteristic
378C2×C22⋊C4C22×D311yescharacteristic
379C2×C22⋊C4C22×D312yes—
380C2×C42—32no—
381C2×C42C22×D311yescharacteristic
382C2×C42C22×D311yescharacteristic
383C2×C42C22×D312yes—
384#95—32no—
385#95—32no—
386#95—31no—
387#95—31no—
388#95—32no—
389#95—34no—
390#95—34no—
391#95—32no—
392#95C22×D314yes—
393#95C22×D314yes—
394C23×C3—21no—
395C23×C3—22no—
396C23×C3—21no—
397C23×C3—22no—
398C23×C3C22×D412yes—
399C23×C3C2511yescharacteristic
400C22×D3—22no—
401C22×D3—22no—
402C22×D3—22no—
403C22×D3—22no—
404C22×D3—22no—
405C22×D3—22no—
406C22×D3—24no—
407C22×D3—24no—
408C22×D3—24no—
409C22×D3—24no—
410C22×D3—24no—
411C22×D3—24no—
412C22×D3—24no—
413C22×D3H(5,𝔽2)14yes—
414C2×C3×C4—21no—
415C2×C3×C4—24no—
416C2×C3×C4—21no—
417C2×C3×C4—24no—
418C2×C3×C4—28no—
419C2×C3×C4—24no—
420C2×C3×C4—21no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—21no—
424C2×C3×C4—22no—
425C2×C3×C4—24no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—84no—
431C2×C3×C4C22×D418yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4C22×D412yes—
434C2×C3×C4H(5,𝔽2)12yes—
435C2×C3×C42−1+412yes—
436#67—48no—
437#67—88no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—24no—
442C2×Dic3—22no—
443C2×Dic3—24no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—24no—
450D12—416no—
451D12—44no—
452D12—48no—
453D12—88no—
454D12—84no—
455D12—84no—
456C4×D3—48no—
457C4×D3—88no—
458C4×D3—88no—
459C24—32no—
460C24—34no—
461C24C23×D311yescharacteristic
462C2×D4—68no—
463C2×D4—64no—
464C2×D4—68no—
465C2×D4—64no—
466C2×D4—64no—
467C2×D4—68no—
468C2×D4—68no—
469C2×D4—64no—
470C2×D4—64no—
471C2×D4—64no—
472C2×D4—121no—
473C2×D4—124no—
474C2×D4—124no—
475C2×D4—124no—
476C2×D4—124no—
477C2×D4—122no—
478C2×D4—122no—
479C2×D4—124no—
480C2×D4—122no—
481C2×D4—124no—
482C2×D4—124no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—34no—
488C22×C4—32no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—64no—
492C22×C4—64no—
493C22×C4—64no—
494C22×C4—68no—
495C22×C4—68no—
496C22×C4—124no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4—124no—
500C22×C4C2×D1214yes—
501C22×C4C23×D311yescharacteristic
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D314yes—
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D311yescharacteristic
506C22×C4C23×D314yes—
507C22×C4C23×D311yescharacteristic
508C22×C4C23×D311yescharacteristic
509C4⋊C4—24no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—64no—
516C4⋊C4—68no—
517C4⋊C4—68no—
518C4⋊C4—64no—
519C4⋊C4—68no—
520C4⋊C4—121no—
521C4⋊C4—124no—
522C4⋊C4—122no—
523C4⋊C4C2×D1218yes—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—68no—
527C22⋊C4—64no—
528C22⋊C4—64no—
529C22⋊C4—64no—
530C22⋊C4—68no—
531C22⋊C4—68no—
532C22⋊C4—68no—
533C22⋊C4—122no—
534C22⋊C4—124no—
535C22⋊C4—124no—
536C22⋊C4—124no—
537C22⋊C4—124no—
538C22⋊C4—124no—
539C22⋊C4—122no—
540C22⋊C4C2×D1218yes—
541C42—22no—
542C42—22no—
543C42—64no—
544C42C2×D1218yes—
545C22×C3—22no—
546C22×C3—24no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—42no—
550C22×C3C23×D411yescharacteristic
551C22×C3C2×H(5,𝔽2)11yescharacteristic
552C22×C3C2×H(5,𝔽2)12yes—
553C22×C3C2×2−1+411yescharacteristic
554C22×C3#58512yes—
555C2×D3—24no—
556C2×D3—24no—
557C2×D3—24no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—44no—
563C2×D3—44no—
564C2×D3—44no—
565C2×D3—44no—
566C2×D3—44no—
567C2×D3—44no—
568C2×D3—44no—
569C3×C4—22no—
570C3×C4—28no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—44no—
574C3×C4—41no—
575C3×C4—41no—
576C3×C4#53714yes—
577Dic3—42no—
578Dic3—42no—
579Dic3—44no—
580Dic3—42no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—34no—
586C23—62no—
587C23—62no—
588C23—62no—
589C23—62no—
590C23—62no—
591C23—62no—
592C23—64no—
593C23—64no—
594C23—64no—
595C23—64no—
596C23—64no—
597C23—64no—
598C23—64no—
599C23C22×D1212yes—
600C23C24×D311yescharacteristic
601D4—1216no—
602D4—124no—
603D4—128no—
604D4—128no—
605D4—248no—
606D4—244no—
607D4—244no—
608D4—248no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—64no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—64no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—64no—
636C2×C4—62no—
637C2×C4—84no—
638C2×C4—128no—
639C2×C4—248no—
640C2×C4—248no—
641C2×C4C22×D1212yes—
642C2×C4C22×D1218yes—
643C2×C4C22×D1212yes—
644C2×C4#100412yes—
645C2×C4#100712yes—
646C2×C3—42no—
647C2×C3C2×#53711yescharacteristic
648C2×C3#348812yes—
649C2×C3#353711yescharacteristic
650C2×C3#355111yescharacteristic
651C2×C3#355912yes—
652D3—44no—
653D3—84no—
654C22—24no—
655C22—22no—
656C22—41no—
657C22—42no—
658C22—41no—
659C22—64no—
660C22—64no—
661C22—64no—
662C22—122no—
663C22—122no—
664C22—122no—
665C22—122no—
666C22—124no—
667C22—124no—
668C22—124no—
669C22—124no—
670C22—124no—
671C22—124no—
672C22—124no—
673C22C23×D1211yescharacteristic
674C22C2×#100411yescharacteristic
675C22C2×#100711yescharacteristic
676C22D3×H(5,𝔽2)12yes—
677C22#596812yes—
678C4—28no—
679C4—22no—
680C4—44no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—122no—
686C4—124no—
687C4—122no—
688C4—122no—
689C4#551014yes—
690C3#4295211yescharacteristic, complemented, Sylow
691C2—42no—
692C2—124no—
693C2—244no—
694C2C2×#551011yescharacteristic
695C2#8590012yes—
696C2#8633411yescharacteristic
697C2#8635011yescharacteristic
698C2#8660312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2317C2
hypercenter#2317D3
hypocenterC3#42952
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#5530C2semidirect
2. C3×#2317C2semidirect
3. #82286C2semidirect
4. #82273C2semidirect
5. #82193C2semidirect
6. #76220C2semidirect
7. #76211C2semidirect
8. #76016C2semidirect
9. #76003C2semidirect
10. #74630C2semidirect
11. #74628C2semidirect
12. #74517C2semidirect
13. #74374C2semidirect
14. #74368C2semidirect
15. #74245C2semidirect
91. #2317D3semidirect
106. C3×#117D4semidirect
345. #117D12semidirect
690. C3#42952semidirect

Maximal subgroups

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

subgroupindex
C2×#55302
C3×#23172
#822862
#822732
#821932
#762202
#762112
#760162
#760032
#746302
#746282
#745172
#743742
#743682
#742452
#429523