Finite Group Databasebeta

← #11166157

Subgroups — #11166157

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×#5614C212yesmaximal, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3#83186C212yesmaximal, complemented
4#82705C212yesmaximal, complemented
5#82273C211yesmaximal, characteristic, complemented
6#82234C211yesmaximal, characteristic, complemented
7#77089C212yesmaximal, complemented
8#76981C214yesmaximal, complemented
9#76944C212yesmaximal, complemented
10#76770C212yesmaximal, complemented
11#76718C212yesmaximal, complemented
12#76710C214yesmaximal, complemented
13#76439C212yesmaximal, complemented
14#76325C212yesmaximal, complemented
15#74163C212yesmaximal, complemented
16#42952—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×#117C2211yescharacteristic
20C2×#939C2214yes—
21C2×#939C2214yes—
22C2×#938C2212yes—
23C2×#938C2214yes—
24C2×#938C2212yes—
25C2×#935C2214yes—
26C2×D4×Dic3C2212yes—
27C2×C4×#67C2212yes—
28C2×#926C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30#5614—216no—
31C2×D3×C4⋊C4C2212yes—
32C2×#890C2212yes—
33C2×#884C2214yes—
34C2×#67⋊C4C2214yes—
35C2×#879C2214yes—
36C2×C4×D12C2211yescharacteristic
37C3×#394C2212yes—
38C3×#394C2212yes—
39C3×#392C2212yes—
40C3×#392C2212yes—
41C3×#390C2212yes—
42C3×#382C2211yescharacteristic
43#5223C2214yes—
44#5222C2214yes—
45#5221C2212yes—
46#5220C2214yes—
47#5220C2214yes—
48#4994C2212yes—
49#4991C2211yescharacteristic
50#4987C2212yes—
51#4981C2211yescharacteristic
52#4977C2212yes—
53#4975C2212yes—
54#4972C2212yes—
55#4961C2214yes—
56#4961C2212yes—
57#4960C2211yescharacteristic
58#4959C2214yes—
59#4956C2214yes—
60#4953C2211yescharacteristic
61#4950C2214yes—
62#4949C2214yes—
63#4948C2214yes—
64#4948C2214yes—
65#4948C2214yes—
66#4946C2212yes—
67#4946C2212yes—
68#4942C2214yes—
69C4×(C2×D3)⋊C4C2212yes—
70C4×C4⋊Dic3C2211yescharacteristic
71#4928C2212yes—
72#4928C2212yes—
73#4668C2214yes—
74#4667C2214yes—
75#4658C2214yes—
76#4657C2214yes—
77C2×#553—32no—
78#2823—32no—
79#2693—32no—
80#2675—32no—
81#2671—34no—
82#2636—34no—
83#2619D311yescharacteristic, complemented
84#2537—32no—
85#2454—32no—
86#2442—32no—
87#2428—32no—
88#2416—32no—
89#2416—32no—
90#2392—31no—
91#2317—31no—
92C22×C3×D4C2312yes—
93C22×C3×D4C2314yes—
94C22×C3×D4C2312yes—
95C22×C3×D4C2312yes—
96C23×C3×C4C2311yescharacteristic
97C22×#67C2314yes—
98C22×#67C2312yes—
99C22×#67C2314yes—
100C23×Dic3C2314yes—
101C23×Dic3C2312yes—
102C22×D12C2311yescharacteristic
103C22×C4×D3C2312yes—
104C22×C4×D3C2314yes—
105C3×#127—42no—
106C3×#121—28no—
107C3×#117—42no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124C2×#217C2312yes—
125#939—48no—
126#939—48no—
127#938—216no—
128#938—28no—
129#938—44no—
130#935—216no—
131D4×Dic3—28no—
132C4×#67—28no—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2312yes—
136C2×(C2×D3)⋊C4C2312yes—
137C2×(C2×D3)⋊C4C2312yes—
138#926D418yescomplemented
139C2×C4⋊Dic3—42no—
140C2×C4⋊Dic3C2314yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2314yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2312yes—
154D3×C4⋊C4—28no—
155#890—28no—
156#884—48no—
157#67⋊C4—216no—
158#879—216no—
159C4×D12—42no—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2311yescharacteristic
169C3⋊#95C2311yescharacteristic
170#553—616no—
171C2×#127C2×D311yescharacteristic
172C2×#126—34no—
173C2×#125—32no—
174C2×#123—34no—
175C2×#121—34no—
176C2×#121—32no—
177C2×#121—34no—
178C2×#121—34no—
179C2×#121—32no—
180C2×#121—34no—
181C2×#121C2×D312yes—
182C2×C4×D4—32no—
183C2×C4×D4—32no—
184C2×C4×D4—34no—
185C2×C4×D4—31no—
186C2×#117—31no—
187C2×#117C2×D311yescharacteristic
188C22×C4⋊C4—31no—
189C22×C4⋊C4—32no—
190#400—34no—
191#399—34no—
192#398—31no—
193#398—34no—
194#397—31no—
195#396—32no—
196#396—34no—
197#396—34no—
198#396—34no—
199#394—34no—
200#394—31no—
201#394—34no—
202#394—34no—
203#394—32no—
204#394C2×D312yes—
205#394C2×D312yes—
206#392—34no—
207#392—31no—
208#392C2×D312yes—
209#392C2×D312yes—
210#390—32no—
211#390C2×D312yes—
212#389—32no—
213#389—32no—
214#389—32no—
215#386—34no—
216#386—34no—
217#386—34no—
218#385—32no—
219#385—32no—
220#385—34no—
221#385—34no—
222#384—32no—
223#382—32no—
224#382C2×D311yescharacteristic
225#380—32no—
226#380—34no—
227#380—34no—
228C4×C4⋊C4—31no—
229C4×C22⋊C4—32no—
230C24×C3C2414yes—
231C24×C3C2411yescharacteristic
232C23×D3C2412yes—
233C2×C3×D4—28no—
234C2×C3×D4—24no—
235C2×C3×D4—24no—
236C2×C3×D4—24no—
237C2×C3×D4—44no—
238C2×C3×D4—44no—
239C2×C3×D4—42no—
240C2×C3×D4—42no—
241C2×C3×D4—44no—
242C2×C3×D4—44no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4C2×D414yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2414yes—
250C22×C3×C4C2411yescharacteristic
251C22×C3×C4C2411yescharacteristic
252C22×C3×C4C2411yescharacteristic
253C2×#67—24no—
254C2×#67—28no—
255C2×#67—24no—
256C2×#67—28no—
257C2×#67—28no—
258C2×#67—44no—
259C2×#67—42no—
260C2×#67—44no—
261C2×#67—44no—
262C22×Dic3—28no—
263C22×Dic3—42no—
264C22×Dic3—44no—
265C22×Dic3—44no—
266C22×Dic3—44no—
267C22×Dic3C2412yes—
268C22×Dic3C2414yes—
269C22×Dic3C2412yes—
270C22×Dic3C2412yes—
271C22×Dic3C2414yes—
272C2×D12—24no—
273C2×D12—41no—
274C2×C4×D3—24no—
275C2×C4×D3—24no—
276C2×C4×D3—24no—
277C2×C4×D3—28no—
278C2×C4×D3—44no—
279C2×C4×D3—44no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—41no—
282C3×C4⋊C4—41no—
283C3×C22⋊C4—28no—
284C3×C22⋊C4—44no—
285C3×C22⋊C4—44no—
286C3×C22⋊C4—41no—
287C3×C22⋊C4—41no—
288C3×C42—22no—
289C3×C42—22no—
290#217—24no—
291#217—28no—
292#217—28no—
293#217—44no—
294#217—44no—
295#217—44no—
296#217C2×D418yes—
297(C2×D3)⋊C4—28no—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—24no—
300(C2×D3)⋊C4—42no—
301(C2×D3)⋊C4—42no—
302C4⋊Dic3—24no—
303C4⋊Dic3—28no—
304C4⋊Dic3—41no—
305C4⋊Dic3—42no—
306C4⋊Dic3—42no—
307C4⋊Dic3C2×D418yes—
308Dic3⋊C4—24no—
309Dic3⋊C4—28no—
310Dic3⋊C4—28no—
311Dic3⋊C4—44no—
312Dic3⋊C4—44no—
313Dic3⋊C4—44no—
314C4×Dic3—24no—
315C4×Dic3C2×D418yes—
316C22×D4—34no—
317C22×D4—32no—
318C22×D4—34no—
319C22×D4—31no—
320C22×D4C22×D312yes—
321C22×D4C22×D314yes—
322C22×D4C22×D312yes—
323C22×D4C22×D312yes—
324C23×C4—34no—
325C23×C4—32no—
326C23×C4—32no—
327C23×C4—34no—
328C23×C4C22×D311yescharacteristic
329#127—42no—
330#126—616no—
331#125—68no—
332#123—616no—
333#121—28no—
334#121—616no—
335#121—68no—
336#121—128no—
337#121—128no—
338#121—128no—
339#121—124no—
340C4×D4—616no—
341C4×D4—68no—
342C4×D4—68no—
343C4×D4—122no—
344#117—38no—
345#117—42no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—34no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—68no—
359C2×C4⋊C4—122no—
360C2×C4⋊C4C22×D312yes—
361C2×C4⋊C4C22×D311yescharacteristic
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—32no—
368C2×C22⋊C4—32no—
369C2×C22⋊C4—34no—
370C2×C22⋊C4—34no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—32no—
375C2×C22⋊C4C22×D314yes—
376C2×C22⋊C4C22×D311yescharacteristic
377C2×C22⋊C4C22×D311yescharacteristic
378C2×C22⋊C4C22×D314yes—
379C2×C22⋊C4C22×D314yes—
380C2×C42—32no—
381C2×C42—32no—
382C2×C42C22×D311yescharacteristic
383C2×C42C22×D311yescharacteristic
384#95—32no—
385#95—32no—
386#95—31no—
387#95—31no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95C22×D312yes—
393#95C22×D312yes—
394C23×C3—21no—
395C23×C3—22no—
396C23×C3—21no—
397C23×C3—22no—
398C23×C3—24no—
399C23×C3—24no—
400C23×C3—24no—
401C23×C3—24no—
402C23×C3—24no—
403C23×C3—24no—
404C23×C3—24no—
405C23×C3C22×D412yes—
406C23×C3C2511yescharacteristic
407C22×D3—22no—
408C22×D3—22no—
409C22×D3—22no—
410C22×D3—22no—
411C22×D3—22no—
412C22×D3—22no—
413C22×D3H(5,𝔽2)14yes—
414C3×D4—48no—
415C3×D4—88no—
416C3×D4—84no—
417C3×D4—84no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—28no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—24no—
432C2×C3×C4C22×D412yes—
433C2×C3×C4C22×D412yes—
434C2×C3×C4H(5,𝔽2)12yes—
435C2×C3×C42−1+412yes—
436#67—416no—
437#67—48no—
438#67—88no—
439C2×Dic3—24no—
440C2×Dic3—24no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—24no—
446C2×Dic3—24no—
447C2×Dic3—24no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—88no—
454C2×Dic3—84no—
455C2×Dic3C22×D418yes—
456D12—44no—
457C4×D3—48no—
458C4×D3—88no—
459C24—32no—
460C24C23×D314yes—
461C24C23×D311yescharacteristic
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—64no—
473C2×D4—68no—
474C2×D4—68no—
475C2×D4—64no—
476C2×D4—64no—
477C2×D4—68no—
478C2×D4—121no—
479C2×D4—124no—
480C2×D4—122no—
481C2×D4—124no—
482C2×D4—124no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—34no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—34no—
489C22×C4—32no—
490C22×C4—68no—
491C22×C4—68no—
492C22×C4—64no—
493C22×C4—64no—
494C22×C4—64no—
495C22×C4—124no—
496C22×C4—124no—
497C22×C4—122no—
498C22×C4—124no—
499C22×C4—124no—
500C22×C4—124no—
501C22×C4D3×D414yes—
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D311yescharacteristic
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D314yes—
506C22×C4C23×D311yescharacteristic
507C22×C4C23×D311yescharacteristic
508C22×C4C23×D311yescharacteristic
509C4⋊C4—24no—
510C4⋊C4—38no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—68no—
514C4⋊C4—68no—
515C4⋊C4—68no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—122no—
519C4⋊C4—121no—
520C4⋊C4—124no—
521C4⋊C4—124no—
522C4⋊C4—124no—
523C4⋊C4—122no—
524C22⋊C4—28no—
525C22⋊C4—38no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—68no—
531C22⋊C4—64no—
532C22⋊C4—68no—
533C22⋊C4—68no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—122no—
537C22⋊C4—122no—
538C22⋊C4—124no—
539C22⋊C4—124no—
540C22⋊C4—124no—
541C42—22no—
542C42—22no—
543C42—38no—
544C42—64no—
545C22×C3—22no—
546C22×C3—24no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—42no—
550C22×C3—44no—
551C22×C3—44no—
552C22×C3—44no—
553C22×C3—44no—
554C22×C3—44no—
555C22×C3—44no—
556C22×C3—44no—
557C22×C3C23×D411yescharacteristic
558C22×C3C2×H(5,𝔽2)11yescharacteristic
559C22×C3C2×H(5,𝔽2)12yes—
560C22×C3C2×2−1+411yescharacteristic
561C22×C3#58512yes—
562C2×D3—24no—
563C2×D3—24no—
564C2×D3—24no—
565C2×D3—42no—
566C2×D3—42no—
567C2×D3—42no—
568C2×D3—42no—
569C3×C4—22no—
570C3×C4—44no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4#53714yes—
576Dic3—28no—
577Dic3—42no—
578Dic3—42no—
579Dic3—42no—
580Dic3—44no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—24no—
587C23—24no—
588C23—24no—
589C23—24no—
590C23—24no—
591C23—24no—
592C23—34no—
593C23—62no—
594C23—62no—
595C23—62no—
596C23—62no—
597C23—62no—
598C23—62no—
599C23C2×D3×D412yes—
600C23C24×D311yescharacteristic
601D4—48no—
602D4—88no—
603D4—84no—
604D4—84no—
605D4—124no—
606D4—1216no—
607D4—128no—
608D4—248no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—38no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—64no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—64no—
634C2×C4—62no—
635C2×C4—64no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—128no—
639C2×C4—248no—
640C2×C4—248no—
641C2×C4—244no—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D412yes—
644C2×C4#100412yes—
645C2×C4#101012yes—
646C2×C3—42no—
647C2×C3—84no—
648C2×C3C2×#53711yescharacteristic
649C2×C3#348812yes—
650C2×C3#353711yescharacteristic
651C2×C3#355111yescharacteristic
652C2×C3#355912yes—
653D3—44no—
654C22—24no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—64no—
667C22—64no—
668C22—64no—
669C22—122no—
670C22—122no—
671C22—122no—
672C22—122no—
673C22C22×D3×D411yescharacteristic
674C22C2×#100411yescharacteristic
675C22C2×#101011yescharacteristic
676C22D3×H(5,𝔽2)12yes—
677C22#596612yes—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—44no—
682C4—41no—
683C4—41no—
684C4—68no—
685C4—122no—
686C4—122no—
687C4—124no—
688C4—122no—
689C4#559714yes—
690C3#4295211yescharacteristic, complemented, Sylow
691C2—42no—
692C2—84no—
693C2—124no—
694C2C2×#559711yescharacteristic
695C2#8632711yescharacteristic
696C2#8633112yes—
697C2#8633411yescharacteristic
698C2#8649412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2619C2
hypercenter#2619D3
hypocenterC3#42952
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5614C2semidirect
2. C3×#2619C2semidirect
3. #83186C2semidirect
4. #82705C2semidirect
5. #82273C2semidirect
6. #82234C2semidirect
7. #77089C2semidirect
8. #76981C2semidirect
9. #76944C2semidirect
10. #76770C2semidirect
11. #76718C2semidirect
12. #76710C2semidirect
13. #76439C2semidirect
14. #76325C2semidirect
15. #74163C2semidirect
83. #2619D3semidirect
138. #926D4semidirect
690. C3#42952semidirect

Maximal subgroups

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

subgroupindex
C2×#56142
C3×#26192
#831862
#827052
#822732
#822342
#770892
#769812
#769442
#767702
#767182
#767102
#764392
#763252
#741632
#429523