Finite Group Databasebeta

← #11233657

Subgroups — #11233657

Order 768.

2498 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2C2×#5553C211yesmaximal, characteristic
3C3×#2580C211yesmaximal, characteristic, complemented
4#82467C211yesmaximal, characteristic, complemented
5#82328C211yesmaximal, characteristic, complemented
6#78103C214yesmaximal, complemented
7#76877C214yesmaximal, complemented
8#76350C212yesmaximal, complemented
9#76316C212yesmaximal, complemented
10#76202C212yesmaximal, complemented
11#76006C212yesmaximal, complemented
12#74639C214yesmaximal, complemented
13#74600C214yesmaximal, complemented
14#74191C211yesmaximal, characteristic, complemented
15#74153C211yesmaximal, characteristic, complemented
16#45354—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#947C2212yes—
21C2×#940C2212yes—
22C2×#936C2211yescharacteristic
23C2×#930C2212yes—
24#5585—28no—
25#5553—28no—
26C2×#894C2214yes—
27C2×#893C2212yes—
28C2×#893C2212yes—
29C2×#893C2212yes—
30C2×D3×C4⋊C4C2214yes—
31C2×#884C2214yes—
32C2×#883C2214yes—
33C2×#875C2212yes—
34C2×#874C2211yescharacteristic
35C2×C4×D12C2212yes—
36C2×C4×Dic6C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38C3×#399C2212yes—
39C3×#393C2212yes—
40C3×#390C2212yes—
41C3×#389C2212yes—
42C3×#385C2214yes—
43C3×#382C2211yescharacteristic
44#5230C2212yes—
45#5221C2211yescharacteristic
46#5210C2214yes—
47#4992C2214yes—
48#4989C2214yes—
49#4989C2214yes—
50#4988C2212yes—
51#4984C2212yes—
52#4983C2212yes—
53#4981C2212yes—
54#4979C2212yes—
55#4975C2211yescharacteristic
56#4973C2212yes—
57#4973C2211yescharacteristic
58#4961C2214yes—
59#4959C2214yes—
60#4959C2214yes—
61#4959C2214yes—
62#4956C2214yes—
63#4956C2214yes—
64#4956C2214yes—
65#4944C2214yes—
66#4932C2211yescharacteristic
67C4×Dic3⋊C4C2212yes—
68C4×Dic3⋊C4C2212yes—
69#4675C2214yes—
70#4668C2214yes—
71#4667C2214yes—
72#4665C2214yes—
73#4665C2214yes—
74#4664C2214yes—
75#4659C2212yes—
76#4656C2212yes—
77#4653C2214yes—
78C2×#559—31no—
79C2×#550—31no—
80#2708—34no—
81#2686—34no—
82#2581—31no—
83#2580D311yescharacteristic, complemented
84#2544—32no—
85#2447—32no—
86#2445—32no—
87#2443—34no—
88#2406—34no—
89#2399—31no—
90D4×C4⋊C4—32no—
91#2328—31no—
92#2326—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C22×#67C2314yes—
98C2×#235—28no—
99C2×#235—48no—
100C22×D12C2312yes—
101C22×D12C2312yes—
102C22×D12C2312yes—
103C22×C4×D3C2314yes—
104C22×C4×D3C2314yes—
105C22×C4×D3C2314yes—
106C22×Dic6C2311yescharacteristic
107C22×Dic6C2311yescharacteristic
108C3×C4×Q8—24no—
109C3×C4×D4—24no—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C42C2312yes—
119C2×C3×C42C2311yescharacteristic
120C2×#217C2314yes—
121C2×#217C2314yes—
122#947—28no—
123#940—28no—
124#936—24no—
125#930—28no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147#894—216no—
148#893—28no—
149#893—28no—
150#893—28no—
151D3×C4⋊C4—48no—
152#884—216no—
153#883—48no—
154#875—28no—
155#874D418yes—
156C4×D12—28no—
157C4×Dic6—24no—
158C4×Dic6—24no—
159C3×#95C2312yes—
160C3×#95C2312yes—
161C3×#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2312yes—
172C22×#41—32no—
173#559—68no—
174#550—68no—
175C2×#127—32no—
176C2×#127—31no—
177C2×#124—32no—
178C2×#124—31no—
179C2×#124—32no—
180C2×#123—34no—
181C2×#123—34no—
182C2×#121—32no—
183C2×#121—34no—
184C2×C4×Q8—31no—
185C2×C4×Q8—31no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×D4—32no—
188C2×C4×D4—32no—
189C2×C4×D4—32no—
190C2×C4×D4—32no—
191C2×C4×D4C2×D311yescharacteristic
192C22×C4⋊C4—34no—
193#400—32no—
194#399C2×D312yes—
195#398—32no—
196#398—32no—
197#397—34no—
198#397—34no—
199#395—32no—
200#395—32no—
201#395—32no—
202#394—34no—
203#394—34no—
204#394—34no—
205#394—34no—
206#393C2×D312yes—
207#391—32no—
208#391—31no—
209#390—31no—
210#390—32no—
211#390C2×D312yes—
212#389—32no—
213#389C2×D312yes—
214#388—34no—
215#387—34no—
216#386—34no—
217#386—34no—
218#386—34no—
219#386—34no—
220#386—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#385C2×D314yes—
226#384—31no—
227#382C2×D311yescharacteristic
228#381—34no—
229#380—34no—
230C4×C4⋊C4—32no—
231C4×C4⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C23×D3C2414yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—42no—
237C2×C3×D4—22no—
238C2×C3×D4—22no—
239C2×C3×D4—22no—
240C22×C3×C4—24no—
241C22×C3×C4—44no—
242C22×C3×C4C2414yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C2×#67—28no—
250C2×#67—28no—
251C2×#67—28no—
252C2×#67—44no—
253C2×#67—44no—
254C2×#67—44no—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2412yes—
258C22×Dic3C2412yes—
259C22×Dic3C2412yes—
260C22×Dic3C2412yes—
261#235—432no—
262C2×D12—24no—
263C2×D12—24no—
264C2×D12—44no—
265C2×D12—44no—
266C2×D12C2×D418yes—
267C2×D12C2×D418yes—
268C2×D12C2×D418yes—
269C2×C4×D3—28no—
270C2×C4×D3—28no—
271C2×C4×D3—28no—
272C2×C4×D3—28no—
273C2×C4×D3—44no—
274C2×C4×D3—44no—
275C2×C4×D3—44no—
276C2×C4×D3—44no—
277C2×C4×D3—44no—
278C2×Dic6—22no—
279C2×Dic6—24no—
280C2×Dic6—22no—
281C2×Dic6—22no—
282C2×Dic6—22no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4—24no—
286C3×C4⋊C4—24no—
287C3×C4⋊C4—44no—
288C3×C22⋊C4—24no—
289C3×C22⋊C4—44no—
290C3×C22⋊C4—44no—
291C3×C42—24no—
292C3×C42C2×D414yes—
293#217—28no—
294#217—44no—
295(C2×D3)⋊C4—28no—
296(C2×D3)⋊C4—28no—
297(C2×D3)⋊C4—28no—
298(C2×D3)⋊C4—28no—
299(C2×D3)⋊C4—28no—
300(C2×D3)⋊C4—44no—
301(C2×D3)⋊C4—44no—
302C4⋊Dic3—22no—
303C4⋊Dic3—22no—
304C4⋊Dic3—28no—
305Dic3⋊C4—24no—
306Dic3⋊C4—24no—
307Dic3⋊C4—24no—
308Dic3⋊C4—24no—
309Dic3⋊C4—44no—
310Dic3⋊C4—42no—
311Dic3⋊C4—42no—
312C4×Dic3—24no—
313C4×Dic3—24no—
314C4×Dic3—22no—
315C4×Dic3#4114yescomplemented
316C2×#41—68no—
317C2×#41—128no—
318C22×Q8—31no—
319C22×Q8—31no—
320C22×Q8C22×D311yescharacteristic
321C22×D4—34no—
322C22×D4—34no—
323C22×D4—32no—
324C22×D4—32no—
325C22×D4—32no—
326C22×D4C22×D311yescharacteristic
327C23×C4—34no—
328C23×C4—34no—
329C23×C4—34no—
330C23×C4C22×D312yes—
331#127—38no—
332#127—68no—
333#124—68no—
334#124—64no—
335#124—68no—
336#123—616no—
337#123—128no—
338#121—616no—
339#121—68no—
340C4×Q8—24no—
341C4×Q8—64no—
342C4×Q8—64no—
343C4×D4—24no—
344C4×D4—68no—
345C4×D4—68no—
346C4×D4—68no—
347C4×D4—68no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—34no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—128no—
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C4⋊C4C22×D311yescharacteristic
361C2×C4⋊C4C22×D312yes—
362C2×C4⋊C4C22×D312yes—
363C2×C4⋊C4C22×D314yes—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C22⋊C4—34no—
370C2×C22⋊C4—34no—
371C2×C22⋊C4—34no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4C22×D312yes—
374C2×C22⋊C4C22×D314yes—
375C2×C22⋊C4C22×D314yes—
376C2×C42—31no—
377C2×C42—32no—
378C2×C42—32no—
379C2×C42—31no—
380C2×C42C22×D312yes—
381C2×C42C22×D311yescharacteristic
382#95—32no—
383#95—32no—
384#95—32no—
385#95—32no—
386#95—32no—
387#95—32no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95C22×D314yes—
393#95C22×D312yes—
394#95C22×D312yes—
395C23×C3—22no—
396C23×C3—22no—
397C23×C3—24no—
398C23×C3—22no—
399C23×C3—24no—
400C23×C3C2511yescharacteristic
401C22×D3—24no—
402C22×D3—24no—
403C22×D3—24no—
404C22×D3—24no—
405C22×D3—24no—
406C22×D3—24no—
407C22×D3C22×D418yes—
408C3×Q8—44no—
409C3×D4—44no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—24no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—48no—
424C2×C3×C4C22×D412yes—
425C2×C3×C4C22×D412yes—
426C2×C3×C4C22×D412yes—
427C2×C3×C4C2×#4112yes—
428C2×C3×C42−1+412yes—
429#67—416no—
430#67—88no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3C2×#4114yes—
447C2×Dic3H(5,𝔽2)14yes—
448D12—216no—
449D12—48no—
450D12—48no—
451C4×D3—416no—
452C4×D3—416no—
453C4×D3—88no—
454Dic6—28no—
455Dic6—44no—
456C24—34no—
457C24C23×D312yes—
458#41—1232no—
459C2×Q8—22no—
460C2×Q8—42no—
461C2×Q8—64no—
462C2×Q8—62no—
463C2×Q8—62no—
464C2×Q8—62no—
465C2×Q8—62no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—38no—
470C2×D4—38no—
471C2×D4—38no—
472C2×D4—68no—
473C2×D4—68no—
474C2×D4—68no—
475C2×D4—64no—
476C2×D4—64no—
477C2×D4—124no—
478C2×D4—124no—
479C2×D4—124no—
480C2×D4—124no—
481C2×D4—124no—
482C22×C4—24no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—32no—
489C22×C4—44no—
490C22×C4—68no—
491C22×C4—68no—
492C22×C4—68no—
493C22×C4—68no—
494C22×C4—124no—
495C22×C4—124no—
496C22×C4—124no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4C23×D314yes—
500C22×C4C23×D312yes—
501C22×C4C23×D312yes—
502C22×C4C23×D312yes—
503C22×C4C23×D311yescharacteristic
504C22×C4C23×D311yescharacteristic
505C22×C4C23×D311yescharacteristic
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—44no—
511C4⋊C4—64no—
512C4⋊C4—68no—
513C4⋊C4—64no—
514C4⋊C4—62no—
515C4⋊C4—64no—
516C4⋊C4—62no—
517C4⋊C4—64no—
518C4⋊C4—124no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—68no—
525C22⋊C4—68no—
526C22⋊C4—68no—
527C22⋊C4—68no—
528C22⋊C4—68no—
529C22⋊C4—68no—
530C22⋊C4—124no—
531C22⋊C4—124no—
532C22⋊C4—124no—
533C42—24no—
534C42—34no—
535C42—64no—
536C42—64no—
537C42—62no—
538C42D3×D414yes—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—44no—
543C22×C3—44no—
544C22×C3C23×D411yescharacteristic
545C22×C3C22×#4111yescharacteristic
546C22×C3C2×H(5,𝔽2)12yes—
547C22×C3C2×2−1+411yescharacteristic
548C22×C3#58512yes—
549C2×D3—28no—
550C2×D3—28no—
551C2×D3—28no—
552C2×D3—44no—
553C2×D3—44no—
554C2×D3—44no—
555C2×D3—44no—
556C3×C4—22no—
557C3×C4—22no—
558C3×C4—42no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—44no—
562C3×C4#54714yes—
563Dic3—24no—
564Dic3—24no—
565Dic3—42no—
566Dic3—42no—
567Dic3—42no—
568Dic3—42no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—38no—
575C23—64no—
576C23—64no—
577C23—64no—
578C23—64no—
579C23—64no—
580C23—64no—
581C23C24×D311yescharacteristic
582Q8—44no—
583Q8—68no—
584Q8—124no—
585D4—44no—
586D4—616no—
587D4—128no—
588D4—128no—
589D4—1216no—
590D4—248no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—34no—
605C2×C4—34no—
606C2×C4—48no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—64no—
622C2×C4—1216no—
623C2×C4—1216no—
624C2×C4—248no—
625C2×C4C2×D3×D412yes—
626C2×C4C2×D3×D412yes—
627C2×C4C2×D3×D412yes—
628C2×C4#100712yes—
629C2×C4D3×#4112yes—
630C2×C3—82no—
631C2×C3C2×#54711yescharacteristic
632C2×C3#348411yescharacteristic
633C2×C3#348511yescharacteristic
634C2×C3#350612yes—
635C2×C3#355412yes—
636D3—48no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—68no—
643C22—68no—
644C22—68no—
645C22—124no—
646C22—124no—
647C22—124no—
648C22—124no—
649C22C22×D3×D411yescharacteristic
650C22C2×#100711yescharacteristic
651C22C2×D3×#4111yescharacteristic
652C22#596412yes—
653C22D3×H(5,𝔽2)12yes—
654C4—22no—
655C4—22no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—64no—
661C4—64no—
662C4—122no—
663C4—122no—
664C4—122no—
665C4—122no—
666C4#563114yes—
667C3#4535411yescharacteristic, complemented, Sylow
668C2—82no—
669C2—128no—
670C2C2×#563111yescharacteristic
671C2#8604011yescharacteristic
672C2#8633011yescharacteristic
673C2#8695512yes—
674C2#8706312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2580C2
prime radicalC2×#5553C2
hypercenter#2580D3
hypocenterC3#45354
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
3. C3×#2580C2semidirect
4. #82467C2semidirect
5. #82328C2semidirect
6. #78103C2semidirect
7. #76877C2semidirect
8. #76350C2semidirect
9. #76316C2semidirect
10. #76202C2semidirect
11. #76006C2semidirect
12. #74639C2semidirect
13. #74600C2semidirect
14. #74191C2semidirect
15. #74153C2semidirect
83. #2580D3semidirect
315. C4×Dic3#41semidirect
667. C3#45354semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×#55532
C3×#25802
#824672
#823282
#781032
#768772
#763502
#763162
#762022
#760062
#746392
#746002
#741912
#741532
#453543