Finite Group Databasebeta

← #11660877

Subgroups — #11660877

Order 768.

1820 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86248C211yesmaximal, characteristic
2C3×#2931C211yesmaximal, characteristic, complemented
3#83794C212yesmaximal, complemented
4#83786C212yesmaximal, complemented
5#83279C211yesmaximal, characteristic, complemented
6#79545C214yesmaximal, complemented
7#79528C214yesmaximal, complemented
8#79421C214yesmaximal, complemented
9#79400C214yesmaximal, complemented
10#79054C212yesmaximal, complemented
11#78668C212yesmaximal, complemented
12#78666C212yesmaximal, complemented
13#78448C212yesmaximal, complemented
14#60307—31nomaximal, Sylow
15C2×C3×#131C2211yescharacteristic
16C2×C3×#131C2211yescharacteristic, complemented
17C3×#532C2211yescharacteristic
18#5832C2212yes—
19#5828C2212yes—
20C2×#901C2212yes—
21C22×D3×C8C2212yescomplemented
22#5735C412yes—
23#5720C412yes—
24#5684C412yes—
25#5683C411yescharacteristic
26#5672C411yescharacteristic
27#5621C414yes—
28#5620C414yes—
29#5547C2214yes—
30#5538C2214yes—
31#5479C2212yes—
32C3×#436C2214yescomplemented
33C3×C8×D4C2214yescomplemented
34C3×#423C2211yescharacteristic, complemented
35C3×#408C2211yescharacteristic
36C3×#408C2211yescharacteristic, complemented
37C2×C3×C4×C8C2211yescharacteristic, complemented
38#5234C2212yes—
39#5234C2212yescomplemented
40#5129C2212yes—
41#5128C2214yescomplemented
42#5128C2214yescomplemented
43#5127C2214yes—
44#5127C2214yescomplemented
45#5126C2212yescomplemented
46#5119C2212yescomplemented
47#5118C2212yescomplemented
48#5111C2214yescomplemented
49C8×#67C2214yes—
50C8×#67C2214yescomplemented
51C8×#67C2214yescomplemented
52#5101C2212yescomplemented
53C2×C8×Dic3C2212yescomplemented
54#5011C2214yescomplemented
55#5011C2214yescomplemented
56#4926C2212yescomplemented
57#4840C2214yescomplemented
58#4839C2214yescomplemented
59#4836C2214yes—
60#4828C2214yescomplemented
61#4730C2214yes—
62#4730C2214yescomplemented
63#4729C2214yes—
64#4728C2214yescomplemented
65#4727C2214yes—
66#4726C2214yescomplemented
67#4720C2214yescomplemented
68#4719C2214yescomplemented
69#4690C2214yescomplemented
70#4689C2214yescomplemented
71#4688C2212yes—
72C8×D12C2212yes—
73#4679C2212yescomplemented
74C8×Dic6C2212yescomplemented
75#3529—31no—
76#2995—34no—
77#2995—32no—
78#2980—32no—
79#2980—31no—
80#2978—34no—
81#2978—32no—
82#2966—34no—
83#2966—32no—
84#2931—32no—
85#2931D311yescharacteristic, complemented
86#2929—34no—
87#2929—32no—
88C2×C3×#41C2311yescharacteristic
89C2×C3×#41C2311yescharacteristic
90C2×#235C2312yes—
91C22×C4×D3C2312yes—
92C3×#131—42no—
93C3×#131—42no—
94C2×C3×M16C2312yes—
95C2×C3×M16C2311yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C2×C3×M16C2312yes—
98C22×C3×C8C2311yescharacteristic
99C22×C3×C8C2312yes—
100C22×C3×C8C2312yes—
101C22×C3×C8C2311yescharacteristic
102C3×C4⋊Q8C2×C411yescharacteristic
103C3×#127C2×C411yescharacteristic
104C3×#124C2×C411yescharacteristic
105C3×#124C2×C411yescharacteristic
106C3×#121C2×C414yes—
107C3×C4×D4C2314yes—
108C2×C3×C42C2311yescharacteristic
109#947C2×C412yes—
110#946C2×C414yes—
111#944C2×C412yes—
112#940C2×C412yes—
113#939C2×C414yes—
114#939C2×C414yes—
115#939C2×C414yes—
116#939C2×C414yes—
117#938C2×C414yes—
118#937C2×C412yes—
119#937C2×C414yes—
120#937C2×C412yes—
121#936C2×C412yes—
122#935C2×C412yes—
123#935C2×C414yes—
124#935C2×C412yes—
125#930C2×C412yes—
126#929C2×C414yes—
127C4×#67C2314yes—
128C4×#67C2314yes—
129C4×#67C2314yes—
130#926C2312yes—
131#924C2×C412yes—
132C2×C4×Dic3C2312yes—
133C2×#208C2312yes—
134C2×#208C2312yes—
135C22×C3⋊C8C2312yes—
136C22×C3⋊C8C2312yes—
137#901—44no—
138C2×ΓL(1,𝔽25)C2314yes—
139C2×ΓL(1,𝔽25)C2314yes—
140C2×D3×C8—44no—
141C2×D3×C8C2314yes—
142C2×D3×C8C2314yes—
143#898C2×C414yes—
144#896C2×C414yes—
145#894C2×C414yes—
146#889C2×C414yes—
147#886C2×C414yes—
148#885C2×C414yes—
149#884C2×C414yes—
150#883C2×C414yes—
151#882C2×C414yes—
152#879C2×C414yes—
153#878C2×C414yes—
154#878C2×C414yes—
155#876C2×C414yes—
156#875C2×C412yes—
157C4×D12C2312yes—
158#872C2314yes—
159#870C2×C412yes—
160C4×Dic6C2312yes—
161C3×C4⋊C8C2314yes—
162C3×C22⋊C8C2314yes—
163C3×C22⋊C8C2314yes—
164C3×C4×C8C2312yes—
165C3×C4×C8C2312yes—
166#830C2314yes—
167#830C2314yes—
168#830C2314yes—
169(C2×D3)⋊C8C2314yes—
170(C2×D3)⋊C8C2314yes—
171(C2×D3)⋊C8C2314yes—
172(C2×D3)⋊C8C2314yes—
173#815C2314yes—
174#815C2314yes—
175Dic3⋊C8C2314yes—
176Dic3⋊C8C2314yes—
177Dic3⋊C8C2314yes—
178Dic3⋊C8C2314yes—
179C8×Dic3C2314yes—
180C8×Dic3C2314yes—
181C4⋊(C3⋊C8)C2312yes—
182C4⋊(C3⋊C8)C2312yes—
183#803C2314yes—
184C2×#131—32no—
185C2×#131C2×D311yescharacteristic
186C2×#131C2×D311yescharacteristic, complemented
187C23×C8—32no—
188#563—32no—
189#563—31no—
190#562—32no—
191#562—34no—
192H(3,𝔽4)—31no—
193#560—32no—
194#560—34no—
195#532—32no—
196#532C2×D311yescharacteristic
197#529—34no—
198#529—32no—
199#525—32no—
200#525—34no—
201#446—34no—
202#446—32no—
203C8×Q8—32no—
204#436—34no—
205#436—34no—
206#436—34no—
207#436—32no—
208#436C2×D314yescomplemented
209#435—34no—
210#435—34no—
211#435—34no—
212#435—34no—
213#435—34no—
214#435—34no—
215#435—34no—
216#435—34no—
217#435—34no—
218C8×D4—34no—
219C8×D4—34no—
220C8×D4—34no—
221C8×D4—32no—
222C8×D4C2×D314yescomplemented
223#433—34no—
224#433—34no—
225#433—34no—
226#432—32no—
227#432—34no—
228#424—32no—
229#424—34no—
230#423—32no—
231#423C2×D311yescharacteristic, complemented
232#408—32no—
233#408—32no—
234#408—32no—
235#408—34no—
236#408C2×D311yescharacteristic
237#408C2×D311yescharacteristic, complemented
238#407—32no—
239#407—34no—
240#405—32no—
241#405—34no—
242C2×C4×C8—32no—
243C2×C4×C8C2×D311yescharacteristic, complemented
244C23×D3C22×C412yes—
245C3×#41—41no—
246C3×#41—41no—
247C3×#41—41no—
248C3×#41—41no—
249C2×C3×Q8C22×C411yescharacteristic
250C2×C3×Q8C22×C411yescharacteristic
251C2×C3×D4C22×C411yescharacteristic
252C2×C3×D4C22×C412yes—
253C2×C3×D4C22×C414yes—
254C2×C3×D4C22×C412yes—
255C2×C3×D4C22×C411yescharacteristic
256C22×C3×C4C22×C412yes—
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2412yes—
259C22×C3×C4C2412yes—
260C2×#67C22×C414yes—
261C2×#67C22×C414yes—
262C2×#67C22×C414yes—
263C2×#67C22×C414yes—
264C22×Dic3C22×C412yes—
265C22×Dic3C22×C414yes—
266#235—42no—
267#235—42no—
268C2×D12C22×C412yes—
269C2×D12C22×C412yes—
270C2×C4×D3—42no—
271C2×C4×D3—42no—
272C2×C4×D3C22×C414yes—
273C2×C4×D3C2414yes—
274C2×Dic6C22×C412yes—
275C2×Dic6C22×C412yes—
276C3×M16—41no—
277C3×M16—42no—
278C3×M16—41no—
279C3×M16—42no—
280C2×C3×C8—24no—
281C2×C3×C8—24no—
282C2×C3×C8—22no—
283C2×C3×C8—41no—
284C2×C3×C8C2412yes—
285C2×C3×C8C2411yescharacteristic
286C2×C3×C8C2412yes—
287C2×C3×C8C2412yes—
288C2×C3×C8C2411yescharacteristic
289C3×C4⋊C4C22×C414yes—
290C3×C22⋊C4C22×C414yes—
291C3×C22⋊C4C22×C414yes—
292C3×C42C22×C411yescharacteristic
293C3×C42C22×C411yescharacteristic
294C3×C42C2412yes—
295#217C22×C414yes—
296#217C22×C414yes—
297#217C22×C412yes—
298#217C22×C412yes—
299#217C22×C414yes—
300(C2×D3)⋊C4C22×C414yes—
301(C2×D3)⋊C4C22×C414yes—
302(C2×D3)⋊C4C22×C414yes—
303(C2×D3)⋊C4C22×C414yes—
304(C2×D3)⋊C4C22×C414yes—
305(C2×D3)⋊C4C22×C414yes—
306C4⋊Dic3C22×C412yes—
307C4⋊Dic3C22×C412yes—
308C4⋊Dic3C22×C414yes—
309Dic3⋊C4C22×C414yes—
310Dic3⋊C4C22×C414yes—
311Dic3⋊C4C22×C414yes—
312Dic3⋊C4C22×C414yes—
313Dic3⋊C4C22×C414yes—
314Dic3⋊C4C22×C414yes—
315C4×Dic3C22×C412yes—
316C4×Dic3C22×C412yes—
317C4×Dic3C2414yes—
318#208—42no—
319#208—42no—
320C2×C3⋊C8—42no—
321C2×C3⋊C8—42no—
322C2×C3⋊C8C2414yes—
323C2×C3⋊C8C2414yes—
324ΓL(1,𝔽25)—44no—
325ΓL(1,𝔽25)—44no—
326D3×C8—28no—
327D3×C8—44no—
328C2×#41—32no—
329C2×#41C22×D311yescharacteristic
330C2×#41C22×D311yescharacteristic
331C23×C4—32no—
332#131—42no—
333#131—42no—
334#131—124no—
335C2×M16—32no—
336C2×M16—32no—
337C2×M16—34no—
338C2×M16—34no—
339C2×M16C22×D312yes—
340C2×M16C22×D311yescharacteristic
341C2×M16C22×D312yes—
342C2×M16C22×D311yescharacteristic
343C22×C8—32no—
344C22×C8—32no—
345C22×C8—34no—
346C22×C8—34no—
347C22×C8—124no—
348C22×C8C22×D312yes—
349C22×C8C22×D312yes—
350C22×C8C22×D311yescharacteristic
351C22×C8C22×D311yescharacteristic
352C4⋊Q8—32no—
353C4⋊Q8C4×D311yescharacteristic
354#127—32no—
355#127C4×D311yescharacteristic
356#126—34no—
357#126—34no—
358#126—34no—
359#125—34no—
360#125—32no—
361#124—32no—
362#124—32no—
363#124—34no—
364#124—32no—
365#124C4×D311yescharacteristic
366#124C4×D311yescharacteristic
367#123—32no—
368#123—32no—
369#123—34no—
370#123—34no—
371#123—34no—
372#123—34no—
373#123—34no—
374C22⋊Q8—34no—
375C22⋊Q8—32no—
376C22⋊Q8—34no—
377C22⋊Q8—34no—
378C22⋊Q8—34no—
379#121—34no—
380#121—34no—
381#121—34no—
382#121—34no—
383#121—34no—
384#121—32no—
385#121—34no—
386#121C4×D314yes—
387#120—32no—
388#120—34no—
389#120—32no—
390#120—34no—
391C4×Q8—32no—
392C4×D4—34no—
393C4×D4—34no—
394C4×D4—34no—
395C4×D4—32no—
396C4×D4C22×D314yes—
397#117—32no—
398#117—34no—
399C2×C42—32no—
400C2×C42C22×D311yescharacteristic
401C4⋊C8—34no—
402C4⋊C8—34no—
403C4⋊C8—34no—
404C4⋊C8—34no—
405C4⋊C8—32no—
406C4⋊C8—32no—
407C4⋊C8C22×D314yes—
408C22⋊C8—34no—
409C22⋊C8—34no—
410C22⋊C8—34no—
411C22⋊C8—34no—
412C22⋊C8—34no—
413C22⋊C8—34no—
414C22⋊C8—34no—
415C22⋊C8C22×D314yes—
416C22⋊C8C22×D314yes—
417#97—34no—
418#97—34no—
419#97—34no—
420C4×C8—34no—
421C4×C8—34no—
422C4×C8C22×D312yes—
423C4×C8C22×D312yes—
424C23×C3C23×C412yes—
425C23×C3C23×C411yescharacteristic
426C23×C3C23×C412yes—
427C22×D3—42no—
428C22×D3—44no—
429C22×D3C23×C414yes—
430C3×Q8—41no—
431C3×Q8—41no—
432C3×D4—44no—
433C3×D4—42no—
434C3×D4—42no—
435C3×D4—41no—
436C3×D4—41no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—21no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—21no—
443C2×C3×C4—42no—
444C2×C3×C4C23×C412yes—
445C2×C3×C4C23×C411yescharacteristic
446C2×C3×C4C23×C412yes—
447C2×C3×C4C23×C412yes—
448C2×C3×C4C23×C412yes—
449C2×C3×C4C2511yescharacteristic
450#67—44no—
451#67—44no—
452#67—44no—
453#67—44no—
454C2×Dic3—42no—
455C2×Dic3—44no—
456C2×Dic3C23×C414yes—
457C2×Dic3C23×C414yes—
458C2×Dic3C23×C414yes—
459D12—42no—
460D12—42no—
461C4×D3—24no—
462C4×D3—24no—
463C4×D3—44no—
464Dic6—42no—
465Dic6—42no—
466C3×C8—22no—
467C3×C8—22no—
468C3×C8—21no—
469C3×C8—22no—
470C3×C8H(5,𝔽2)12yes—
471C3⋊C8—24no—
472C3⋊C8—24no—
473C24—32no—
474#41—41no—
475#41—41no—
476#41—41no—
477#41—41no—
478#41—122no—
479#41—122no—
480C2×Q8—32no—
481C2×Q8—32no—
482C2×Q8C2×C4×D311yescharacteristic
483C2×Q8C2×C4×D311yescharacteristic
484C2×D4—34no—
485C2×D4—34no—
486C2×D4—34no—
487C2×D4—34no—
488C2×D4—32no—
489C2×D4—32no—
490C2×D4C2×C4×D314yes—
491C2×D4C2×C4×D311yescharacteristic
492C2×D4C2×C4×D312yes—
493C2×D4C2×C4×D311yescharacteristic
494C2×D4C2×C4×D312yes—
495C22×C4—34no—
496C22×C4—32no—
497C22×C4—34no—
498C22×C4—34no—
499C22×C4—122no—
500C22×C4—122no—
501C22×C4C2×C4×D312yes—
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D312yes—
504C22×C4C23×D312yes—
505M16—41no—
506M16—42no—
507M16—42no—
508M16—41no—
509M16—124no—
510M16—124no—
511M16—122no—
512M16—122no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—34no—
517C2×C8—34no—
518C2×C8—41no—
519C2×C8—68no—
520C2×C8—124no—
521C2×C8—122no—
522C2×C8—122no—
523C2×C8C23×D312yes—
524C2×C8C23×D312yes—
525C2×C8C23×D311yescharacteristic
526C2×C8C23×D312yes—
527C2×C8C23×D311yescharacteristic
528C4⋊C4—34no—
529C4⋊C4—34no—
530C4⋊C4—34no—
531C4⋊C4—34no—
532C4⋊C4—34no—
533C4⋊C4—34no—
534C4⋊C4—32no—
535C4⋊C4—34no—
536C4⋊C4—32no—
537C4⋊C4C2×C4×D314yes—
538C22⋊C4—32no—
539C22⋊C4—32no—
540C22⋊C4—34no—
541C22⋊C4—34no—
542C22⋊C4—34no—
543C22⋊C4—34no—
544C22⋊C4—34no—
545C22⋊C4—34no—
546C22⋊C4—34no—
547C22⋊C4—34no—
548C22⋊C4—34no—
549C22⋊C4C2×C4×D314yes—
550C22⋊C4C2×C4×D314yes—
551C42—32no—
552C42—32no—
553C42—34no—
554C42C2×C4×D311yescharacteristic
555C42C2×C4×D311yescharacteristic
556C42C23×D312yes—
557C22×C3—22no—
558C22×C3—22no—
559C22×C3—21no—
560C22×C3—24no—
561C22×C3—22no—
562C22×C3—24no—
563C22×C3C24×C411yescharacteristic
564C2×D3—24no—
565C2×D3—24no—
566C2×D3—24no—
567C2×D3—48no—
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—21no—
571C3×C4—22no—
572C3×C4—22no—
573C3×C4C2×H(5,𝔽2)11yescharacteristic
574C3×C4#58511yescharacteristic
575Dic3—24no—
576Dic3—24no—
577Dic3—24no—
578C23—34no—
579C23—122no—
580C23—124no—
581C23C22×C4×D312yes—
582C23C22×C4×D311yescharacteristic
583C23C22×C4×D312yes—
584Q8—41no—
585Q8—41no—
586Q8—122no—
587Q8—122no—
588D4—42no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—44no—
593D4—122no—
594D4—122no—
595D4—124no—
596D4—124no—
597D4—124no—
598D4—124no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—34no—
606C2×C4—34no—
607C2×C4—34no—
608C2×C4—42no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—124no—
612C2×C4—122no—
613C2×C4—124no—
614C2×C4C22×C4×D312yes—
615C2×C4C22×C4×D311yescharacteristic
616C2×C4C22×C4×D312yes—
617C2×C4C22×C4×D312yes—
618C2×C4C22×C4×D312yes—
619C2×C4C24×D311yescharacteristic
620C8—22no—
621C8—21no—
622C8—22no—
623C8—22no—
624C8—64no—
625C8—64no—
626C8#100412yes—
627C2×C3—42no—
628C2×C3—42no—
629C2×C3—41no—
630C2×C3C4×H(5,𝔽2)11yescharacteristic
631C2×C3#357312yes—
632D3—44no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—22no—
637C22—21no—
638C22—22no—
639C22—64no—
640C22—64no—
641C22—64no—
642C22—128no—
643C22C23×C4×D311yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—21no—
648C4—22no—
649C4—64no—
650C4—64no—
651C4—64no—
652C4C2×#100411yescharacteristic
653C4#596411yescharacteristic
654C3#6030711yescharacteristic, complemented, Sylow
655C2—42no—
656C2—41no—
657C2—42no—
658C2—124no—
659C2C4×#100411yescharacteristic
660C2#8716012yes—
subgroupG/H
centerC2×C8C23×D3
commutator, socleC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2931C2
prime radical#86248C2
cosocleC2×C3×C4C25
hypercenter#2931D3
hypocenterC3#60307
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ #2931

Complements

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

NH ≅ G/Ntype
2. C3×#2931C2semidirect
3. #83794C2semidirect
4. #83786C2semidirect
5. #83279C2semidirect
6. #79545C2semidirect
7. #79528C2semidirect
8. #79421C2semidirect
9. #79400C2semidirect
10. #79054C2semidirect
11. #78668C2semidirect
12. #78666C2semidirect
13. #78448C2semidirect
16. C2×C3×#131C22semidirect
21. C22×D3×C8C22semidirect
32. C3×#436C22semidirect
33. C3×C8×D4C22semidirect
34. C3×#423C22semidirect
36. C3×#408C22semidirect
37. C2×C3×C4×C8C22semidirect
39. #5234C22semidirect
41. #5128C22semidirect
42. #5128C22semidirect
44. #5127C22semidirect
45. #5126C22semidirect
46. #5119C22semidirect
47. #5118C22semidirect
48. #5111C22semidirect
50. C8×#67C22semidirect
51. C8×#67C22semidirect
52. #5101C22semidirect
53. C2×C8×Dic3C22semidirect
54. #5011C22semidirect
55. #5011C22semidirect
56. #4926C22semidirect
57. #4840C22semidirect
58. #4839C22semidirect
60. #4828C22semidirect
62. #4730C22semidirect
64. #4728C22semidirect
66. #4726C22semidirect
67. #4720C22semidirect
68. #4719C22semidirect
69. #4690C22semidirect
70. #4689C22semidirect
73. #4679C22semidirect
74. C8×Dic6C22semidirect
85. #2931D3semidirect
186. C2×#131C2×D3semidirect
208. #436C2×D3semidirect
222. C8×D4C2×D3semidirect
231. #423C2×D3semidirect
237. #408C2×D3semidirect
243. C2×C4×C8C2×D3semidirect
654. C3#60307semidirect

Maximal subgroups

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

subgroupindex
#862482
C3×#29312
#837942
#837862
#832792
#795452
#795282
#794212
#794002
#790542
#786682
#786662
#784482
#603073