Finite Group Databasebeta

← #11660880

Subgroups — #11660880

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×#2980C211yesmaximal, characteristic, complemented
3#83275C211yesmaximal, characteristic, complemented
4#79490C214yesmaximal, complemented
5#79478C214yesmaximal, complemented
6#79467C214yesmaximal, complemented
7#79454C214yesmaximal, complemented
8#79342C212yesmaximal, complemented
9#79325C212yesmaximal, complemented
10#78668C212yesmaximal, complemented
11#78666C212yesmaximal, complemented
12#78657C212yesmaximal, complemented
13#78655C212yesmaximal, complemented
14#60307—31nomaximal, Sylow
15C3×#532C2211yescharacteristic
16#5832C2212yes—
17#5828C2212yes—
18C2×#948C2211yescharacteristic
19C2×#948C2211yescharacteristic, complemented
20C2×#907C2212yes—
21C22×D3×C8C2212yescomplemented
22#5735C412yes—
23#5720C412yes—
24#5684C412yes—
25#5683C411yescharacteristic
26#5672C411yescharacteristic
27#5621C414yes—
28#5620C414yes—
29#5547C2214yes—
30#5538C2214yes—
31#5479C2212yes—
32C3×#435C2214yescomplemented
33C3×#435C2214yescomplemented
34C3×#432C2212yescomplemented
35C3×#408C2212yes—
36C3×#408C2212yescomplemented
37#5234C2211yescharacteristic
38#5234C2211yescharacteristic, complemented
39#5128C2214yes—
40#5127C2214yescomplemented
41#5126C2212yescomplemented
42#5122C2212yescomplemented
43#5118C2212yescomplemented
44#5111C2214yes—
45#5111C2214yescomplemented
46#5110C2212yes—
47#5102C2212yescomplemented
48C2×C8×Dic3C2212yescomplemented
49#5012C2214yescomplemented
50D4×C3⋊C8C2214yescomplemented
51#4924C2211yescharacteristic, complemented
52C2×C4×C3⋊C8C2211yescharacteristic, complemented
53#4840C2214yescomplemented
54#4839C2214yescomplemented
55#4838C2212yes—
56#4834C2212yes—
57#4831C2212yescomplemented
58#4830C2212yescomplemented
59#4730C2214yescomplemented
60#4729C2214yescomplemented
61#4729C2214yescomplemented
62#4728C2214yes—
63#4728C2214yescomplemented
64#4727C2214yes—
65#4726C2214yescomplemented
66#67⋊C8C2214yes—
67#67⋊C8C2214yescomplemented
68#67⋊C8C2214yescomplemented
69#4720C2214yescomplemented
70#4719C2214yescomplemented
71#4709C2214yescomplemented
72#4708C2214yescomplemented
73#4706C2214yes—
74#4701C2214yescomplemented
75#3529—31no—
76#2995—34no—
77#2995—32no—
78#2980—32no—
79#2980D311yescharacteristic, complemented
80#2978—34no—
81#2978—32no—
82#2966—34no—
83#2966—32no—
84#2931—32no—
85#2931—31no—
86#2929—34no—
87#2929—32no—
88C2×C3×#41C2311yescharacteristic
89C2×C3×#41C2311yescharacteristic
90C2×#235C2312yes—
91C22×C4×D3C2312yes—
92C2×C3×M16C2312yes—
93C2×C3×M16C2312yes—
94C22×C3×C8C2312yes—
95C22×C3×C8C2312yes—
96C3×C4⋊Q8C2×C411yescharacteristic
97C3×#127C2×C411yescharacteristic
98C3×#124C2×C411yescharacteristic
99C3×#124C2×C411yescharacteristic
100C3×#121C2×C414yes—
101C3×C4×D4C2314yes—
102C2×C3×C42C2311yescharacteristic
103#948—42no—
104#948—42no—
105#947C2×C412yes—
106#946C2×C414yes—
107#944C2×C412yes—
108#940C2×C412yes—
109#939C2×C414yes—
110#939C2×C414yes—
111#939C2×C414yes—
112#939C2×C414yes—
113#938C2×C414yes—
114#937C2×C412yes—
115#937C2×C414yes—
116#937C2×C412yes—
117#936C2×C412yes—
118#935C2×C412yes—
119#935C2×C414yes—
120#935C2×C412yes—
121#930C2×C412yes—
122#929C2×C414yes—
123C4×#67C2314yes—
124C4×#67C2314yes—
125C4×#67C2314yes—
126#926C2312yes—
127#924C2×C412yes—
128C2×C4×Dic3C2312yes—
129C2×#208C2312yes—
130C2×#208C2311yescharacteristic
131C2×#208C2311yescharacteristic
132C2×#208C2312yes—
133C22×C3⋊C8C2312yes—
134C22×C3⋊C8C2311yescharacteristic
135C22×C3⋊C8C2312yes—
136C22×C3⋊C8C2311yescharacteristic
137#907—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⋊C8C2312yes—
162C3×C4⋊C8C2312yes—
163C3×C22⋊C8C2314yes—
164C3×C22⋊C8C2314yes—
165C3×C22⋊C8C2314yes—
166C3×#97C2314yes—
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)C2314yes—
182C4×C3⋊C8C2312yes—
183C4×C3⋊C8C2312yes—
184C2×#131—31no—
185C2×#131—31no—
186C2×#131—32no—
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—32no—
202#446—34no—
203C8×Q8—32no—
204#436—34no—
205#436—34no—
206#436—32no—
207#436—34no—
208#436—34no—
209#435—34no—
210#435—34no—
211#435—34no—
212#435—34no—
213#435—34no—
214#435—34no—
215#435—34no—
216#435C2×D314yescomplemented
217#435C2×D314yescomplemented
218C8×D4—34no—
219C8×D4—32no—
220C8×D4—34no—
221C8×D4—34no—
222C8×D4—34no—
223#433—34no—
224#433—34no—
225#433—34no—
226#432—34no—
227#432C2×D312yescomplemented
228#424—32no—
229#424—34no—
230#423—32no—
231#423—31no—
232#408—31no—
233#408—31no—
234#408—32no—
235#408—34no—
236#408C2×D312yes—
237#408C2×D312yescomplemented
238#407—32no—
239#407—34no—
240#405—32no—
241#405—34no—
242C2×C4×C8—32no—
243C2×C4×C8—31no—
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×C4C2412yes—
258C22×C3×C4C2412yes—
259C22×C3×C4C2411yescharacteristic
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—42no—
277C3×M16—42no—
278C2×C3×C8—42no—
279C2×C3×C8—42no—
280C2×C3×C8C2414yes—
281C2×C3×C8C2414yes—
282C3×C4⋊C4C22×C414yes—
283C3×C22⋊C4C22×C414yes—
284C3×C22⋊C4C22×C414yes—
285C3×C42C22×C411yescharacteristic
286C3×C42C22×C411yescharacteristic
287C3×C42C2412yes—
288#217C22×C414yes—
289#217C22×C414yes—
290#217C22×C412yes—
291#217C22×C412yes—
292#217C22×C414yes—
293(C2×D3)⋊C4C22×C414yes—
294(C2×D3)⋊C4C22×C414yes—
295(C2×D3)⋊C4C22×C414yes—
296(C2×D3)⋊C4C22×C414yes—
297(C2×D3)⋊C4C22×C414yes—
298(C2×D3)⋊C4C22×C414yes—
299C4⋊Dic3C22×C412yes—
300C4⋊Dic3C22×C412yes—
301C4⋊Dic3C22×C414yes—
302Dic3⋊C4C22×C414yes—
303Dic3⋊C4C22×C414yes—
304Dic3⋊C4C22×C414yes—
305Dic3⋊C4C22×C414yes—
306Dic3⋊C4C22×C414yes—
307Dic3⋊C4C22×C414yes—
308C4×Dic3C22×C412yes—
309C4×Dic3C22×C412yes—
310C4×Dic3C2414yes—
311#208—41no—
312#208—42no—
313#208—41no—
314#208—42no—
315C2×C3⋊C8—24no—
316C2×C3⋊C8—24no—
317C2×C3⋊C8—22no—
318C2×C3⋊C8—41no—
319C2×C3⋊C8C2411yescharacteristic
320C2×C3⋊C8C2412yes—
321C2×C3⋊C8C2412yes—
322C2×C3⋊C8C2412yes—
323C2×C3⋊C8C2411yescharacteristic
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—124no—
333#131—122no—
334#131—122no—
335C2×M16—32no—
336C2×M16—31no—
337C2×M16—32no—
338C2×M16—31no—
339C2×M16—34no—
340C2×M16—34no—
341C2×M16C22×D312yes—
342C2×M16C22×D312yes—
343C22×C8—32no—
344C22×C8—32no—
345C22×C8—31no—
346C22×C8—31no—
347C22×C8—34no—
348C22×C8—34no—
349C22×C8—124no—
350C22×C8C22×D312yes—
351C22×C8C22×D312yes—
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—34no—
406C4⋊C8C22×D312yes—
407C4⋊C8C22×D312yes—
408C22⋊C8—34no—
409C22⋊C8—34no—
410C22⋊C8—34no—
411C22⋊C8—34no—
412C22⋊C8—34no—
413C22⋊C8—34no—
414C22⋊C8C22×D314yes—
415C22⋊C8C22×D314yes—
416C22⋊C8C22×D314yes—
417#97—34no—
418#97—34no—
419#97C22×D314yes—
420C4×C8—34no—
421C4×C8—34no—
422C4×C8—32no—
423C4×C8—32no—
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—21no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—21no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
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—24no—
467C3×C8—24no—
468C3⋊C8—22no—
469C3⋊C8—22no—
470C3⋊C8—21no—
471C3⋊C8—22no—
472C3⋊C8H(5,𝔽2)12yes—
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×D312yes—
503C22×C4C23×D312yes—
504C22×C4C23×D311yescharacteristic
505M16—42no—
506M16—42no—
507M16—124no—
508M16—124no—
509M16—121no—
510M16—122no—
511M16—122no—
512M16—121no—
513C2×C8—31no—
514C2×C8—32no—
515C2×C8—32no—
516C2×C8—31no—
517C2×C8—32no—
518C2×C8—42no—
519C2×C8—42no—
520C2×C8—68no—
521C2×C8—64no—
522C2×C8—64no—
523C2×C8—62no—
524C2×C8—124no—
525C2×C8—121no—
526C2×C8C23×D314yes—
527C2×C8C23×D314yes—
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—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
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—24no—
621C8—24no—
622C8—32no—
623C8—62no—
624C8—62no—
625C8—61no—
626C8—62no—
627C2×C3—42no—
628C2×C3—42no—
629C2×C3—41no—
630C2×C3C4×H(5,𝔽2)11yescharacteristic
631C2×C3#357312yes—
632D3—44no—
633C22—22no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—64no—
640C22—64no—
641C22—64no—
642C22—128no—
643C22C23×C4×D311yescharacteristic
644C4—22no—
645C4—21no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—64no—
650C4—64no—
651C4—64no—
652C4D3×H(5,𝔽2)11yescharacteristic
653C4#596611yescharacteristic
654C3#6030711yescharacteristic, complemented, Sylow
655C2—42no—
656C2—41no—
657C2—42no—
658C2—124no—
659C2#8585711yescharacteristic
660C2#8716612yes—
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2980C2
prime radical#86248C2
cosocleC2×C3×C4C25
hypercenter#2980D3
hypocenterC3#60307
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. C3×#2980C2semidirect
3. #83275C2semidirect
4. #79490C2semidirect
5. #79478C2semidirect
6. #79467C2semidirect
7. #79454C2semidirect
8. #79342C2semidirect
9. #79325C2semidirect
10. #78668C2semidirect
11. #78666C2semidirect
12. #78657C2semidirect
13. #78655C2semidirect
19. C2×#948C22semidirect
21. C22×D3×C8C22semidirect
32. C3×#435C22semidirect
33. C3×#435C22semidirect
34. C3×#432C22semidirect
36. C3×#408C22semidirect
38. #5234C22semidirect
40. #5127C22semidirect
41. #5126C22semidirect
42. #5122C22semidirect
43. #5118C22semidirect
45. #5111C22semidirect
47. #5102C22semidirect
48. C2×C8×Dic3C22semidirect
49. #5012C22semidirect
50. D4×C3⋊C8C22semidirect
51. #4924C22semidirect
52. C2×C4×C3⋊C8C22semidirect
53. #4840C22semidirect
54. #4839C22semidirect
57. #4831C22semidirect
58. #4830C22semidirect
59. #4730C22semidirect
60. #4729C22semidirect
61. #4729C22semidirect
63. #4728C22semidirect
65. #4726C22semidirect
67. #67⋊C8C22semidirect
68. #67⋊C8C22semidirect
69. #4720C22semidirect
70. #4719C22semidirect
71. #4709C22semidirect
72. #4708C22semidirect
74. #4701C22semidirect
79. #2980D3semidirect
216. #435C2×D3semidirect
217. #435C2×D3semidirect
227. #432C2×D3semidirect
237. #408C2×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×#29802
#832752
#794902
#794782
#794672
#794542
#793422
#793252
#786682
#786662
#786572
#786552
#603073