Finite Group Databasebeta

← #10844843

Subgroups — #10844843

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4786C211yesmaximal, characteristic
2C2×#4701C211yesmaximal, characteristic, complemented
3#76353C211yesmaximal, characteristic, complemented
4#76075C211yesmaximal, characteristic, complemented
5C3×#1936C211yesmaximal, characteristic, complemented
6#70190C211yesmaximal, characteristic, complemented
7#70179C211yesmaximal, characteristic, complemented
8#70160C211yesmaximal, characteristic, complemented
9#69825C211yesmaximal, characteristic, complemented
10#69650C211yesmaximal, characteristic, complemented
11#69630C211yesmaximal, characteristic, complemented
12#68862C211yesmaximal, characteristic, complemented
13#68852C211yesmaximal, characteristic, complemented
14#68797C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#25304—31nomaximal, Sylow
17C2×#940C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#885—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4993—21no—
38#4989—21no—
39#4988C2211yescharacteristic
40#4988C2211yescharacteristic
41#4977C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4961—21no—
49#4959—21no—
50#4957—21no—
51#4939C2211yescharacteristic
52#4939C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4786—24no—
55#4701—24no—
56#4673—21no—
57#4668—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#459—31no—
63C2×#446—31no—
64#2580—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2054—31no—
69#2054—31no—
70#1972—31no—
71#1970—31no—
72#1961—31no—
73#1949—31no—
74#1936D311yescharacteristic, complemented
75#1934—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6D411yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#940—22no—
96#940—22no—
97C2×#213—22no—
98C2×#213—22no—
99C2×#213—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D24—22no—
121C2×D24—22no—
122C2×D24—41no—
123#893—22no—
124#893—22no—
125#885—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#97—22no—
131C3×#95—21no—
132#832—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138#821—22no—
139#817—41no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144#809—22no—
145#809—22no—
146#809—41no—
147#807—41no—
148C4⋊(C3⋊C8)—22no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#459—64no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97C2×D311yescharacteristic
172C2×C4×C8—31no—
173#399—61no—
174#394—61no—
175#394—61no—
176#393—61no—
177#390—31no—
178#390—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—61no—
183#390C2×D311yescharacteristic
184#389—31no—
185#389C2×D311yescharacteristic
186#385—61no—
187#385—61no—
188#382—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C23×D3—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—81no—
214C2×#67—81no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12#4112yes—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—41no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—81no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256#217—42no—
257#217—81no—
258#213—44no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268C4⋊Dic3—22no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—81no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—42no—
279C4×Dic3C2×D412yes—
280C4×Dic3#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
287D24—44no—
288C22×Q8—31no—
289C22×D4—31no—
290C22×D4—31no—
291C22×D4—61no—
292C22×D4—61no—
293C22×D4C22×D311yescharacteristic
294C23×C4—61no—
295C2×D8—62no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—62no—
299C2×D8—121no—
300C2×D8—121no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—62no—
306#127—62no—
307#124—124no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—62no—
311C4×D4—62no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—121no—
340#106—121no—
341C4⋊C8—34no—
342C4⋊C8—62no—
343C4⋊C8—62no—
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—121no—
351#102—121no—
352#97—22no—
353#97—62no—
354C4×C8—62no—
355#95—21no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—84no—
408#67—84no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3#12112yes—
425C2×Dic3#12112yes—
426C2×Dic3AGL(1,ℤ/8)12yes—
427D12—22no—
428D12—22no—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—41no—
435C4×D3—84no—
436Dic6—44no—
437C3×C8—22no—
438C3×C8—22no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24—21no—
442C24—61no—
443C24—61no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—121no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—32no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—122no—
469C2×D4—241no—
470C2×D4—241no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—124no—
487D8—124no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—32no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8#23512yes—
499C2×C8D3×D412yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—81no—
514C4⋊C4—241no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—32no—
531C42—32no—
532C42—61no—
533C42—61no—
534C42—122no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3C2×#12111yescharacteristic
543C22×C3#54811yescharacteristic
544C22×C3#55011yescharacteristic
545C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
546C22×C3#57511yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57611yescharacteristic
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4#52911yescharacteristic
569C3×C4#54211yescharacteristic
570C3×C4#54611yescharacteristic
571C3×C4#54811yescharacteristic
572Dic3—22no—
573Dic3—22no—
574Dic3—22no—
575Dic3—41no—
576Dic3—81no—
577Dic3—81no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—62no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23C2×D3×D411yescharacteristic
600Q8—124no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—62no—
606D4—62no—
607D4—121no—
608D4—121no—
609D4—121no—
610D4—121no—
611D4—121no—
612D4—121no—
613D4—244no—
614D4—244no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—32no—
622C2×C4—32no—
623C2×C4—32no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—244no—
649C2×C4C2×#23511yescharacteristic
650C2×C4C2×D3×D411yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4#100911yescharacteristic
654C2×C4#101011yescharacteristic
655C8—22no—
656C8—22no—
657C8—62no—
658C8—121no—
659C2×C3—161no—
660C2×C3#242611yescharacteristic
661C2×C3#305511yescharacteristic
662C2×C3#305811yescharacteristic
663C2×C3#336611yescharacteristic
664C2×C3#337011yescharacteristic
665C2×C3#339911yescharacteristic
666C2×C3#340311yescharacteristic
667D3—82no—
668D3—161no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—122no—
677C22—122no—
678C22—122no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22#555311yescharacteristic
691C22D3×#12111yescharacteristic
692C22#567711yescharacteristic
693C22D3×AGL(1,ℤ/8)11yescharacteristic
694C22#577411yescharacteristic
695C22#577411yescharacteristic
696C22#577711yescharacteristic
697C4—41no—
698C4—41no—
699C4—62no—
700C4—62no—
701C4—62no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—121no—
706C4—241no—
707C4—241no—
708C4#552811yescharacteristic
709C4#555211yescharacteristic
710C4#565411yescharacteristic
711C4#571011yescharacteristic
712C3#2530411yescharacteristic, complemented, Sylow
713C2—161no—
714C2—242no—
715C2—481no—
716C2#7549911yescharacteristic
717C2#7976611yescharacteristic
718C2#7985711yescharacteristic
719C2#8080511yescharacteristic
720C2#8086511yescharacteristic
721C2#8167211yescharacteristic
722C2#8170811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1936C2
socleC23×C3C22×D4
prime radicalC2×#4786C2
hypercenter#1936D3
hypocenterC3#25304
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. C2×#4701C2semidirect
3. #76353C2semidirect
4. #76075C2semidirect
5. C3×#1936C2semidirect
6. #70190C2semidirect
7. #70179C2semidirect
8. #70160C2semidirect
9. #69825C2semidirect
10. #69650C2semidirect
11. #69630C2semidirect
12. #68862C2semidirect
13. #68852C2semidirect
14. #68797C2semidirect
15. #68782C2semidirect
74. #1936D3semidirect
712. C3#25304semidirect

Maximal subgroups

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

subgroupindex
C2×#47862
C2×#47012
#763532
#760752
C3×#19362
#701902
#701792
#701602
#698252
#696502
#696302
#688622
#688522
#687972
#687822
#253043