Finite Group Databasebeta

← #10844785

Subgroups — #10844785

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5150C211yesmaximal, characteristic
2C2×#4828C211yesmaximal, characteristic, complemented
3#76353C211yesmaximal, characteristic, complemented
4#76346C211yesmaximal, characteristic, complemented
5C3×#2054C211yesmaximal, characteristic, complemented
6#72039C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70212C211yesmaximal, characteristic, complemented
9#70193C211yesmaximal, characteristic, complemented
10#70157C211yesmaximal, characteristic, complemented
11Dic3×#102C211yesmaximal, characteristic, complemented
12#69648C211yesmaximal, characteristic, complemented
13#69645C211yesmaximal, characteristic, complemented
14#68857C211yesmaximal, characteristic, complemented
15#68796C211yesmaximal, characteristic, complemented
16#25300—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×#885—21no—
22C2×#878—21no—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5150—24no—
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#815C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42#4988C2211yescharacteristic
43#4984C2211yescharacteristic
44#4975C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4961—21no—
49#4959—21no—
50#4957—21no—
51#4948—21no—
52#4948—21no—
53#4942—21no—
54#4939C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×#803C2211yescharacteristic
57#4828—24no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#459—31no—
63C2×#446—31no—
64#2548—31no—
65#2538—31no—
66#2065—31no—
67#2065—31no—
68#2054—31no—
69#2054D311yescharacteristic, complemented
70#1972—31no—
71#1972—31no—
72#1957—31no—
73#1948—31no—
74#1936—31no—
75#1936—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C23×Dic3—21no—
82C22×D12C2311yescharacteristic
83C22×Dic6D411yescharacteristic
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#940—22no—
102#940—22no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—41no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4D411yescharacteristic
121C2×Dic3⋊C4D411yescharacteristic
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#885—44no—
128#878—44no—
129C4×Dic6—22no—
130C4×Dic6—22no—
131C3×C4⋊C8—22no—
132C3×#102—22no—
133C3×#102—41no—
134#832—22no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—41no—
142#817—41no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8D414yes—
146C8×Dic3—22no—
147#809—22no—
148#808—41no—
149#803—22no—
150C22×D8—31no—
151C22×D8C2×D311yescharacteristic
152C2×#127—31no—
153C2×#124—61no—
154C2×C22⋊Q8—61no—
155C2×C4×Q8—31no—
156C2×C4×D4—31no—
157#459—64no—
158#446—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173C2×#97—31no—
174C2×C4×C8—31no—
175#396—61no—
176#396—61no—
177#394—61no—
178#394—61no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#390C2×D311yescharacteristic
185#390C2×D311yescharacteristic
186#389—31no—
187#389—31no—
188#384—31no—
189#380—61no—
190C4×C4⋊C4—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195C24×C3—21no—
196C24×C3—21no—
197C23×D3—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4#4112yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—81no—
220C2×#67—81no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3—42no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—22no—
240C3×D8—44no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8C2×D412yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C22⋊C4—81no—
255C3×C42—21no—
256C3×C42—21no—
257#217—42no—
258#217—42no—
259#217—42no—
260#217—81no—
261#217—81no—
262#213—44no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—42no—
275Dic3⋊C4—22no—
276Dic3⋊C4—22no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—42no—
283C4×Dic3C2×D412yes—
284C4×Dic3#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8C2×D412yes—
290C2×C3⋊C8#4112yes—
291C22×Q8—31no—
292C22×D4—31no—
293C22×D4—61no—
294C22×D4—61no—
295C22×D4C22×D311yescharacteristic
296C22×D4C22×D311yescharacteristic
297C23×C4—61no—
298C2×D8—22no—
299C2×D8—22no—
300C2×D8—41no—
301C2×D8—62no—
302C2×D8—62no—
303C2×D8—121no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308#127—62no—
309#127—62no—
310#124—124no—
311C22⋊Q8—124no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×D4—62no—
315C4×D4—62no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345#106—121no—
346#106—121no—
347C4⋊C8—22no—
348C4⋊C8—34no—
349C4⋊C8—62no—
350#102—22no—
351#102—41no—
352#102—62no—
353#102—62no—
354#102—62no—
355#102—62no—
356#102—62no—
357#102—121no—
358#97—62no—
359#97—62no—
360C4×C8—62no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—22no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3C22×D411yescharacteristic
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—41no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412#67—84no—
413#67—84no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—84no—
432C2×Dic3#12112yes—
433C2×Dic3#12112yes—
434C2×Dic3AGL(1,ℤ/8)12yes—
435C2×Dic3AGL(1,ℤ/8)12yes—
436D12—41no—
437D12—41no—
438D12—41no—
439D12—41no—
440Dic6—44no—
441C3×C8—22no—
442C3×C8—41no—
443C3⋊C8—22no—
444C3⋊C8—22no—
445C24—21no—
446C24—21no—
447C24—61no—
448C2×Q8—62no—
449C2×Q8—62no—
450C2×Q8—62no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—122no—
469C2×D4—122no—
470C2×D4—122no—
471C2×D4—122no—
472C2×D4—241no—
473C2×D4—241no—
474C2×D4#23712yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490D8—44no—
491D8—124no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—32no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8D3×D412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4—241no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—241no—
531C22⋊C4—241no—
532C22⋊C4—241no—
533C22⋊C4—241no—
534C42—21no—
535C42—21no—
536C42—32no—
537C42—32no—
538C42—61no—
539C42—61no—
540C42—122no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3—81no—
552C22×C3—81no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3C2×#12111yescharacteristic
556C22×C3#54811yescharacteristic
557C22×C3#55011yescharacteristic
558C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
559C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
560C22×C3#57611yescharacteristic
561C22×C3#57711yescharacteristic
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C3×C4—41no—
570C3×C4—41no—
571C3×C4—81no—
572C3×C4—81no—
573C3×C4#52911yescharacteristic
574C3×C4#54211yescharacteristic
575C3×C4#54611yescharacteristic
576C3×C4#54811yescharacteristic
577Dic3—22no—
578Dic3—41no—
579Dic3—41no—
580Dic3—81no—
581Dic3—81no—
582Dic3—81no—
583Dic3#46914yes—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23C2×D3×D411yescharacteristic
606Q8—124no—
607D4—22no—
608D4—22no—
609D4—41no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—121no—
616D4—121no—
617D4—121no—
618D4—121no—
619D4—244no—
620D4—244no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—32no—
628C2×C4—32no—
629C2×C4—32no—
630C2×C4—32no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—244no—
655C2×C4C2×D3×D411yescharacteristic
656C2×C4C2×#23711yescharacteristic
657C2×C4#100411yescharacteristic
658C2×C4D3×#4111yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4#101011yescharacteristic
661C8—22no—
662C8—41no—
663C8—62no—
664C8—62no—
665C2×C3—82no—
666C2×C3—161no—
667C2×C3#242611yescharacteristic
668C2×C3C2×#46911yescharacteristic
669C2×C3#306211yescharacteristic
670C2×C3#336611yescharacteristic
671C2×C3#337311yescharacteristic
672C2×C3#339711yescharacteristic
673C2×C3#339911yescharacteristic
674D3—161no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22#559811yescharacteristic
697C22#559911yescharacteristic
698C22D3×#12111yescharacteristic
699C22#575711yescharacteristic
700C22#577011yescharacteristic
701C22D3×AGL(1,ℤ/8)11yescharacteristic
702C22D3×AGL(1,ℤ/8)11yescharacteristic
703C4—34no—
704C4—41no—
705C4—41no—
706C4—62no—
707C4—81no—
708C4—81no—
709C4—121no—
710C4—121no—
711C4—241no—
712C4—241no—
713C4—241no—
714C4#559111yescharacteristic
715C4#564911yescharacteristic
716C4#566211yescharacteristic
717C4#567711yescharacteristic
718C3#2530011yescharacteristic, complemented, Sylow
719C2—82no—
720C2—161no—
721C2—481no—
722C2#7690111yescharacteristic
723C2#8032111yescharacteristic
724C2#8032811yescharacteristic
725C2#8052111yescharacteristic
726C2#8052611yescharacteristic
727C2#8071811yescharacteristic
728C2D3×#46911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2054C2
socleC23×C3C22×D4
prime radicalC2×#5150C2
hypercenter#2054D3
hypocenterC3#25300
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
2. C2×#4828C2semidirect
3. #76353C2semidirect
4. #76346C2semidirect
5. C3×#2054C2semidirect
6. #72039C2semidirect
7. #70483C2semidirect
8. #70212C2semidirect
9. #70193C2semidirect
10. #70157C2semidirect
11. Dic3×#102C2semidirect
12. #69648C2semidirect
13. #69645C2semidirect
14. #68857C2semidirect
15. #68796C2semidirect
69. #2054D3semidirect
718. C3#25300semidirect

Maximal subgroups

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

subgroupindex
C2×#51502
C2×#48282
#763532
#763462
C3×#20542
#720392
#704832
#702122
#701932
#701572
Dic3×#1022
#696482
#696452
#688572
#687962
#253003