Finite Group Databasebeta

← #10855217

Subgroups — #10855217

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83176C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×#4756C211yesmaximal, characteristic, complemented
4#76203C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6#70208C211yesmaximal, characteristic, complemented
7#70208C211yesmaximal, characteristic, complemented
8#70208C211yesmaximal, characteristic, complemented
9#70185C211yesmaximal, characteristic, complemented
10#70182C211yesmaximal, characteristic, complemented
11#70170C211yesmaximal, characteristic, complemented
12#70166C211yesmaximal, characteristic, complemented
13#69623C211yesmaximal, characteristic, complemented
14#68671C211yesmaximal, characteristic, complemented
15#68611C211yesmaximal, characteristic, complemented
16#26162—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#935—21no—
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#893C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C3×#389C2211yescharacteristic
30#5222—21no—
31#5222—21no—
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5220—21no—
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42#5072—24no—
43#4989—21no—
44#4987—21no—
45#4984C2211yescharacteristic
46#4977C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C4×Dic3⋊C4C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4756—24no—
52#4668—21no—
53D3×#95—21no—
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#459—31no—
63C2×C4×D8—31no—
64#2445—31no—
65#2349—31no—
66#2092—31no—
67#2087—31no—
68#2073—31no—
69#2041D311yescharacteristic, complemented
70#1934—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1879—31no—
75#1864—31no—
76#1864—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#127—22no—
89C3×#127—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#935—44no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104C2×#213—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213D414yes—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#893—22no—
125#893—22no—
126#893—22no—
127#893—41no—
128C4×D12—22no—
129C4×D12—41no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#102—22no—
133C3×#102—41no—
134C3×#95—21no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#821—22no—
144#821—22no—
145#817—41no—
146C8×Dic3—22no—
147#807—22no—
148C4⋊(C3⋊C8)—22no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127C2×D311yescharacteristic
152C2×#123—61no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#459—64no—
158C4×D8—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389C2×D311yescharacteristic
179#386—61no—
180#385—61no—
181#385—61no—
182#382—31no—
183#380—61no—
184C4×C4⋊C4—31no—
185C2×#95—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C24×C3—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—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4—81no—
213C2×C3×D4—81no—
214C2×C3×D4C2×D412yes—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—81no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12C2×D412yes—
234C2×D12#4112yes—
235C2×D12#4112yes—
236C2×C4×D3—24no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×C4×D3—81no—
242C2×C4×D3—81no—
243C2×C4×D3—81no—
244C2×C4×D3—81no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—81no—
255C3×C4⋊C4#4112yes—
256C3×C42—21no—
257C3×C42—21no—
258#217—42no—
259#217—42no—
260#217—42no—
261#217—42no—
262#217—81no—
263#217—81no—
264#217—81no—
265#213—28no—
266#213—44no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—81no—
277Dic3⋊C4—22no—
278Dic3⋊C4—22no—
279Dic3⋊C4—42no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8C2×D412yes—
290C22×D4—21no—
291C22×D4—21no—
292C22×D4—31no—
293C22×D4C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—61no—
296C23×C4—61no—
297C23×C4—61no—
298C23×C4—61no—
299C2×D8—34no—
300C2×D8—62no—
301C2×D8—62no—
302C2×D8—62no—
303C2×D8—62no—
304C2×D8—62no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#127—22no—
310#127—41no—
311#123—124no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—121no—
318C4×D4—121no—
319C4×D4—121no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
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×C42—31no—
341C2×C42—31no—
342C2×C42C22×D311yescharacteristic
343#107—62no—
344#106—121no—
345C4⋊C8—62no—
346#102—22no—
347#102—22no—
348#102—22no—
349#102—41no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354C4×C8—62no—
355#95—21no—
356#95—31no—
357#95—31no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—22no—
363C23×C3—41no—
364C23×C3—41no—
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×C3C22×D411yescharacteristic
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—84no—
393C3×D4—84no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
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#13512yes—
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
412C2×Dic3—21no—
413C2×Dic3—21no—
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—21no—
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—41no—
432C2×Dic3—84no—
433C2×Dic3AGL(1,ℤ/8)12yes—
434D12—22no—
435D12—22no—
436D12—22no—
437D12#12314yescomplemented
438C4×D3—84no—
439C4×D3—84no—
440C4×D3—84no—
441C3×C8—41no—
442C3×C8—41no—
443C3⋊C8—22no—
444C3⋊C8—41no—
445C24—21no—
446C24—21no—
447C24—61no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
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—32no—
460C2×D4—32no—
461C2×D4—32no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—81no—
470C2×D4—81no—
471C2×D4D3×D412yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—64no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—241no—
489C22×C4—241no—
490C22×C4—241no—
491C22×C4—241no—
492C22×C4—241no—
493C22×C4D3×D411yescharacteristic
494C22×C4D3×D411yescharacteristic
495C22×C4C23×D311yescharacteristic
496D8—68no—
497D8—124no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—32no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—62no—
514C4⋊C4—62no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—81no—
522C4⋊C4—122no—
523C4⋊C4—241no—
524C4⋊C4#23712yes—
525C22⋊C4—64no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—241no—
533C22⋊C4—241no—
534C22⋊C4—241no—
535C42—21no—
536C42—21no—
537C42—61no—
538C42—61no—
539C42—61no—
540C42—61no—
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×C3#52511yescharacteristic
556C22×C3D4211yescharacteristic
557C22×C3#54811yescharacteristic
558C22×C3C2×#13511yescharacteristic
559C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
560C22×C3#57611yescharacteristic
561C22×C3#57611yescharacteristic
562C2×D3—24no—
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—42no—
567C2×D3—42no—
568C2×D3—42no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—41no—
572C3×C4—81no—
573C3×C4C2×#12311yescharacteristic
574C3×C4#53911yescharacteristic
575C3×C4#54711yescharacteristic
576C3×C4#55311yescharacteristic
577Dic3—22no—
578Dic3—41no—
579Dic3—41no—
580Dic3—41no—
581Dic3—81no—
582Dic3—81no—
583Dic3—81no—
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—62no—
599C23—62no—
600C23—62no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23C2×D3×D411yescharacteristic
606D4—22no—
607D4—22no—
608D4—34no—
609D4—41no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—62no—
616D4—62no—
617D4—62no—
618D4—84no—
619D4—84no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—32no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—61no—
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—61no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—244no—
653C2×C4—244no—
654C2×C4—244no—
655C2×C4—244no—
656C2×C4#91612yes—
657C2×C4C2×D3×D411yescharacteristic
658C2×C4C2×#23711yescharacteristic
659C2×C4#100411yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C2×C4D3×#4111yescharacteristic
662C2×C4#101011yescharacteristic
663C8—41no—
664C8—41no—
665C8—62no—
666C8—121no—
667C2×C3—82no—
668C2×C3—161no—
669C2×C3#244311yescharacteristic
670C2×C3#310611yescharacteristic
671C2×C3#312211yescharacteristic
672C2×C3#326411yescharacteristic
673C2×C3#328011yescharacteristic
674C2×C3#336111yescharacteristic
675C2×C3#336611yescharacteristic
676D3—44no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—64no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—122no—
693C22—122no—
694C22—122no—
695C22—122no—
696C22—122no—
697C22—122no—
698C22#559011yescharacteristic
699C22#560611yescharacteristic
700C22#567711yescharacteristic
701C22#575711yescharacteristic
702C22C2×#91611yescharacteristic
703C22D3×AGL(1,ℤ/8)11yescharacteristic
704C22#577411yescharacteristic
705C4—22no—
706C4—41no—
707C4—42no—
708C4—62no—
709C4—81no—
710C4—121no—
711C4—121no—
712C4—121no—
713C4—241no—
714C4—241no—
715C4—241no—
716C4#561111yescharacteristic
717C4#561411yescharacteristic
718C4D3×#12311yescharacteristic
719C4#566011yescharacteristic
720C3#2616211yescharacteristic, complemented, Sylow
721C2—82no—
722C2—124no—
723C2—161no—
724C2#7711711yescharacteristic
725C2#8001211yescharacteristic
726C2#8016811yescharacteristic
727C2#8031711yescharacteristic
728C2#8036111yescharacteristic
729C2#8158511yescharacteristic
730C2#8170811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenter#2041D3
hypocenterC3#26162
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
1. #83176C2semidirect
3. C2×#4756C2semidirect
4. #76203C2semidirect
5. C3×#2041C2semidirect
6. #70208C2semidirect
7. #70208C2semidirect
8. #70208C2semidirect
9. #70185C2semidirect
10. #70182C2semidirect
11. #70170C2semidirect
12. #70166C2semidirect
13. #69623C2semidirect
14. #68671C2semidirect
15. #68611C2semidirect
69. #2041D3semidirect
437. D12#123semidirect
720. C3#26162semidirect

Maximal subgroups

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

subgroupindex
#831762
C2×#50722
C2×#47562
#762032
C3×#20412
#702082
#701852
#701822
#701702
#701662
#696232
#686712
#686112
#261623