Finite Group Databasebeta

← #10844631

Subgroups — #10844631

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4871C211yesmaximal, characteristic, complemented
2C2×#4764C211yesmaximal, characteristic, complemented
3#76348C211yesmaximal, characteristic
4#75989C211yesmaximal, characteristic, complemented
5C3×#2065C211yesmaximal, characteristic, complemented
6#70569C211yesmaximal, characteristic, complemented
7#70457C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70211C211yesmaximal, characteristic, complemented
10#70161C211yesmaximal, characteristic, complemented
11#70161C211yesmaximal, characteristic, complemented
12#69636C211yesmaximal, characteristic, complemented
13#69627C211yesmaximal, characteristic, complemented
14#68901C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#25288—31nomaximal, Sylow
17C2×#940C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#890C2211yescharacteristic
21C2×#884—21no—
22C2×#884—21no—
23C2×#874C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4988C2211yescharacteristic
40#4978—21no—
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43#4972—21no—
44#4972C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4960—21no—
51#4960—21no—
52C4×C4⋊Dic3C2211yescharacteristic
53C4×Dic3⋊C4C2211yescharacteristic
54#4871—24no—
55#4764—24no—
56#4653—21no—
57#4649—21no—
58C3×#340C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#478—31no—
64#2541—31no—
65#2370—31no—
66#2067—31no—
67#2065D311yescharacteristic, complemented
68#2041—31no—
69#2041—31no—
70#1982—31no—
71#1982—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C22×#67—21no—
80C22×#67—21no—
81C22×#67—21no—
82C22×D12—21no—
83C22×D12—21no—
84C22×D12C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96#940—22no—
97#940—22no—
98#940—22no—
99#940—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#893—22no—
118#893—22no—
119#890—22no—
120#890D414yescomplemented
121#884—44no—
122#884—44no—
123#874—22no—
124#874—41no—
125C3×#106—22no—
126C3×C4⋊C8—41no—
127C3×#102—22no—
128C3×#102—41no—
129C3×#95—21no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#821—22no—
139#821—22no—
140#821—22no—
141#821—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—41no—
145#809—22no—
146#809—22no—
147#809—22no—
148#809—41no—
149#807—22no—
150C2×#127—31no—
151C2×#127—31no—
152C2×#127—31no—
153C2×#125—31no—
154C2×#121—61no—
155C2×#121—61no—
156C2×C4×D4—31no—
157#490—64no—
158#478—64no—
159C2×#107—31no—
160C2×#106C2×D311yescharacteristic
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×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102C2×D311yescharacteristic
175C2×#102C2×D311yescharacteristic
176C2×#97—31no—
177#392—61no—
178#392—61no—
179#390—31no—
180#390C2×D311yescharacteristic
181#389—31no—
182#389—61no—
183#389C2×D311yescharacteristic
184#387—61no—
185#387—61no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—31no—
190C4×C4⋊C4—61no—
191#340—31no—
192#340C2×D311yescharacteristic
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C23×D3—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—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—42no—
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—81no—
219C2×#67—81no—
220C2×#67—81no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—42no—
240C2×D12—42no—
241C2×D12—42no—
242C2×D12—42no—
243C2×D12—81no—
244C2×D12—81no—
245C2×D12#4112yes—
246C2×C4×D3—42no—
247C2×C4×D3—42no—
248C2×C4×D3—42no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—81no—
261C3×C4⋊C4C2×D412yes—
262C3×C22⋊C4—42no—
263C3×C22⋊C4—42no—
264C3×C42—21no—
265C3×C42—21no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—24no—
272C4⋊Dic3C2×D412yes—
273C4⋊Dic3#4112yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—42no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C4×Dic3—81no—
282C4×Dic3—81no—
283C4×Dic3C2×D412yes—
284C4×Dic3#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C22×D4—31no—
292C22×D4—31no—
293C22×D4—61no—
294C22×D4—61no—
295C22×D4—61no—
296C22×D4—61no—
297C22×D4—61no—
298C22×D4—61no—
299C22×D4C22×D311yescharacteristic
300C23×C4—61no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—62no—
306#127—62no—
307#127—62no—
308#127—62no—
309#127—121no—
310#127—121no—
311#125—34no—
312#125—62no—
313#121—124no—
314#121—124no—
315C4×D4—62no—
316C4×D4—62no—
317C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—22no—
341C4⋊C8—41no—
342C4⋊C8—62no—
343C4⋊C8—121no—
344#102—22no—
345#102—41no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#102—62no—
355#102—121no—
356#97—62no—
357#95—21no—
358#95—31no—
359#95—31no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3C22×D411yescharacteristic
372C22×D3—22no—
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—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4AGL(1,ℤ/8)12yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411#67—84no—
412#67—84no—
413#67—84no—
414#67—84no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3#12112yes—
431C2×Dic3#12112yes—
432C2×Dic3AGL(1,ℤ/8)12yes—
433D12—22no—
434D12—22no—
435D12—41no—
436D12—41no—
437D12—41no—
438D12—41no—
439D12—41no—
440D12—41no—
441D12—84no—
442D12—84no—
443C4×D3—84no—
444C3×C8—41no—
445C3×C8—41no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C24—21no—
449C24—61no—
450C24—61no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—32no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—122no—
470C2×D4—122no—
471C2×D4—122no—
472C2×D4—122no—
473C2×D4—122no—
474C2×D4—122no—
475C2×D4—122no—
476C2×D4—122no—
477C2×D4—122no—
478C2×D4—122no—
479C2×D4—122no—
480C2×D4—122no—
481C2×D4—241no—
482C2×D4—241no—
483C2×D4—241no—
484C2×D4—241no—
485C2×D4—241no—
486C2×D4—241no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—122no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4D3×D411yescharacteristic
500C22×C4D3×D411yescharacteristic
501C22×C4C23×D311yescharacteristic
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—32no—
520C4⋊C4—32no—
521C4⋊C4—62no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—62no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—81no—
528C4⋊C4—122no—
529C4⋊C4D3×D412yes—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—21no—
536C42—21no—
537C42—32no—
538C42—32no—
539C42—61no—
540C42—61no—
541C42—122no—
542C42—241no—
543C42—241no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3C2×#12111yescharacteristic
552C22×C3#54611yescharacteristic
553C22×C3#54811yescharacteristic
554C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
555C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
556C22×C3#57611yescharacteristic
557C22×C3#57711yescharacteristic
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C2×D3—81no—
572C3×C4—22no—
573C3×C4—42no—
574C3×C4—41no—
575C3×C4—81no—
576C3×C4—81no—
577C3×C4#52911yescharacteristic
578C3×C4#54211yescharacteristic
579C3×C4#54811yescharacteristic
580C3×C4#55011yescharacteristic
581Dic3—22no—
582Dic3—22no—
583Dic3—22no—
584Dic3—42no—
585Dic3—42no—
586Dic3—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—62no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23—121no—
607C23—121no—
608C23C2×D3×D411yescharacteristic
609D4—41no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—62no—
614D4—62no—
615D4—121no—
616D4—121no—
617D4—121no—
618D4—121no—
619D4—121no—
620D4—121no—
621D4—244no—
622D4—244no—
623D4—244no—
624D4—244no—
625D4—244no—
626D4—244no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—32no—
634C2×C4—32no—
635C2×C4—32no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—62no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—62no—
654C2×C4—61no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—244no—
660C2×C4#91412yes—
661C2×C4C2×D3×D411yescharacteristic
662C2×C4#100411yescharacteristic
663C2×C4#100711yescharacteristic
664C2×C4D3×#4111yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4D3×#4111yescharacteristic
667C8—41no—
668C8—41no—
669C8—121no—
670C8—121no—
671C2×C3—161no—
672C2×C3#242611yescharacteristic
673C2×C3#303811yescharacteristic
674C2×C3#304511yescharacteristic
675C2×C3#331011yescharacteristic
676C2×C3#331811yescharacteristic
677C2×C3#336611yescharacteristic
678C2×C3#336811yescharacteristic
679D3—82no—
680D3—161no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—122no—
689C22—122no—
690C22—122no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—241no—
701C22—241no—
702C22D3×#12111yescharacteristic
703C22#561011yescharacteristic
704C22#563311yescharacteristic
705C22C2×#91411yescharacteristic
706C22#577011yescharacteristic
707C22D3×AGL(1,ℤ/8)11yescharacteristic
708C22#577411yescharacteristic
709C4—22no—
710C4—42no—
711C4—41no—
712C4—62no—
713C4—62no—
714C4—62no—
715C4—81no—
716C4—81no—
717C4—122no—
718C4—121no—
719C4—122no—
720C4#559911yescharacteristic
721C4#563311yescharacteristic
722C4#564011yescharacteristic
723C4#570611yescharacteristic
724C3#2528811yescharacteristic, complemented, Sylow
725C2—161no—
726C2—242no—
727C2—481no—
728C2#7728611yescharacteristic
729C2#8038411yescharacteristic
730C2#8053811yescharacteristic
731C2#8068511yescharacteristic
732C2#8087611yescharacteristic
733C2#8108811yescharacteristic
734C2#8115311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2065C2
socleC23×C3C22×D4
prime radical#76348C2
hypercenter#2065D3
hypocenterC3#25288
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. C2×#4871C2semidirect
2. C2×#4764C2semidirect
4. #75989C2semidirect
5. C3×#2065C2semidirect
6. #70569C2semidirect
7. #70457C2semidirect
8. #70452C2semidirect
9. #70211C2semidirect
10. #70161C2semidirect
11. #70161C2semidirect
12. #69636C2semidirect
13. #69627C2semidirect
14. #68901C2semidirect
15. #68782C2semidirect
67. #2065D3semidirect
120. #890D4semidirect
724. C3#25288semidirect

Maximal subgroups

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

subgroupindex
C2×#48712
C2×#47642
#763482
#759892
C3×#20652
#705692
#704572
#704522
#702112
#701612
#696362
#696272
#689012
#687822
#252883