Finite Group Databasebeta

← #10844633

Subgroups — #10844633

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
1#83177C211yesmaximal, characteristic
2C2×#4764C211yesmaximal, characteristic, complemented
3C2×#4763C211yesmaximal, characteristic, complemented
4#73661C211yesmaximal, characteristic, complemented
5C3×#1982C211yesmaximal, characteristic, complemented
6#71762C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic, complemented
10#70177C211yesmaximal, characteristic, complemented
11#70161C211yesmaximal, characteristic, complemented
12#70161C211yesmaximal, characteristic, complemented
13#70156C211yesmaximal, characteristic, complemented
14#70152C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic, complemented
16#25288—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#940C2211yescharacteristic
20C2×#939—21no—
21C2×#939—21no—
22C2×D4×Dic3C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30#5223—21no—
31#5223—21no—
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#815C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4939C2211yescharacteristic
50C4×C4⋊Dic3—21no—
51#4933—21no—
52C4×Dic3⋊C4C2211yescharacteristic
53#4928C2211yescharacteristic
54#4764—24no—
55#4763—24no—
56#4654—21no—
57#4647—21no—
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#478—31no—
64#2541—31no—
65#2370—31no—
66#2067—31no—
67#2065—31no—
68#2041—31no—
69#2041—31no—
70#1982—31no—
71#1982—31no—
72#1982D311yescharacteristic, complemented
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—21no—
83C22×#67—21no—
84C22×#67—21no—
85C23×Dic3—21no—
86C22×D12C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#127—22no—
90C3×#127—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#940—22no—
99#940—22no—
100#940—22no—
101#940—41no—
102#939—44no—
103#939—44no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#890—22no—
124#890D414yescomplemented
125C3×C4⋊C8—41no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#95—21no—
131#832—22no—
132#832—22no—
133#832—22no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—22no—
142#821—41no—
143#817—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—41no—
147#809—22no—
148#809—22no—
149#807—22no—
150C2×#127—31no—
151C2×#127—31no—
152C2×#127C2×D311yescharacteristic
153C2×#125—31no—
154C2×#121—61no—
155C2×#121—61no—
156C2×C4×D4—31no—
157#490—64no—
158#478—64no—
159C2×#107—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×#102—31no—
171C2×#102—31no—
172C2×#102C2×D311yescharacteristic
173C2×#102C2×D311yescharacteristic
174C2×#102C2×D311yescharacteristic
175C2×#102C2×D311yescharacteristic
176C2×#97—31no—
177#392—61no—
178#392—61no—
179#390—31no—
180#390—31no—
181#389—31no—
182#389—31no—
183#389—61no—
184#387—61no—
185#387—61no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C4×C4⋊C4C2×D311yescharacteristic
191#340—31no—
192#340—31no—
193#336—31no—
194#336—31no—
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—42no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4—42no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4#4112yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C2×#67—42no—
222C2×#67—42no—
223C2×#67—42no—
224C2×#67—42no—
225C2×#67—42no—
226C2×#67—42no—
227C2×#67—42no—
228C2×#67—42no—
229C2×#67—81no—
230C2×#67—81no—
231C2×#67—81no—
232C2×#67—81no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C22×Dic3—21no—
236C22×Dic3—42no—
237C22×Dic3—42no—
238C22×Dic3—42no—
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
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⋊C4C2×D412yes—
258C3×C4⋊C4#4112yes—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262#217—42no—
263#217—42no—
264#217—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267C4⋊Dic3—21no—
268C4⋊Dic3—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—42no—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—22no—
276Dic3⋊C4—24no—
277Dic3⋊C4—22no—
278Dic3⋊C4—81no—
279C4×Dic3—21no—
280C4×Dic3—21no—
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—21no—
292C22×D4—21no—
293C22×D4—31no—
294C22×D4—61no—
295C22×D4—61no—
296C22×D4—61no—
297C22×D4—61no—
298C22×D4C22×D311yescharacteristic
299C22×D4C22×D311yescharacteristic
300C23×C4—61no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—22no—
306#127—41no—
307#127—62no—
308#127—62no—
309#127—62no—
310#127—121no—
311#125—34no—
312#125—62no—
313#121—124no—
314#121—124no—
315C4×D4—62no—
316C4×D4—62no—
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—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—41no—
342C4⋊C8—62no—
343C4⋊C8—121no—
344#102—22no—
345#102—22no—
346#102—22no—
347#102—22no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#102—121no—
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—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—
394C3×D4—84no—
395C3×D4—84no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4AGL(1,ℤ/8)12yes—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414#67—84no—
415#67—84no—
416#67—84no—
417#67—84no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—22no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—84no—
437C2×Dic3#12112yes—
438C2×Dic3#12112yes—
439C2×Dic3AGL(1,ℤ/8)12yes—
440D12—41no—
441D12—41no—
442D12—41no—
443D12—41no—
444C3×C8—41no—
445C3×C8—41no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C24—21no—
449C24—21no—
450C24—61no—
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—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—61no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—81no—
473C2×D4—81no—
474C2×D4—122no—
475C2×D4—122no—
476C2×D4—122no—
477C2×D4—122no—
478C2×D4—122no—
479C2×D4—122no—
480C2×D4—122no—
481C2×D4—122no—
482C2×D4—241no—
483C2×D4—241no—
484C2×D4—241no—
485C2×D4—241no—
486C2×D4#23512yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—61no—
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—32no—
517C4⋊C4—62no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—62no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—122no—
527C4⋊C4—241no—
528C4⋊C4D3×D412yes—
529C4⋊C4#23712yes—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—21no—
536C42—21no—
537C42—32no—
538C42—32no—
539C42—42no—
540C42—61no—
541C42—61no—
542C42—241no—
543C42—241no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
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—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3C2×#12111yescharacteristic
559C22×C3#54611yescharacteristic
560C22×C3#54811yescharacteristic
561C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
562C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
563C22×C3#57611yescharacteristic
564C22×C3#57711yescharacteristic
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—42no—
575C3×C4—41no—
576C3×C4#52911yescharacteristic
577C3×C4#54211yescharacteristic
578C3×C4#54811yescharacteristic
579C3×C4#55011yescharacteristic
580Dic3—22no—
581Dic3—22no—
582Dic3—22no—
583Dic3—42no—
584Dic3—41no—
585Dic3—81no—
586Dic3—81no—
587C23—22no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23—121no—
607C23—121no—
608C23C2×D3×D411yescharacteristic
609D4—22no—
610D4—22no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—84no—
618D4—84no—
619D4—121no—
620D4—121no—
621D4—121no—
622D4—121no—
623D4—244no—
624D4—244no—
625D4—244no—
626D4—244no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—32no—
635C2×C4—32no—
636C2×C4—32no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—62no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—244no—
660C2×C4#91112yes—
661C2×C4C2×#23511yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4C2×#23711yescharacteristic
664C2×C4#100411yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4#101011yescharacteristic
667C8—41no—
668C8—41no—
669C8—121no—
670C8—121no—
671C2×C3—82no—
672C2×C3—161no—
673C2×C3#242611yescharacteristic
674C2×C3#303811yescharacteristic
675C2×C3#304511yescharacteristic
676C2×C3#331011yescharacteristic
677C2×C3#331811yescharacteristic
678C2×C3#336611yescharacteristic
679C2×C3#336811yescharacteristic
680D3—161no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—241no—
701C22—241no—
702C22#549011yescharacteristic
703C22#559811yescharacteristic
704C22D3×#12111yescharacteristic
705C22C2×#91111yescharacteristic
706C22#576211yescharacteristic
707C22D3×AGL(1,ℤ/8)11yescharacteristic
708C22#577411yescharacteristic
709C4—22no—
710C4—41no—
711C4—42no—
712C4—42no—
713C4—62no—
714C4—62no—
715C4—62no—
716C4—122no—
717C4—121no—
718C4—241no—
719C4—241no—
720C4#553811yescharacteristic
721C4#555211yescharacteristic
722C4#566911yescharacteristic
723C4#571811yescharacteristic
724C3#2528811yescharacteristic, complemented, Sylow
725C2—82no—
726C2—161no—
727C2—481no—
728C2#7631211yescharacteristic
729C2#7876211yescharacteristic
730C2#7893611yescharacteristic
731C2#8032011yescharacteristic
732C2#8033311yescharacteristic
733C2#8037211yescharacteristic
734C2#8038611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radical#83177C2
hypercenter#1982D3
hypocenterC3#25288
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#4764C2semidirect
3. C2×#4763C2semidirect
4. #73661C2semidirect
5. C3×#1982C2semidirect
6. #71762C2semidirect
7. #70483C2semidirect
8. #70452C2semidirect
9. #70194C2semidirect
10. #70177C2semidirect
11. #70161C2semidirect
12. #70161C2semidirect
13. #70156C2semidirect
14. #70152C2semidirect
15. #69462C2semidirect
72. #1982D3semidirect
124. #890D4semidirect
724. C3#25288semidirect

Maximal subgroups

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

subgroupindex
#831772
C2×#47642
C2×#47632
#736612
C3×#19822
#717622
#704832
#704522
#701942
#701772
#701612
#701562
#701522
#694622
#252883