Finite Group Databasebeta

← #10824738

Subgroups — #10824738

Order 768.

1042 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5152C211yesmaximal, characteristic, complemented
2Dic3×#121C211yesmaximal, characteristic, complemented
3Dic3×#121C211yesmaximal, characteristic, complemented
4C2×#4719C211yesmaximal, characteristic, complemented
5C3×#1999C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic, complemented
7#71680C211yesmaximal, characteristic, complemented
8#71679C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic, complemented
10#70193C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12#70152C211yesmaximal, characteristic, complemented
13#69940C211yesmaximal, characteristic, complemented
14#68840C211yesmaximal, characteristic, complemented
15#68786C211yesmaximal, characteristic, complemented
16#23708—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×D4×Dic3—21no—
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2211yescharacteristic
23C2×D4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30#5220—21no—
31#5220—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#5152—24no—
40C2×#817C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×#815C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984C2211yescharacteristic
46#4984C2211yescharacteristic
47Dic3×C4⋊C4C2211yescharacteristic
48Dic3×C4⋊C4C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4951—21no—
52#4951—21no—
53Dic3×C22⋊C4—21no—
54Dic3×C22⋊C4—21no—
55#4719—24no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×#432—31no—
64C4×#121—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999D311yescharacteristic
74#1776—31no—
75#1762—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C23×Dic3—21no—
84C23×Dic3D411yescharacteristic
85C23×Dic3D411yescharacteristic
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—22no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#121—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107D4×Dic3—22no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110D4×Dic3—22no—
111D4×Dic3—44no—
112D4×Dic3—44no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4×Dic3—22no—
120C2×C4×Dic3—22no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3D411yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C3×#102—22no—
130C3×#102—22no—
131C3×C22⋊C8—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#830—22no—
143#817—22no—
144#815—22no—
145#815D414yescomplemented
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148#808—22no—
149#807—22no—
150C22×D8C2×D311yescharacteristic
151C2×#121C2×D311yescharacteristic
152C2×#121C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4—61no—
157C22×C42—31no—
158#442—64no—
159#432—64no—
160C2×#107—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×C4⋊C8—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×#102C2×D311yescharacteristic
171C2×C22⋊C8—31no—
172C2×C22⋊C8C2×D311yescharacteristic
173C2×#97—31no—
174C2×#97—31no—
175#390—31no—
176#390—31no—
177#389—31no—
178#389—31no—
179#388—61no—
180#388—61no—
181#386—61no—
182#386—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C22⋊C4—61no—
188C4×C22⋊C4—61no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195C24×C3—21no—
196C24×C3—21no—
197C24×C3C2×D411yescharacteristic
198C2×C3×D4—24no—
199C2×C3×D4—24no—
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—21no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4—42no—
213C2×C3×D4—42no—
214C2×C3×D4#4112yes—
215C2×C3×D4#4112yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—22no—
227C22×Dic3—22no—
228C22×Dic3—22no—
229C22×Dic3—22no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—41no—
235C22×Dic3—41no—
236C22×Dic3—42no—
237C22×Dic3—42no—
238C22×Dic3—42no—
239C22×Dic3—42no—
240C22×Dic3—42no—
241C22×Dic3—42no—
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C22×Dic3C2×D411yescharacteristic
245C22×Dic3C2×D411yescharacteristic
246C3×D8—28no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8C2×D412yes—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4#4112yes—
256C3×C4⋊C4#4112yes—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—24no—
259#217—42no—
260#217—42no—
261#217—42no—
262#217—42no—
263#217—42no—
264#217—42no—
265#217—42no—
266#217—42no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—42no—
276C4⋊Dic3—42no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C4×Dic3—42no—
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⋊C8—21no—
290C2×C3⋊C8C2×D412yes—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—61no—
298C23×C4—61no—
299C23×C4C22×D311yescharacteristic
300C2×D8—22no—
301C2×D8—22no—
302C2×D8—22no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—124no—
316C4×D4—124no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
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—31no—
338C2×C42—31no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42—61no—
342C2×C42—61no—
343#107—62no—
344#106—62no—
345#106—62no—
346C4⋊C8—62no—
347C4⋊C8—62no—
348#102—22no—
349#102—22no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354C22⋊C8—22no—
355C22⋊C8—62no—
356#97—34no—
357#97—62no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3AGL(1,ℤ/8)12yes—
385C23×C3C22×D411yescharacteristic
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—48no—
391C3×D4—48no—
392C3×D4—41no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
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—21no—
424C2×Dic3—44no—
425C2×Dic3—44no—
426C2×Dic3—41no—
427C2×Dic3—41no—
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—41no—
437C2×Dic3—41no—
438C2×Dic3—84no—
439C2×Dic3—84no—
440C2×Dic3#12112yes—
441C2×Dic3#12112yes—
442C2×Dic3AGL(1,ℤ/8)12yes—
443C3×C8—22no—
444C3×C8—41no—
445C3⋊C8—22no—
446C3⋊C8—41no—
447C24—21no—
448C24—21no—
449C24D3×D411yescharacteristic
450C2×D4—24no—
451C2×D4—24no—
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#23712yes—
467C2×D4#23712yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4D3×D411yescharacteristic
497C22×C4C23×D311yescharacteristic
498D8—28no—
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—
510C2×C8D3×D412yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—122no—
522C4⋊C4—122no—
523C4⋊C4#23712yes—
524C4⋊C4#23712yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—32no—
536C42—32no—
537C42—62no—
538C42—62no—
539C42—61no—
540C42—61no—
541C42—122no—
542C42—122no—
543C22×C3—22no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3—42no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3—81no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3C2×#12111yescharacteristic
565C22×C3#52511yescharacteristic
566C22×C3#52511yescharacteristic
567C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
568C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
569C22×C3#57511yescharacteristic
570C22×C3#57511yescharacteristic
571C3×C4—42no—
572C3×C4—42no—
573C3×C4—41no—
574C3×C4#52511yescharacteristic
575C3×C4#52911yescharacteristic
576C3×C4#52911yescharacteristic
577C3×C4#53211yescharacteristic
578Dic3—22no—
579Dic3—22no—
580Dic3—22no—
581Dic3—41no—
582Dic3—81no—
583Dic3—81no—
584Dic3—81no—
585Dic3—81no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—22no—
593C23—22no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23#91112yes—
607C23C2×D3×D411yescharacteristic
608D4—22no—
609D4—22no—
610D4—22no—
611D4—22no—
612D4—48no—
613D4—48no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—41no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
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—124no—
644C2×C4—124no—
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—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—244no—
658C2×C4—244no—
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×#23711yescharacteristic
661C2×C4C2×#23711yescharacteristic
662C2×C4C2×#23711yescharacteristic
663C2×C4C2×#23711yescharacteristic
664C2×C4D3×#4111yescharacteristic
665C8—22no—
666C8—41no—
667C8—62no—
668C8—121no—
669C2×C3—42no—
670C2×C3—82no—
671C2×C3—82no—
672C2×C3#239211yescharacteristic
673C2×C3#305511yescharacteristic
674C2×C3#305511yescharacteristic
675C2×C3#310911yescharacteristic
676C2×C3#311111yescharacteristic
677C2×C3#311411yescharacteristic
678C2×C3#311511yescharacteristic
679C22—22no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22#558611yescharacteristic
701C22#558611yescharacteristic
702C22D3×#12111yescharacteristic
703C22C2×#91111yescharacteristic
704C22#576811yescharacteristic
705C22D3×AGL(1,ℤ/8)11yescharacteristic
706C22#577311yescharacteristic
707C4—42no—
708C4—42no—
709C4—41no—
710C4—62no—
711C4—62no—
712C4—62no—
713C4—121no—
714C4—241no—
715C4—241no—
716C4—241no—
717C4—241no—
718C4#558611yescharacteristic
719C4#560011yescharacteristic
720C4#560011yescharacteristic
721C4#571611yescharacteristic
722C3#2370811yescharacteristic, complemented, Sylow
723C2—42no—
724C2—82no—
725C2—82no—
726C2#7666611yescharacteristic
727C2#7999011yescharacteristic
728C2#7999111yescharacteristic
729C2#8007011yescharacteristic
730C2#8007111yescharacteristic
731C2#8071511yescharacteristic
732C2#8086411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23708
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×#5152C2semidirect
2. Dic3×#121C2semidirect
3. Dic3×#121C2semidirect
4. C2×#4719C2semidirect
6. #72040C2semidirect
7. #71680C2semidirect
8. #71679C2semidirect
9. #70194C2semidirect
10. #70193C2semidirect
11. #70152C2semidirect
12. #70152C2semidirect
13. #69940C2semidirect
14. #68840C2semidirect
15. #68786C2semidirect
145. #815D4semidirect
722. C3#23708semidirect

Maximal subgroups

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

subgroupindex
C2×#51522
Dic3×#1212
C2×#47192
C3×#19992
#720402
#716802
#716792
#701942
#701932
#701522
#699402
#688402
#687862
#237083