Finite Group Databasebeta

← #10824478

Subgroups — #10824478

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D8C211yesmaximal, characteristic, complemented
2C2×#4785C211yesmaximal, characteristic, complemented
3#76298C211yesmaximal, characteristic, complemented
4#76295C211yesmaximal, characteristic, complemented
5C3×#2054C211yesmaximal, characteristic, complemented
6#72044C211yesmaximal, characteristic
7#70484C211yesmaximal, characteristic, complemented
8#70442C211yesmaximal, characteristic, complemented
9#70211C211yesmaximal, characteristic, complemented
10#70166C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12#69652C211yesmaximal, characteristic, complemented
13#69650C211yesmaximal, characteristic, complemented
14#68837C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#23688—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32Dic3×D8—24no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4958—21no—
44#4958—21no—
45#4956—21no—
46#4956—21no—
47#4945—21no—
48#4944—21no—
49#4943—21no—
50#4942—21no—
51#4939C2211yescharacteristic
52#4932C2211yescharacteristic
53#4928C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4785—24no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×C4×D8—31no—
64#2328—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2054D311yescharacteristic, complemented
72#2041—31no—
73#2011—31no—
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#940—22no—
102#940—22no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3D414yescomplemented
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#893—22no—
124#893—22no—
125C3×C4⋊C8—22no—
126C3×#102—22no—
127C3×#102—22no—
128#832—22no—
129#832—22no—
130#832—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142#818—22no—
143#815—22no—
144C8×Dic3—22no—
145#809—22no—
146#808—22no—
147C4⋊(C3⋊C8)—22no—
148C22×D8—31no—
149C22×D8C2×D311yescharacteristic
150C2×#127—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#442—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#102—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8—31no—
170#390—31no—
171#390—31no—
172#390C2×D311yescharacteristic
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—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×D4C2×D412yes—
207C2×C3×D4#4112yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—24no—
214C2×#67—24no—
215C2×#67—81no—
216C2×#67—81no—
217C2×#67—81no—
218C2×#67—81no—
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3—81no—
226C22×Dic3—81no—
227C22×Dic3—81no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12#4112yes—
238C2×C4×D3—42no—
239C2×C4×D3—81no—
240C2×C4×D3—81no—
241C3×D8—28no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8#4112yes—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C22⋊C4—81no—
255C3×C22⋊C4—81no—
256C3×C42—21no—
257C3×C42—21no—
258#217—24no—
259#217—42no—
260#217—81no—
261#217—81no—
262#213—28no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—81no—
273Dic3⋊C4—81no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—81no—
277C4×Dic3C2×D412yes—
278C4×Dic3#4112yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8#4112yes—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4—61no—
288C22×D4C22×D311yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—61no—
291C23×C4—61no—
292C23×C4—61no—
293C2×D8—22no—
294C2×D8—22no—
295C2×D8—22no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—62no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#127—62no—
304#127—62no—
305C4×D4—34no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—22no—
336C4⋊C8—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#97—62no—
346C4×C8—62no—
347#95—31no—
348#95—31no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3C22×D411yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404#67—84no—
405#67—84no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—84no—
426C2×Dic3—84no—
427C2×Dic3#12112yes—
428C2×Dic3#12112yes—
429D12—22no—
430D12—22no—
431D12—41no—
432D12—41no—
433C4×D3—84no—
434C3×C8—22no—
435C3×C8—41no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C24—21no—
439C24—21no—
440C24—61no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—32no—
452C2×D4—64no—
453C2×D4—64no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—241no—
460C2×D4—241no—
461C2×D4—241no—
462C2×D4—241no—
463C2×D4D3×D412yes—
464C2×D4#23712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—64no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4—241no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—28no—
487D8—68no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8#23712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—32no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—241no—
511C4⋊C4—241no—
512C22⋊C4—64no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—21no—
524C42—21no—
525C42—32no—
526C42—32no—
527C42—61no—
528C42—61no—
529C42—241no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3C2×#12111yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#53211yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57611yescharacteristic
549C22×C3#57611yescharacteristic
550C22×C3#57711yescharacteristic
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4#52511yescharacteristic
563C3×C4#52511yescharacteristic
564C3×C4#52911yescharacteristic
565C3×C4#52911yescharacteristic
566Dic3—22no—
567Dic3—22no—
568Dic3—41no—
569Dic3—41no—
570Dic3—42no—
571Dic3—81no—
572Dic3—81no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C2×D3×D411yescharacteristic
595D4—22no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—62no—
604D4—62no—
605D4—121no—
606D4—121no—
607D4—244no—
608D4—244no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—32no—
616C2×C4—32no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—62no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—244no—
643C2×C4—244no—
644C2×C4—244no—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C8—22no—
652C8—41no—
653C8—62no—
654C8—121no—
655C2×C3—82no—
656C2×C3—82no—
657C2×C3#239211yescharacteristic
658C2×C3#304111yescharacteristic
659C2×C3#304511yescharacteristic
660C2×C3#311811yescharacteristic
661C2×C3#312211yescharacteristic
662C2×C3#317611yescharacteristic
663C2×C3#317911yescharacteristic
664D3—82no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—122no—
680C22—122no—
681C22—122no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22#559611yescharacteristic
687C22D3×#12111yescharacteristic
688C22#560611yescharacteristic
689C22#575711yescharacteristic
690C22#576911yescharacteristic
691C22#577411yescharacteristic
692C22#577511yescharacteristic
693C4—41no—
694C4—41no—
695C4—62no—
696C4—62no—
697C4—81no—
698C4—81no—
699C4—121no—
700C4—122no—
701C4—121no—
702C4—241no—
703C4—241no—
704C4#558611yescharacteristic
705C4#564911yescharacteristic
706C4#565611yescharacteristic
707C4#568011yescharacteristic
708C3#2368811yescharacteristic, complemented, Sylow
709C2—82no—
710C2—82no—
711C2—242no—
712C2#7689011yescharacteristic
713C2#8031711yescharacteristic
714C2#8033511yescharacteristic
715C2#8050811yescharacteristic
716C2#8053611yescharacteristic
717C2#8069911yescharacteristic
718C2#8085111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2054C2
socleC23×C3C22×D4
prime radical#72044C2
hypercenter#2054D3
hypocenterC3#23688
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
1. C2×Dic3×D8C2semidirect
2. C2×#4785C2semidirect
3. #76298C2semidirect
4. #76295C2semidirect
5. C3×#2054C2semidirect
7. #70484C2semidirect
8. #70442C2semidirect
9. #70211C2semidirect
10. #70166C2semidirect
11. #70152C2semidirect
12. #69652C2semidirect
13. #69650C2semidirect
14. #68837C2semidirect
15. #68780C2semidirect
71. #2054D3semidirect
106. D4×Dic3D4semidirect
708. C3#23688semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
C2×#47852
#762982
#762952
C3×#20542
#720442
#704842
#704422
#702112
#701662
#701522
#696522
#696502
#688372
#687802
#236883