Finite Group Databasebeta

← #10900985

Subgroups — #10900985

Order 768.

990 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5729C211yesmaximal, characteristic, complemented
2C2×#4774C211yesmaximal, characteristic
3C2×#4779C211yesmaximal, characteristic
4#76458C211yesmaximal, characteristic
5C3×#2023C211yesmaximal, characteristic, complemented
6#71782C211yesmaximal, characteristic, complemented
7#70164C211yesmaximal, characteristic, complemented
8#70163C211yesmaximal, characteristic, complemented
9#69639C211yesmaximal, characteristic, complemented
10#68873C211yesmaximal, characteristic, complemented
11#68872C211yesmaximal, characteristic, complemented
12#68828C211yesmaximal, characteristic, complemented
13#68826C211yesmaximal, characteristic, complemented
14#68665C211yesmaximal, characteristic, complemented
15#68634C211yesmaximal, characteristic, complemented
16#29646—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C2×#944—21no—
20C2×#938C2211yescharacteristic
21#5729—24no—
22#5729—24no—
23C2×#896—21no—
24C2×#892C2211yescharacteristic
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#888C2211yescharacteristic
29C2×#872C2211yescharacteristic
30C2×#870C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C3×#390C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41#4987C2211yescharacteristic
42#4984C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4961—21no—
46#4959—21no—
47#4952—21no—
48#4949—21no—
49#4928C2211yescharacteristic
50#4779—24no—
51#4774—24no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#544—31no—
65C2×#482—31no—
66C2×#480—31no—
67#2624—31no—
68#2059—31no—
69#2059—31no—
70#2057—31no—
71#2057—31no—
72#2025—31no—
73#2025—31no—
74#2023—31no—
75#2023D311yescharacteristic, complemented
76#1825—31no—
77#1824—31no—
78#1818—31no—
79C22×C3×Q8—21no—
80C22×C3×D4C2311yescharacteristic
81C22×#67—21no—
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8—22no—
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
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99#946—28no—
100#944—28no—
101#938—22no—
102#938—22no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4D411yescharacteristic
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#896—28no—
126#892—22no—
127#892D414yes—
128D3×C4⋊C4—22no—
129D3×C4⋊C4D414yes—
130#890—22no—
131#890—22no—
132#889—28no—
133#888—22no—
134#888—22no—
135#872—22no—
136#872—22no—
137#870—22no—
138#870—22no—
139C3×#102—22no—
140C3×#102—22no—
141#832—22no—
142#832—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146(C2×D3)⋊C8—22no—
147(C2×D3)⋊C8—22no—
148#818—22no—
149#817—22no—
150#807—22no—
151#807—22no—
152#544—64no—
153#544—64no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—61no—
156C2×C4⋊Q8C2×D311yescharacteristic
157C2×#125—31no—
158C2×#125—31no—
159C2×#125—61no—
160C2×C22⋊Q8—61no—
161C2×C22⋊Q8—61no—
162C2×#121—31no—
163C2×#117—31no—
164C2×#117—31no—
165C22×C4⋊C4—31no—
166#482—64no—
167#480—64no—
168C2×#107—31no—
169C2×#107—31no—
170C2×#107—31no—
171C2×#106—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102C2×D311yescharacteristic
175C2×#102C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178#399—61no—
179#397—61no—
180#394—61no—
181#394—61no—
182#390—31no—
183#390C2×D311yescharacteristic
184#389—31no—
185#389—31no—
186#380—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194#340C2×D311yescharacteristic
195#336—31no—
196#336—31no—
197#336—31no—
198#336C2×D311yescharacteristic
199C24×C3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—24no—
202C2×C3×Q8—81no—
203C2×C3×Q8—81no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—42no—
217C2×#67—81no—
218C2×#67—81no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—24no—
231C2×C4×D3—81no—
232C2×C4×D3—81no—
233C2×C4×D3C2×D412yes—
234C2×C4×D3C2×D412yes—
235C2×Dic6—24no—
236C2×Dic6—81no—
237C2×Dic6—81no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255#217—42no—
256#217—81no—
257#217—81no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—81no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—24no—
279Dic3⋊C4—81no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—24no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C22×Q8—21no—
290C22×Q8—61no—
291C22×D4—61no—
292C22×D4C22×D311yescharacteristic
293C23×C4—31no—
294C23×C4—61no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—62no—
302C4⋊Q8—68no—
303C4⋊Q8—62no—
304#125—62no—
305#125—62no—
306#125—62no—
307#125—68no—
308#125—62no—
309C22⋊Q8—68no—
310C22⋊Q8—68no—
311#121—62no—
312#121—62no—
313#117—34no—
314#117—62no—
315#117—62no—
316#117—62no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—62no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—31no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349#107—62no—
350#107—62no—
351#107—62no—
352#106—62no—
353#102—22no—
354#102—22no—
355#102—62no—
356#102—62no—
357C22⋊C8—62no—
358C22⋊C8—62no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C22×D411yescharacteristic
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3#12012yes—
377C3×Q8—84no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C42−1+411yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403#67—84no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3#12012yes—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—84no—
430Dic6—84no—
431C3×C8—41no—
432C3×C8—41no—
433C3⋊C8—41no—
434C3⋊C8—41no—
435C24—21no—
436C24—31no—
437C2×Q8—24no—
438C2×Q8—64no—
439C2×Q8—81no—
440C2×Q8—81no—
441C2×Q8—241no—
442C2×Q8—241no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—122no—
450C2×D4—241no—
451C2×D4—241no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—64no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—241no—
470C22×C4—241no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—32no—
492C4⋊C4—32no—
493C4⋊C4—64no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—64no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4D3×D412yes—
510C4⋊C4D3×D412yes—
511C22⋊C4—62no—
512C22⋊C4—62no—
513C22⋊C4—64no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—122no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C42—21no—
521C42—21no—
522C42—64no—
523C42—61no—
524C42—61no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3C2×#12011yescharacteristic
533C22×C3#53611yescharacteristic
534C22×C3#53611yescharacteristic
535C22×C3#57611yescharacteristic
536C22×C3#57611yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57811yescharacteristic
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C2×D3—41no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4#53611yescharacteristic
552C3×C4#53611yescharacteristic
553C3×C4#53611yescharacteristic
554C3×C4#53611yescharacteristic
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558Dic3—41no—
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—32no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23C2×D3×D411yescharacteristic
577Q8—84no—
578Q8—244no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—244no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—32no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—244no—
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4#100711yescharacteristic
628C2×C4#101011yescharacteristic
629C2×C4#101011yescharacteristic
630C2×C4#101011yescharacteristic
631C8—41no—
632C8—41no—
633C8—121no—
634C8—121no—
635C2×C3—161no—
636C2×C3#261411yescharacteristic
637C2×C3#300811yescharacteristic
638C2×C3#301311yescharacteristic
639C2×C3#324111yescharacteristic
640C2×C3#324111yescharacteristic
641C2×C3#324311yescharacteristic
642C2×C3#324311yescharacteristic
643D3—42no—
644C22—62no—
645C22—62no—
646C22—62no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—121no—
655C22—121no—
656C22—121no—
657C22—121no—
658C22D3×#12011yescharacteristic
659C22#562811yescharacteristic
660C22#567511yescharacteristic
661C22#575711yescharacteristic
662C22#576211yescharacteristic
663C22#577411yescharacteristic
664C22#577911yescharacteristic
665C4—41no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—62no—
670C4—81no—
671C4—121no—
672C4—122no—
673C4—121no—
674C4—122no—
675C4—241no—
676C4—241no—
677C4#562811yescharacteristic
678C4#562811yescharacteristic
679C4#567511yescharacteristic
680C4#567511yescharacteristic
681C3#2964611yescharacteristic, complemented, Sylow
682C2—122no—
683C2—161no—
684C2#7687011yescharacteristic
685C2#8000411yescharacteristic
686C2#8004211yescharacteristic
687C2#8097411yescharacteristic
688C2#8106811yescharacteristic
689C2#8157511yescharacteristic
690C2#8169411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2023D3
hypocenterC3#29646
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#5729C2semidirect
5. C3×#2023C2semidirect
6. #71782C2semidirect
7. #70164C2semidirect
8. #70163C2semidirect
9. #69639C2semidirect
10. #68873C2semidirect
11. #68872C2semidirect
12. #68828C2semidirect
13. #68826C2semidirect
14. #68665C2semidirect
15. #68634C2semidirect
75. #2023D3semidirect
681. C3#29646semidirect

Maximal subgroups

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

subgroupindex
C2×#57292
C2×#47742
C2×#47792
#764582
C3×#20232
#717822
#701642
#701632
#696392
#688732
#688722
#688282
#688262
#686652
#686342
#296463