Finite Group Databasebeta

← #10901367

Subgroups — #10901367

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5680C211yesmaximal, characteristic
2C2×#4791C211yesmaximal, characteristic, complemented
3C2×#4759C211yesmaximal, characteristic, complemented
4#76288C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic, complemented
6#71722C211yesmaximal, characteristic, complemented
7#70275C211yesmaximal, characteristic, complemented
8#70244C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#70207C211yesmaximal, characteristic
11#70176C211yesmaximal, characteristic
12#70153C211yesmaximal, characteristic, complemented
13#69606C211yesmaximal, characteristic, complemented
14#68662C211yesmaximal, characteristic, complemented
15#68634C211yesmaximal, characteristic, complemented
16#29689—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5680—24no—
23#5680—24no—
24C2×#896—21no—
25C2×#888C2211yescharacteristic
26C2×#883—21no—
27C2×#67⋊C4—21no—
28C2×#879—21no—
29C2×#877—21no—
30C2×C42×D3C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×(C2×D3)⋊C8C2211yescharacteristic
42C2×(C2×D3)⋊C8C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#817C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×Dic3⋊C8C2211yescharacteristic
47#4987C2211yescharacteristic
48#4987C2211yescharacteristic
49#4976—21no—
50#4975C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4791—24no—
55#4759—24no—
56#4667—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#529—31no—
65C2×#475—31no—
66C2×#474—31no—
67#2857—31no—
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042D311yescharacteristic, complemented
72#2036—31no—
73#2036—31no—
74#2034—31no—
75#2034—31no—
76#1803—31no—
77#1802—31no—
78#1797—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×#67—21no—
82C23×Dic3—21no—
83C22×C4×D3D411yescharacteristic
84C22×C4×D3D411yescharacteristic
85C22×C4×D3C2311yescharacteristic
86C22×Dic6—21no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#124—22no—
90C3×#124—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99#946—22no—
100#946—22no—
101Q8×Dic3—22no—
102Q8×Dic3D414yes—
103#938—22no—
104#938—22no—
105D4×Dic3—22no—
106D4×Dic3D414yes—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3D411yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#896—44no—
129#888—22no—
130#888—22no—
131#883—28no—
132#67⋊C4—28no—
133#879—28no—
134#877—28no—
135C42×D3—22no—
136C42×D3—22no—
137#870—22no—
138#870—22no—
139C3×#103—22no—
140C3×#103—41no—
141C3×#102—22no—
142C3×#102—41no—
143C3×#95—21no—
144#835—22no—
145#835—41no—
146#832—22no—
147#832—41no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151(C2×D3)⋊C8—41no—
152(C2×D3)⋊C8—41no—
153#818—22no—
154#817—22no—
155Dic3⋊C8—22no—
156Dic3⋊C8—22no—
157#808—22no—
158#807—22no—
159C2×C4⋊Q8—31no—
160#529—64no—
161#529—64no—
162C2×#126—61no—
163C2×#125—31no—
164C2×#124C2×D311yescharacteristic
165C2×#123—61no—
166C2×C22⋊Q8—31no—
167C2×C22⋊Q8—61no—
168C2×#121—31no—
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×C4×D4—61no—
172C2×#117—61no—
173C22×C42—31no—
174#475—64no—
175#474—64no—
176C2×#107—31no—
177C2×#107—31no—
178C2×#106—31no—
179C2×#106—31no—
180C2×C4⋊C8—31no—
181C2×C4⋊C8—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103C2×D311yescharacteristic
185C2×#103C2×D311yescharacteristic
186C2×#102—31no—
187C2×#102—31no—
188C2×#102C2×D311yescharacteristic
189C2×#102C2×D311yescharacteristic
190C2×C22⋊C8—31no—
191C2×C22⋊C8—31no—
192#391—61no—
193#389—31no—
194#389—61no—
195#389C2×D311yescharacteristic
196#384—31no—
197#380—31no—
198#380—31no—
199#380—61no—
200#340—31no—
201#340—31no—
202#340—31no—
203#340—31no—
204#336—31no—
205#336C2×D311yescharacteristic
206C24×C3—21no—
207C23×D3C2×D411yescharacteristic
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D412yes—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4C2×D412yes—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C2×#67—24no—
227C2×#67—42no—
228C2×#67—42no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—24no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—42no—
235C22×Dic3—42no—
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C2×C4×D3—22no—
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3—21no—
244C2×C4×D3—21no—
245C2×C4×D3—21no—
246C2×C4×D3—22no—
247C2×C4×D3—22no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—41no—
251C2×Dic6—42no—
252C2×Dic6—42no—
253C2×Dic6—42no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—42no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C42—21no—
268C3×C42—21no—
269#217—24no—
270#217—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—24no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3C2×D412yes—
282C4⋊Dic3C2×D412yes—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—42no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289Dic3⋊C4—42no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292C4×Dic3—24no—
293C4×Dic3C2×D412yes—
294C4×Dic3C2×D412yes—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C22×Q8—61no—
302C22×Q8C22×D311yescharacteristic
303C22×D4—61no—
304C22×D4C22×D311yescharacteristic
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—31no—
308C23×C4—61no—
309C22×C8—31no—
310C22×C8—31no—
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C4⋊Q8—62no—
314C4⋊Q8—62no—
315#126—68no—
316#125—62no—
317#125—62no—
318#124—22no—
319#124—22no—
320#123—68no—
321C22⋊Q8—62no—
322C22⋊Q8—62no—
323C22⋊Q8—124no—
324#121—62no—
325#121—62no—
326C4×Q8—34no—
327C4×Q8—62no—
328C4×D4—34no—
329C4×D4—62no—
330C4×D4—68no—
331#117—68no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—62no—
358C2×C42—62no—
359C2×C42—61no—
360C2×C42C22×D311yescharacteristic
361#107—62no—
362#107—62no—
363#106—62no—
364#106—62no—
365C4⋊C8—62no—
366C4⋊C8—62no—
367#103—22no—
368#103—41no—
369#103—62no—
370#103—121no—
371#102—22no—
372#102—41no—
373#102—62no—
374#102—121no—
375C22⋊C8—121no—
376C22⋊C8—121no—
377#95—21no—
378#95—61no—
379#95—61no—
380#95—61no—
381C23×C3—22no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3C22×D411yescharacteristic
389C22×D3—21no—
390C22×D3—21no—
391C22×D3—21no—
392C22×D3—21no—
393C22×D3—21no—
394C22×D3—21no—
395C22×D3#13712yes—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—41no—
399C3×Q8—41no—
400C3×D4—22no—
401C3×D4—22no—
402C3×D4—41no—
403C3×D4—41no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4H(5,𝔽2)11yescharacteristic
422C2×C3×C42−1+411yescharacteristic
423C2×C3×C42−1+411yescharacteristic
424C2×C3×C42−1+411yescharacteristic
425#67—48no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—48no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3#12012yes—
445C2×Dic3#12012yes—
446C2×Dic3#13512yes—
447C2×Dic3AGL(1,ℤ/8)12yes—
448C4×D3—44no—
449C4×D3—41no—
450C4×D3—41no—
451C4×D3—41no—
452C4×D3—41no—
453C4×D3—44no—
454Dic6—84no—
455C3×C8—41no—
456C3×C8—41no—
457C3⋊C8—41no—
458C3⋊C8—41no—
459C24—21no—
460C24—31no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—122no—
467C2×Q8—122no—
468C2×Q8—122no—
469C2×Q8D3×D412yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—64no—
476C2×D4—122no—
477C2×D4—122no—
478C2×D4D3×D412yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—61no—
500C22×C4—61no—
501C22×C4—121no—
502C22×C4—122no—
503C22×C4—122no—
504C22×C4D3×D411yescharacteristic
505C22×C4D3×D411yescharacteristic
506C22×C4C23×D311yescharacteristic
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—32no—
524C4⋊C4—32no—
525C4⋊C4—42no—
526C4⋊C4—64no—
527C4⋊C4—64no—
528C4⋊C4—64no—
529C4⋊C4—64no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—122no—
537C4⋊C4—122no—
538C4⋊C4—122no—
539C4⋊C4—122no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—122no—
546C22⋊C4—122no—
547C42—21no—
548C42—21no—
549C42—32no—
550C42—32no—
551C42—62no—
552C42—62no—
553C42—64no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3—42no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3C2×#12011yescharacteristic
562C22×C3#53611yescharacteristic
563C22×C3#53711yescharacteristic
564C22×C3C2×#13511yescharacteristic
565C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
566C22×C3C2×#13711yescharacteristic
567C22×C3#57511yescharacteristic
568C2×D3—22no—
569C2×D3—22no—
570C2×D3—22no—
571C2×D3—41no—
572C2×D3—41no—
573C2×D3—41no—
574C2×D3—41no—
575C3×C4—42no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—81no—
579C3×C4—81no—
580C3×C4#53611yescharacteristic
581C3×C4#53611yescharacteristic
582C3×C4#53711yescharacteristic
583C3×C4#53711yescharacteristic
584Dic3—22no—
585Dic3—41no—
586Dic3—42no—
587Dic3—42no—
588Dic3—81no—
589Dic3—81no—
590Dic3#44914yes—
591C23—22no—
592C23—32no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—61no—
600C23—61no—
601C23—61no—
602C23—61no—
603C23—61no—
604C23—61no—
605C23C2×D3×D411yescharacteristic
606Q8—22no—
607Q8—22no—
608Q8—41no—
609Q8—41no—
610Q8—244no—
611D4—22no—
612D4—22no—
613D4—41no—
614D4—41no—
615D4—128no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—61no—
642C2×C4—124no—
643C2×C4—128no—
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—121no—
658C2×C4—121no—
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4#100411yescharacteristic
662C2×C4#101011yescharacteristic
663C2×C4#101011yescharacteristic
664C2×C4#101011yescharacteristic
665C8—41no—
666C8—41no—
667C8—121no—
668C8—121no—
669C2×C3—82no—
670C2×C3#261611yescharacteristic
671C2×C3C2×#44911yescharacteristic
672C2×C3#300711yescharacteristic
673C2×C3#323311yescharacteristic
674C2×C3#323911yescharacteristic
675C2×C3#324811yescharacteristic
676C2×C3#325111yescharacteristic
677D3—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—62no—
682C22—62no—
683C22—62no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—121no—
689C22—121no—
690C22—121no—
691C22—121no—
692C22D3×#12011yescharacteristic
693C22#560811yescharacteristic
694C22#567511yescharacteristic
695C22D3×#13511yescharacteristic
696C22#576811yescharacteristic
697C22D3×AGL(1,ℤ/8)11yescharacteristic
698C22D3×#13711yescharacteristic
699C4—34no—
700C4—41no—
701C4—42no—
702C4—41no—
703C4—62no—
704C4—81no—
705C4—81no—
706C4—122no—
707C4—121no—
708C4—122no—
709C4—241no—
710C4—241no—
711C4#559711yescharacteristic
712C4#559711yescharacteristic
713C4#564811yescharacteristic
714C4#564811yescharacteristic
715C3#2968911yescharacteristic, complemented, Sylow
716C2—82no—
717C2—122no—
718C2#7697011yescharacteristic
719C2D3×#44911yescharacteristic
720C2#8022811yescharacteristic
721C2#8051311yescharacteristic
722C2#8057611yescharacteristic
723C2#8164911yescharacteristic
724C2#8169111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2042D3
hypocenterC3#29689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
3. C2×#4759C2semidirect
4. #76288C2semidirect
5. C3×#2042C2semidirect
6. #71722C2semidirect
7. #70275C2semidirect
8. #70244C2semidirect
9. #70221C2semidirect
12. #70153C2semidirect
13. #69606C2semidirect
14. #68662C2semidirect
15. #68634C2semidirect
71. #2042D3semidirect
715. C3#29689semidirect

Maximal subgroups

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

subgroupindex
C2×#56802
C2×#47912
C2×#47592
#762882
C3×#20422
#717222
#702752
#702442
#702212
#702072
#701762
#701532
#696062
#686622
#686342
#296893