Finite Group Databasebeta

← #10900956

Subgroups — #10900956

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#125C211yesmaximal, characteristic
2C2×#4884C211yesmaximal, characteristic
3C2×#4862C211yesmaximal, characteristic
4#76287C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic, complemented
6#71744C211yesmaximal, characteristic, complemented
7#70618C211yesmaximal, characteristic, complemented
8#70566C211yesmaximal, characteristic, complemented
9#69606C211yesmaximal, characteristic, complemented
10#68911C211yesmaximal, characteristic, complemented
11#68911C211yesmaximal, characteristic, complemented
12#68872C211yesmaximal, characteristic
13#68871C211yesmaximal, characteristic
14#68690C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic
16#29643—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18D3×#125—24no—
19D3×#125—24no—
20C2×#897C2211yescharacteristic
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4—21no—
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×D3×C4⋊C4C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#889—21no—
29C2×C42×D3C2211yescharacteristic
30C2×#870C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4989—21no—
39#4987C2211yescharacteristic
40#4980—21no—
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4929C2211yescharacteristic
48#4884—24no—
49#4862—24no—
50#4664—21no—
51#4655—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C22×#125—31no—
65C2×#484—31no—
66C2×#483—31no—
67#2855—31no—
68#2059—31no—
69#2059—31no—
70#2059—31no—
71#2059D311yescharacteristic, complemented
72#2026—31no—
73#2026—31no—
74#2026—31no—
75#2026—31no—
76#1824—31no—
77#1823—31no—
78#1817—31no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3D411yescharacteristic
82C22×C4×D3D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#125—22no—
88C3×#125—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#897—22no—
118#897—22no—
119D3×C4⋊C4—28no—
120D3×C4⋊C4—28no—
121D3×C4⋊C4—22no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4D414yes—
124D3×C4⋊C4D414yes—
125#890—22no—
126#890—22no—
127#890—22no—
128#890—22no—
129#889—28no—
130#889—28no—
131C42×D3—22no—
132C42×D3—22no—
133#870—22no—
134#870—22no—
135C3×#107—22no—
136C3×#106—22no—
137C3×#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8—22no—
146#816—22no—
147#816—22no—
148#810—22no—
149#810—22no—
150#808—22no—
151#807—22no—
152C2×#125—31no—
153C2×#125—31no—
154C2×#125—31no—
155C2×#125—61no—
156C2×#125—61no—
157C2×#125—64no—
158C2×#125—64no—
159C2×#125C2×D311yescharacteristic
160C2×C22⋊Q8—31no—
161C22×C4⋊C4—31no—
162C22×C4⋊C4—31no—
163C22×C4⋊C4—61no—
164C22×C4⋊C4—61no—
165C22×C42—31no—
166#484—64no—
167#483—64no—
168C2×#107—31no—
169C2×#107C2×D311yescharacteristic
170C2×#106—31no—
171C2×#106C2×D311yescharacteristic
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178#391—31no—
179#391—31no—
180#389—31no—
181#389C2×D311yescharacteristic
182#387—61no—
183#387—61no—
184#385—61no—
185#381—61no—
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
199C23×D3C2×D411yescharacteristic
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—24no—
221C2×C4×D3—24no—
222C2×C4×D3—22no—
223C2×C4×D3—22no—
224C2×C4×D3—41no—
225C2×C4×D3—41no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3C2×D412yes—
231C2×C4×D3C2×D412yes—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
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—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4C2×D412yes—
254C3×C4⋊C4C2×D412yes—
255C3×C42—21no—
256C3×C42—21no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266C4⋊Dic3C2×D412yes—
267C4⋊Dic3C2×D412yes—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—42no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—22no—
278C4×Dic3—22no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C22×Q8—31no—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4—61no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#125—22no—
296#125—22no—
297#125—62no—
298#125—62no—
299#125—62no—
300#125—62no—
301#125—68no—
302#125—68no—
303#125—62no—
304#125—62no—
305C22⋊Q8—62no—
306C22⋊Q8—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—68no—
318C2×C4⋊C4—68no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—62no—
339C2×C42—62no—
340C2×C42C22×D311yescharacteristic
341#107—22no—
342#107—62no—
343#106—22no—
344#106—62no—
345#103—62no—
346#103—62no—
347#103—62no—
348#103—62no—
349C22⋊C8—62no—
350C22⋊C8—62no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3C22×D411yescharacteristic
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3#12012yes—
364C22×D3#13512yes—
365C22×D3#13712yes—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3#12012yes—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—44no—
416C4×D3—48no—
417C4×D3—44no—
418C4×D3—48no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—31no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—32no—
487C4⋊C4—32no—
488C4⋊C4—42no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4D3×D412yes—
504C4⋊C4D3×D412yes—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C42—21no—
508C42—21no—
509C42—62no—
510C42—62no—
511C42—61no—
512C42—61no—
513C22×C3C2×#12011yescharacteristic
514C22×C3#53611yescharacteristic
515C22×C3#53611yescharacteristic
516C22×C3C2×#13511yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57511yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3#45214yes—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#53611yescharacteristic
534C3×C4#53611yescharacteristic
535C3×C4#53611yescharacteristic
536C3×C4#53611yescharacteristic
537Dic3—22no—
538Dic3—41no—
539Dic3—42no—
540Dic3—41no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—32no—
545C23—32no—
546C23—32no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23C2×D3×D411yescharacteristic
552Q8—121no—
553Q8—121no—
554Q8—121no—
555Q8—121no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—124no—
590C2×C4—128no—
591C2×C4—124no—
592C2×C4—128no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4#100711yescharacteristic
606C2×C4#100711yescharacteristic
607C2×C4#100711yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C2×C3#261411yescharacteristic
614C2×C3C2×#45211yescharacteristic
615C2×C3#300511yescharacteristic
616C2×C3#323211yescharacteristic
617C2×C3#323211yescharacteristic
618C2×C3#323611yescharacteristic
619C2×C3#323611yescharacteristic
620D3—24no—
621C22—34no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22D3×#12011yescharacteristic
629C22#562811yescharacteristic
630C22#572411yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22#577311yescharacteristic
634C22D3×#13711yescharacteristic
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—62no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—122no—
644C4—121no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4#562811yescharacteristic
649C4#562811yescharacteristic
650C4#572411yescharacteristic
651C4#572411yescharacteristic
652C3#2964311yescharacteristic, complemented, Sylow
653C2—64no—
654C2#7725611yescharacteristic
655C2#8008311yescharacteristic
656C2D3×#45211yescharacteristic
657C2#8117311yescharacteristic
658C2#8123811yescharacteristic
659C2#8192011yescharacteristic
660C2#8198011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2059D3
hypocenterC3#29643
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. #71744C2semidirect
7. #70618C2semidirect
8. #70566C2semidirect
9. #69606C2semidirect
10. #68911C2semidirect
11. #68911C2semidirect
14. #68690C2semidirect
71. #2059D3semidirect
652. C3#29643semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1252
C2×#48842
C2×#48622
#762872
C3×#20592
#717442
#706182
#705662
#696062
#689112
#688722
#688712
#686902
#686642
#296433