Finite Group Databasebeta

← #10831780

Subgroups — #10831780

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic6C211yesmaximal, characteristic
2C2×#5184C211yesmaximal, characteristic
3C2×#4791C211yesmaximal, characteristic
4#76547C211yesmaximal, characteristic
5C3×#1878C211yesmaximal, characteristic
6#72065C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic
8#70238C211yesmaximal, characteristic
9#70231C211yesmaximal, characteristic
10#70222C211yesmaximal, characteristic
11#69668C211yesmaximal, characteristic
12#68827C211yesmaximal, characteristic
13#68806C211yesmaximal, characteristic
14#68633C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic
16#24345—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22Q8×Dic6—24no—
23Q8×Dic6—24no—
24C2×#888—21no—
25C2×#888—21no—
26C2×#888C2211yescharacteristic
27C2×#888C2211yescharacteristic
28C2×#869—21no—
29C2×#869C2211yescharacteristic
30C2×C4×Dic6—21no—
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C3×#391C2211yescharacteristic
35#5229C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40#5184—24no—
41C2×#817C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984C2211yescharacteristic
46#4977C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4928C2211yescharacteristic
49#4791—24no—
50#4660—21no—
51#4656—21no—
52#4655—21no—
53#4654—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×Q82—31no—
65C2×#475—31no—
66C2×#441—31no—
67#2581—31no—
68#2058—31no—
69#2058—31no—
70#2024—31no—
71#2024—31no—
72#1993—31no—
73#1985—31no—
74#1878—31no—
75#1878D311yescharacteristic
76#1865—31no—
77#1865—31no—
78#1805—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×Dic6—21no—
82C22×Dic6—21no—
83C22×Dic6—21no—
84C22×Dic6D411yescharacteristic
85C2×C3×Q16—22no—
86C2×C3×Q16—22no—
87C2×C3×Q16Q814yescomplemented
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102Q8×Dic3D414yes—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
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⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#888—28no—
126#888—28no—
127#888—22no—
128#888—22no—
129#888—22no—
130#888D414yes—
131#869—28no—
132#869—22no—
133#869—22no—
134C4×Dic6—28no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#95—21no—
140C3×#95—21no—
141#835—22no—
142#835—22no—
143#835—22no—
144#835—22no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#817—22no—
152#817—22no—
153#815—22no—
154Dic3⋊C8—22no—
155#808—22no—
156C22×Q16C2×D311yescharacteristic
157Q82—64no—
158Q82—64no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—61no—
163C2×C4⋊Q8—61no—
164C2×C4⋊Q8—61no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—61no—
170C2×C4×Q8C2×D311yescharacteristic
171#475—64no—
172#441—64no—
173C2×#106—31no—
174C2×#106—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103C2×D311yescharacteristic
182C2×#103C2×D311yescharacteristic
183C2×#97—31no—
184#400—61no—
185#395—61no—
186#391—31no—
187#391C2×D311yescharacteristic
188#389—31no—
189#389—31no—
190#387—61no—
191#387—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C2×C3×Q8C2×Q812yes—
213C2×C3×Q8C2×Q812yes—
214C2×C3×Q8#4112yes—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C2×Dic6—24no—
234C2×Dic6—24no—
235C2×Dic6—24no—
236C2×Dic6—81no—
237C2×Dic6—81no—
238C2×Dic6—81no—
239C2×Dic6—81no—
240C2×Dic6—81no—
241C2×Dic6—81no—
242C3×Q16—28no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×Q812yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4C2×D412yes—
254C3×C42—24no—
255C3×C42—21no—
256C3×C42—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3—24no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—22no—
262C4⋊Dic3—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—42no—
268C4⋊Dic3—42no—
269C4⋊Dic3C2×D412yes—
270C4⋊Dic3C2×D412yes—
271Dic3⋊C4—24no—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—24no—
275Dic3⋊C4—42no—
276C4×Dic3—24no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—24no—
280C4×Dic3—42no—
281C4×Dic3C2×D412yes—
282C4×Dic3C2×D412yes—
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—31no—
290C22×Q8—61no—
291C22×Q8—61no—
292C22×Q8—61no—
293C22×Q8C22×D311yescharacteristic
294C22×Q8C22×D311yescharacteristic
295C2×Q16—22no—
296C2×Q16—22no—
297C2×Q16Dic614yescomplemented
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C4⋊Q8—34no—
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305C4⋊Q8—68no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—68no—
309C4⋊Q8—68no—
310C4⋊Q8—62no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—34no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—68no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C42—21no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—61no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42C22×D311yescharacteristic
347#106—62no—
348#106—62no—
349#106—62no—
350C4⋊C8—62no—
351#103—22no—
352#103—22no—
353#103—62no—
354#103—62no—
355#103—62no—
356#103—62no—
357#97—62no—
358#95—21no—
359#95—21no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3C22×D411yescharacteristic
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—22no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8C22⋊Q814yes—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×Q811yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3#12012yes—
421C2×Dic3#12012yes—
422C2×Dic3#13712yes—
423Dic6—28no—
424Dic6—84no—
425Dic6—84no—
426Dic6—84no—
427C3×C8—22no—
428C3×C8—41no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—62no—
440C2×Q8—62no—
441C2×Q8—64no—
442C2×Q8—64no—
443C2×Q8—64no—
444C2×Q8—62no—
445C2×Q8—241no—
446C2×Q8—241no—
447C2×Q8—241no—
448C2×Q8—241no—
449C2×Q8—241no—
450C2×Q8—241no—
451C2×Q8C2×Dic612yes—
452C2×Q8C2×Dic612yes—
453C2×Q8D3×D412yes—
454C2×Q8#23712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470Q16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8C2×Dic612yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—32no—
488C4⋊C4—32no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
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—61no—
502C4⋊C4—61no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4D3×D412yes—
507C42—24no—
508C42—21no—
509C42—21no—
510C42—32no—
511C42—32no—
512C42—64no—
513C42—64no—
514C42—61no—
515C42—61no—
516C42—122no—
517C22×C3C2×#12011yescharacteristic
518C22×C3#54811yescharacteristic
519C22×C3D4×Q811yescharacteristic
520C22×C3C2×#13711yescharacteristic
521C22×C3C2×#13711yescharacteristic
522C22×C3#57711yescharacteristic
523C22×C3#57711yescharacteristic
524C3×C4—24no—
525C3×C4—22no—
526C3×C4—42no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—81no—
530C3×C4C2×C22⋊Q811yescharacteristic
531C3×C4#53611yescharacteristic
532C3×C4#54811yescharacteristic
533C3×C4D4×Q811yescharacteristic
534Dic3—22no—
535Dic3—22no—
536Dic3—22no—
537Dic3—41no—
538Dic3—42no—
539Dic3—42no—
540Dic3—81no—
541Dic3—81no—
542C23C2×D3×D411yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—68no—
551Q8—244no—
552Q8—244no—
553Q8—244no—
554Q8#87814yes—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—32no—
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—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4#91812yes—
597C2×C4C22×Dic611yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4#101011yescharacteristic
602C2×C4#101011yescharacteristic
603C8—22no—
604C8—41no—
605C8—121no—
606C8—121no—
607C2×C3#240911yescharacteristic
608C2×C3#300411yescharacteristic
609C2×C3#300911yescharacteristic
610C2×C3#336911yescharacteristic
611C2×C3#336911yescharacteristic
612C2×C3#338711yescharacteristic
613C2×C3#338711yescharacteristic
614C22D4×Dic611yescharacteristic
615C22D3×#12011yescharacteristic
616C22#559811yescharacteristic
617C22C2×#91811yescharacteristic
618C22#577011yescharacteristic
619C22D3×#13711yescharacteristic
620C22#577811yescharacteristic
621C4—22no—
622C4—24no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—62no—
627C4—62no—
628C4—62no—
629C4—81no—
630C4—121no—
631C4—122no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4C2×#87811yescharacteristic
636C4D4×Dic611yescharacteristic
637C4#559811yescharacteristic
638C4#564811yescharacteristic
639C3#2434511yescharacteristic, complemented, Sylow
640C2#7468911yescharacteristic
641C2#7970811yescharacteristic
642C2#7981911yescharacteristic
643C2#8018311yescharacteristic
644C2#8022911yescharacteristic
645C2#8046411yescharacteristic
646C2#8058211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socle, prime radicalC23×C3C22×D4
hypercenter#1878D3
hypocenterC3#24345
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
87. C2×C3×Q16Q8semidirect
297. C2×Q16Dic6semidirect
639. C3#24345semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic62
C2×#51842
C2×#47912
#765472
C3×#18782
#720652
#716292
#702382
#702312
#702222
#696682
#688272
#688062
#686332
#686202
#243453