Finite Group Databasebeta

← #10824582

Subgroups — #10824582

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
2#82455C211yesmaximal, characteristic
3#82454C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic
5#72072C211yesmaximal, characteristic
6#72068C211yesmaximal, characteristic, complemented
7#71631C211yesmaximal, characteristic, complemented
8#70639C211yesmaximal, characteristic
9#70619C211yesmaximal, characteristic, complemented
10#70280C211yesmaximal, characteristic
11#70262C211yesmaximal, characteristic, complemented
12#70245C211yesmaximal, characteristic
13#70233C211yesmaximal, characteristic, complemented
14C4×#821C211yesmaximal, characteristic
15#69755C211yesmaximal, characteristic, complemented
16#23695—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×C4×Q16—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#5026—24no—
39#4989—21no—
40#4989—21no—
41#4988C2211yescharacteristic
42#4984C2211yescharacteristic
43#4984C2211yescharacteristic
44#4980—21no—
45#4980—21no—
46#4975C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4940—21no—
50#4940—21no—
51#4937—21no—
52#4936—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54#4932C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440—31no—
63C2×C4×Q16D311yescharacteristic, complemented
64#2326—31no—
65#2320—31no—
66#2083—31no—
67#2072—31no—
68#2068—31no—
69#2067—31no—
70#2056—31no—
71#2055—31no—
72#2042—31no—
73#2015—31no—
74#1768—31no—
75#1763—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97#947—22no—
98#947—22no—
99C2×#215—22no—
100C2×#215—22no—
101C2×#215—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C4×D12—22no—
120C4×D12—22no—
121C3×#107—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×C4×C8—22no—
125#835—22no—
126#835—22no—
127#835—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—22no—
142#821—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#809—22no—
146#807—22no—
147#803—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154#440—64no—
155C4×Q16—24no—
156C2×#107—31no—
157C2×#107C2×D311yescharacteristic
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#391—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#383—61no—
184#382—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C2×C4×D3—81no—
229C3×Q16—28no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C42—24no—
243C3×C42—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#215—28no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251(C2×D3)⋊C4—81no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—81no—
259C4⋊Dic3—81no—
260Dic3⋊C4—81no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8C22×D311yescharacteristic
274C22×Q8C22×D311yescharacteristic
275C22×D4—31no—
276C23×C4—61no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—22no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—62no—
288#124—62no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8#6714yes—
293C4×D4—62no—
294C4×D4—62no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—22no—
318#107—62no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—22no—
322#103—22no—
323#103—62no—
324#103—62no—
325#103—62no—
326#102—62no—
327#102—62no—
328#102—62no—
329#97—62no—
330C4×C8—22no—
331#95—31no—
332#95—31no—
333#95—31no—
334#95—31no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—84no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—61no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#23512yes—
426C2×Q8C2×#6712yes—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—28no—
451SD16—68no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8#23512yes—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—241no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C4⋊C4C2×#6712yes—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—24no—
486C42—42no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—241no—
492C42#23712yes—
493C42C2×#6712yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53211yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57811yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—81no—
514C3×C4#52511yescharacteristic
515C3×C4#52511yescharacteristic
516C3×C4#52911yescharacteristic
517C3×C4#52911yescharacteristic
518Dic3—41no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#93912yes—
584C2×C4#93912yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×#23711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4C22×#6711yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—121no—
595C2×C3#239211yescharacteristic
596C2×C3#304411yescharacteristic
597C2×C3#304511yescharacteristic
598C2×C3#312011yescharacteristic
599C2×C3#312411yescharacteristic
600C2×C3#317511yescharacteristic
601C2×C3#317811yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#554711yescharacteristic
611C22#554811yescharacteristic
612C22#576611yescharacteristic
613C22#576611yescharacteristic
614C22#576911yescharacteristic
615C22#577011yescharacteristic
616C22C2×#93911yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—81no—
623C4—121no—
624C4—121no—
625C4—121no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#553811yescharacteristic
631C4#555711yescharacteristic
632C4#579111yescharacteristic
633C4#582811yescharacteristic
634C3#2369511yescharacteristic, complemented, Sylow
635C2—242no—
636C2#7972711yescharacteristic
637C2#7972911yescharacteristic
638C2#7974811yescharacteristic
639C2#7974911yescharacteristic
640C2#8234511yescharacteristic
641C2#8387711yescharacteristic
642C2#8390111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×C4×Q16D3
hypocenterC3#23695
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q16C2semidirect
3. #82454C2semidirect
6. #72068C2semidirect
7. #71631C2semidirect
9. #70619C2semidirect
11. #70262C2semidirect
13. #70233C2semidirect
15. #69755C2semidirect
63. C2×C4×Q16D3semidirect
634. C3#23695semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
#824552
#824542
C2×#50262
#720722
#720682
#716312
#706392
#706192
#702802
#702622
#702452
#702332
C4×#8212
#697552
#236953