Finite Group Databasebeta

← #10844740

Subgroups — #10844740

Order 768.

800 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×#5185C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#76555C211yesmaximal, characteristic
4#76547C211yesmaximal, characteristic, complemented
5C3×#2056C211yesmaximal, characteristic, complemented
6#72066C211yesmaximal, characteristic, complemented
7#70484C211yesmaximal, characteristic
8#70280C211yesmaximal, characteristic
9#70261C211yesmaximal, characteristic, complemented
10#70226C211yesmaximal, characteristic
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69676C211yesmaximal, characteristic
13#69674C211yesmaximal, characteristic, complemented
14#68853C211yesmaximal, characteristic
15#68798C211yesmaximal, characteristic, complemented
16#25297—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5185—24no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#4988C2211yescharacteristic
41#4984C2211yescharacteristic
42#4977C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4939C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4×C3⋊C8C2211yescharacteristic
49#4830—24no—
50#4675—21no—
51#4673—21no—
52#4670—21no—
53#4669—21no—
54#4660—21no—
55#4656—21no—
56#4655—21no—
57#4654—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×C8×Q8—31no—
64#2592—31no—
65#2581—31no—
66#2067—31no—
67#2066—31no—
68#2056D311yescharacteristic, complemented
69#2055—31no—
70#1969—31no—
71#1968—31no—
72#1958—31no—
73#1949—31no—
74#1934—31no—
75#1933—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×Dic6D411yescharacteristic
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—41no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×C4⋊C8—22no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126#835—22no—
127#835—22no—
128#835—22no—
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⋊#95—21no—
138#821—22no—
139#821—41no—
140#818—41no—
141Dic3⋊C8—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144C8×Dic3—22no—
145#809—22no—
146#808—41no—
147C4×C3⋊C8—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153#464—64no—
154C8×Q8—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
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×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171#400—61no—
172#397—61no—
173#395—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391C2×D311yescharacteristic
178#391C2×D311yescharacteristic
179#390—31no—
180#390—31no—
181#389—31no—
182#389—31no—
183#388—61no—
184#388—61no—
185#387—61no—
186#387—61no—
187#382—31no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
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×Q8—21no—
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—21no—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C3×Q16—44no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D412yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C42—21no—
241C3×C42—21no—
242#215—44no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—81no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—42no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—42no—
264C4×Dic3—81no—
265C4×Dic3C2×D412yes—
266C4×Dic3#4112yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×Q8C22×D311yescharacteristic
276C22×D4—31no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—41no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—121no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—62no—
288#124—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—121no—
317C4⋊C8—22no—
318C4⋊C8—34no—
319C4⋊C8—62no—
320#103—22no—
321#103—41no—
322#103—62no—
323#103—62no—
324#103—62no—
325#102—62no—
326#102—62no—
327#102—121no—
328C4×C8—62no—
329C4×C8—62no—
330C4×C8—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
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—41no—
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—41no—
355C3×Q8—41no—
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—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
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—41no—
377C2×C3×C4#13512yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
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—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3#12112yes—
403C2×Dic3#12112yes—
404C2×Dic3#13512yes—
405D12—41no—
406D12—41no—
407D12—41no—
408D12—41no—
409Dic6—28no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
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—21no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8#23712yes—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—44no—
451SD16—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8D3×D412yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—241no—
485C22⋊C4—241no—
486C42—21no—
487C42—21no—
488C42—32no—
489C42—32no—
490C42—61no—
491C42—61no—
492C42—122no—
493C42—241no—
494C22×C3C2×#12111yescharacteristic
495C22×C3#54811yescharacteristic
496C22×C3#55011yescharacteristic
497C22×C3C2×#13511yescharacteristic
498C22×C3C2×#13511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#54211yescharacteristic
516C3×C4#54611yescharacteristic
517C3×C4#54811yescharacteristic
518Dic3—22no—
519Dic3—22no—
520Dic3—22no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—68no—
542D4—121no—
543D4—121no—
544D4—121no—
545D4—121no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4#91912yes—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4#100411yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4#101011yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—62no—
595C2×C3#242611yescharacteristic
596C2×C3#305011yescharacteristic
597C2×C3#306311yescharacteristic
598C2×C3#336411yescharacteristic
599C2×C3#336511yescharacteristic
600C2×C3#339311yescharacteristic
601C2×C3#339411yescharacteristic
602D3—161no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#559811yescharacteristic
611C22#559911yescharacteristic
612C22D3×#12111yescharacteristic
613C22C2×#91911yescharacteristic
614C22D3×#13511yescharacteristic
615C22#577811yescharacteristic
616C22#577911yescharacteristic
617C4—22no—
618C4—41no—
619C4—42no—
620C4—62no—
621C4—62no—
622C4—62no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#559111yescharacteristic
631C4#564911yescharacteristic
632C4#566211yescharacteristic
633C4#567711yescharacteristic
634C3#2529711yescharacteristic, complemented, Sylow
635C2—481no—
636C2#7690111yescharacteristic
637C2#8046311yescharacteristic
638C2#8047211yescharacteristic
639C2#8058111yescharacteristic
640C2#8058811yescharacteristic
641C2#8075811yescharacteristic
642C2#8090711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2056D3
hypocenterC3#25297
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
4. #76547C2semidirect
5. C3×#2056C2semidirect
6. #72066C2semidirect
9. #70261C2semidirect
11. Dic3×#103C2semidirect
13. #69674C2semidirect
15. #68798C2semidirect
68. #2056D3semidirect
634. C3#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#51852
C2×#48302
#765552
#765472
C3×#20562
#720662
#704842
#702802
#702612
#702262
Dic3×#1032
#696762
#696742
#688532
#687982
#252973