Finite Group Databasebeta

← #10858218

Subgroups — #10858218

Order 768.

864 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×#4851C211yesmaximal, characteristic, complemented
2C2×C4×#204C211yesmaximal, characteristic
3#73675C211yesmaximal, characteristic
4C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
5C3×#1916C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic
7#69769C211yesmaximal, characteristic, complemented
8#68686C211yesmaximal, characteristic
9#68686C211yesmaximal, characteristic
10#68684C211yesmaximal, characteristic, complemented
11#68684C211yesmaximal, characteristic, complemented
12#68682C211yesmaximal, characteristic
13#68671C211yesmaximal, characteristic
14#68640C211yesmaximal, characteristic, complemented
15#68633C211yesmaximal, characteristic, complemented
16#26373—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#896—21no—
19C2×C4×D12C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4994—21no—
39#4975—21no—
40Dic3×C4⋊C4—21no—
41#4939C2211yescharacteristic
42#4939C2211yescharacteristic
43#4935C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4851—24no—
49C4×#204—24no—
50#4673—21no—
51#4668—21no—
52#4647—21no—
53#4647—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#475—31no—
63C2×C4×SD16—31no—
64#2452—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2084—31no—
68#2065—31no—
69#2058—31no—
70#1941—31no—
71#1916D311yescharacteristic, complemented
72#1880—31no—
73#1880—31no—
74#1865—31no—
75#1865—31no—
76#1864—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
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×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204Q814yescomplemented
117#896—44no—
118C4×D12—22no—
119C4×D12—41no—
120#869—22no—
121#869—41no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C4×Dic6—41no—
128C3×C4⋊C8—22no—
129C3×C4×C8—22no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—22no—
138#821—41no—
139#817—22no—
140#817—22no—
141#817—22no—
142#817—22no—
143#817—41no—
144#816—22no—
145#816—22no—
146#816D414yes—
147C22×SD16—31no—
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#475—64no—
155C4×SD16—64no—
156C2×#106—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8C2×D311yescharacteristic
169#396—61no—
170#393—61no—
171#390—31no—
172#390—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#385—61no—
180#384—31no—
181#384—61no—
182C4×C4⋊C4—61no—
183C4×C4⋊C4C2×D311yescharacteristic
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#336C2×D311yescharacteristic
189#336C2×D311yescharacteristic
190#336C2×D311yescharacteristic
191#336C2×D311yescharacteristic
192C23×D3—21no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×Q812yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—42no—
220C2×Dic6—42no—
221C2×Dic6—81no—
222C2×Dic6C2×D412yes—
223C2×Dic6C2×Q812yes—
224C2×Dic6#4112yes—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D412yes—
236C2×C3×C8C2×Q812yes—
237C3×C4⋊C4—41no—
238C3×C4⋊C4SD1614yescomplemented
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262C4⋊Dic3—81no—
263C4⋊Dic3C2×D412yes—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—24no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271Dic3⋊C4—81no—
272C4×Dic3—24no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275#204—28no—
276C22×Q8—31no—
277C22×Q8—61no—
278C22×D4—31no—
279C23×C4—61no—
280C2×SD16—34no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—121no—
289C22⋊Q8—124no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—121no—
295C4×Q8—121no—
296C4×D4—62no—
297C4×D4—121no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4D1211yescharacteristic
315C2×C4⋊C4D1211yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—62no—
329#106—62no—
330#106—62no—
331#106—121no—
332C4⋊C8—22no—
333#103—34no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—121no—
339C4×C8—22no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95D1211yescharacteristic
345#95D1211yescharacteristic
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4C2×SD1612yes—
370C2×C3×C4C2×SD1612yes—
371C2×C3×C4C2×SD1612yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×Q811yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399D12—22no—
400D12—22no—
401D12—41no—
402D12—41no—
403C4×D3—84no—
404Dic6—22no—
405Dic6—22no—
406Dic6—22no—
407Dic6—84no—
408Dic6C22⋊Q814yescomplemented
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—41no—
412C3×C8—41no—
413C24—61no—
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—122no—
421C2×Q8—122no—
422C2×Q8—241no—
423C2×D4—32no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4—241no—
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×D1211yescharacteristic
442C22×C4C2×D1211yescharacteristic
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C23×D311yescharacteristic
446SD16—68no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8D3×D412yes—
458C2×C8D3×Q812yes—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—41no—
462C4⋊C4—64no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4#20414yescomplemented
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—241no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—64no—
495C42—122no—
496C42—122no—
497C22×C3#52511yescharacteristic
498C22×C3D4211yescharacteristic
499C22×C3D4×Q811yescharacteristic
500C22×C3C22×SD1611yescharacteristic
501C22×C3#57511yescharacteristic
502C22×C3#57711yescharacteristic
503C22×C3#57711yescharacteristic
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4C2×C22⋊Q811yescharacteristic
518C3×C4#53911yescharacteristic
519C3×C4#54711yescharacteristic
520C3×C4#55411yescharacteristic
521Dic3—24no—
522Dic3—42no—
523Dic3—42no—
524Dic3—81no—
525Dic3—81no—
526Dic3—81no—
527Dic3—81no—
528C23—62no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23C22×D1211yescharacteristic
536Q8—34no—
537Q8—62no—
538Q8—62no—
539Q8—62no—
540Q8—244no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
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—244no—
582C2×C4C2×#20412yes—
583C2×C4C2×#20412yes—
584C2×C4C2×#20412yes—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×Q811yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4#101011yescharacteristic
591C8—22no—
592C8—22no—
593C8—41no—
594C8—41no—
595C2×C3#244911yescharacteristic
596C2×C3#310311yescharacteristic
597C2×C3#312511yescharacteristic
598C2×C3#327111yescharacteristic
599C2×C3D4×SD1611yescharacteristic
600C2×C3Q8×SD1611yescharacteristic
601C2×C3#338811yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#552411yescharacteristic
611C22D4×D1211yescharacteristic
612C22Q8×D1211yescharacteristic
613C22C22×#20411yescharacteristic
614C22#574811yescharacteristic
615C22#575111yescharacteristic
616C22#575111yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—64no—
624C4—122no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4D3×C22⋊Q811yescharacteristic
631C4#565511yescharacteristic
632C4#569011yescharacteristic
633C4#570711yescharacteristic
634C3#2637311yescharacteristic, complemented, Sylow
635C2—242no—
636C2#7532111yescharacteristic
637C2#7930911yescharacteristic
638C2#7938911yescharacteristic
639C2D4×#20411yescharacteristic
640C2#7950811yescharacteristic
641C2Q8×#20411yescharacteristic
642C2#7997811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1916D3
hypocenterC3#26373
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
1. C2×#4851C2semidirect
4. C4⋊C4×Dic6C2semidirect
5. C3×#1916C2semidirect
7. #69769C2semidirect
10. #68684C2semidirect
11. #68684C2semidirect
14. #68640C2semidirect
15. #68633C2semidirect
71. #1916D3semidirect
116. C2×#204Q8semidirect
238. C3×C4⋊C4SD16semidirect
408. Dic6C22⋊Q8semidirect
482. C4⋊C4#204semidirect
634. C3#26373semidirect

Maximal subgroups

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

subgroupindex
C2×#48512
C2×C4×#2042
#736752
C4⋊C4×Dic62
C3×#19162
#704832
#697692
#686862
#686842
#686822
#686712
#686402
#686332
#263733