Finite Group Databasebeta

← #10858220

Subgroups — #10858220

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×#4854C211yesmaximal, characteristic, complemented
2C2×C4×#204C211yesmaximal, characteristic
3#76010C211yesmaximal, characteristic
4#75981C211yesmaximal, characteristic, complemented
5C3×#1941C211yesmaximal, characteristic, complemented
6#70588C211yesmaximal, characteristic, complemented
7#70587C211yesmaximal, characteristic
8#70585C211yesmaximal, characteristic
9#69784C211yesmaximal, characteristic, complemented
10#69632C211yesmaximal, characteristic, complemented
11#69626C211yesmaximal, characteristic
12#68909C211yesmaximal, characteristic
13#68909C211yesmaximal, characteristic
14#68835C211yesmaximal, characteristic, complemented
15#68827C211yesmaximal, characteristic, complemented
16#26373—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#896—21no—
19C2×#888C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4989—21no—
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37#4976—21no—
38#4975—21no—
39#4975C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4854—24no—
49C4×#204—24no—
50#4676—21no—
51#4673—21no—
52#4649—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
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#1941D311yescharacteristic, complemented
71#1916—31no—
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
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
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⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×Dic3⋊C4D411yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×#204—22no—
115C2×#204—22no—
116C2×#204Q814yescomplemented
117#896—44no—
118#888—22no—
119#888—41no—
120#887—22no—
121#887—22no—
122#887—22no—
123#887—41no—
124C4×D12—22no—
125C4×D12—41no—
126C4×Dic6—22no—
127C4×Dic6—41no—
128C3×#106—22no—
129C3×#106—22no—
130C3×C4×C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#817—22no—
139#816D414yes—
140Dic3⋊C8—22no—
141#810—22no—
142#810—22no—
143#809—22no—
144#809—41no—
145#808—22no—
146#808—41no—
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×#106C2×D311yescharacteristic
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
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—61no—
176#389—61no—
177#389C2×D311yescharacteristic
178#389C2×D311yescharacteristic
179#385—61no—
180#384—31no—
181#384—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184#340—31no—
185#340—31no—
186#340C2×D311yescharacteristic
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C23×D3—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
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×C8C2×D412yes—
230C2×C3×C8C2×Q812yes—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4#4112yes—
241C3×C42—21no—
242C3×C42—21no—
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—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—81no—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—41no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—81no—
261Dic3⋊C4SD1614yescomplemented
262C4×Dic3—24no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
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×C8—31no—
284C22×C8—31no—
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—21no—
299C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—22no—
328#106—22no—
329#106—62no—
330#106—62no—
331#106—121no—
332C4⋊C8—62no—
333#103—34no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—121no—
339C4×C8—22no—
340#95—21no—
341#95—21no—
342#95—31no—
343#95—31no—
344#95—61no—
345#95—61no—
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—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
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—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3C2×SD1612yes—
398C2×Dic3C2×SD1612yes—
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—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—241no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446SD16—68no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8D3×D412yes—
458C2×C8D3×Q812yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—34no—
467C4⋊C4—32no—
468C4⋊C4—42no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—81no—
474C4⋊C4—121no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4#23512yes—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—241no—
488C42—21no—
489C42—21no—
490C42—64no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—61no—
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—42no—
513C3×C4—41no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4C2×C22⋊Q811yescharacteristic
518C3×C4#53911yescharacteristic
519C3×C4#54711yescharacteristic
520C3×C4#55411yescharacteristic
521Dic3—22no—
522Dic3—24no—
523Dic3—22no—
524Dic3—41no—
525Dic3—41no—
526Dic3—42no—
527Dic3—81no—
528C23—62no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23C2×D3×D411yescharacteristic
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—22no—
550C2×C4—21no—
551C2×C4—32no—
552C2×C4—32no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4D3×SD1612yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×D3×Q811yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4#100911yescharacteristic
591C8—22no—
592C8—22no—
593C8—121no—
594C8—121no—
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#554711yescharacteristic
611C22#560911yescharacteristic
612C22#572711yescharacteristic
613C22C2×D3×SD1611yescharacteristic
614C22#576211yescharacteristic
615C22#576811yescharacteristic
616C22#577011yescharacteristic
617C4—22no—
618C4—42no—
619C4—41no—
620C4—62no—
621C4—62no—
622C4—64no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—122no—
627C4—121no—
628C4—121no—
629C4—241no—
630C4#550711yescharacteristic
631C4#557911yescharacteristic
632C4D3×C22⋊Q811yescharacteristic
633C4#569211yescharacteristic
634C3#2637311yescharacteristic, complemented, Sylow
635C2—242no—
636C2#7622511yescharacteristic
637C2#7965311yescharacteristic
638C2#7966011yescharacteristic
639C2#8067611yescharacteristic
640C2#8068811yescharacteristic
641C2#8190311yescharacteristic
642C2#8193011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1941C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1941D3
hypocenterC3#26373
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
1. C2×#4854C2semidirect
4. #75981C2semidirect
5. C3×#1941C2semidirect
6. #70588C2semidirect
9. #69784C2semidirect
10. #69632C2semidirect
14. #68835C2semidirect
15. #68827C2semidirect
70. #1941D3semidirect
116. C2×#204Q8semidirect
261. Dic3⋊C4SD16semidirect
408. Dic6C22⋊Q8semidirect
634. C3#26373semidirect

Maximal subgroups

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

subgroupindex
C2×#48542
C2×C4×#2042
#760102
#759812
C3×#19412
#705882
#705872
#705852
#697842
#696322
#696262
#689092
#688352
#688272
#263733