Finite Group Databasebeta

← #10967454

Subgroups — #10967454

Order 768.

3270 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic, complemented
2C2×C4×#235C211yesmaximal, characteristic, complemented
3C3×#2283C211yesmaximal, characteristic, complemented
4#74307C214yesmaximal, complemented
5#74239C212yesmaximal, complemented
6#74190C212yesmaximal, complemented
7#74129C212yesmaximal, complemented
8#74125C212yesmaximal, complemented
9#74028C212yesmaximal, complemented
10#74016C212yesmaximal, complemented
11C4×#895C214yesmaximal, complemented
12C4×#882C214yesmaximal, complemented
13#73419C212yesmaximal, complemented
14#73418C212yesmaximal, complemented
15#34963—31nomaximal, Sylow
16C22×#235C2211yescharacteristic
17C23×D12C2211yescharacteristic
18C23×C4×D3C2212yes—
19C22×C3×C42C2211yescharacteristic
20C2×C4×#67C2212yes—
21C2×C4×#67C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C2×#895C2214yes—
26C2×#882C2214yes—
27C4×#235—28no—
28C2×C4×D12—28no—
29C2×C4×D12C2212yes—
30C2×C4×D12C2212yes—
31C2×C4×D12C2212yes—
32C2×C4×D12C2212yes—
33C2×C4×D12C2211yescharacteristic, complemented
34C2×C4×D12C2211yescharacteristic, complemented
35C2×C4×D12C2218yescomplemented
36C2×#872C2212yescomplemented
37C2×C42×D3C2212yescomplemented
38C2×C4×Dic6C2211yescharacteristic, complemented
39C2×C4×Dic6C2211yescharacteristic, complemented
40C3×#383C2212yescomplemented
41C3×#382C2212yescomplemented
42C3×#381C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2212yescomplemented
45C3×C4×C22⋊C4C2212yes—
46C3×#376C2212yescomplemented
47#4988C2214yes—
48#4987C2214yescomplemented
49#4984C2214yescomplemented
50#4958C2214yes—
51#4955C2214yes—
52#4950C2214yes—
53C4×(C2×D3)⋊C4C2218yescomplemented
54#4677C2214yescomplemented
55#4677C2214yescomplemented
56#4674C2214yescomplemented
57#4673C2214yescomplemented
58#4672C2214yes—
59#4671C2214yes—
60#4671C2214yescomplemented
61#4670C2218yescomplemented
62#4668C2218yescomplemented
63#4665C2218yescomplemented
64#4665C2218yescomplemented
65#4659C2214yescomplemented
66C2×C4×#41—31no—
67C22×C4×D4—31no—
68#2598—34no—
69#2580—32no—
70#2535—32no—
71#2487—32no—
72#2482—32no—
73#2408—32no—
74#2404—32no—
75#2398—32no—
76#2397—32no—
77C4×#121—34no—
78C4×#120—34no—
79#2283D311yescharacteristic, complemented
80C24×D3C2312yes—
81C23×C3×C4C2311yescharacteristic
82C23×C3×C4C2311yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C22×#67C2312yes—
85C23×Dic3C2312yes—
86C2×#235—24no—
87C2×#235—28no—
88C22×D12—24no—
89C22×D12—28no—
90C22×D12C2312yes—
91C22×D12C2311yescharacteristic
92C22×D12C2318yes—
93C22×D12C2312yes—
94C22×C4×D3—28no—
95C22×C4×D3—48no—
96C22×C4×D3C2312yes—
97C22×C4×D3C2314yes—
98C22×C4×D3C2312yes—
99C22×C4×D3C2312yes—
100C22×C4×D3C2314yes—
101C22×Dic6C2311yescharacteristic
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C42D418yescomplemented
105C2×C3×C42C2314yes—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×#217C2312yes—
111C4×#67—28no—
112C4×#67—28no—
113C2×(C2×D3)⋊C4—28no—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2318yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2318yes—
121C2×(C2×D3)⋊C4C2318yes—
122#926—24no—
123C2×C4⋊Dic3—24no—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2318yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2312yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2312yes—
134#895—216no—
135#882—216no—
136C4×D12—28no—
137C4×D12—28no—
138C4×D12—28no—
139C4×D12—28no—
140C4×D12—24no—
141C4×D12—24no—
142C4×D12—416no—
143#872—28no—
144C42×D3—28no—
145C4×Dic6—24no—
146C4×Dic6—24no—
147C3×#95C2314yes—
148C3×#95C2314yes—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3⋊#95C2318yes—
152C3⋊#95C2318yes—
153C22×#41—31no—
154C23×D4—31no—
155C24×C4—32no—
156C2×#121—34no—
157C2×#120—34no—
158C4×#41—68no—
159C2×C4×Q8—31no—
160C2×C4×Q8—31no—
161C2×C4×D4—32no—
162C2×C4×D4—32no—
163C2×C4×D4—32no—
164C2×C4×D4—32no—
165C2×C4×D4—31no—
166C2×C4×D4—32no—
167C2×C4×D4—32no—
168C2×C4×D4—31no—
169C2×C4×D4—38no—
170C2×C4×D4—68no—
171C2×#117—31no—
172C2×#117—32no—
173C22×C4⋊C4—31no—
174C22×C22⋊C4—32no—
175C22×C42—32no—
176C22×C42C2×D311yescharacteristic
177#400—34no—
178#399—34no—
179#399—34no—
180#397—34no—
181#394—34no—
182#393—34no—
183#392—34no—
184#392—34no—
185#390—34no—
186#389—34no—
187#388—38no—
188#386—34no—
189#386—38no—
190#386—38no—
191#385—34no—
192#385—38no—
193#383C2×D312yescomplemented
194#382C2×D312yescomplemented
195#381C2×D312yes—
196#380—34no—
197#380C2×D312yes—
198#379—34no—
199C4×C4⋊C4C2×D312yescomplemented
200C4×C22⋊C4—38no—
201C4×C22⋊C4C2×D312yes—
202#376C2×D312yescomplemented
203C24×C3C2411yescharacteristic
204C23×D3—24no—
205C23×D3—28no—
206C23×D3—28no—
207C23×D3—44no—
208C23×D3C2414yes—
209C23×D3C2412yes—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4C2×D414yes—
214C22×C3×C4C2×D414yes—
215C22×C3×C4C2×D414yes—
216C22×C3×C4C2418yes—
217C22×C3×C4C2411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C2×#67—24no—
224C2×#67—24no—
225C2×#67#4118yes—
226C22×Dic3—24no—
227C22×Dic3—44no—
228C22×Dic3C2412yes—
229C22×Dic3C2414yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232#235—232no—
233C2×D12—232no—
234C2×D12—28no—
235C2×D12—24no—
236C2×D12—24no—
237C2×D12—24no—
238C2×D12—22no—
239C2×D12—24no—
240C2×D12—216no—
241C2×D12—216no—
242C2×D12—48no—
243C2×D12#4118yescomplemented
244C2×C4×D3—24no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—24no—
249C2×C4×D3—28no—
250C2×C4×D3—28no—
251C2×C4×D3—28no—
252C2×C4×D3—28no—
253C2×C4×D3—216no—
254C2×C4×D3—48no—
255C2×C4×D3—432no—
256C2×Dic6—22no—
257C2×Dic6—24no—
258C3×C4⋊C4—28no—
259C3×C22⋊C4—28no—
260C3×C42—28no—
261C3×C42C2×D414yes—
262C3×C42C2×D414yes—
263C3×C42C2×D414yes—
264C3×C42C2×D414yes—
265#217—24no—
266(C2×D3)⋊C4—216no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—28no—
271(C2×D3)⋊C4—216no—
272(C2×D3)⋊C4—48no—
273(C2×D3)⋊C4—48no—
274C4⋊Dic3—22no—
275C4⋊Dic3—22no—
276C4⋊Dic3—24no—
277C4⋊Dic3—24no—
278C4⋊Dic3—48no—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C25—32no—
285C2×#41—68no—
286C2×#41—64no—
287C22×Q8—31no—
288C22×D4—32no—
289C22×D4—32no—
290C22×D4—32no—
291C22×D4—31no—
292C22×D4—38no—
293C22×D4—68no—
294C22×D4—64no—
295C23×C4—32no—
296C23×C4—32no—
297C23×C4—32no—
298C23×C4—32no—
299C23×C4—34no—
300C23×C4—34no—
301C23×C4—68no—
302C23×C4—128no—
303C23×C4C22×D311yescharacteristic
304C23×C4C22×D311yescharacteristic
305C23×C4C22×D311yescharacteristic
306#121—616no—
307#120—616no—
308C4×Q8—64no—
309C4×Q8—64no—
310C4×D4—68no—
311C4×D4—68no—
312C4×D4—68no—
313C4×D4—68no—
314C4×D4—68no—
315C4×D4—68no—
316C4×D4—64no—
317C4×D4—64no—
318C4×D4—1216no—
319#117—64no—
320#117—68no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—38no—
328C2×C4⋊C4—64no—
329C2×C4⋊C4C22×D314yes—
330C2×C22⋊C4—32no—
331C2×C22⋊C4—34no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—38no—
336C2×C22⋊C4—38no—
337C2×C22⋊C4—38no—
338C2×C22⋊C4—38no—
339C2×C22⋊C4—68no—
340C2×C22⋊C4C22×D314yes—
341C2×C42—32no—
342C2×C42—32no—
343C2×C42—32no—
344C2×C42—68no—
345C2×C42D1218yescomplemented
346C2×C42C22×D314yes—
347C2×C42C22×D311yescharacteristic
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#95—38no—
352#95—38no—
353#95C22×D314yes—
354#95C22×D314yes—
355#95C22×D314yes—
356#95C22×D314yes—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3C22×D412yes—
360C23×C3C2×#4114yes—
361C23×C3C2511yescharacteristic
362C22×D3—216no—
363C22×D3—22no—
364C22×D3—24no—
365C22×D3—28no—
366C22×D3—24no—
367C22×D3—22no—
368C22×D3—24no—
369C22×D3—24no—
370C22×D3—28no—
371C22×D3—432no—
372C22×D3—48no—
373C22×D3—48no—
374C22×D3C2×#4118yes—
375C22×D3C2×#4114yes—
376C2×C3×C4—28no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—28no—
382C2×C3×C4—28no—
383C2×C3×C4—28no—
384C2×C3×C4—28no—
385C2×C3×C4—28no—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C2×#4114yes—
393C2×C3×C4H(5,𝔽2)14yes—
394#67—216no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—28no—
403C2×Dic3—48no—
404C2×Dic3C2×#4114yes—
405D12—216no—
406D12—28no—
407D12—216no—
408D12—432no—
409C4×D3—216no—
410C4×D3—416no—
411C4×D3—48no—
412C4×D3—48no—
413C4×D3—416no—
414Dic6—28no—
415C24—34no—
416C24—32no—
417C24—68no—
418C24—64no—
419C24—68no—
420C24—124no—
421C24C23×D311yescharacteristic
422#41—632no—
423C2×Q8—64no—
424C2×Q8—62no—
425C2×D4—38no—
426C2×D4—38no—
427C2×D4—68no—
428C2×D4—64no—
429C2×D4—632no—
430C2×D4—616no—
431C2×D4—616no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—62no—
437C2×D4—64no—
438C2×D4—128no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—34no—
445C22×C4—32no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—616no—
457C22×C4—128no—
458C22×C4—1232no—
459C22×C4—124no—
460C22×C4C2×D1214yes—
461C22×C4C2×D1214yes—
462C22×C4C2×D1214yes—
463C22×C4C23×D318yes—
464C22×C4C23×D311yescharacteristic
465C22×C4C23×D311yescharacteristic
466C22×C4C23×D311yescharacteristic
467C22×C4C23×D311yescharacteristic
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C4⋊C4—28no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—64no—
474C4⋊C4—64no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—128no—
478C22⋊C4—28no—
479C22⋊C4—616no—
480C22⋊C4—616no—
481C22⋊C4—64no—
482C22⋊C4—68no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4—64no—
486C22⋊C4—128no—
487C22⋊C4—128no—
488C42—28no—
489C42—64no—
490C42—64no—
491C42—64no—
492C42C2×D1214yes—
493C42C2×D1214yes—
494C42C2×D1214yes—
495C42C2×D1214yes—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—28no—
499C22×C3#52514yes—
500C22×C3C23×D411yescharacteristic
501C22×C3C22×#4112yes—
502C22×C3C22×#4112yes—
503C22×C3C2×H(5,𝔽2)12yes—
504C2×D3—24no—
505C2×D3—28no—
506C2×D3—28no—
507C2×D3—216no—
508C2×D3—432no—
509C2×D3—42no—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—42no—
513C2×D3—44no—
514C2×D3—48no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—48no—
520C3×C4#52514yes—
521C3×C4#53514yes—
522Dic3—24no—
523Dic3—42no—
524Dic3—44no—
525Dic3—42no—
526C23—22no—
527C23—22no—
528C23—38no—
529C23—34no—
530C23—616no—
531C23—62no—
532C23—64no—
533C23—68no—
534C23—64no—
535C23—62no—
536C23—64no—
537C23—64no—
538C23—68no—
539C23—1232no—
540C23—128no—
541C23—128no—
542C23C22×D1212yes—
543C23D3×#4114yes—
544C23C24×D311yescharacteristic
545Q8—68no—
546D4—616no—
547D4—68no—
548D4—616no—
549D4—616no—
550D4—1232no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—34no—
562C2×C4—616no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—68no—
571C2×C4—1216no—
572C2×C4—128no—
573C2×C4—128no—
574C2×C4—1216no—
575C2×C4—128no—
576C2×C4C22×D1212yes—
577C2×C4C22×D1212yes—
578C2×C4C22×D1212yes—
579C2×C4C22×D1212yes—
580C2×C4C22×D1212yes—
581C2×C4C22×D1212yes—
582C2×C4D3×#4114yes—
583C2×C4#100914yes—
584C2×C3—24no—
585C2×C3C2×#52511yescharacteristic
586C2×C3C2×#52511yescharacteristic
587C2×C3C2×#53511yescharacteristic
588C2×C3#348114yes—
589D3—44no—
590D3—48no—
591C22—28no—
592C22—22no—
593C22—22no—
594C22—68no—
595C22—616no—
596C22—64no—
597C22—68no—
598C22—1232no—
599C22—128no—
600C22—124no—
601C22—122no—
602C22—124no—
603C22—124no—
604C22—122no—
605C22#552414yes—
606C22C23×D1211yescharacteristic
607C22C2×D3×#4112yes—
608C22C2×D3×#4112yes—
609C22C2×#100912yes—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—48no—
615C4—64no—
616C4—122no—
617C4—124no—
618C4—122no—
619C4#552414yes—
620C4#552514yes—
621C3#3496311yescharacteristic, complemented, Sylow
622C2—24no—
623C2—128no—
624C2—124no—
625C2C2×#552411yescharacteristic
626C2C2×#552411yescharacteristic
627C2C2×#552511yescharacteristic
628C2#8615314yes—
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC3×#2283C2
hypercenter#2283D3
hypocenterC3#34963
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #2283

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. C2×C4×#235C2semidirect
3. C3×#2283C2semidirect
4. #74307C2semidirect
5. #74239C2semidirect
6. #74190C2semidirect
7. #74129C2semidirect
8. #74125C2semidirect
9. #74028C2semidirect
10. #74016C2semidirect
11. C4×#895C2semidirect
12. C4×#882C2semidirect
13. #73419C2semidirect
14. #73418C2semidirect
33. C2×C4×D12C22semidirect
34. C2×C4×D12C22semidirect
35. C2×C4×D12C22semidirect
36. C2×#872C22semidirect
37. C2×C42×D3C22semidirect
38. C2×C4×Dic6C22semidirect
39. C2×C4×Dic6C22semidirect
40. C3×#383C22semidirect
41. C3×#382C22semidirect
44. C3×C4×C4⋊C4C22semidirect
46. C3×#376C22semidirect
48. #4987C22semidirect
49. #4984C22semidirect
53. C4×(C2×D3)⋊C4C22semidirect
54. #4677C22semidirect
55. #4677C22semidirect
56. #4674C22semidirect
57. #4673C22semidirect
60. #4671C22semidirect
61. #4670C22semidirect
62. #4668C22semidirect
63. #4665C22semidirect
64. #4665C22semidirect
65. #4659C22semidirect
79. #2283D3semidirect
104. C2×C3×C42D4semidirect
193. #383C2×D3semidirect
194. #382C2×D3semidirect
199. C4×C4⋊C4C2×D3semidirect
202. #376C2×D3semidirect
243. C2×D12#41semidirect
345. C2×C42D12semidirect
621. C3#34963semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C2×C4×#2352
C3×#22832
#743072
#742392
#741902
#741292
#741252
#740282
#740162
C4×#8952
C4×#8822
#734192
#734182
#349633