Finite Group Databasebeta

← #11234113

Subgroups — #11234113

Order 768.

2194 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2487C211yesmaximal, characteristic, complemented
2#82403C211yesmaximal, characteristic
3#82348C211yesmaximal, characteristic, complemented
4#82163C212yesmaximal, complemented
5#78391C214yesmaximal, complemented
6#78120C212yesmaximal, complemented
7#78102C212yesmaximal, complemented
8#76059C212yesmaximal, complemented
9#75969C212yesmaximal, complemented
10#74901C214yesmaximal, complemented
11#74613C212yesmaximal, complemented
12#74517C212yesmaximal, complemented
13#74007C212yesmaximal, complemented
14#73924C212yesmaximal, complemented
15#73893C212yesmaximal, complemented
16#45377—31nomaximal, Sylow
17C23×D12C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic, complemented
20C2×#937C2212yes—
21C2×#930C2212yes—
22C2×C4×#67C2214yescomplemented
23C2×#896C2214yescomplemented
24C2×#893C2212yes—
25C2×#893C2212yes—
26C2×#883C2214yescomplemented
27C2×D3×C22⋊C4C2214yes—
28C2×C4×Dic6C2211yescharacteristic, complemented
29C3×#400C2212yescomplemented
30C3×#399C2212yes—
31C3×#389C2212yescomplemented
32C3×#386C2212yes—
33C3×#385C2212yescomplemented
34C3×#380C2212yes—
35C3×#380C2211yescharacteristic, complemented
36#5222C2212yes—
37#5213C2214yescomplemented
38#5212C2211yescharacteristic
39C4×#217C2212yes—
40C4×#217C2212yescomplemented
41#4994C2214yescomplemented
42#4991C2212yes—
43#4990C2214yescomplemented
44#4990C2214yescomplemented
45#4989C2214yescomplemented
46#4988C2212yes—
47#4988C2212yes—
48#4987C2214yescomplemented
49#4985C2212yescomplemented
50#4984C2212yescomplemented
51#4975C2212yescomplemented
52Dic3×C4⋊C4C2212yescomplemented
53#4972C2212yescomplemented
54#4960C2212yes—
55#4959C2214yescomplemented
56#4955C2212yes—
57#4955C2214yes—
58#4953C2211yescharacteristic
59#4953C2211yescharacteristic, complemented
60#4948C2214yes—
61#4947C2211yescharacteristic
62#4947C2211yescharacteristic, complemented
63#4944C2214yes—
64#4939C2212yes—
65#4932C2211yescharacteristic, complemented
66#4676C2214yescomplemented
67#4675C2214yescomplemented
68#4670C2214yescomplemented
69#4668C2214yescomplemented
70#4667C2214yescomplemented
71#4665C2214yes—
72#4665C2214yes—
73#4664C2214yescomplemented
74#4659C2214yescomplemented
75#4652C2214yescomplemented
76#2750—34no—
77#2735—32no—
78#2693—32no—
79#2538—32no—
80#2534—34no—
81#2497—32no—
82#2487D311yescharacteristic, complemented
83#2463—32no—
84#2437—32no—
85#2422—32no—
86#2408—31no—
87#2398—31no—
88#2361—32no—
89D4×C22⋊C4—32no—
90#2319—32no—
91C24×D3C2312yes—
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C22×D12—28no—
98C22×D12—42no—
99C22×D12C2312yes—
100C22×D12C2312yes—
101C22×C4×D3C2314yes—
102C22×C4×D3C2314yes—
103C22×Dic6C2311yescharacteristic
104C3×C4×D4—42no—
105C2×C3×C4⋊C4—42no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2312yes—
119C2×#217C2312yes—
120C2×#217C2312yes—
121#937—44no—
122#930—28no—
123C4×#67—48no—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2314yes—
143#896—48no—
144#893—28no—
145#893—44no—
146#883—48no—
147D3×C22⋊C4—48no—
148C4×Dic6—42no—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3×#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2311yescharacteristic
162C3⋊#95C2311yescharacteristic
163C23×D4—31no—
164C2×#123—34no—
165C2×C22⋊Q8—34no—
166C2×#121—32no—
167C2×#120—32no—
168C2×C4×Q8—31no—
169C2×C4×D4—34no—
170C2×C4×D4—32no—
171C2×C4×D4—32no—
172C2×C4×D4C2×D311yescharacteristic
173C22×C4⋊C4C2×D311yescharacteristic, complemented
174C22×C22⋊C4—34no—
175#400—32no—
176#400—34no—
177#400C2×D312yescomplemented
178#399C2×D312yes—
179#397—31no—
180#397—31no—
181#397—34no—
182#396—34no—
183#396—34no—
184#396—34no—
185#394—32no—
186#394—32no—
187#394—34no—
188#393—31no—
189#393—31no—
190#392—32no—
191#390—32no—
192#390—32no—
193#390—32no—
194#389—32no—
195#389—32no—
196#389C2×D312yescomplemented
197#388—34no—
198#388—34no—
199#388—34no—
200#388—34no—
201#387—34no—
202#386—34no—
203#386—34no—
204#386—34no—
205#386C2×D312yes—
206#385—34no—
207#385—34no—
208#385C2×D312yescomplemented
209#384—32no—
210#381—34no—
211#380—31no—
212#380—34no—
213#380—34no—
214#380C2×D312yes—
215#380C2×D311yescharacteristic, complemented
216#379—34no—
217#379—32no—
218C4×C4⋊C4—32no—
219C4×C22⋊C4—32no—
220C4×C22⋊C4—32no—
221#376—31no—
222C24×C3C2411yescharacteristic
223C24×C3C2411yescharacteristic
224C23×D3—24no—
225C23×D3—28no—
226C23×D3—44no—
227C23×D3—42no—
228C23×D3—42no—
229C23×D3C2414yes—
230C2×C3×D4—22no—
231C2×C3×D4—22no—
232C2×C3×D4—41no—
233C22×C3×C4—24no—
234C22×C3×C4—24no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—44no—
238C22×C3×C4—42no—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2414yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2414yes—
245C22×C3×C4C2411yescharacteristic
246C2×#67—28no—
247C2×#67—28no—
248C2×#67—44no—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2414yes—
252C22×Dic3C2412yes—
253C22×Dic3C2412yes—
254C2×D12—24no—
255C2×D12—24no—
256C2×D12—28no—
257C2×D12—416no—
258C2×D12—42no—
259C2×D12#4118yes—
260C2×C4×D3—28no—
261C2×C4×D3—28no—
262C2×C4×D3—44no—
263C2×C4×D3—44no—
264C2×C4×D3—44no—
265C2×C4×D3—44no—
266C2×Dic6—24no—
267C2×Dic6—41no—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—24no—
270C3×C4⋊C4—44no—
271C3×C4⋊C4—42no—
272C3×C4⋊C4—41no—
273C3×C22⋊C4—24no—
274C3×C22⋊C4—44no—
275C3×C22⋊C4—41no—
276C3×C22⋊C4—41no—
277C3×C42—22no—
278#217—24no—
279#217—24no—
280#217—24no—
281#217—24no—
282#217—42no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—28no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288(C2×D3)⋊C4—44no—
289(C2×D3)⋊C4—44no—
290(C2×D3)⋊C4—44no—
291(C2×D3)⋊C4—44no—
292C4⋊Dic3—24no—
293C4⋊Dic3—41no—
294C4⋊Dic3—44no—
295Dic3⋊C4—28no—
296Dic3⋊C4—42no—
297Dic3⋊C4—44no—
298Dic3⋊C4—44no—
299C4×Dic3—24no—
300C4×Dic3—24no—
301C4×Dic3—44no—
302C25—32no—
303C22×Q8—31no—
304C22×D4—34no—
305C22×D4—32no—
306C22×D4—32no—
307C22×D4—68no—
308C22×D4—122no—
309C22×D4C22×D311yescharacteristic
310C23×C4—34no—
311C23×C4—34no—
312C23×C4C22×D311yescharacteristic
313C23×C4C22×D311yescharacteristic
314C23×C4C22×D312yes—
315#123—128no—
316C22⋊Q8—128no—
317#121—68no—
318#120—124no—
319C4×Q8—122no—
320C4×D4—42no—
321C4×D4—68no—
322C4×D4—124no—
323C4×D4—128no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4C22×D312yes—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—128no—
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D314yes—
354C2×C22⋊C4C22×D311yescharacteristic
355C2×C22⋊C4C22×D311yescharacteristic
356C2×C42—32no—
357C2×C42—32no—
358C2×C42—34no—
359C2×C42C22×D311yescharacteristic
360#95—31no—
361#95—32no—
362#95—31no—
363#95—32no—
364#95—32no—
365#95—32no—
366#95—32no—
367#95—34no—
368#95—34no—
369#95—32no—
370#95—34no—
371#95C22×D314yes—
372#95C22×D314yes—
373#95C22×D312yes—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3—22no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3H(5,𝔽2)12yes—
383C23×C32−1+412yes—
384C23×C3C2511yescharacteristic
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C22×D3—24no—
390C22×D3—24no—
391C22×D3—48no—
392C22×D3—416no—
393C22×D3—88no—
394C22×D3—84no—
395C22×D3—84no—
396C22×D3C2×#4118yes—
397C3×D4—44no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4—24no—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—24no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—44no—
414C2×C3×C4—44no—
415C2×C3×C4—84no—
416C2×C3×C4C2×#4114yes—
417C2×C3×C4H(5,𝔽2)12yes—
418#67—416no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3H(5,𝔽2)14yes—
434D12—216no—
435D12—48no—
436C4×D3—88no—
437C4×D3—88no—
438Dic6—44no—
439C24—34no—
440C24—68no—
441C24—64no—
442C24—124no—
443C24—122no—
444C24—122no—
445C24C23×D311yescharacteristic
446C24C23×D311yescharacteristic
447C2×Q8—64no—
448C2×Q8—121no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—38no—
452C2×D4—41no—
453C2×D4—68no—
454C2×D4—68no—
455C2×D4—68no—
456C2×D4—64no—
457C2×D4—64no—
458C2×D4—122no—
459C2×D4—1216no—
460C2×D4—124no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—34no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—68no—
473C22×C4—68no—
474C22×C4—124no—
475C22×C4—124no—
476C22×C4—124no—
477C22×C4—124no—
478C22×C4C23×D311yescharacteristic
479C22×C4C23×D311yescharacteristic
480C22×C4C23×D311yescharacteristic
481C22×C4C23×D314yes—
482C22×C4C23×D311yescharacteristic
483C22×C4C23×D314yes—
484C22×C4C23×D311yescharacteristic
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—44no—
490C4⋊C4—68no—
491C4⋊C4—64no—
492C4⋊C4—122no—
493C4⋊C4—121no—
494C4⋊C4—124no—
495C4⋊C4—124no—
496C4⋊C4—124no—
497C22⋊C4—24no—
498C22⋊C4—44no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—68no—
502C22⋊C4—68no—
503C22⋊C4—68no—
504C22⋊C4—64no—
505C22⋊C4—64no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—124no—
509C22⋊C4—124no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C22⋊C4—124no—
514C22⋊C4—122no—
515C42—22no—
516C42—64no—
517C42—64no—
518C42—124no—
519C22×C3—24no—
520C22×C3—24no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3C22×#4112yes—
530C22×C3C2×H(5,𝔽2)11yescharacteristic
531C22×C3C2×H(5,𝔽2)11yescharacteristic
532C22×C3C2×H(5,𝔽2)12yes—
533C22×C3C2×2−1+411yescharacteristic
534C2×D3—28no—
535C2×D3—28no—
536C2×D3—28no—
537C2×D3—44no—
538C2×D3—44no—
539C2×D3—44no—
540C2×D3—44no—
541C2×D3—816no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—44no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—44no—
548C3×C4#55114yes—
549Dic3—24no—
550Dic3—42no—
551Dic3—44no—
552Dic3—42no—
553Dic3—42no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—21no—
560C23—22no—
561C23—22no—
562C23—38no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—1216no—
570C23—128no—
571C23—244no—
572C23—248no—
573C23—244no—
574C23#100712yes—
575C23#100912yes—
576C23C24×D311yescharacteristic
577Q8—124no—
578D4—44no—
579D4—616no—
580D4—128no—
581D4—1216no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—34no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—62no—
614C2×C4—84no—
615C2×C4—248no—
616C2×C4—248no—
617C2×C4#100412yes—
618C2×C4D3×#4114yes—
619C2×C3—42no—
620C2×C3—42no—
621C2×C3C2×#55111yescharacteristic
622C2×C3#351912yes—
623C2×C3#352012yes—
624C2×C3#353611yescharacteristic
625C2×C3#353711yescharacteristic
626D3—48no—
627C22—24no—
628C22—24no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—41no—
637C22—68no—
638C22—68no—
639C22—68no—
640C22—124no—
641C22—124no—
642C22—124no—
643C22—124no—
644C22—2416no—
645C22C2×#100411yescharacteristic
646C22C2×#100711yescharacteristic
647C22C2×D3×#4112yes—
648C22C2×#100911yescharacteristic
649C22D3×H(5,𝔽2)12yes—
650C4—44no—
651C4—41no—
652C4—44no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—64no—
657C4—122no—
658C4—124no—
659C4—122no—
660C4—122no—
661C4#565914yes—
662C3#4537711yescharacteristic, complemented, Sylow
663C2—42no—
664C2—42no—
665C2—128no—
666C2#8599011yescharacteristic
667C2C2×#565911yescharacteristic
668C2#8633411yescharacteristic
669C2#8688812yes—
670C2#8693412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2487C2
prime radical#82403C2
hypercenter#2487D3
hypocenterC3#45377
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2487

Complements

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

NH ≅ G/Ntype
1. C3×#2487C2semidirect
3. #82348C2semidirect
4. #82163C2semidirect
5. #78391C2semidirect
6. #78120C2semidirect
7. #78102C2semidirect
8. #76059C2semidirect
9. #75969C2semidirect
10. #74901C2semidirect
11. #74613C2semidirect
12. #74517C2semidirect
13. #74007C2semidirect
14. #73924C2semidirect
15. #73893C2semidirect
19. C22×C3×C4⋊C4C22semidirect
22. C2×C4×#67C22semidirect
23. C2×#896C22semidirect
26. C2×#883C22semidirect
28. C2×C4×Dic6C22semidirect
29. C3×#400C22semidirect
31. C3×#389C22semidirect
33. C3×#385C22semidirect
35. C3×#380C22semidirect
37. #5213C22semidirect
40. C4×#217C22semidirect
41. #4994C22semidirect
43. #4990C22semidirect
44. #4990C22semidirect
45. #4989C22semidirect
48. #4987C22semidirect
49. #4985C22semidirect
50. #4984C22semidirect
51. #4975C22semidirect
52. Dic3×C4⋊C4C22semidirect
53. #4972C22semidirect
55. #4959C22semidirect
59. #4953C22semidirect
62. #4947C22semidirect
65. #4932C22semidirect
66. #4676C22semidirect
67. #4675C22semidirect
68. #4670C22semidirect
69. #4668C22semidirect
70. #4667C22semidirect
73. #4664C22semidirect
74. #4659C22semidirect
75. #4652C22semidirect
82. #2487D3semidirect
173. C22×C4⋊C4C2×D3semidirect
177. #400C2×D3semidirect
196. #389C2×D3semidirect
208. #385C2×D3semidirect
215. #380C2×D3semidirect
662. C3#45377semidirect

Maximal subgroups

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

subgroupindex
C3×#24872
#824032
#823482
#821632
#783912
#781202
#781022
#760592
#759692
#749012
#746132
#745172
#740072
#739242
#738932
#453773