Finite Group Databasebeta

← #11234166

Subgroups — #11234166

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2398C211yesmaximal, characteristic, complemented
2#82501C212yesmaximal, complemented
3#82403C211yesmaximal, characteristic
4#82373C211yesmaximal, characteristic, complemented
5#78388C214yesmaximal, complemented
6#77677C212yesmaximal, complemented
7#77534C212yesmaximal, complemented
8#77111C214yesmaximal, complemented
9#76881C212yesmaximal, complemented
10#76842C212yesmaximal, complemented
11#73584C212yesmaximal, complemented
12C22⋊C4×D12C212yesmaximal, complemented
13#73510C212yesmaximal, complemented
14#73467C212yesmaximal, complemented
15#73415C212yesmaximal, complemented
16#45380—31nomaximal, Sylow
17C23×D12C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic, complemented
19C2×C3×C4×D4C2211yescharacteristic
20C2×#937C2212yes—
21C2×#935C2212yes—
22C2×#930C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic, complemented
24C2×#67⋊C4C2214yescomplemented
25C2×D3×C22⋊C4C2214yes—
26C2×C4×D12C2212yes—
27C2×C4×D12C2212yes—
28C3×#400C2212yescomplemented
29C3×#397C2211yescharacteristic
30C3×#397C2211yescharacteristic, complemented
31C3×#394C2212yes—
32C3×#393C2211yescharacteristic
33C3×#393C2211yescharacteristic, complemented
34C3×C4×C22⋊C4C2212yes—
35C3×C4×C22⋊C4C2212yescomplemented
36C3×#376C2211yescharacteristic, complemented
37#5231C2214yescomplemented
38#5212C2211yescharacteristic
39#5212C2212yes—
40#5212C2211yescharacteristic, complemented
41#4993C2214yescomplemented
42#4991C2212yes—
43#4989C2214yescomplemented
44#4989C2214yescomplemented
45#4981C2212yescomplemented
46#4960C2212yes—
47#4959C2214yescomplemented
48#4958C2214yescomplemented
49#4956C2214yescomplemented
50#4955C2214yes—
51#4955C2212yes—
52#4953C2212yes—
53#4952C2214yes—
54#4945C2214yes—
55#4943C2212yescomplemented
56#4939C2212yes—
57#4939C2212yes—
58#4939C2212yes—
59#4936C2212yescomplemented
60C4×C4⋊Dic3C2212yescomplemented
61#4928C2212yescomplemented
62#4928C2212yescomplemented
63#4928C2212yescomplemented
64#4675C2214yescomplemented
65#4670C2214yescomplemented
66#4669C2214yescomplemented
67#4669C2214yescomplemented
68#4668C2214yescomplemented
69#4667C2214yescomplemented
70#4665C2214yes—
71#4665C2214yes—
72#4664C2214yescomplemented
73#4664C2214yescomplemented
74C3⋊#401C2214yescomplemented
75#4652C2214yescomplemented
76#2760—32no—
77#2746—34no—
78#2736—32no—
79#2600—32no—
80#2580—32no—
81#2494—32no—
82#2486—31no—
83#2483—32no—
84#2420—32no—
85#2418—34no—
86#2408—31no—
87#2398D311yescharacteristic, complemented
88#2361—32no—
89D4×C22⋊C4—32no—
90#2319—32no—
91C24×D3C2312yes—
92C22×C3×Q8C2311yescharacteristic
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C22×#67C2314yes—
97C23×Dic3C2312yes—
98C22×D12—28no—
99C22×D12—42no—
100C22×D12C2312yes—
101C22×D12C2312yes—
102C22×C4×D3C2314yes—
103C22×C4×D3C2314yes—
104C3×C4×Q8—42no—
105C3×C4×D4—42no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2312yes—
117C2×C3×C42C2312yes—
118C2×#217C2314yes—
119C2×#217C2312yes—
120C2×#217C2312yes—
121#937—44no—
122#935—44no—
123#930—28no—
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⋊Dic3—42no—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2314yes—
142#67⋊C4—48no—
143D3×C22⋊C4—48no—
144C4×D12—28no—
145C4×D12—44no—
146C3×#95C2312yes—
147C3×#95C2312yes—
148C3×#95C2311yescharacteristic
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2311yescharacteristic
152C3×#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2312yes—
161C3⋊#95C2314yes—
162C23×D4—31no—
163C2×#123—32no—
164C2×#121—32no—
165C2×#120—32no—
166C2×C4×Q8C2×D311yescharacteristic, complemented
167C2×C4×D4—34no—
168C2×C4×D4—32no—
169C2×C4×D4—32no—
170C2×C4×D4C2×D311yescharacteristic
171C22×C4⋊C4—31no—
172C22×C22⋊C4—34no—
173#401—34no—
174#400C2×D312yescomplemented
175#399—34no—
176#398—32no—
177#397—34no—
178#397—32no—
179#397—34no—
180#397C2×D311yescharacteristic
181#397C2×D311yescharacteristic, complemented
182#394—32no—
183#394—34no—
184#394C2×D312yes—
185#393—34no—
186#393C2×D311yescharacteristic
187#393C2×D311yescharacteristic, complemented
188#392—32no—
189#390—32no—
190#390—32no—
191#390—32no—
192#389—32no—
193#389—32no—
194#389—32no—
195#388—34no—
196#388—34no—
197#388—34no—
198#388—34no—
199#387—34no—
200#386—34no—
201#386—34no—
202#386—34no—
203#386—34no—
204#385—34no—
205#385—34no—
206#385—34no—
207#382—32no—
208#381—32no—
209#381—34no—
210#381—34no—
211#380—32no—
212#380—31no—
213#380—31no—
214#380—34no—
215#379—34no—
216#379—32no—
217C4×C4⋊C4—32no—
218C4×C22⋊C4C2×D312yes—
219C4×C22⋊C4C2×D312yescomplemented
220#376C2×D311yescharacteristic, complemented
221C24×C3C2411yescharacteristic
222C24×C3C2411yescharacteristic
223C23×D3—24no—
224C23×D3—28no—
225C23×D3—42no—
226C23×D3—44no—
227C23×D3—42no—
228C23×D3C2414yes—
229C2×C3×Q8—41no—
230C2×C3×Q8—42no—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4—41no—
234C22×C3×C4—24no—
235C22×C3×C4—24no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2412yes—
240C22×C3×C4C2412yes—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C2×#67—28no—
248C2×#67—28no—
249C2×#67—44no—
250C22×Dic3—24no—
251C22×Dic3—42no—
252C22×Dic3—42no—
253C22×Dic3C2414yes—
254C22×Dic3C2414yes—
255C22×Dic3C2414yes—
256C2×D12—24no—
257C2×D12—24no—
258C2×D12—28no—
259C2×D12—416no—
260C2×D12—42no—
261C2×D12#4118yes—
262C2×C4×D3—28no—
263C2×C4×D3—44no—
264C2×C4×D3—44no—
265C2×C4×D3—44no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4—42no—
270C3×C4⋊C4—41no—
271C3×C22⋊C4—24no—
272C3×C22⋊C4—24no—
273C3×C22⋊C4—42no—
274C3×C22⋊C4—42no—
275C3×C22⋊C4—41no—
276C3×C22⋊C4—41no—
277C3×C42—24no—
278C3×C42—24no—
279C3×C42—22no—
280#217—28no—
281#217—42no—
282#217—42no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—44no—
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—24no—
294C4⋊Dic3—24no—
295C4⋊Dic3—42no—
296Dic3⋊C4—44no—
297Dic3⋊C4—44no—
298Dic3⋊C4—44no—
299C4×Dic3—44no—
300C25—32no—
301C22×Q8C22×D311yescharacteristic
302C22×D4—34no—
303C22×D4—32no—
304C22×D4—32no—
305C22×D4—68no—
306C22×D4—122no—
307C22×D4C22×D311yescharacteristic
308C23×C4—32no—
309C23×C4—34no—
310C23×C4—34no—
311C23×C4C22×D311yescharacteristic
312C23×C4C22×D311yescharacteristic
313#123—124no—
314#121—68no—
315#120—124no—
316C4×Q8—42no—
317C4×D4—42no—
318C4×D4—68no—
319C4×D4—124no—
320C4×D4—128no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—122no—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D312yes—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—128no—
345C2×C22⋊C4C22×D312yes—
346C2×C22⋊C4C22×D312yes—
347C2×C22⋊C4C22×D311yescharacteristic
348C2×C22⋊C4C22×D311yescharacteristic
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D312yes—
351C2×C42—34no—
352C2×C42C22×D312yes—
353C2×C42C22×D312yes—
354C2×C42C22×D311yescharacteristic
355#95—34no—
356#95—32no—
357#95—34no—
358#95—32no—
359#95—34no—
360#95—34no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95C22×D312yes—
365#95C22×D311yescharacteristic
366#95C22×D311yescharacteristic
367#95C22×D312yes—
368#95C22×D312yes—
369#95C22×D312yes—
370#95C22×D312yes—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3H(5,𝔽2)12yes—
380C23×C3H(5,𝔽2)12yes—
381C23×C3C2511yescharacteristic
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—48no—
389C22×D3—416no—
390C22×D3—88no—
391C22×D3—84no—
392C22×D3—84no—
393C22×D3C2×#4118yes—
394C3×Q8—44no—
395C3×D4—44no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—44no—
418C2×C3×C4—44no—
419C2×C3×C4C2×#4114yes—
420C2×C3×C4H(5,𝔽2)12yes—
421#67—416no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—24no—
431C2×Dic3—84no—
432D12—216no—
433D12—48no—
434C4×D3—88no—
435C4×D3—88no—
436C24—34no—
437C24—68no—
438C24—64no—
439C24—122no—
440C24—124no—
441C24—122no—
442C24C23×D311yescharacteristic
443C24C23×D311yescharacteristic
444C2×Q8—41no—
445C2×Q8—42no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—38no—
449C2×D4—41no—
450C2×D4—68no—
451C2×D4—68no—
452C2×D4—68no—
453C2×D4—64no—
454C2×D4—64no—
455C2×D4—122no—
456C2×D4—1216no—
457C2×D4—124no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—68no—
466C22×C4—64no—
467C22×C4—124no—
468C22×C4—124no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4—124no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D312yes—
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D312yes—
478C22×C4C23×D311yescharacteristic
479C22×C4C23×D312yes—
480C22×C4C23×D312yes—
481C22×C4C23×D311yescharacteristic
482C22×C4C23×D311yescharacteristic
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—122no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C4⋊C4—124no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—68no—
500C22⋊C4—68no—
501C22⋊C4—68no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C22⋊C4—124no—
508C22⋊C4—124no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—124no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3C22×#4112yes—
526C22×C3C2×H(5,𝔽2)11yescharacteristic
527C22×C3C2×H(5,𝔽2)11yescharacteristic
528C22×C3C2×H(5,𝔽2)11yescharacteristic
529C22×C3#58512yes—
530C2×D3—28no—
531C2×D3—28no—
532C2×D3—28no—
533C2×D3—44no—
534C2×D3—44no—
535C2×D3—44no—
536C2×D3—44no—
537C2×D3—816no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—41no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4#55114yes—
547Dic3—44no—
548Dic3—44no—
549Dic3—44no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—38no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—1216no—
566C23—128no—
567C23—244no—
568C23—248no—
569C23—244no—
570C23#100912yes—
571C23#100912yes—
572C23C24×D311yescharacteristic
573Q8—44no—
574D4—44no—
575D4—616no—
576D4—128no—
577D4—1216no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—248no—
611C2×C4—248no—
612C2×C4—244no—
613C2×C4#100412yes—
614C2×C4D3×#4114yes—
615C2×C3—42no—
616C2×C3—42no—
617C2×C3C2×#55111yescharacteristic
618C2×C3#351312yes—
619C2×C3#351312yes—
620C2×C3#353711yescharacteristic
621C2×C3#354011yescharacteristic
622D3—48no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—24no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22—42no—
633C22—68no—
634C22—68no—
635C22—68no—
636C22—124no—
637C22—124no—
638C22—124no—
639C22—124no—
640C22—2416no—
641C22C2×#100411yescharacteristic
642C22C2×D3×#4112yes—
643C22C2×#100911yescharacteristic
644C22C2×#100911yescharacteristic
645C22#596412yes—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—124no—
655C4—124no—
656C4—124no—
657C4#567914yes—
658C3#4538011yescharacteristic, complemented, Sylow
659C2—42no—
660C2—42no—
661C2—128no—
662C2#8605211yescharacteristic
663C2#8605511yescharacteristic
664C2C2×#567911yescharacteristic
665C2#8616112yes—
666C2#8616112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2398C2
prime radical#82403C2
hypercenter#2398D3
hypocenterC3#45380
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. C3×#2398C2semidirect
2. #82501C2semidirect
4. #82373C2semidirect
5. #78388C2semidirect
6. #77677C2semidirect
7. #77534C2semidirect
8. #77111C2semidirect
9. #76881C2semidirect
10. #76842C2semidirect
11. #73584C2semidirect
12. C22⋊C4×D12C2semidirect
13. #73510C2semidirect
14. #73467C2semidirect
15. #73415C2semidirect
18. C2×C3×C4×Q8C22semidirect
23. C22×C4⋊Dic3C22semidirect
24. C2×#67⋊C4C22semidirect
28. C3×#400C22semidirect
30. C3×#397C22semidirect
33. C3×#393C22semidirect
35. C3×C4×C22⋊C4C22semidirect
36. C3×#376C22semidirect
37. #5231C22semidirect
40. #5212C22semidirect
41. #4993C22semidirect
43. #4989C22semidirect
44. #4989C22semidirect
45. #4981C22semidirect
47. #4959C22semidirect
48. #4958C22semidirect
49. #4956C22semidirect
55. #4943C22semidirect
59. #4936C22semidirect
60. C4×C4⋊Dic3C22semidirect
61. #4928C22semidirect
62. #4928C22semidirect
63. #4928C22semidirect
64. #4675C22semidirect
65. #4670C22semidirect
66. #4669C22semidirect
67. #4669C22semidirect
68. #4668C22semidirect
69. #4667C22semidirect
72. #4664C22semidirect
73. #4664C22semidirect
74. C3⋊#401C22semidirect
75. #4652C22semidirect
87. #2398D3semidirect
166. C2×C4×Q8C2×D3semidirect
174. #400C2×D3semidirect
181. #397C2×D3semidirect
187. #393C2×D3semidirect
219. C4×C22⋊C4C2×D3semidirect
220. #376C2×D3semidirect
658. C3#45380semidirect

Maximal subgroups

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

subgroupindex
C3×#23982
#825012
#824032
#823732
#783882
#776772
#775342
#771112
#768812
#768422
#735842
C22⋊C4×D122
#735102
#734672
#734152
#453803