Finite Group Databasebeta

← #11234112

Subgroups — #11234112

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×#2398C211yesmaximal, characteristic, complemented
2#82500C212yesmaximal, complemented
3#82403C211yesmaximal, characteristic
4#82374C211yesmaximal, characteristic, complemented
5#78392C214yesmaximal, complemented
6#77705C212yesmaximal, complemented
7#77574C212yesmaximal, complemented
8#77119C214yesmaximal, complemented
9#76865C212yesmaximal, complemented
10#76787C212yesmaximal, complemented
11#73585C212yesmaximal, complemented
12C22⋊C4×D12C212yesmaximal, complemented
13#73513C212yesmaximal, complemented
14#73467C212yesmaximal, complemented
15#73415C212yesmaximal, complemented
16#45377—31nomaximal, Sylow
17C23×D12C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic, complemented
19C2×C3×C4×D4C2211yescharacteristic
20C2×#946C2214yescomplemented
21C2×#937C2212yes—
22C2×#930C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic, complemented
24C2×#883C2214yescomplemented
25C2×#67⋊C4C2214yescomplemented
26C2×D3×C22⋊C4C2214yes—
27C2×C4×D12C2212yes—
28C2×C4×D12C2212yes—
29C3×#400C2212yescomplemented
30C3×#397C2211yescharacteristic
31C3×#397C2211yescharacteristic, complemented
32C3×#394C2212yes—
33C3×#393C2211yescharacteristic
34C3×#393C2211yescharacteristic, complemented
35C3×C4×C22⋊C4C2212yes—
36C3×C4×C22⋊C4C2212yescomplemented
37C3×#376C2211yescharacteristic, complemented
38#5220C2212yes—
39#5212C2211yescharacteristic
40#5212C2212yes—
41#5212C2211yescharacteristic, complemented
42#4994C2214yescomplemented
43#4994C2214yescomplemented
44#4991C2212yes—
45#4987C2214yescomplemented
46#4986C2212yescomplemented
47#4961C2214yescomplemented
48#4960C2212yes—
49#4958C2214yescomplemented
50#4955C2214yes—
51#4955C2212yes—
52#4951C2214yes—
53#4949C2212yes—
54#4948C2214yes—
55#4946C2212yescomplemented
56#4939C2212yes—
57#4939C2212yes—
58#4939C2212yes—
59#4935C2212yescomplemented
60C4×C4⋊Dic3C2212yescomplemented
61#4928C2212yescomplemented
62#4928C2212yescomplemented
63#4928C2212yescomplemented
64#4674C2214yescomplemented
65#4670C2214yescomplemented
66#4669C2214yescomplemented
67#4669C2214yescomplemented
68#4668C2214yescomplemented
69#4668C2214yescomplemented
70#4667C2214yescomplemented
71#4665C2214yes—
72#4665C2214yes—
73#4664C2214yescomplemented
74#4660C2214yescomplemented
75#4655C2214yescomplemented
76#2750—34no—
77#2735—32no—
78#2693—32no—
79#2538—32no—
80#2534—34no—
81#2497—32no—
82#2487—31no—
83#2463—32no—
84#2437—32no—
85#2422—32no—
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⋊C4C2312yes—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2312yes—
117C2×C3×C42C2312yes—
118C2×#217C2312yes—
119C2×#217C2312yes—
120C2×#217C2314yes—
121#946—48no—
122#937—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⋊Dic3C2314yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×C4×Dic3C2314yes—
144#883—48no—
145#67⋊C4—48no—
146D3×C22⋊C4—48no—
147C4×D12—28no—
148C4×D12—44no—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2312yes—
152C3×#95C2311yescharacteristic
153C3×#95C2312yes—
154C3×#95C2311yescharacteristic
155C3×#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C23×D4—31no—
164C2×#123—34no—
165C2×C22⋊Q8—34no—
166C2×#121—32no—
167C2×#120—32no—
168C2×C4×Q8C2×D311yescharacteristic, complemented
169C2×C4×D4—34no—
170C2×C4×D4—32no—
171C2×C4×D4—32no—
172C2×C4×D4C2×D311yescharacteristic
173C22×C4⋊C4—31no—
174C22×C22⋊C4—34no—
175#400—32no—
176#400—34no—
177#400C2×D312yescomplemented
178#399—32no—
179#397—34no—
180#397C2×D311yescharacteristic
181#397C2×D311yescharacteristic, complemented
182#396—34no—
183#396—34no—
184#396—34no—
185#394—32no—
186#394—34no—
187#394C2×D312yes—
188#393C2×D311yescharacteristic
189#393C2×D311yescharacteristic, complemented
190#392—32no—
191#390—32no—
192#390—32no—
193#390—32no—
194#389—32no—
195#389—32no—
196#389—32no—
197#388—34no—
198#388—34no—
199#388—34no—
200#388—34no—
201#387—34no—
202#386—32no—
203#386—34no—
204#386—34no—
205#386—34no—
206#385—32no—
207#385—34no—
208#385—34no—
209#384—32no—
210#381—34no—
211#380—32no—
212#380—31no—
213#380—31no—
214#380—34no—
215#380—34no—
216#379—34no—
217#379—32no—
218C4×C4⋊C4—32no—
219C4×C22⋊C4C2×D312yes—
220C4×C22⋊C4C2×D312yescomplemented
221#376C2×D311yescharacteristic, complemented
222C24×C3C2411yescharacteristic
223C24×C3C2411yescharacteristic
224C23×D3—24no—
225C23×D3—28no—
226C23×D3—44no—
227C23×D3—42no—
228C23×D3—42no—
229C23×D3C2414yes—
230C2×C3×Q8—24no—
231C2×C3×Q8—41no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—41no—
235C22×C3×C4—24no—
236C22×C3×C4—24no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2412yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2412yes—
247C22×C3×C4C2411yescharacteristic
248C2×#67—28no—
249C2×#67—28no—
250C2×#67—44no—
251C22×Dic3—44no—
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—28no—
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—24no—
274C3×C22⋊C4—24no—
275C3×C22⋊C4—41no—
276C3×C22⋊C4—41no—
277C3×C42—24no—
278C3×C42—24no—
279C3×C42—22no—
280#217—24no—
281#217—44no—
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—24no—
294C4⋊Dic3—24no—
295C4⋊Dic3—44no—
296C4⋊Dic3—42no—
297C4⋊Dic3—44no—
298Dic3⋊C4—28no—
299Dic3⋊C4—44no—
300Dic3⋊C4—44no—
301C4×Dic3—44no—
302C25—32no—
303C22×Q8C22×D311yescharacteristic
304C22×D4—34no—
305C22×D4—32no—
306C22×D4—32no—
307C22×D4—68no—
308C22×D4—122no—
309C22×D4C22×D311yescharacteristic
310C23×C4—32no—
311C23×C4—34no—
312C23×C4—34no—
313C23×C4C22×D311yescharacteristic
314C23×C4C22×D311yescharacteristic
315#123—128no—
316C22⋊Q8—128no—
317#121—68no—
318#120—124no—
319C4×Q8—42no—
320C4×D4—42no—
321C4×D4—68no—
322C4×D4—124no—
323C4×D4—128no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—122no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—32no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—128no—
350C2×C22⋊C4C22×D312yes—
351C2×C22⋊C4C22×D312yes—
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D312yes—
354C2×C22⋊C4C22×D311yescharacteristic
355C2×C22⋊C4C22×D311yescharacteristic
356C2×C42—34no—
357C2×C42C22×D312yes—
358C2×C42C22×D312yes—
359C2×C42C22×D311yescharacteristic
360#95—34no—
361#95—34no—
362#95—32no—
363#95—34no—
364#95—34no—
365#95—32no—
366#95—34no—
367#95C22×D311yescharacteristic
368#95C22×D312yes—
369#95C22×D311yescharacteristic
370#95C22×D312yes—
371#95C22×D312yes—
372#95C22×D312yes—
373#95C22×D312yes—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3—21no—
378C23×C3—22no—
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×Q8—44no—
398C3×D4—44no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—44no—
420C2×C3×C4—44no—
421C2×C3×C4C2×#4114yes—
422C2×C3×C4H(5,𝔽2)12yes—
423C2×C3×C4H(5,𝔽2)14yes—
424#67—416no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—24no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—84no—
435D12—216no—
436D12—48no—
437C4×D3—88no—
438C4×D3—88no—
439C24—34no—
440C24—68no—
441C24—64no—
442C24—124no—
443C24—122no—
444C24—122no—
445C24C23×D311yescharacteristic
446C24C23×D311yescharacteristic
447C2×Q8—24no—
448C2×Q8—41no—
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—34no—
466C22×C4—34no—
467C22×C4—34no—
468C22×C4—68no—
469C22×C4—68no—
470C22×C4—124no—
471C22×C4—124no—
472C22×C4—124no—
473C22×C4—124no—
474C22×C4—122no—
475C22×C4—124no—
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D312yes—
479C22×C4C23×D312yes—
480C22×C4C23×D311yescharacteristic
481C22×C4C23×D312yes—
482C22×C4C23×D311yescharacteristic
483C22×C4C23×D312yes—
484C22×C4C23×D311yescharacteristic
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—68no—
492C4⋊C4—122no—
493C4⋊C4—124no—
494C4⋊C4—124no—
495C4⋊C4—124no—
496C4⋊C4—124no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—68no—
504C22⋊C4—68no—
505C22⋊C4—64no—
506C22⋊C4—68no—
507C22⋊C4—124no—
508C22⋊C4—124no—
509C22⋊C4—124no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C22⋊C4—124no—
514C22⋊C4—122no—
515C42—22no—
516C42—24no—
517C42—24no—
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—24no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4#55114yes—
551Dic3—44no—
552Dic3—44no—
553Dic3—44no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—21no—
561C23—21no—
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#100912yes—
575C23#101012yes—
576C23C24×D311yescharacteristic
577Q8—44no—
578D4—44no—
579D4—616no—
580D4—128no—
581D4—1216no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—248no—
614C2×C4—248no—
615C2×C4—244no—
616C2×C4#100412yes—
617C2×C4D3×#4114yes—
618C2×C4#100914yes—
619C2×C3—42no—
620C2×C3—42no—
621C2×C3C2×#55111yescharacteristic
622C2×C3#351912yes—
623C2×C3#352012yes—
624C2×C3#353611yescharacteristic
625C2×C3#353711yescharacteristic
626D3—48no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—68no—
638C22—68no—
639C22—68no—
640C22—124no—
641C22—124no—
642C22—124no—
643C22—124no—
644C22—2416no—
645C22C2×#100411yescharacteristic
646C22C2×D3×#4112yes—
647C22C2×#100911yescharacteristic
648C22C2×#100912yes—
649C22C2×#101011yescharacteristic
650C4—24no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—124no—
659C4—124no—
660C4—124no—
661C4#567914yes—
662C3#4537711yescharacteristic, complemented, Sylow
663C2—42no—
664C2—42no—
665C2—128no—
666C2#8605111yescharacteristic
667C2#8605211yescharacteristic
668C2C2×#567911yescharacteristic
669C2#8616612yes—
670C2#8616712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2398C2
prime radical#82403C2
hypercenter#2398D3
hypocenterC3#45377
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. #82500C2semidirect
4. #82374C2semidirect
5. #78392C2semidirect
6. #77705C2semidirect
7. #77574C2semidirect
8. #77119C2semidirect
9. #76865C2semidirect
10. #76787C2semidirect
11. #73585C2semidirect
12. C22⋊C4×D12C2semidirect
13. #73513C2semidirect
14. #73467C2semidirect
15. #73415C2semidirect
18. C2×C3×C4×Q8C22semidirect
20. C2×#946C22semidirect
23. C22×C4⋊Dic3C22semidirect
24. C2×#883C22semidirect
25. C2×#67⋊C4C22semidirect
29. C3×#400C22semidirect
31. C3×#397C22semidirect
34. C3×#393C22semidirect
36. C3×C4×C22⋊C4C22semidirect
37. C3×#376C22semidirect
41. #5212C22semidirect
42. #4994C22semidirect
43. #4994C22semidirect
45. #4987C22semidirect
46. #4986C22semidirect
47. #4961C22semidirect
49. #4958C22semidirect
55. #4946C22semidirect
59. #4935C22semidirect
60. C4×C4⋊Dic3C22semidirect
61. #4928C22semidirect
62. #4928C22semidirect
63. #4928C22semidirect
64. #4674C22semidirect
65. #4670C22semidirect
66. #4669C22semidirect
67. #4669C22semidirect
68. #4668C22semidirect
69. #4668C22semidirect
70. #4667C22semidirect
73. #4664C22semidirect
74. #4660C22semidirect
75. #4655C22semidirect
87. #2398D3semidirect
168. C2×C4×Q8C2×D3semidirect
177. #400C2×D3semidirect
181. #397C2×D3semidirect
189. #393C2×D3semidirect
220. C4×C22⋊C4C2×D3semidirect
221. #376C2×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×#23982
#825002
#824032
#823742
#783922
#777052
#775742
#771192
#768652
#767872
#735852
C22⋊C4×D122
#735132
#734672
#734152
#453773