Finite Group Databasebeta

← #11002191

Subgroups — #11002191

Order 768.

2450 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5634C211yesmaximal, characteristic, complemented
2C22×#886C211yesmaximal, characteristic
3C3×#2561C211yesmaximal, characteristic, complemented
4#82775C211yesmaximal, characteristic, complemented
5#82116C211yesmaximal, characteristic, complemented
6#77568C214yesmaximal, complemented
7#76878C214yesmaximal, complemented
8#75922C214yesmaximal, complemented
9#75668C214yesmaximal, complemented
10#74130C212yesmaximal, complemented
11#74090C212yesmaximal, complemented
12#73853C212yesmaximal
13#73846C212yesmaximal, complemented
14#73567C211yesmaximal, characteristic, complemented
15#73517C211yesmaximal, characteristic, complemented
16#36720—31nomaximal, Sylow
17C23×#67C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C2×#946C2212yescomplemented
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C2×#924C2211yescharacteristic
27#5634—28no—
28C2×#897C2212yescomplemented
29C2×#896C2212yescomplemented
30C2×#896C2212yescomplemented
31C2×#886—28no—
32C2×#886C2214yes—
33C2×#886C2214yes—
34C2×#883C2214yes—
35C2×D3×C22⋊C4C2212yes—
36C2×#878C2212yes—
37C2×#878C2214yescomplemented
38C3×#397C2212yescomplemented
39C3×#393C2212yescomplemented
40C3×#388C2214yescomplemented
41C3×#381C2212yes—
42C3×#379C2212yes—
43C3×C4×C22⋊C4C2211yescharacteristic
44#5247C2212yes—
45#5229C2212yescomplemented
46#5211C2212yes—
47C4×#217C2211yescharacteristic
48#4981C2214yescomplemented
49#4975C2212yescomplemented
50#4961C2214yes—
51#4959C2214yes—
52#4956C2214yes—
53#4955C2212yes—
54#4953C2212yescomplemented
55#4950C2212yes—
56#4950C2212yes—
57#4950C2214yescomplemented
58#4947C2212yescomplemented
59#4946C2212yes—
60#4946C2212yes—
61#4946C2214yescomplemented
62#4945C2214yescomplemented
63#4942C2212yes—
64#4942C2214yescomplemented
65#4940C2214yescomplemented
66#4935C2212yescomplemented
67#4929C2212yescomplemented
68#4675C2214yescomplemented
69#4674C2214yescomplemented
70#4669C2214yescomplemented
71#4668C2214yescomplemented
72#4664C2214yescomplemented
73#4661C2214yescomplemented
74#4661C2214yescomplemented
75#4660C2214yescomplemented
76#4657C2214yescomplemented
77#4656C2214yescomplemented
78C2×#543—31no—
79C22×#123—31no—
80#2848—32no—
81#2687—34no—
82#2687—34no—
83#2680—34no—
84#2680—34no—
85#2627—32no—
86#2561—31no—
87#2561D311yescharacteristic, complemented
88#2502—31no—
89#2502—31no—
90#2483—32no—
91#2405—32no—
92#2312—31no—
93C25×C3C2311yescharacteristic
94C24×D3C2311yescharacteristic
95C22×C3×Q8C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C22×#67—24no—
99C22×#67—44no—
100C22×#67C2314yes—
101C23×Dic3—24no—
102C23×Dic3—44no—
103C23×Dic3C2314yes—
104C23×Dic3C2311yescharacteristic
105C23×Dic3C2311yescharacteristic
106C23×Dic3C2311yescharacteristic
107C22×C4×D3C2312yes—
108C22×C4×D3C2312yes—
109C22×Dic6C2312yes—
110C22×Dic6C2311yescharacteristic
111C3×C22⋊Q8—24no—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C22⋊C4—24no—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C42C2312yes—
122C2×#217C2312yes—
123C2×#217C2314yes—
124C2×#217C2312yes—
125C2×#217C2314yes—
126#946—28no—
127C2×(C2×D3)⋊C4—24no—
128C2×(C2×D3)⋊C4D418yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2312yes—
136C2×C4⋊Dic3—24no—
137C2×C4⋊Dic3—24no—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2311yescharacteristic
143C2×C4⋊Dic3C2314yes—
144C2×C4⋊Dic3C2314yes—
145#924—24no—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2314yes—
152C2×C4×Dic3C2312yes—
153#897—28no—
154#896—28no—
155#896—28no—
156#886—216no—
157#886—216no—
158#883—48no—
159D3×C22⋊C4—28no—
160#878—28no—
161#878—48no—
162C3×#95C2314yes—
163C3×#95C2314yes—
164C3×#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C23×D4—31no—
174C24×C4—31no—
175#543—68no—
176C2×#123—34no—
177C2×#123—34no—
178C2×#123—34no—
179C2×#123—68no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8—31no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—32no—
184C2×C22⋊Q8—32no—
185C2×C22⋊Q8—32no—
186C2×C22⋊Q8—34no—
187C2×C22⋊Q8C2×D311yescharacteristic
188C22×C4⋊C4—31no—
189C22×C4⋊C4—31no—
190C22×C22⋊C4—31no—
191C22×C22⋊C4—31no—
192C22×C22⋊C4—32no—
193C22×C22⋊C4C2×D311yescharacteristic
194#400—34no—
195#398—34no—
196#398—34no—
197#398—34no—
198#398—34no—
199#397—32no—
200#397—34no—
201#397—34no—
202#397C2×D312yescomplemented
203#395—34no—
204#394—34no—
205#394—34no—
206#393—32no—
207#393C2×D312yescomplemented
208#391—32no—
209#391—32no—
210#388—34no—
211#388—34no—
212#388—34no—
213#388C2×D314yescomplemented
214#386—34no—
215#385—32no—
216#385—34no—
217#385—32no—
218#385—34no—
219#385—32no—
220#385—32no—
221#385—34no—
222#384—32no—
223#384—32no—
224#381—32no—
225#381—34no—
226#381C2×D312yes—
227#380—32no—
228#380—34no—
229#379—32no—
230#379—32no—
231#379C2×D312yes—
232C4×C22⋊C4—31no—
233C4×C22⋊C4C2×D311yescharacteristic
234C24×C3—22no—
235C24×C3—24no—
236C24×C3—24no—
237C24×C3—42no—
238C24×C3C2412yes—
239C24×C3C2411yescharacteristic
240C23×D3—24no—
241C23×D3—42no—
242C23×D3—42no—
243C23×D3C2×D414yes—
244C23×D3C2412yes—
245C2×C3×Q8—22no—
246C2×C3×Q8—42no—
247C22×C3×C4—22no—
248C22×C3×C4—24no—
249C22×C3×C4—42no—
250C22×C3×C4C2×D414yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2412yes—
253C22×C3×C4C2414yes—
254C22×C3×C4C2412yes—
255C22×C3×C4C2411yescharacteristic
256C22×C3×C4C2411yescharacteristic
257C2×#67—28no—
258C2×#67—28no—
259C2×#67—44no—
260C2×#67—416no—
261C22×Dic3—22no—
262C22×Dic3—22no—
263C22×Dic3—28no—
264C22×Dic3—28no—
265C22×Dic3—42no—
266C22×Dic3—44no—
267C22×Dic3—42no—
268C22×Dic3—42no—
269C22×Dic3—416no—
270C22×Dic3C2×D414yes—
271C22×Dic3C2414yes—
272C22×Dic3C2412yes—
273C22×Dic3C2412yes—
274C22×Dic3C2412yes—
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C2×C4×D3—24no—
278C2×C4×D3—24no—
279C2×C4×D3—44no—
280C2×C4×D3—44no—
281C2×Dic6—22no—
282C2×Dic6—24no—
283C2×Dic6—44no—
284C2×Dic6—42no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4—24no—
287C3×C22⋊C4—24no—
288C3×C22⋊C4—28no—
289C3×C22⋊C4—24no—
290C3×C22⋊C4—24no—
291C3×C22⋊C4—24no—
292C3×C22⋊C4—44no—
293C3×C42—42no—
294#217—24no—
295#217—28no—
296#217—24no—
297#217—44no—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—24no—
300(C2×D3)⋊C4—28no—
301(C2×D3)⋊C4—44no—
302(C2×D3)⋊C4—44no—
303(C2×D3)⋊C4C2×D418yes—
304(C2×D3)⋊C4C2×D418yes—
305C4⋊Dic3—24no—
306C4⋊Dic3—24no—
307C4⋊Dic3—22no—
308C4⋊Dic3—24no—
309C4⋊Dic3—28no—
310C4⋊Dic3—44no—
311C4⋊Dic3—44no—
312Dic3⋊C4—24no—
313Dic3⋊C4—24no—
314Dic3⋊C4—24no—
315Dic3⋊C4—44no—
316Dic3⋊C4—44no—
317Dic3⋊C4—44no—
318C4×Dic3—42no—
319C25—31no—
320C25C22×D311yescharacteristic
321C22×Q8—32no—
322C22×Q8—31no—
323C22×Q8C22×D311yescharacteristic
324C22×D4—34no—
325C22×D4—64no—
326C22×D4—124no—
327C23×C4—31no—
328C23×C4—34no—
329C23×C4—31no—
330C23×C4—31no—
331C23×C4—32no—
332C23×C4—32no—
333C23×C4—64no—
334C23×C4—124no—
335C23×C4C22×D311yescharacteristic
336C23×C4C22×D311yescharacteristic
337#123—616no—
338#123—616no—
339#123—128no—
340C22⋊Q8—24no—
341C22⋊Q8—68no—
342C22⋊Q8—68no—
343C22⋊Q8—68no—
344C22⋊Q8—68no—
345C22⋊Q8—68no—
346C22⋊Q8—64no—
347C22⋊Q8—128no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—34no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—34no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—64no—
362C2×C4⋊C4—64no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C4⋊C4C22×D312yes—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—38no—
367C2×C22⋊C4—32no—
368C2×C22⋊C4—32no—
369C2×C22⋊C4—32no—
370C2×C22⋊C4—34no—
371C2×C22⋊C4—32no—
372C2×C22⋊C4—32no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—34no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—68no—
379C2×C22⋊C4—64no—
380C2×C22⋊C4C22×D312yes—
381C2×C22⋊C4C22×D312yes—
382C2×C22⋊C4C22×D312yes—
383C2×C22⋊C4C22×D314yes—
384C2×C22⋊C4C22×D312yes—
385C2×C22⋊C4C22×D314yes—
386C2×C42—32no—
387C2×C42C22×D312yes—
388#95—32no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—34no—
395#95—34no—
396#95C22×D312yes—
397#95C22×D314yes—
398#95C22×D314yes—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—28no—
402C23×C3—22no—
403C23×C3—22no—
404C23×C3—24no—
405C23×C3—24no—
406C23×C3—416no—
407C23×C3—44no—
408C23×C3—44no—
409C23×C3C22×D412yes—
410C23×C3H(5,𝔽2)14yes—
411C23×C32−1+414yes—
412C23×C3C2511yescharacteristic
413C22×D3—24no—
414C22×D3—24no—
415C22×D3—24no—
416C22×D3—28no—
417C22×D3—44no—
418C22×D3—44no—
419C22×D3—88no—
420C22×D3C22×D414yes—
421C3×Q8—82no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—24no—
428C2×C3×C4—24no—
429C2×C3×C4—24no—
430C2×C3×C4—24no—
431C2×C3×C4—24no—
432C2×C3×C4—24no—
433C2×C3×C4—44no—
434C2×C3×C4—44no—
435C2×C3×C4C22×D412yes—
436C2×C3×C4C22×D412yes—
437#67—416no—
438C2×Dic3—24no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—24no—
444C2×Dic3—24no—
445C2×Dic3—24no—
446C2×Dic3—24no—
447C2×Dic3—24no—
448C2×Dic3—24no—
449C2×Dic3—24no—
450C2×Dic3—44no—
451C2×Dic3—416no—
452C2×Dic3—44no—
453C2×Dic3—44no—
454C2×Dic3C22×D414yes—
455C4×D3—84no—
456C4×D3—84no—
457Dic6—84no—
458Dic6—82no—
459C24—22no—
460C24—24no—
461C24—24no—
462C24—32no—
463C24—34no—
464C24—42no—
465C24—64no—
466C24—122no—
467C24—122no—
468C24C23×D312yes—
469C24C23×D311yescharacteristic
470C2×Q8—22no—
471C2×Q8—42no—
472C2×Q8—64no—
473C2×Q8—62no—
474C2×Q8—124no—
475C2×Q8—122no—
476C2×D4—68no—
477C2×D4—68no—
478C2×D4—124no—
479C2×D4—1216no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—32no—
483C22×C4—34no—
484C22×C4—34no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—32no—
489C22×C4—42no—
490C22×C4—64no—
491C22×C4—64no—
492C22×C4—68no—
493C22×C4—62no—
494C22×C4—62no—
495C22×C4—68no—
496C22×C4—124no—
497C22×C4—124no—
498C22×C4—122no—
499C22×C4—124no—
500C22×C4—122no—
501C22×C4—122no—
502C22×C4—1216no—
503C22×C4D3×D414yes—
504C22×C4C23×D312yes—
505C22×C4C23×D312yes—
506C22×C4C23×D314yes—
507C22×C4C23×D312yes—
508C22×C4C23×D311yescharacteristic
509C22×C4C23×D311yescharacteristic
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—68no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—62no—
520C4⋊C4—124no—
521C4⋊C4—124no—
522C4⋊C4—124no—
523C4⋊C4—124no—
524C4⋊C4—124no—
525C22⋊C4—24no—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—38no—
531C22⋊C4—38no—
532C22⋊C4—44no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—68no—
536C22⋊C4—64no—
537C22⋊C4—68no—
538C22⋊C4—64no—
539C22⋊C4—124no—
540C22⋊C4—124no—
541C22⋊C4—124no—
542C42—42no—
543C42—122no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—28no—
547C22×C3—24no—
548C22×C3—28no—
549C22×C3—416no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—44no—
553C22×C3#53614yes—
554C22×C3C23×D411yescharacteristic
555C22×C3C2×H(5,𝔽2)12yes—
556C22×C3C2×2−1+412yes—
557C22×C3#58512yes—
558C2×D3—24no—
559C2×D3—24no—
560C2×D3—24no—
561C2×D3—48no—
562C2×D3—88no—
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—44no—
569Dic3—24no—
570Dic3—42no—
571Dic3—42no—
572Dic3—42no—
573Dic3—42no—
574Dic3—44no—
575C23—22no—
576C23—22no—
577C23—28no—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—24no—
582C23—34no—
583C23—416no—
584C23—44no—
585C23—44no—
586C23—64no—
587C23—64no—
588C23—64no—
589C23—68no—
590C23—124no—
591C23—124no—
592C23—248no—
593C23C2×D3×D412yes—
594C23#100914yes—
595C23#101014yes—
596C23C24×D311yescharacteristic
597Q8—82no—
598Q8—242no—
599Q8—244no—
600D4—1216no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—34no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—64no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—124no—
628C2×C4—1216no—
629C2×C4—124no—
630C2×C4—124no—
631C2×C4—244no—
632C2×C4—244no—
633C2×C4C2×D3×D412yes—
634C2×C4C2×D3×D412yes—
635C2×C3—24no—
636C2×C3—44no—
637C2×C3C2×#53611yescharacteristic
638C2×C3#348411yescharacteristic
639C2×C3#348511yescharacteristic
640C2×C3#353514yes—
641D3—44no—
642C22—28no—
643C22—22no—
644C22—22no—
645C22—28no—
646C22—24no—
647C22—416no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—64no—
652C22—64no—
653C22—64no—
654C22—128no—
655C22—248no—
656C22#567514yes—
657C22C22×D3×D411yescharacteristic
658C22C2×#100912yes—
659C22C2×#101012yes—
660C22#596612yes—
661C4—22no—
662C4—22no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—64no—
668C4—124no—
669C4—122no—
670C4—122no—
671C4—122no—
672C4—122no—
673C3#3672011yescharacteristic, complemented, Sylow
674C2—24no—
675C2—44no—
676C2—124no—
677C2#8603911yescharacteristic
678C2C2×#567511yescharacteristic
679C2#8629611yescharacteristic
680C2#8680214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2561C2
prime radicalC23×#67C22
hypercenter#2561D3
hypocenterC3#36720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2561

Complements

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

NH ≅ G/Ntype
1. C2×#5634C2semidirect
3. C3×#2561C2semidirect
4. #82775C2semidirect
5. #82116C2semidirect
6. #77568C2semidirect
7. #76878C2semidirect
8. #75922C2semidirect
9. #75668C2semidirect
10. #74130C2semidirect
11. #74090C2semidirect
13. #73846C2semidirect
14. #73567C2semidirect
15. #73517C2semidirect
21. C2×#946C22semidirect
28. C2×#897C22semidirect
29. C2×#896C22semidirect
30. C2×#896C22semidirect
37. C2×#878C22semidirect
38. C3×#397C22semidirect
39. C3×#393C22semidirect
40. C3×#388C22semidirect
45. #5229C22semidirect
48. #4981C22semidirect
49. #4975C22semidirect
54. #4953C22semidirect
57. #4950C22semidirect
58. #4947C22semidirect
61. #4946C22semidirect
62. #4945C22semidirect
64. #4942C22semidirect
65. #4940C22semidirect
66. #4935C22semidirect
67. #4929C22semidirect
68. #4675C22semidirect
69. #4674C22semidirect
70. #4669C22semidirect
71. #4668C22semidirect
72. #4664C22semidirect
73. #4661C22semidirect
74. #4661C22semidirect
75. #4660C22semidirect
76. #4657C22semidirect
77. #4656C22semidirect
87. #2561D3semidirect
202. #397C2×D3semidirect
207. #393C2×D3semidirect
213. #388C2×D3semidirect
673. C3#36720semidirect

Maximal subgroups

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

subgroupindex
C2×#56342
C22×#8862
C3×#25612
#827752
#821162
#775682
#768782
#759222
#756682
#741302
#740902
#738532
#738462
#735672
#735172
#367203