Finite Group Databasebeta

← #11233677

Subgroups — #11233677

Order 768.

2582 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic, complemented
2C2×#5551C211yesmaximal, characteristic, complemented
3C3×#2326C211yesmaximal, characteristic, complemented
4#82998C212yesmaximal, complemented
5#82499C211yesmaximal, characteristic, complemented
6#82416C211yesmaximal, characteristic, complemented
7#76621C212yesmaximal, complemented
8#76439C212yesmaximal, complemented
9#75674C214yesmaximal, complemented
10#75504C214yesmaximal, complemented
11#74541C214yesmaximal, complemented
12#74404C214yesmaximal, complemented
13#74360C212yesmaximal, complemented
14#73418C211yesmaximal, characteristic, complemented
15#73391C211yesmaximal, characteristic, complemented
16#45356—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#946C2212yes—
22C2×#946C2212yes—
23C2×#938C2212yes—
24C2×#938C2212yes—
25C2×#924C2212yes—
26#5579—28no—
27#5551—28no—
28C2×#896C2214yes—
29C2×#893C2214yes—
30C2×D3×C4⋊C4C2212yes—
31C2×D3×C4⋊C4C2212yes—
32C2×#884C2214yes—
33C2×#883C2214yes—
34C2×C4×D12C2211yescharacteristic
35C2×C4×D12C2212yes—
36C2×C4×D12C2211yescharacteristic
37C2×C4×D12C2212yes—
38C2×C4×D12C2212yes—
39C2×#870C2212yes—
40C2×#869C2211yescharacteristic
41C3×#391C2211yescharacteristic
42C3×#390C2211yescharacteristic
43C3×#388C2214yes—
44C3×#385C2214yes—
45C3×#384C2211yescharacteristic
46C3×#376C2211yescharacteristic
47#5213C2214yes—
48#4991C2212yes—
49#4989C2214yes—
50#4989C2214yes—
51#4987C2212yes—
52#4987C2212yes—
53#4981C2212yes—
54#4972C2214yes—
55#4960C2214yes—
56#4960C2212yes—
57#4959C2214yes—
58#4957C2214yes—
59#4957C2214yes—
60#4956C2214yes—
61#4953C2212yes—
62#4949C2214yes—
63#4948C2214yes—
64#4940C2214yes—
65#4936C2211yescharacteristic
66C4×C4⋊Dic3C2212yes—
67#4928C2212yes—
68#4677C2212yes—
69#4677C2212yes—
70#4676C2214yes—
71#4671C2212yes—
72#4671C2212yes—
73#4667C2214yes—
74#4665C2214yes—
75#4657C2214yes—
76#4654C2214yes—
77#4646C2214yes—
78C2×#554—31no—
79C2×#547—31no—
80#2732—34no—
81#2694—34no—
82#2540—32no—
83#2536—31no—
84#2535—31no—
85#2456—34no—
86#2417—32no—
87#2416—32no—
88#2407—34no—
89#2397—31no—
90D4×C4⋊C4—32no—
91#2331—31no—
92#2326D311yescharacteristic, complemented
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C22×#67C2314yes—
98C2×#235—28no—
99C2×#235—48no—
100C22×D12C2311yescharacteristic
101C22×D12C2311yescharacteristic
102C22×D12C2312yes—
103C22×D12C2312yes—
104C22×C4×D3C2312yes—
105C22×C4×D3C2312yes—
106C22×C4×D3C2314yes—
107C22×C4×D3C2314yes—
108C22×Dic6C2312yes—
109C3×#124—24no—
110C3×C4×Q8—24no—
111C3×C4×D4—24no—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2312yes—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C2×#217C2314yes—
124C2×#217C2314yes—
125#946—28no—
126#946—44no—
127#938—28no—
128#938—44no—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2312yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2311yescharacteristic
142C2×C4⋊Dic3C2311yescharacteristic
143C2×C4⋊Dic3C2312yes—
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2312yes—
146#924—28no—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2314yes—
152C2×C4×Dic3C2314yes—
153#896—216no—
154#893—48no—
155D3×C4⋊C4—28no—
156D3×C4⋊C4—28no—
157#884—48no—
158#883—216no—
159C4×D12—24no—
160C4×D12—24no—
161C4×D12—28no—
162C4×D12—28no—
163C4×D12—44no—
164#870—28no—
165#869D418yescomplemented
166C3×#95C2312yes—
167C3×#95C2312yes—
168C3×#95C2314yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C22×#41—32no—
176#554—68no—
177#547—68no—
178C2×C4⋊Q8—31no—
179C2×#125—32no—
180C2×#124C2×D311yescharacteristic
181C2×#123—34no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—32no—
184C2×C22⋊Q8—32no—
185C2×C22⋊Q8—34no—
186C2×#121—32no—
187C2×#121—32no—
188C2×#121—34no—
189C2×C4×Q8C2×D311yescharacteristic
190C2×C4×D4—34no—
191C2×C4×D4—32no—
192C2×C4×D4—31no—
193C2×C4×D4—31no—
194C2×C4×D4—32no—
195C2×C4×D4—32no—
196C2×C4×D4C2×D311yescharacteristic
197C22×C4⋊C4—32no—
198C22×C4⋊C4—32no—
199#399—34no—
200#399—32no—
201#399—32no—
202#398—32no—
203#398—34no—
204#397—32no—
205#396—34no—
206#396—34no—
207#394—32no—
208#394—34no—
209#392—34no—
210#392—32no—
211#392—32no—
212#392—32no—
213#391—34no—
214#391C2×D311yescharacteristic
215#390—34no—
216#390—34no—
217#390C2×D311yescharacteristic
218#389—34no—
219#389—32no—
220#388—34no—
221#388C2×D314yes—
222#387—34no—
223#386—34no—
224#386—34no—
225#386—34no—
226#385—34no—
227#385—34no—
228#385C2×D314yes—
229#384C2×D311yescharacteristic
230#382—31no—
231#380—32no—
232#380—32no—
233#380—34no—
234C4×C4⋊C4—32no—
235#376C2×D311yescharacteristic
236C24×C3C2412yes—
237C23×D3C2412yes—
238C23×D3C2412yes—
239C2×C3×Q8—22no—
240C2×C3×Q8—22no—
241C2×C3×Q8—22no—
242C2×C3×D4—22no—
243C2×C3×D4—22no—
244C2×C3×D4—22no—
245C22×C3×C4—24no—
246C22×C3×C4—44no—
247C22×C3×C4C2412yes—
248C22×C3×C4C2412yes—
249C22×C3×C4C2412yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2411yescharacteristic
252C22×C3×C4C2411yescharacteristic
253C22×C3×C4C2412yes—
254C22×C3×C4C2411yescharacteristic
255C2×#67—28no—
256C2×#67—28no—
257C2×#67—28no—
258C2×#67—44no—
259C2×#67—44no—
260C2×#67—44no—
261C22×Dic3C2414yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2414yes—
265#235—432no—
266C2×D12—24no—
267C2×D12—24no—
268C2×D12—24no—
269C2×D12—24no—
270C2×D12—22no—
271C2×D12—22no—
272C2×D12—24no—
273C2×D12—24no—
274C2×D12—24no—
275C2×D12—42no—
276C2×C4×D3—28no—
277C2×C4×D3—24no—
278C2×C4×D3—24no—
279C2×C4×D3—24no—
280C2×C4×D3—24no—
281C2×C4×D3—24no—
282C2×C4×D3—24no—
283C2×C4×D3—28no—
284C2×C4×D3—28no—
285C2×C4×D3—44no—
286C2×C4×D3—44no—
287C2×C4×D3—44no—
288C2×Dic6—44no—
289C2×Dic6C2×D418yes—
290C3×C4⋊C4—22no—
291C3×C4⋊C4—22no—
292C3×C4⋊C4—24no—
293C3×C4⋊C4—44no—
294C3×C22⋊C4—24no—
295C3×C22⋊C4—28no—
296C3×C22⋊C4—44no—
297C3×C42—24no—
298C3×C42—22no—
299C3×C42C2×D414yes—
300C3×C42#4114yescomplemented
301#217—28no—
302#217—44no—
303(C2×D3)⋊C4—28no—
304(C2×D3)⋊C4—24no—
305(C2×D3)⋊C4—24no—
306(C2×D3)⋊C4—24no—
307(C2×D3)⋊C4—24no—
308(C2×D3)⋊C4—28no—
309(C2×D3)⋊C4—42no—
310(C2×D3)⋊C4—42no—
311(C2×D3)⋊C4—44no—
312(C2×D3)⋊C4—44no—
313C4⋊Dic3—22no—
314C4⋊Dic3—22no—
315C4⋊Dic3—24no—
316C4⋊Dic3—24no—
317C4⋊Dic3—42no—
318C4⋊Dic3C2×D418yes—
319C4⋊Dic3C2×D418yes—
320Dic3⋊C4—28no—
321Dic3⋊C4—28no—
322Dic3⋊C4—28no—
323Dic3⋊C4—44no—
324Dic3⋊C4—44no—
325C4×Dic3—44no—
326C2×#41—68no—
327C2×#41—128no—
328C22×Q8—32no—
329C22×Q8C22×D311yescharacteristic
330C22×D4—34no—
331C22×D4—34no—
332C22×D4—32no—
333C22×D4—31no—
334C22×D4—31no—
335C22×D4—32no—
336C22×D4C22×D311yescharacteristic
337C23×C4—32no—
338C23×C4—32no—
339C23×C4—34no—
340C23×C4—34no—
341C23×C4C22×D312yes—
342C4⋊Q8—38no—
343#125—68no—
344#124—24no—
345#123—616no—
346C22⋊Q8—616no—
347C22⋊Q8—68no—
348C22⋊Q8—68no—
349C22⋊Q8—124no—
350#121—68no—
351#121—128no—
352#121—124no—
353C4×Q8—24no—
354C4×D4—24no—
355C4×D4—68no—
356C4×D4—68no—
357C4×D4—64no—
358C4×D4—64no—
359C4×D4—124no—
360C4×D4—128no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—34no—
364C2×C4⋊C4—34no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—34no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—68no—
374C2×C4⋊C4—68no—
375C2×C4⋊C4C22×D311yescharacteristic
376C2×C4⋊C4C22×D311yescharacteristic
377C2×C4⋊C4C22×D312yes—
378C2×C4⋊C4C22×D314yes—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—34no—
381C2×C22⋊C4—32no—
382C2×C22⋊C4—34no—
383C2×C22⋊C4—32no—
384C2×C22⋊C4—32no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—34no—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—34no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—34no—
391C2×C22⋊C4C22×D312yes—
392C2×C22⋊C4C22×D314yes—
393C2×C22⋊C4C22×D314yes—
394C2×C42—34no—
395C2×C42C22×D311yescharacteristic
396C2×C42C22×D312yes—
397C2×C42C22×D311yescharacteristic
398C2×C42C22×D311yescharacteristic
399#95—34no—
400#95—34no—
401#95—34no—
402#95—32no—
403#95—32no—
404#95—34no—
405#95C22×D312yes—
406#95C22×D312yes—
407#95C22×D314yes—
408C23×C3—22no—
409C23×C3—22no—
410C23×C3—22no—
411C23×C3—24no—
412C23×C3—24no—
413C23×C3C2511yescharacteristic
414C22×D3—22no—
415C22×D3—22no—
416C22×D3—22no—
417C22×D3—22no—
418C22×D3—22no—
419C22×D3—22no—
420C22×D3—22no—
421C22×D3—22no—
422C22×D3—22no—
423C22×D3—22no—
424C22×D3—22no—
425C22×D3—22no—
426C22×D3H(5,𝔽2)14yes—
427C22×D32−1+414yes—
428C3×Q8—44no—
429C3×D4—44no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—22no—
444C2×C3×C4—48no—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×D412yes—
447C2×C3×C4C22×D412yes—
448C2×C3×C4C2×#4114yes—
449C2×C3×C4C2×#4112yes—
450C2×C3×C4H(5,𝔽2)12yes—
451#67—416no—
452#67—88no—
453C2×Dic3—24no—
454C2×Dic3—24no—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—24no—
460C2×Dic3—22no—
461C2×Dic3—24no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3C22×D418yes—
465D12—28no—
466D12—216no—
467D12—28no—
468D12—48no—
469C4×D3—416no—
470C4×D3—48no—
471C4×D3—48no—
472C4×D3—88no—
473Dic6—48no—
474C24—32no—
475C24—32no—
476C24C23×D312yes—
477#41—1232no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—38no—
482C2×Q8—124no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—64no—
487C2×D4—64no—
488C2×D4—64no—
489C2×D4—64no—
490C2×D4—64no—
491C2×D4—64no—
492C2×D4—68no—
493C2×D4—68no—
494C2×D4—68no—
495C2×D4—64no—
496C2×D4—62no—
497C2×D4—62no—
498C2×D4—122no—
499C2×D4—124no—
500C2×D4—124no—
501C2×D4—124no—
502C22×C4—24no—
503C22×C4—34no—
504C22×C4—34no—
505C22×C4—32no—
506C22×C4—32no—
507C22×C4—44no—
508C22×C4—68no—
509C22×C4—68no—
510C22×C4—68no—
511C22×C4—64no—
512C22×C4—64no—
513C22×C4—64no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—124no—
518C22×C4—124no—
519C22×C4—124no—
520C22×C4C23×D312yes—
521C22×C4C23×D312yes—
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D312yes—
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D311yescharacteristic
527C22×C4C23×D311yescharacteristic
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—38no—
532C4⋊C4—38no—
533C4⋊C4—44no—
534C4⋊C4—68no—
535C4⋊C4—68no—
536C4⋊C4—68no—
537C4⋊C4—64no—
538C4⋊C4—62no—
539C4⋊C4—62no—
540C4⋊C4—64no—
541C4⋊C4—122no—
542C4⋊C4—124no—
543C4⋊C4—124no—
544C22⋊C4—28no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—68no—
548C22⋊C4—68no—
549C22⋊C4—68no—
550C22⋊C4—64no—
551C22⋊C4—64no—
552C22⋊C4—64no—
553C22⋊C4—64no—
554C22⋊C4—122no—
555C22⋊C4—122no—
556C22⋊C4—124no—
557C22⋊C4—124no—
558C22⋊C4—124no—
559C42—24no—
560C42—22no—
561C42—124no—
562C42#23514yescomplemented
563C42D3×D414yes—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—42no—
567C22×C3—44no—
568C22×C3—44no—
569C22×C3C23×D411yescharacteristic
570C22×C3C22×#4111yescharacteristic
571C22×C3C2×H(5,𝔽2)11yescharacteristic
572C22×C3C2×H(5,𝔽2)12yes—
573C22×C3C2×2−1+412yes—
574C2×D3—24no—
575C2×D3—24no—
576C2×D3—24no—
577C2×D3—24no—
578C2×D3—24no—
579C2×D3—24no—
580C2×D3—42no—
581C2×D3—42no—
582C2×D3—42no—
583C2×D3—42no—
584C2×D3—42no—
585C2×D3—42no—
586C2×D3—42no—
587C2×D3—42no—
588C3×C4—22no—
589C3×C4—22no—
590C3×C4—24no—
591C3×C4—42no—
592C3×C4—42no—
593C3×C4—42no—
594C3×C4—42no—
595C3×C4#55014yes—
596Dic3—28no—
597Dic3—42no—
598Dic3—42no—
599Dic3—44no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—24no—
605C23—34no—
606C23—34no—
607C23—62no—
608C23—62no—
609C23—62no—
610C23—62no—
611C23—62no—
612C23—62no—
613C23—62no—
614C23—62no—
615C23—62no—
616C23—62no—
617C23—62no—
618C23—62no—
619C23C24×D311yescharacteristic
620Q8—44no—
621Q8—128no—
622D4—44no—
623D4—68no—
624D4—68no—
625D4—616no—
626D4—128no—
627D4—1216no—
628D4—248no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—38no—
644C2×C4—48no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—64no—
653C2×C4—62no—
654C2×C4—64no—
655C2×C4—64no—
656C2×C4—1216no—
657C2×C4—128no—
658C2×C4—128no—
659C2×C4—248no—
660C2×C4C2×#23514yes—
661C2×C4C2×#23512yes—
662C2×C4C2×D3×D412yes—
663C2×C4C2×D3×D412yes—
664C2×C4C2×D3×D412yes—
665C2×C4#100912yes—
666C2×C3—82no—
667C2×C3C2×#55011yescharacteristic
668C2×C3#348411yescharacteristic
669C2×C3#348611yescharacteristic
670C2×C3#349212yes—
671C2×C3#353612yes—
672D3—44no—
673D3—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—42no—
679C22—64no—
680C22—64no—
681C22—64no—
682C22—64no—
683C22—64no—
684C22—64no—
685C22—122no—
686C22—122no—
687C22—122no—
688C22—122no—
689C22—122no—
690C22—122no—
691C22—122no—
692C22—122no—
693C22C22×#23511yescharacteristic
694C22C22×D3×D411yescharacteristic
695C22C2×#100911yescharacteristic
696C22D3×H(5,𝔽2)12yes—
697C22D3×2−1+412yes—
698C4—22no—
699C4—22no—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—68no—
706C4—124no—
707C4—122no—
708C4—122no—
709C4#555314yes—
710C3#4535611yescharacteristic, complemented, Sylow
711C2—82no—
712C2—124no—
713C2—124no—
714C2C2×#555311yescharacteristic
715C2#8590212yes—
716C2#8633011yescharacteristic
717C2#8635511yescharacteristic
718C2#8656112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2326C2
hypercenter#2326D3
hypocenterC3#45356
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C2×#5551C2semidirect
3. C3×#2326C2semidirect
4. #82998C2semidirect
5. #82499C2semidirect
6. #82416C2semidirect
7. #76621C2semidirect
8. #76439C2semidirect
9. #75674C2semidirect
10. #75504C2semidirect
11. #74541C2semidirect
12. #74404C2semidirect
13. #74360C2semidirect
14. #73418C2semidirect
15. #73391C2semidirect
92. #2326D3semidirect
165. #869D4semidirect
300. C3×C42#41semidirect
562. C42#235semidirect
710. C3#45356semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×#55512
C3×#23262
#829982
#824992
#824162
#766212
#764392
#756742
#755042
#745412
#744042
#743602
#734182
#733912
#453563