Finite Group Databasebeta

← #10826814

Subgroups — #10826814

Order 768.

1744 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3485C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C2×#5243C212yesmaximal, complemented
4#82864C212yesmaximal, complemented
5#72139C211yesmaximal, characteristic, complemented
6#71849C214yesmaximal, complemented
7#71705C212yesmaximal, complemented
8#71655C212yesmaximal, complemented
9#23905—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic, complemented
13C3×#547—24no—
14C3×#546—24no—
15C3×#536—21no—
16C3×#534—21no—
17C3×#534C2212yes—
18C3×#525—21no—
19C2×C3×#123—21no—
20C2×C3×#123C2212yes—
21C3×#518C2211yescharacteristic
22#5825—28no—
23C2×#944—21no—
24C2×#936—21no—
25C2×#935C2214yes—
26C2×#935C2212yes—
27C2×#935C2212yescomplemented
28C2×#926C2211yescharacteristic, complemented
29C2×#924—21no—
30C2×#924—21no—
31#5243—28no—
32C2×#834C2214yes—
33C2×#834C2212yes—
34C2×#834C2212yescomplemented
35#5211C2212yescomplemented
36#4979—24no—
37#4947—24no—
38#4943C2214yescomplemented
39#4526C2214yescomplemented
40#3485D311yescharacteristic
41C2×#536—31no—
42C2×#458—32no—
43#2614—32no—
44#2022—34no—
45#1899—32no—
46#1890—32no—
47#1782—31no—
48C3×H(5,𝔽2)—24no—
49C3×H(5,𝔽2)—42no—
50C2×C3×#41—24no—
51C2×C3×#41—24no—
52C2×C3×#41—21no—
53C2×C3×#41—22no—
54C2×C3×#41—42no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×Dic3C2312yes—
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—24no—
69C22×Dic6—24no—
70C22×Dic6—42no—
71C3×C4⋊Q8—22no—
72C3×#124—22no—
73C3×#123—24no—
74C3×#123—22no—
75C3×#123—22no—
76C3×#123—22no—
77C3×#123—22no—
78C3×#123—24no—
79C3×#123—22no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—24no—
82C3×#121—22no—
83C3×#121—24no—
84C3×#121—24no—
85C3×#120—22no—
86C3×#120—22no—
87C3×#120—22no—
88C3×C4×D4—24no—
89C3×C4×D4—24no—
90C3×#117—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D414yes—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4D414yes—
96C2×C3×C22⋊C4C2312yes—
97C2×#217—22no—
98C2×#217—22no—
99C2×#217—24no—
100C2×#217C2314yes—
101C2×#217C2312yes—
102#944—28no—
103#936—28no—
104#935—24no—
105#935—24no—
106#935—24no—
107#935—28no—
108#935—44no—
109#935D418yescomplemented
110#926—24no—
111C2×C4⋊Dic3—21no—
112#924—28no—
113#924—28no—
114C2×Dic3⋊C4—22no—
115C2×Dic3⋊C4—22no—
116C2×Dic3⋊C4—24no—
117C2×C4×Dic3—21no—
118#834—24no—
119#834—28no—
120#834—42no—
121C3⋊#95—28no—
122C3⋊#95—24no—
123C2×H(5,𝔽2)C2×D311yescharacteristic
124C22×#41C2×D311yescharacteristic
125C23×Q8—31no—
126#547—24no—
127#546—24no—
128#536—21no—
129#536—68no—
130#534—21no—
131#534C2×D312yes—
132C2×C4⋊Q8—61no—
133C2×#124—61no—
134#525—21no—
135C2×#123—21no—
136C2×#123—32no—
137C2×#123—32no—
138C2×#123—34no—
139C2×#123C2×D312yes—
140C2×C22⋊Q8—61no—
141C2×C22⋊Q8—61no—
142#518C2×D311yescharacteristic
143C2×#117—31no—
144#458—68no—
145C2×#99—32no—
146C2×#99—34no—
147C2×#99—32no—
148#395—64no—
149#393—64no—
150#381—32no—
151#381—34no—
152#342—34no—
153C24×C3—21no—
154C24×C3C2×D412yes—
155C24×C3C2×D411yescharacteristic
156C24×C3C2×D411yescharacteristic
157C24×C3C2411yescharacteristic
158C3×#41—416no—
159C3×#41—48no—
160C3×#41—42no—
161C3×#41—42no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—41no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—41no—
181C2×C3×D4—48no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—41no—
195C22×C3×C4—41no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D414yes—
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3C2×D414yes—
210C2×Dic6—24no—
211C2×Dic6—24no—
212C2×Dic6—416no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C2×Dic6—42no—
216C2×Dic6—42no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C42—22no—
229#217—28no—
230#217—28no—
231#217—22no—
232#217—24no—
233#217—24no—
234#217—48no—
235#217C2×D414yes—
236C4⋊Dic3—24no—
237Dic3⋊C4—28no—
238Dic3⋊C4—28no—
239Dic3⋊C4—48no—
240C4×Dic3—24no—
241H(5,𝔽2)—24no—
242H(5,𝔽2)—42no—
243C2×#41—24no—
244C2×#41—21no—
245C2×#41—24no—
246C2×#41—22no—
247C2×#41—42no—
248C22×Q8—64no—
249C22×Q8—61no—
250C22×Q8—61no—
251C22×Q8—64no—
252C22×Q8—122no—
253C22×Q8#6711yescharacteristic
254C22×D4—21no—
255C22×D4—22no—
256C22×D4#6711yescharacteristic
257C22×D4C22×D312yes—
258C22×D4C22×D311yescharacteristic
259C22×D4C22×D311yescharacteristic
260C23×C4—32no—
261C23×C4#6711yescharacteristic
262C23×C4#6711yescharacteristic
263C23×C4C22×D311yescharacteristic
264C4⋊Q8—22no—
265C4⋊Q8—68no—
266#124—22no—
267#124—68no—
268#123—24no—
269#123—22no—
270#123—22no—
271#123—22no—
272#123—22no—
273#123—24no—
274#123—22no—
275#123—38no—
276#123—64no—
277#123—64no—
278#123—64no—
279#123—68no—
280#123—124no—
281C22⋊Q8—22no—
282C22⋊Q8—24no—
283C22⋊Q8—68no—
284C22⋊Q8—68no—
285#121—22no—
286#121—24no—
287#121—24no—
288#120—22no—
289#120—22no—
290#120—22no—
291C4×D4—24no—
292C4×D4—24no—
293#117—21no—
294#117—64no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—62no—
299C2×C4⋊C4—64no—
300C2×C4⋊C4#6714yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—34no—
305C2×C22⋊C4—62no—
306C2×C22⋊C4—62no—
307C2×C22⋊C4—64no—
308C2×C22⋊C4#6714yes—
309C2×C22⋊C4C22×D312yes—
310C2×C42—61no—
311#99—68no—
312#99—64no—
313#99—122no—
314#95—68no—
315#95—64no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—24no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—44no—
328C23×C3—82no—
329C23×C3#12012yes—
330C23×C3C22×D412yes—
331C23×C3C22×D411yescharacteristic
332C23×C3C22×D412yes—
333C23×C3H(5,𝔽2)12yes—
334C3×Q8—44no—
335C3×Q8—42no—
336C3×D4—28no—
337C3×D4—44no—
338C3×D4—42no—
339C3×D4—44no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—44no—
343C3×D4—44no—
344C3×D4—84no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—28no—
352C2×C3×C4—21no—
353C2×C3×C4—24no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—24no—
358C2×C3×C4—44no—
359C2×C3×C4—44no—
360C2×C3×C4—44no—
361C2×C3×C4—42no—
362C2×C3×C4—44no—
363C2×C3×C4—82no—
364C2×C3×C4#12012yes—
365C2×C3×C4#12012yes—
366C2×C3×C4#12012yes—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—28no—
370C2×Dic3—44no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—44no—
374C2×Dic3—48no—
375C2×Dic3—44no—
376Dic6—48no—
377Dic6—48no—
378C24—21no—
379C24C2×#6712yes—
380C24C2×#6711yescharacteristic
381C24C2×#6711yescharacteristic
382C24C23×D311yescharacteristic
383#41—416no—
384#41—48no—
385#41—42no—
386#41—42no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—41no—
391C2×Q8—64no—
392C2×Q8—64no—
393C2×Q8—1216no—
394C2×Q8—122no—
395C2×Q8—122no—
396C2×Q8—122no—
397C2×Q8—122no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—41no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4D3×D414yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—34no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—62no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4C2×#6712yes—
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6714yes—
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6714yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—64no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—128no—
450C22⋊C4—22no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—34no—
458C22⋊C4—68no—
459C22⋊C4—68no—
460C22⋊C4—62no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—128no—
464C42—22no—
465C42—64no—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—81no—
479C22×C3—84no—
480C22×C3—81no—
481C22×C3—162no—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12011yescharacteristic
484C22×C3#53411yescharacteristic
485C22×C3D4214yes—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—44no—
492Dic3—44no—
493Dic3—44no—
494Dic3—84no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—21no—
500C23—24no—
501C23—21no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—82no—
508C23#93712yes—
509C23C2×D3×D412yes—
510C23#100412yes—
511C23C22×#6711yescharacteristic
512C23C22×#6712yes—
513Q8—44no—
514Q8—42no—
515Q8—128no—
516Q8—128no—
517D4—28no—
518D4—44no—
519D4—42no—
520D4—44no—
521D4—42no—
522D4—44no—
523D4—44no—
524D4—42no—
525D4—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—68no—
547C2×C4—82no—
548C2×C4—124no—
549C2×C4—124no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—128no—
553C2×C4—124no—
554C2×C4#93712yes—
555C2×C4#95312yes—
556C2×C4#95312yes—
557C2×C3—21no—
558C2×C3—24no—
559C2×C3—21no—
560C2×C3—42no—
561C2×C3—81no—
562C2×C3—82no—
563C2×C3#240311yescharacteristic
564C2×C3#302812yes—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—44no—
577C22—81no—
578C22—84no—
579C22—81no—
580C22—162no—
581C22C2×#93711yescharacteristic
582C22D4×#6714yes—
583C22#580411yescharacteristic
584C22C2×#95311yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—41no—
590C4—44no—
591C4—124no—
592C4—124no—
593C4—244no—
594C3#2390511yescharacteristic, complemented, Sylow
595C2—21no—
596C2—24no—
597C2—21no—
598C2—42no—
599C2—81no—
600C2—82no—
601C2#8405211yescharacteristic
602C2#8412412yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC3×#3485C2
socleC22×C3C2×#120
hypercenter#3485D3
hypocenterC3#23905
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C2×#5243C2semidirect
4. #82864C2semidirect
5. #72139C2semidirect
6. #71849C2semidirect
7. #71705C2semidirect
8. #71655C2semidirect
12. C23×Dic6C22semidirect
27. C2×#935C22semidirect
28. C2×#926C22semidirect
34. C2×#834C22semidirect
35. #5211C22semidirect
38. #4943C22semidirect
39. #4526C22semidirect
109. #935D4semidirect
594. C3#23905semidirect

Maximal subgroups

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

subgroupindex
C3×#34852
C2×#58252
C2×#52432
#828642
#721392
#718492
#717052
#716552
#239053