Finite Group Databasebeta

← #10828801

Subgroups — #10828801

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5565C211yesmaximal, characteristic
2C3×#2487C211yesmaximal, characteristic, complemented
3C2×#4968C211yesmaximal, characteristic, complemented
4C22×#807C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic, complemented
6#69625C212yesmaximal, complemented
7#69606C212yesmaximal, complemented
8#69523C212yesmaximal, complemented
9#69515C211yesmaximal, characteristic, complemented
10#69506C212yesmaximal, complemented
11#69502C211yesmaximal, characteristic, complemented
12#24092—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#939—21no—
16C2×#930C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21#5565—24no—
22#5565—24no—
23C2×#898—21no—
24C2×#883—21no—
25C2×#67⋊C4—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#870C2211yescharacteristic
28C3×#400—21no—
29C3×#399—21no—
30C3×#389C2212yes—
31C3×#386—21no—
32C3×#385—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37#4968—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2212yes—
41C2×#807—24no—
42C2×#807C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#553—31no—
49C2×#481—31no—
50C22×#107—31no—
51#2487D311yescharacteristic, complemented
52#2059—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—21no—
83#939—28no—
84#930—22no—
85#930—22no—
86#930—22no—
87#930D414yes—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91#926—22no—
92#926—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3D414yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106#898—28no—
107#883—28no—
108#67⋊C4—28no—
109C4×D12—22no—
110C4×D12—22no—
111#870—22no—
112#870—22no—
113C3×#95—22no—
114C3×#95—21no—
115C3×#95—22no—
116#830—22no—
117#830—24no—
118#809—24no—
119#809—24no—
120#808—24no—
121#807—24no—
122#807—24no—
123C23×C8—31no—
124#553—64no—
125#553—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#123—61no—
129C2×#121—31no—
130C2×#121—31no—
131C2×#121—61no—
132C2×C4×D4—31no—
133C2×C4×D4—61no—
134C2×C4×D4C2×D311yescharacteristic
135C2×#117—31no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#481—64no—
139C2×#107—32no—
140C2×#107—32no—
141C2×#107—64no—
142C2×#106—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—32no—
147#400—21no—
148#399—21no—
149#389C2×D312yes—
150#386—21no—
151#385—21no—
152#380C2×D311yescharacteristic
153#380C2×D312yes—
154#340—32no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4—82no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—24no—
182C2×#67—24no—
183C2×#67—42no—
184C2×#67—42no—
185C2×#67—82no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—24no—
190C22×Dic3—82no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12C2×D412yes—
195C2×C4×D3—24no—
196C2×C4×D3—82no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—82no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—82no—
208C3×C42—21no—
209C3×C42—21no—
210#217—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3C2×D412yes—
219C4⋊Dic3C2×D412yes—
220C4⋊Dic3C2×D412yes—
221Dic3⋊C4—24no—
222Dic3⋊C4—24no—
223C4×Dic3—24no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8#4114yes—
231C22×D4—31no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4#6711yescharacteristic
235C23×C4—21no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8—32no—
242C22×C8—64no—
243C22×C8—62no—
244#126—68no—
245#125—62no—
246#125—62no—
247#123—68no—
248#121—34no—
249#121—62no—
250#121—62no—
251#121—68no—
252#121—62no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—68no—
258#117—62no—
259#117—62no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4#6711yescharacteristic
273C2×C4⋊C4C22×D312yes—
274C2×C4⋊C4C22×D312yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4#6711yescharacteristic
282C2×C22⋊C4#6711yescharacteristic
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#107—64no—
287#106—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—64no—
291C22⋊C8—62no—
292#95—22no—
293#95—21no—
294#95—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3#12012yes—
303C23×C3#13512yes—
304C23×C3#13712yes—
305C23×C3C22×D411yescharacteristic
306C22×D3—22no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—42no—
311C3×D4—44no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—84no—
333C2×C3×C4#12012yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C42−1+412yes—
338#67—48no—
339#67—84no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—84no—
348D12—24no—
349D12—41no—
350D12—41no—
351C4×D3—84no—
352C3⋊C8—24no—
353C3⋊C8—42no—
354C24—61no—
355C24C2×#6711yescharacteristic
356C24C2×#6711yescharacteristic
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—32no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—61no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—242no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—64no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—82no—
384C22×C4—242no—
385C22×C4—242no—
386C22×C4D3×D412yes—
387C22×C4D3×D412yes—
388C22×C4C2×#6711yescharacteristic
389C22×C4C2×#6711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C23×D311yescharacteristic
393C2×C8—34no—
394C2×C8—68no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—32no—
406C4⋊C4—32no—
407C4⋊C4—32no—
408C4⋊C4—42no—
409C4⋊C4—64no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—82no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—42no—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—82no—
424C42—21no—
425C42—21no—
426C42—64no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3#45214yes—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54712yes—
439C22×C3C2×#13511yescharacteristic
440C22×C3C2×#13711yescharacteristic
441C22×C3#57612yes—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—82no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—82no—
452C3×C4—82no—
453C3×C4#52511yescharacteristic
454C3×C4#53611yescharacteristic
455C3×C4#54712yes—
456Dic3—42no—
457Dic3—42no—
458Dic3—42no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—62no—
467C23—121no—
468C23—121no—
469C23—122no—
470C23—122no—
471C23#93712yes—
472C23#94212yes—
473C23#95012yes—
474C23C22×#6711yescharacteristic
475D4—44no—
476D4—64no—
477D4—128no—
478D4—121no—
479D4—121no—
480D4—244no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—84no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—244no—
510C2×C4—244no—
511C2×C4#93712yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4D3×#4112yes—
515C2×C4#101012yes—
516C8—64no—
517C8—122no—
518C2×C3—24no—
519C2×C3—81no—
520C2×C3#240611yescharacteristic
521C2×C3C2×#45211yescharacteristic
522C2×C3#300811yescharacteristic
523C2×C3#333912yes—
524C2×C3#334612yes—
525D3—82no—
526C22—24no—
527C22—22no—
528C22—24no—
529C22—22no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—41no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—242no—
539C22—242no—
540C22#504714yes—
541C22C2×#93711yescharacteristic
542C22C2×#94211yescharacteristic
543C22C2×#95011yescharacteristic
544C22#582412yes—
545C22#583512yes—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—82no—
551C4—82no—
552C4—122no—
553C4—122no—
554C4—122no—
555C4#564811yescharacteristic
556C4#565512yes—
557C4#565611yescharacteristic
558C3#2409211yescharacteristic, complemented, Sylow
559C2—24no—
560C2—81no—
561C2—242no—
562C2C2×#504711yescharacteristic
563C2#8284111yescharacteristic
564C2#8338311yescharacteristic
565C2#8360712yes—
566C2#8362512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2487C2
socleC23×C3C22×D4
prime radicalC2×#5565C2
hypercenter#2487D3
hypocenterC3#24092
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C3×#2487C2semidirect
3. C2×#4968C2semidirect
4. C22×#807C2semidirect
5. #71610C2semidirect
6. #69625C2semidirect
7. #69606C2semidirect
8. #69523C2semidirect
9. #69515C2semidirect
10. #69506C2semidirect
11. #69502C2semidirect
51. #2487D3semidirect
558. C3#24092semidirect

Maximal subgroups

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

subgroupindex
C2×#55652
C3×#24872
C2×#49682
C22×#8072
#716102
#696252
#696062
#695232
#695152
#695062
#695022
#240923