Finite Group Databasebeta

← #10829279

Subgroups — #10829279

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5563C211yesmaximal, characteristic
2C3×#2536C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic, complemented
4C2×#5020C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71693C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic, complemented
8#69556C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69524C212yesmaximal, complemented
11#69506C212yesmaximal, complemented
12#24133—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#215C2211yescharacteristic
17C2×#938—21no—
18C2×#930C2212yes—
19C2×#926C2212yes—
20#5563—24no—
21#5563—24no—
22C2×#892—21no—
23C2×#885—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#398—21no—
27C3×#394—21no—
28C3×#388—22no—
29C3×#382C2211yescharacteristic
30C3×#380C2212yes—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5026—24no—
36#5020—24no—
37C2×#809C2211yescharacteristic
38C2×#809C2212yes—
39C2×#808C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#555—31no—
47C2×#475—31no—
48C2×#440—31no—
49#2536D311yescharacteristic, complemented
50#2034—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1873—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4D411yescharacteristic
59C23×C3×C4C2312yes—
60C22×#67—22no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6—21no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—42no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#215—22no—
82C2×#215—22no—
83C2×#215—22no—
84#938—28no—
85#930—24no—
86#930—24no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—22no—
89#926—24no—
90#926—24no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4×Dic3—22no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#892—28no—
99#885—216no—
100C4×D12—22no—
101C4×D12D414yes—
102#869—22no—
103#869D414yes—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—21no—
107#835—24no—
108#835—22no—
109#830—24no—
110#830—42no—
111#809—22no—
112#809—42no—
113#808—24no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#555—64no—
119#555—64no—
120C2×C4⋊Q8—31no—
121C2×#124—62no—
122C2×C22⋊Q8C2×D312yes—
123C2×#121—32no—
124C2×#121—61no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4—31no—
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—32no—
129C2×#117—61no—
130#475—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#398—21no—
143#394—21no—
144#388—22no—
145#382C2×D311yescharacteristic
146#380C2×D312yes—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C23×D3—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8#4112yes—
158C2×C3×D4—22no—
159C2×C3×D4—42no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—28no—
174C2×#67—82no—
175C2×#67—82no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12C2×D412yes—
184C2×C4×D3—24no—
185C2×C4×D3—44no—
186C2×Dic6—24no—
187C2×Dic6—82no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—44no—
194C3×C4⋊C4—42no—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C42—42no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—28no—
202#215—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—24no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D412yes—
209C4⋊Dic3C2×D414yes—
210C4×Dic3—28no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C22×D4—62no—
224C22×D4#6711yescharacteristic
225C23×C4—61no—
226C23×C4C22×D312yes—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8—32no—
233C4⋊Q8—34no—
234C4⋊Q8—62no—
235#124—616no—
236C22⋊Q8—24no—
237C22⋊Q8—42no—
238#121—64no—
239#121—64no—
240#121—68no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—34no—
246C4×D4—62no—
247#117—68no—
248#117—64no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4#6712yes—
264C2×C42—21no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—62no—
270#103—62no—
271#103—64no—
272#102—62no—
273#102—122no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—22no—
278#95—21no—
279#95—21no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×D4—44no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323#67—88no—
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—22no—
330D12—22no—
331D12—41no—
332D12—41no—
333C4×D3—48no—
334Dic6—84no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C3⋊C8—42no—
338C24—61no—
339C24C2×#6712yes—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—64no—
346C2×Q8—242no—
347C2×Q8#23712yes—
348C2×D4—22no—
349C2×D4—32no—
350C2×D4—42no—
351C2×D4—68no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—242no—
358C2×D4—242no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—124no—
372C22×C4C2×#6712yes—
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C23×D311yescharacteristic
376SD16—68no—
377C2×C8—32no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—62no—
382C2×C8—61no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—61no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—34no—
392C4⋊C4—32no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—64no—
396C4⋊C4—61no—
397C4⋊C4—62no—
398C22⋊C4—24no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—68no—
402C22⋊C4—68no—
403C22⋊C4—64no—
404C42—42no—
405C42—68no—
406C42D3×D412yes—
407C42D3×D412yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#54611yescharacteristic
417C22×C3#54711yescharacteristic
418C22×C3#57512yes—
419C22×C3#57712yes—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—82no—
424C2×D3—82no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—81no—
430C3×C4—82no—
431C3×C4#52511yescharacteristic
432C3×C4#53711yescharacteristic
433C3×C4#54811yescharacteristic
434C3×C4#55011yescharacteristic
435Dic3—42no—
436Dic3—44no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—62no—
445C23—121no—
446C23—122no—
447C23—121no—
448C23—122no—
449C23C22×#6711yescharacteristic
450Q8—22no—
451Q8—22no—
452Q8—41no—
453Q8—41no—
454Q8—244no—
455D4—44no—
456D4—62no—
457D4—62no—
458D4—121no—
459D4—121no—
460D4—248no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—62no—
479C2×C4—64no—
480C2×C4—128no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4#93712yes—
485C2×C4#93712yes—
486C2×C4C2×D3×D411yescharacteristic
487C2×C4C2×D3×D411yescharacteristic
488C2×C4C2×#23711yescharacteristic
489C2×C4#100411yescharacteristic
490C2×C4D3×#4111yescharacteristic
491C2×C4#101011yescharacteristic
492C8—62no—
493C8—122no—
494C8—121no—
495C2×C3—82no—
496C2×C3#240711yescharacteristic
497C2×C3#300511yescharacteristic
498C2×C3#300911yescharacteristic
499C2×C3#332112yes—
500C2×C3#334812yes—
501D3—82no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—242no—
513C22—242no—
514C22C2×#93711yescharacteristic
515C22#580611yescharacteristic
516C22#582212yes—
517C22#582411yescharacteristic
518C22#583712yes—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—124no—
526C4—122no—
527C4#559811yescharacteristic
528C4#559911yescharacteristic
529C4#560611yescharacteristic
530C4#560811yescharacteristic
531C3#2413311yescharacteristic, complemented, Sylow
532C2—82no—
533C2—242no—
534C2#8270111yescharacteristic
535C2#8292811yescharacteristic
536C2#8340411yescharacteristic
537C2#8353912yes—
538C2#8359412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2536C2
socleC23×C3C22×D4
prime radicalC2×#5563C2
hypercenter#2536D3
hypocenterC3#24133
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. C3×#2536C2semidirect
3. C2×#5026C2semidirect
4. C2×#5020C2semidirect
5. #71699C2semidirect
6. #71693C2semidirect
7. #71615C2semidirect
8. #69556C2semidirect
9. #69555C2semidirect
10. #69524C2semidirect
11. #69506C2semidirect
49. #2536D3semidirect
531. C3#24133semidirect

Maximal subgroups

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

subgroupindex
C2×#55632
C3×#25362
C2×#50262
C2×#50202
#716992
#716932
#716152
#695562
#695552
#695242
#695062
#241333