Finite Group Databasebeta

← #10828601

Subgroups — #10828601

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5643C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3#82233C211yesmaximal, characteristic, complemented
4C2×#4735C211yesmaximal, characteristic
5C3×#1873C211yesmaximal, characteristic, complemented
6#71586C211yesmaximal, characteristic
7#70245C212yesmaximal, complemented
8#70238C212yesmaximal, complemented
9#69911C211yesmaximal, characteristic, complemented
10#68895C212yesmaximal, complemented
11#68827C212yesmaximal, complemented
12#24074—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#947C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20#5643—24no—
21#5643—24no—
22C2×#898—21no—
23C2×#897—21no—
24C2×#893C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#885—21no—
27C2×#883—21no—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#835C2212yes—
31C2×#830C2211yescharacteristic
32C2×#821C2212yes—
33C2×#817C2212yes—
34#5043—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#809C2212yes—
38C2×#808C2212yes—
39#4950—21no—
40#4950—21no—
41#4936—21no—
42#4935—21no—
43#4735—24no—
44C3×#340C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#542—31no—
49C2×#481—31no—
50C2×#483—31no—
51#2317—31no—
52#2058—32no—
53#2042—32no—
54#1875—31no—
55#1873D311yescharacteristic, complemented
56#1865—32no—
57#1864—32no—
58#1818—31no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6—21no—
66C22×C3×C8C2312yes—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8D414yes—
69C3×#117—24no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×#217D411yescharacteristic
77C2×#217D411yescharacteristic
78#947—22no—
79#947—22no—
80Q8×Dic3—22no—
81Q8×Dic3—22no—
82#930—22no—
83#930D414yes—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88#926—22no—
89#926—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3D411yescharacteristic
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105#898—28no—
106#897—28no—
107#893—22no—
108#893—22no—
109#888—22no—
110#888—22no—
111#885—28no—
112#883—28no—
113C3×#103—24no—
114C3×C22⋊C8—22no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#821—24no—
120#817—24no—
121#809—24no—
122#808—24no—
123#542—64no—
124#542—64no—
125C2×C4⋊Q8—31no—
126C2×#126—61no—
127C2×#124—31no—
128C2×#124—61no—
129C2×#123—61no—
130C2×C22⋊Q8—61no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×#121—31no—
133C2×C4×Q8—31no—
134C2×C4×D4—31no—
135C2×#117—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138#483—64no—
139#481—64no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103C2×D312yes—
144C2×#102—32no—
145C2×#102—32no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340C2×D312yes—
156#336—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—82no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—42no—
178C22×Dic3—82no—
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—82no—
189C2×Dic6—24no—
190C2×Dic6—82no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4#4114yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C42—24no—
202#217—22no—
203#217—22no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3—82no—
216C4⋊Dic3—82no—
217C4⋊Dic3C2×D412yes—
218Dic3⋊C4—24no—
219C4×Dic3—24no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C22×Q8—61no—
228C22×Q8C22×D311yescharacteristic
229C22×D4—31no—
230C22×D4—61no—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8C22×D312yes—
236C4⋊Q8—62no—
237C4⋊Q8—62no—
238#126—68no—
239#124—68no—
240#124—62no—
241#124—62no—
242#123—68no—
243C22⋊Q8—22no—
244C22⋊Q8—68no—
245C22⋊Q8D1214yes—
246#121—34no—
247#121—62no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×D4—62no—
251C4×D4—62no—
252#117—24no—
253#117—62no—
254#117—62no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C4⋊C4C22×D312yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—31no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—21no—
278C2×C42—31no—
279C2×C42—31no—
280C2×C42—61no—
281#106—64no—
282#106—64no—
283#103—24no—
284#103—64no—
285#102—64no—
286#102—64no—
287C22⋊C8—22no—
288C22⋊C8—62no—
289#95—62no—
290#95—62no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3#12012yes—
296C23×C3C22×D411yescharacteristic
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—42no—
302C3×Q8—24no—
303C3×Q8—42no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C42−1+412yes—
321#67—84no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—22no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—84no—
336D12—24no—
337D12—42no—
338C4×D3—84no—
339Dic6—84no—
340C3×C8—42no—
341C3⋊C8—42no—
342C24—61no—
343C24D3×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—64no—
348C2×Q8—242no—
349C2×Q8C2×D1212yes—
350C2×D4—32no—
351C2×D4—64no—
352C2×D4—61no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—242no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—64no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—122no—
369C22×C4—242no—
370C22×C4—242no—
371C22×C4C2×D1212yes—
372C22×C4D3×D411yescharacteristic
373C22×C4D3×D412yes—
374C22×C4C23×D311yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—62no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—32no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—61no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—122no—
393C4⋊C4—242no—
394C4⋊C4—242no—
395C4⋊C4C2×D1212yes—
396C4⋊C4#23714yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—64no—
405C42—24no—
406C42—64no—
407C42—61no—
408C42—61no—
409C42—61no—
410C42—61no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3C2×#12011yescharacteristic
417C22×C3#52511yescharacteristic
418C22×C3#53611yescharacteristic
419C22×C3#57712yes—
420C22×C3#57712yes—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—82no—
425C2×D3—82no—
426C3×C4—22no—
427C3×C4—44no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4#54712yes—
431C3×C4#54712yes—
432Dic3—41no—
433Dic3—42no—
434Dic3—41no—
435Dic3—41no—
436Dic3—41no—
437Dic3—82no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—21no—
442C23—62no—
443C23—121no—
444C23—122no—
445C23—122no—
446C23—121no—
447C23#88212yes—
448C23C2×D3×D411yescharacteristic
449Q8—24no—
450Q8—42no—
451Q8—244no—
452D4—64no—
453D4—122no—
454D4—244no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—62no—
471C2×C4—61no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—244no—
481C2×C4—244no—
482C2×C4#88212yes—
483C2×C4C22×D1211yescharacteristic
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×#23712yes—
486C2×C4#101012yes—
487C8—42no—
488C8—122no—
489C2×C3—42no—
490C2×C3#240611yescharacteristic
491C2×C3#300911yescharacteristic
492C2×C3#301011yescharacteristic
493C2×C3#312512yes—
494C2×C3#324412yes—
495D3—82no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—242no—
505C22—242no—
506C22C2×#88211yescharacteristic
507C22#558611yescharacteristic
508C22#564811yescharacteristic
509C22#577012yes—
510C22#577812yes—
511C4—22no—
512C4—44no—
513C4—42no—
514C4—42no—
515C4—121no—
516C4—121no—
517C4—121no—
518C4—121no—
519C4—122no—
520C4—242no—
521C4#555512yes—
522C4#561112yes—
523C3#2407411yescharacteristic, complemented, Sylow
524C2—42no—
525C2—242no—
526C2#7468711yescharacteristic
527C2#7883511yescharacteristic
528C2#7896011yescharacteristic
529C2#8021912yes—
530C2#8150712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1873D3
hypocenterC3#24074
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
3. #82233C2semidirect
5. C3×#1873C2semidirect
7. #70245C2semidirect
8. #70238C2semidirect
9. #69911C2semidirect
10. #68895C2semidirect
11. #68827C2semidirect
55. #1873D3semidirect
523. C3#24074semidirect

Maximal subgroups

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

subgroupindex
C2×#56432
C2×#50432
#822332
C2×#47352
C3×#18732
#715862
#702452
#702382
#699112
#688952
#688272
#240743