Finite Group Databasebeta

← #10828596

Subgroups — #10828596

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×#5534C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic, complemented
3C2×#4971C211yesmaximal, characteristic, complemented
4C2×#4968C211yesmaximal, characteristic
5#71586C211yesmaximal, characteristic
6#71585C211yesmaximal, characteristic, complemented
7#71584C211yesmaximal, characteristic, complemented
8#69632C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69626C212yesmaximal, complemented
11#69605C212yesmaximal, complemented
12#24074—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#924C2211yescharacteristic
18#5534—24no—
19#5534—24no—
20C2×#898—21no—
21C2×#896—21no—
22C2×#885—21no—
23C2×#883—21no—
24C2×#875C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#389C2212yes—
29C3×#389C2212yes—
30C3×#385—21no—
31C3×#385—21no—
32C3×#384—21no—
33C3×#382—21no—
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#4971—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39#4968—24no—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×#808C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#542—31no—
49C2×#481—31no—
50C2×#483—31no—
51#2317D311yescharacteristic, complemented
52#2058—32no—
53#2042—32no—
54#1875—31no—
55#1873—31no—
56#1865—32no—
57#1864—32no—
58#1818—31no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×Dic6—21no—
65C22×Dic6C2311yescharacteristic
66C3×#117—22no—
67C3×#117—22no—
68C3×#117—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85#930—22no—
86#930D414yes—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94#924—22no—
95#924D414yes—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2312yes—
101#898—28no—
102#896—28no—
103#885—28no—
104#883—28no—
105#875—22no—
106#875—22no—
107C4×D12—22no—
108C4×D12—22no—
109#869—22no—
110#869—22no—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C3×#95—22no—
114C3×#95—22no—
115#830—22no—
116#830—22no—
117#810—24no—
118#810—24no—
119#809—24no—
120#809—24no—
121#808—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—31no—
131C2×C22⋊Q8—61no—
132C2×#121—31no—
133C2×C4×Q8—31no—
134C2×C4×D4—31no—
135C2×#117C2×D311yescharacteristic
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138#483—64no—
139#481—64no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8—31no—
148#389C2×D312yes—
149#389C2×D312yes—
150#385—21no—
151#385—21no—
152#384—21no—
153#382—21no—
154#340—32no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—82no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—24no—
175C2×#67—82no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12C2×D412yes—
183C2×C4×D3—24no—
184C2×C4×D3—82no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—24no—
189C2×Dic6—82no—
190C2×Dic6C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—82no—
198C3×C4⋊C4—82no—
199C3×C4⋊C4#4114yes—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C42—24no—
204C3×C42—21no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42—21no—
208#217—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213C4⋊Dic3—24no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3C2×D412yes—
217C4⋊Dic3C2×D412yes—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220C4×Dic3—24no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C22×Q8—31no—
228C22×Q8—61no—
229C22×D4—31no—
230C22×D4—61no—
231C23×C4—21no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C4⋊Q8—62no—
237C4⋊Q8—62no—
238#126—68no—
239#124—62no—
240#124—62no—
241#124—68no—
242#123—68no—
243C22⋊Q8—34no—
244C22⋊Q8—68no—
245C22⋊Q8—62no—
246#121—34no—
247#121—62no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×D4—62no—
251C4×D4—62no—
252#117—22no—
253#117—22no—
254#117—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C4⋊C4C22×D312yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4#6711yescharacteristic
276C2×C22⋊C4#6711yescharacteristic
277C2×C42—21no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281#106—64no—
282#106—64no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#95—22no—
290#95—22no—
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—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—84no—
321C2×C3×C4#12012yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4112yes—
325C2×C3×C42−1+412yes—
326#67—84no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334D12—24no—
335D12—42no—
336C4×D3—84no—
337Dic6—24no—
338Dic6—42no—
339Dic6—84no—
340C3⋊C8—42no—
341C3⋊C8—42no—
342C24—61no—
343C24C2×#6711yescharacteristic
344C2×Q8—32no—
345C2×Q8—64no—
346C2×Q8—61no—
347C2×Q8—62no—
348C2×Q8—62no—
349C2×Q8—242no—
350C2×D4—32no—
351C2×D4—64no—
352C2×D4—62no—
353C2×D4—61no—
354C2×D4—62no—
355C2×D4—242no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—64no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—82no—
366C22×C4—242no—
367C22×C4D3×D412yes—
368C22×C4D3×D412yes—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C23×D311yescharacteristic
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—62no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—32no—
387C4⋊C4—32no—
388C4⋊C4—42no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4#23514yes—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—64no—
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—41no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—82no—
433C3×C4#54712yes—
434C3×C4#54712yes—
435Dic3—42no—
436Dic3—42no—
437Dic3—42no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—62no—
443C23—121no—
444C23—121no—
445C23—122no—
446C23—122no—
447C23#93712yes—
448C23C22×#6711yescharacteristic
449Q8—64no—
450Q8—122no—
451Q8—244no—
452D4—64no—
453D4—122no—
454D4—244no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—84no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—244no—
482C2×C4#93712yes—
483C2×C4C2×#23512yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4#100712yes—
487C8—122no—
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—
506C22#579111yescharacteristic
507C22C2×#93711yescharacteristic
508C22#581611yescharacteristic
509C22#583612yes—
510C22#583712yes—
511C4—22no—
512C4—44no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—82no—
518C4—122no—
519C4—122no—
520C4—122no—
521C4#550712yes—
522C4#550712yes—
523C3#2407411yescharacteristic, complemented, Sylow
524C2—42no—
525C2—242no—
526C2#8219211yescharacteristic
527C2#8257612yes—
528C2#8339011yescharacteristic
529C2#8340411yescharacteristic
530C2#8347912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2317D3
hypocenterC3#24074
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C3×#2317C2semidirect
3. C2×#4971C2semidirect
6. #71585C2semidirect
7. #71584C2semidirect
8. #69632C2semidirect
9. #69628C2semidirect
10. #69626C2semidirect
11. #69605C2semidirect
51. #2317D3semidirect
523. C3#24074semidirect

Maximal subgroups

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

subgroupindex
C2×#55342
C3×#23172
C2×#49712
C2×#49682
#715862
#715852
#715842
#696322
#696282
#696262
#696052
#240743