Finite Group Databasebeta

← #10828565

Subgroups — #10828565

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3C2×#4970C211yesmaximal, characteristic
4C2×#4971C211yesmaximal, characteristic
5#71585C211yesmaximal, characteristic
6#71585C211yesmaximal, characteristic
7#71584C211yesmaximal, characteristic
8#69633C212yesmaximal, complemented
9#69632C212yesmaximal, complemented
10#69631C212yesmaximal, complemented
11#69606C212yesmaximal, complemented
12#24071—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C2×#924C2211yescharacteristic
18#5518—24no—
19#5518—24no—
20C2×#889—21no—
21C2×#889—21no—
22C2×#879—21no—
23C2×#879—21no—
24C2×#870C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
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—
37#4970—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42C2×#808C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#541—31no—
49C2×#484—31no—
50C2×#483—31no—
51#2317D311yescharacteristic
52#2059—32no—
53#2043—32no—
54#1873—31no—
55#1873—31no—
56#1865—32no—
57#1863—32no—
58#1818—31no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×Dic6C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C3×#117—22no—
64C3×#117—22no—
65C3×#117—24no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4D411yescharacteristic
77C2×C3×C22⋊C4D411yescharacteristic
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—21no—
82C2×#217—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87#924—22no—
88#924—22no—
89#924D414yes—
90#924D414yes—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2312yes—
101#889—28no—
102#889—28no—
103#879—28no—
104#879—28no—
105#870—22no—
106#870—22no—
107#869—22no—
108#869—22no—
109C4×Dic6—22no—
110C4×Dic6—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#810—24no—
120#810—24no—
121#808—24no—
122#807—24no—
123#541—64no—
124#541—64no—
125C2×C4⋊Q8—31no—
126C2×#126—61no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#125—61no—
130C2×#125—61no—
131C2×C22⋊Q8—31no—
132C2×C22⋊Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×Q8—31no—
135C2×#117C2×D311yescharacteristic
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138#484—64no—
139#483—64no—
140C2×#107—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—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
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—42no—
164C22×C3×C4—82no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6C2×D412yes—
184C2×Dic6C2×D412yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—82no—
192C3×C4⋊C4—82no—
193C3×C4⋊C4#4114yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C42—24no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42—21no—
202#217—24no—
203#217—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D412yes—
209C4⋊Dic3C2×D412yes—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216C4×Dic3—24no—
217C4×Dic3—24no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C22×Q8—31no—
225C22×Q8—31no—
226C23×C4—21no—
227C23×C4C22×D311yescharacteristic
228C22×C8—32no—
229C22×C8—32no—
230C4⋊Q8—62no—
231C4⋊Q8—62no—
232#126—68no—
233#126—68no—
234#125—62no—
235#125—62no—
236#125—68no—
237#125—68no—
238C22⋊Q8—34no—
239C22⋊Q8—34no—
240C22⋊Q8—62no—
241C22⋊Q8—62no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246#117—22no—
247#117—22no—
248#117—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4#6711yescharacteristic
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4#6711yescharacteristic
272C2×C22⋊C4#6711yescharacteristic
273C2×C42—21no—
274C2×C42—61no—
275C2×C42—61no—
276C2×C42C22×D311yescharacteristic
277C2×C42C22×D311yescharacteristic
278#107—64no—
279#106—64no—
280#103—64no—
281#103—64no—
282#103—64no—
283#103—64no—
284C22⋊C8—62no—
285C22⋊C8—62no—
286#95—22no—
287#95—22no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3#12012yes—
293C23×C3C22×D411yescharacteristic
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—84no—
313C2×C3×C4#12012yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4112yes—
317C2×C3×C42−1+412yes—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328Dic6—24no—
329Dic6—24no—
330Dic6—42no—
331Dic6—42no—
332C3⋊C8—42no—
333C3⋊C8—42no—
334C24C2×#6711yescharacteristic
335C2×Q8—32no—
336C2×Q8—32no—
337C2×Q8—61no—
338C2×Q8—61no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—82no—
353C22×C4D3×D412yes—
354C22×C4D3×D412yes—
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C23×D311yescharacteristic
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—62no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—32no—
373C4⋊C4—32no—
374C4⋊C4—42no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4#23514yes—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—64no—
392C22⋊C4—64no—
393C42—24no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—64no—
399C42—64no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3C2×#12011yescharacteristic
406C22×C3#52511yescharacteristic
407C22×C3#53611yescharacteristic
408C22×C3#57712yes—
409C22×C3#57812yes—
410C3×C4—22no—
411C3×C4—44no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—82no—
417C3×C4#54712yes—
418C3×C4#54712yes—
419Dic3—42no—
420Dic3—42no—
421Dic3—42no—
422Dic3—42no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23#93712yes—
428C23C22×#6711yescharacteristic
429Q8—64no—
430Q8—64no—
431Q8—122no—
432Q8—122no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—84no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4#93712yes—
463C2×C4C2×#23512yes—
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×D3×D411yescharacteristic
466C2×C4#100712yes—
467C8—122no—
468C8—122no—
469C2×C3—42no—
470C2×C3#240611yescharacteristic
471C2×C3#301011yescharacteristic
472C2×C3#301111yescharacteristic
473C2×C3#312012yes—
474C2×C3#324212yes—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22#579111yescharacteristic
481C22C2×#93711yescharacteristic
482C22#581611yescharacteristic
483C22#583612yes—
484C22#583812yes—
485C4—22no—
486C4—44no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—122no—
493C4—122no—
494C4—122no—
495C4—122no—
496C4#550712yes—
497C4#550712yes—
498C3#2407111yescharacteristic, complemented, Sylow
499C2—42no—
500C2#8219211yescharacteristic
501C2#8257512yes—
502C2#8339011yescharacteristic
503C2#8341111yescharacteristic
504C2#8348112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2317D3
hypocenterC3#24071
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
8. #69633C2semidirect
9. #69632C2semidirect
10. #69631C2semidirect
11. #69606C2semidirect
498. C3#24071semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C3×#23172
C2×#49702
C2×#49712
#715852
#715842
#696332
#696322
#696312
#696062
#240713