Finite Group Databasebeta

← #10800426

Subgroups — #10800426

Order 768.

1186 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C2×Dic3×M16C211yesmaximal, characteristic
3#73864C211yesmaximal, characteristic
4C3×#1894C211yesmaximal, characteristic
5#71989C211yesmaximal, characteristic
6#71697C211yesmaximal, characteristic
7#70225C212yesmaximal, complemented
8#70220C212yesmaximal, complemented
9#69516C211yesmaximal, characteristic
10#68803C212yesmaximal, complemented
11#68783C212yesmaximal, complemented
12#21409—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C2×#924C2211yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C22×#208C2211yescharacteristic
20#5622—24no—
21#5622C418yescomplemented
22C2×#889—21no—
23C2×#888C2211yescharacteristic
24C2×#887—21no—
25C2×#887C2211yescharacteristic
26C2×#878—21no—
27C2×#877—21no—
28C2×#877—21no—
29C2×C3×#103C2212yes—
30C3×#380C2211yescharacteristic
31C2×#835C2212yes—
32#5212C2211yescharacteristic
33Dic3×M16—24no—
34C2×#816C2212yes—
35C2×#815C2212yes—
36C2×C8×Dic3C2212yes—
37#4975C2212yes—
38Dic3×C4⋊C4C2212yes—
39C2×#810C2212yes—
40#4942—21no—
41#4650—21no—
42#4649—21no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47C2×#533—31no—
48C2×C4×M16—31no—
49#2285—31no—
50#1990—32no—
51#1956—32no—
52#1894—31no—
53#1894—31no—
54#1894—31no—
55#1894D311yescharacteristic
56#1763—32no—
57#1763—32no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3D411yescharacteristic
61C23×Dic3D411yescharacteristic
62C22×Dic6—21no—
63C22×Dic6C2311yescharacteristic
64C2×C3×M16—22no—
65C2×C3×M16—24no—
66C22×C3×C8C2312yes—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8C2×C414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×#217—21no—
75C2×#217—21no—
76Q8×Dic3—22no—
77Q8×Dic3C2×C414yes—
78#944—22no—
79#944C2×C414yes—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83#924—22no—
84#924C2×C414yes—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3C2×C414yes—
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3D411yescharacteristic
98C2×C4×Dic3D414yes—
99C2×C4×Dic3C2311yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C2×#208—22no—
102C2×#208—24no—
103C22×C3⋊C8C2312yes—
104#889—28no—
105#888—22no—
106#888C2×C414yes—
107#887—28no—
108#887—22no—
109#887C2×C414yes—
110#878—28no—
111#877—28no—
112#877—44no—
113C3×#103—24no—
114C3×#95—21no—
115#835—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118#816—24no—
119#815—24no—
120C8×Dic3—24no—
121#810—24no—
122C22×M16—31no—
123C22×M16C2×D311yescharacteristic
124#533—38no—
125#533—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—31no—
128C2×#125—61no—
129C2×C22⋊Q8—31no—
130C2×C22⋊Q8—61no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×Q8—61no—
135C2×#117—61no—
136C2×#117—61no—
137C22×C42—31no—
138C2×#103—32no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103C2×D312yes—
142C4×M16—64no—
143C2×#97—32no—
144C2×C4×C8—32no—
145#384—32no—
146#382—61no—
147#380—31no—
148#380—61no—
149#380C2×D311yescharacteristic
150C4×C4⋊C4—32no—
151C4×C4⋊C4—61no—
152#340—32no—
153#340—32no—
154#340—32no—
155#340C2×D312yes—
156C24×C3C2×D411yescharacteristic
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8C22×C412yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3C22⋊C414yes—
177C22×Dic3C22⋊C414yes—
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—21no—
185C2×Dic6—24no—
186C2×Dic6—42no—
187C2×Dic6—42no—
188C2×Dic6C22×C412yes—
189C3×M16—28no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8#4114yes—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C22×C412yes—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—42no—
200#217—24no—
201#217—42no—
202C4⋊Dic3—21no—
203C4⋊Dic3—24no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3C22×C412yes—
207Dic3⋊C4—24no—
208Dic3⋊C4—24no—
209Dic3⋊C4—24no—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—44no—
213Dic3⋊C4—44no—
214C4×Dic3—24no—
215C4×Dic3—24no—
216C4×Dic3—44no—
217C4×Dic3C22⋊C414yes—
218C4×Dic3C22⋊C414yes—
219C4×Dic3C22×C412yes—
220C4×Dic3C22×C412yes—
221C4×Dic3C2×D412yes—
222C4×Dic3C2×D412yes—
223#208—28no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8#4114yes—
227C22×Q8—31no—
228C22×Q8—61no—
229C22×Q8C22×D311yescharacteristic
230C23×C4—31no—
231C23×C4—31no—
232C23×C4C22×D311yescharacteristic
233C2×M16—22no—
234C2×M16—24no—
235C2×M16—64no—
236C2×M16—62no—
237C22×C8—32no—
238C22×C8C22×D312yes—
239C4⋊Q8—34no—
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243#125—68no—
244C22⋊Q8—22no—
245C22⋊Q8—34no—
246C22⋊Q8—68no—
247C22⋊Q8—62no—
248C22⋊Q8C4×D314yes—
249C4×Q8—34no—
250C4×Q8—34no—
251C4×Q8—62no—
252C4×Q8—68no—
253C4×Q8—62no—
254#117—68no—
255#117—124no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4C22×D312yes—
268C2×C4⋊C4C22×D311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C42—34no—
274C2×C42—31no—
275C2×C42—31no—
276C2×C42—34no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—61no—
280C2×C42—61no—
281C2×C42—62no—
282#103—24no—
283#103—64no—
284#103—64no—
285#103—64no—
286#97—64no—
287C4×C8—64no—
288#95—21no—
289#95—62no—
290#95—61no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3C2×C22⋊C412yes—
296C23×C3C22×D411yescharacteristic
297C3×Q8—24no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4C2×C22⋊C412yes—
313C2×C3×C4C23×C411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4112yes—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—21no—
320C2×Dic3—28no—
321C2×Dic3—28no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3C2×C22⋊C412yes—
331C2×Dic3C2×C22⋊C412yes—
332C2×Dic3C2×C22⋊C412yes—
333C2×Dic3C2×C22⋊C412yes—
334C2×Dic3AGL(1,ℤ/8)12yes—
335C2×Dic3#13712yes—
336C2×Dic3#13714yes—
337Dic6—24no—
338Dic6—41no—
339Dic6—41no—
340Dic6—48no—
341C3×C8—24no—
342C3⋊C8—24no—
343C24D3×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—32no—
348C2×Q8—64no—
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—122no—
353C2×Q8—122no—
354C2×Q8C2×C4×D312yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—34no—
363C22×C4—34no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—62no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4C2×C4×D312yes—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D412yes—
375C22×C4C23×D311yescharacteristic
376M16—28no—
377M16—68no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—34no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8#23714yes—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—61no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—124no—
399C4⋊C4—124no—
400C4⋊C4C2×C4×D312yes—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—64no—
404C22⋊C4—122no—
405C42—34no—
406C42—34no—
407C42—32no—
408C42—32no—
409C42—32no—
410C42—32no—
411C42—64no—
412C42—64no—
413C42—124no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3C22×C22⋊C411yescharacteristic
420C22×C3#53212yes—
421C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
422C22×C3C2×#13712yes—
423C22×C3C2×#13711yescharacteristic
424C3×C4—22no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—82no—
428C3×C4C4×#4112yes—
429C3×C4#52511yescharacteristic
430C3×C4#52511yescharacteristic
431Dic3—22no—
432Dic3—22no—
433Dic3—42no—
434Dic3—42no—
435Dic3—82no—
436Dic3#41714yes—
437Dic3#41914yes—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23D3×C22⋊C412yes—
443C23C2×D3×D411yescharacteristic
444Q8—24no—
445Q8—41no—
446Q8—41no—
447Q8—64no—
448Q8—128no—
449Q8—121no—
450Q8—121no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—34no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—68no—
471C2×C4—68no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4D3×C22⋊C412yes—
484C2×C4C22×C4×D311yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23712yes—
487C2×C4D3×#4112yes—
488C8—24no—
489C8—64no—
490C2×C3—42no—
491C2×C3#229711yescharacteristic
492C2×C3C2×#41911yescharacteristic
493C2×C3C2×#41711yescharacteristic
494C2×C3#318212yes—
495C2×C3#318512yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22C2×D3×C22⋊C411yescharacteristic
502C22#559612yes—
503C22D3×AGL(1,ℤ/8)11yescharacteristic
504C22D3×#13711yescharacteristic
505C22D3×#13712yes—
506C4—22no—
507C4—34no—
508C4—34no—
509C4—42no—
510C4—42no—
511C4—62no—
512C4—62no—
513C4—82no—
514C4—122no—
515C4—122no—
516C4—242no—
517C4C4×#23712yes—
518C4#558611yescharacteristic
519C4#565611yescharacteristic
520C3#2140911yescharacteristic, complemented, Sylow
521C2—42no—
522C2#7465811yescharacteristic
523C2D3×#41711yescharacteristic
524C2D3×#41911yescharacteristic
525C2#8079012yes—
526C2#8089312yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1894C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenter#1894D3
hypocenterC3#21409
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
7. #70225C2semidirect
8. #70220C2semidirect
10. #68803C2semidirect
11. #68783C2semidirect
21. #5622C4semidirect
520. C3#21409semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C2×Dic3×M162
#738642
C3×#18942
#719892
#716972
#702252
#702202
#695162
#688032
#687832
#214093