Finite Group Databasebeta

← #10823425

Subgroups — #10823425

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#4757C212yesmaximal, complemented
3#76291C211yesmaximal, characteristic
4C3×#2023C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70205C212yesmaximal
7#69661C211yesmaximal, characteristic, complemented
8#68825C212yesmaximal, complemented
9#68783C212yesmaximal, complemented
10#68631C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23574—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16C2×#938C2211yescharacteristic
17C2×D4×Dic3C2211yescharacteristic
18C22×#214C2212yes—
19#5728—28no—
20C2×#897C2212yes—
21C2×#892—21no—
22C2×#890—21no—
23C2×#887C2212yes—
24C2×#67⋊C4—21no—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#390C2211yescharacteristic
29C2×(C2×D3)⋊C8C2212yes—
30C2×#816C2212yes—
31C2×#816C2212yes—
32C2×#815C2212yes—
33#4987C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35C2×#810C2212yes—
36C2×#807C2212yes—
37#4956—21no—
38#4945—21no—
39Dic3×C22⋊C4—21no—
40#4928C2211yescharacteristic
41#4757—28no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#440—32no—
48C4×#121—31no—
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2023D311yescharacteristic, complemented
53#2013—31no—
54#2000—32no—
55#1763—32no—
56#1749—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C23×Dic3—21no—
61C22×C4×D3D411yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3C2311yescharacteristic
64C22×Dic6C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—21no—
74C2×#217—21no—
75#946—28no—
76Q8×Dic3—28no—
77#938—22no—
78#938—22no—
79D4×Dic3—24no—
80C2×#214—24no—
81C2×#214—24no—
82C2×#214—24no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98#897—24no—
99#897—24no—
100#892—28no—
101#890—28no—
102#887—24no—
103#887D418yescomplemented
104#67⋊C4—44no—
105C42×D3—24no—
106#869—24no—
107C3×#102—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111(C2×D3)⋊C8—24no—
112#816—24no—
113#816—24no—
114#815—24no—
115#810—24no—
116#807—24no—
117C22×SD16—32no—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—61no—
124C2×#121—31no—
125C2×C4×Q8—32no—
126C2×C4×Q8—61no—
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C2×#117—61no—
130C22×C42—31no—
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#102C2×D312yes—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#390C2×D311yescharacteristic
140#389—31no—
141#388—61no—
142#386—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C22⋊C4—61no—
146#340—32no—
147#340C2×D312yes—
148#336—31no—
149#336C2×D311yescharacteristic
150C24×C3—21no—
151C23×D3C2×D411yescharacteristic
152C2×C3×Q8—24no—
153C2×C3×Q8—42no—
154C2×C3×Q8—42no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4#4112yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—24no—
165C2×#67—42no—
166C2×#67—42no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C22×Dic3—42no—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—22no—
185C2×C4×D3—41no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6C2×D414yes—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C42—22no—
202#217—24no—
203#217—42no—
204#214—216no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—42no—
207C4⋊Dic3—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—28no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3#4112yes—
213Dic3⋊C4—28no—
214Dic3⋊C4—42no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—28no—
218C4×Dic3—42no—
219C4×Dic3C2×D414yes—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8#4114yes—
223C22×Q8—21no—
224C22×Q8—32no—
225C22×D4—61no—
226C22×D4C22×D311yescharacteristic
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—61no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C22×C8—32no—
235C22×C8C22×D312yes—
236C4⋊Q8—24no—
237C4⋊Q8—64no—
238#125—68no—
239C22⋊Q8—64no—
240C22⋊Q8—68no—
241C22⋊Q8—64no—
242#121—62no—
243#121—62no—
244C4×Q8—38no—
245C4×Q8—64no—
246C4×Q8—68no—
247C4×D4—64no—
248C4×D4—124no—
249#117—68no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C4⋊C4C22×D312yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—64no—
270C2×C42—62no—
271C2×C42—61no—
272C2×C42C22×D311yescharacteristic
273#107—64no—
274#103—64no—
275#103—64no—
276#103—64no—
277#102—24no—
278C22⋊C8—64no—
279#97—64no—
280#95—61no—
281#95—61no—
282#95—61no—
283C23×C3—22no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3C22×D411yescharacteristic
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—21no—
297C22×D3#13712yes—
298C3×Q8—48no—
299C3×D4—24no—
300C3×D4—42no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4AGL(1,ℤ/8)12yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4111yescharacteristic
317#67—48no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—84no—
335C2×Dic3#12012yes—
336C2×Dic3#12012yes—
337C4×D3—28no—
338C4×D3—44no—
339C4×D3—42no—
340C4×D3—42no—
341Dic6—24no—
342Dic6—24no—
343Dic6—42no—
344Dic6—42no—
345C3×C8—42no—
346C3⋊C8—24no—
347C24—21no—
348C24—31no—
349C2×Q8—24no—
350C2×Q8—34no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—64no—
362C2×D4—122no—
363C2×D4—122no—
364C2×D4#23912yes—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—121no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4—122no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—616no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—34no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—32no—
400C4⋊C4—68no—
401C4⋊C4—64no—
402C4⋊C4—68no—
403C4⋊C4—64no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C4⋊C4D3×D414yes—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—122no—
413C22⋊C4—122no—
414C42—22no—
415C42—34no—
416C42—62no—
417C42—62no—
418C42—68no—
419C42—122no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3C2×#12011yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#53211yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×#13711yescharacteristic
432C22×C3C2×#13711yescharacteristic
433C22×C3#57511yescharacteristic
434C2×D3—22no—
435C2×D3—22no—
436C2×D3—22no—
437C2×D3—41no—
438C2×D3—41no—
439C2×D3—41no—
440C2×D3—41no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—44no—
444C3×C4—81no—
445C3×C4#52512yes—
446C3×C4#53212yes—
447Dic3—22no—
448Dic3—22no—
449Dic3—42no—
450Dic3—44no—
451Dic3—41no—
452Dic3—81no—
453C23—22no—
454C23—32no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23C2×D3×D411yescharacteristic
468Q8—48no—
469Q8—64no—
470Q8—64no—
471Q8—122no—
472Q8—122no—
473D4—24no—
474D4—42no—
475D4—128no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—68no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—124no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—244no—
509C2×C4#91112yes—
510C2×C4#91512yes—
511C2×C4C2×D3×D412yes—
512C2×C4C2×#23911yescharacteristic
513C2×C4D3×#4111yescharacteristic
514C2×C4D3×#4112yes—
515C8—42no—
516C8—64no—
517C2×C3—82no—
518C2×C3#238811yescharacteristic
519C2×C3#300411yescharacteristic
520C2×C3#300611yescharacteristic
521C2×C3#311011yescharacteristic
522C2×C3#311611yescharacteristic
523C2×C3#318111yescharacteristic
524C2×C3#318211yescharacteristic
525D3—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—62no—
530C22—62no—
531C22—62no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—121no—
537C22—121no—
538C22—121no—
539C22—121no—
540C22D3×#12011yescharacteristic
541C22#562711yescharacteristic
542C22#566811yescharacteristic
543C22C2×#91111yescharacteristic
544C22C2×#91511yescharacteristic
545C22#577311yescharacteristic
546C22D3×#13711yescharacteristic
547C4—22no—
548C4—22no—
549C4—44no—
550C4—62no—
551C4—62no—
552C4—81no—
553C4—122no—
554C4—121no—
555C4—124no—
556C4—241no—
557C4#562712yes—
558C4#566812yes—
559C3#2357411yescharacteristic, complemented, Sylow
560C2—82no—
561C2—122no—
562C2#7677711yescharacteristic
563C2#8000311yescharacteristic
564C2#8003911yescharacteristic
565C2#8095011yescharacteristic
566C2#8104711yescharacteristic
567C2#8155911yescharacteristic
568C2#8166911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2023D3
hypocenterC3#23574
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
2. C2×#4757C2semidirect
4. C3×#2023C2semidirect
5. #71782C2semidirect
7. #69661C2semidirect
8. #68825C2semidirect
9. #68783C2semidirect
10. #68631C2semidirect
11. #68600C2semidirect
52. #2023D3semidirect
103. #887D4semidirect
559. C3#23574semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#47572
#762912
C3×#20232
#717822
#702052
#696612
#688252
#687832
#686312
#686002
#235743