Finite Group Databasebeta

← #10828474

Subgroups — #10828474

Order 768.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2#82069C211yesmaximal, characteristic, complemented
3C2×#4737C212yesmaximal, complemented
4C3×#1817C211yesmaximal, characteristic, complemented
5#69933C212yesmaximal, complemented
6#68671C212yesmaximal, complemented
7#68670C212yesmaximal, complemented
8#68633C211yesmaximal, characteristic, complemented
9#68632C211yesmaximal, characteristic, complemented
10#68629C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#24062—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17C2×#926C2211yescharacteristic
18C2×#926C2211yescharacteristic
19#5477—28no—
20C2×#875—21no—
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C4×D12C2212yes—
24C2×#869C2211yescharacteristic
25C2×C3×C22⋊C8C2212yes—
26#5210—21no—
27#5210—21no—
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#818C2212yes—
33C2×#817C2212yes—
34C4×Dic3⋊C4—21no—
35C4×Dic3⋊C4—21no—
36#4928C2211yescharacteristic
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39#4928C2211yescharacteristic
40#4737—28no—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#532—31no—
47C2×#482—32no—
48#2296—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041—31no—
53#1875—32no—
54#1864—32no—
55#1862—32no—
56#1817D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—44no—
63C2×#235—44no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×Dic6—21no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C2×C3×C42—24no—
72C2×C3×C42D411yescharacteristic
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77C2×#217—21no—
78#930—28no—
79#930—24no—
80#930D418yes—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83#926—22no—
84#926—22no—
85#926—24no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—21no—
98#875—28no—
99#874—24no—
100C4×D12—28no—
101C4×D12—24no—
102C4×D12—24no—
103#869—24no—
104C3×C22⋊C8—24no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109#821—24no—
110#821—24no—
111#821—24no—
112#821—24no—
113#818—24no—
114#817—24no—
115C22×#41—61no—
116#532—68no—
117C2×C4⋊Q8—31no—
118C2×#127—31no—
119C2×#124—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C2×#117—31no—
125C2×#117—31no—
126C22×C42C2×D311yescharacteristic
127#482—68no—
128C2×#107—32no—
129C2×#106—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×C22⋊C8C2×D312yes—
135#389—31no—
136#389—31no—
137#389—31no—
138#389—31no—
139#385—61no—
140#385—61no—
141C4×C4⋊C4—61no—
142C4×C4⋊C4—61no—
143#340—32no—
144#340—32no—
145#336C2×D311yescharacteristic
146#336C2×D311yescharacteristic
147#336C2×D312yes—
148C24×C3C2×D411yescharacteristic
149C23×D3—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—44no—
168C2×#67—44no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174#235—432no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—24no—
180C2×D12—24no—
181C2×D12—22no—
182C2×D12—42no—
183C2×D12—42no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12C2×D414yes—
187C2×C4×D3—28no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C2×Dic6—24no—
191C2×Dic6—42no—
192C2×Dic6—42no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203#217—24no—
204#217—42no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—28no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D414yes—
214C4⋊Dic3#4112yescomplemented
215C4⋊Dic3#4112yescomplemented
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218Dic3⋊C4—42no—
219Dic3⋊C4—42no—
220C4×Dic3—24no—
221C4×Dic3—42no—
222C2×#41—68no—
223C2×#41—124no—
224C2×#41—124no—
225C22×Q8—61no—
226C22×D4—32no—
227C22×D4—62no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—62no—
231C23×C4D1211yescharacteristic
232C23×C4D1211yescharacteristic
233C23×C4C22×D311yescharacteristic
234C22×C8C22×D312yes—
235C22×C8C22×D312yes—
236C4⋊Q8—64no—
237#127—64no—
238#124—68no—
239#121—38no—
240#121—68no—
241#121—64no—
242C4×D4—64no—
243C4×D4—64no—
244C4×D4—68no—
245#117—62no—
246#117—62no—
247#117—64no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C42—24no—
263C2×C42—61no—
264C2×C42—61no—
265C2×C42D1211yescharacteristic
266C2×C42D1211yescharacteristic
267C2×C42C22×D311yescharacteristic
268C2×C42C22×D311yescharacteristic
269#107—64no—
270#106—64no—
271#102—64no—
272#102—64no—
273#102—64no—
274#102—64no—
275C22⋊C8—24no—
276#95—61no—
277#95—61no—
278#95—61no—
279#95—61no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3#12012yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—24no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C2×C3×C4—28no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
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—44no—
310C2×C3×C4#12012yes—
311C2×C3×C4#13512yes—
312C2×C3×C4AGL(1,ℤ/8)12yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C22×D412yes—
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C42−1+412yes—
318#67—416no—
319C2×Dic3—24no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334D12—24no—
335D12—24no—
336D12—48no—
337D12—48no—
338D12—42no—
339D12—42no—
340C4×D3—416no—
341Dic6—48no—
342C3×C8—42no—
343C3×C8—42no—
344C24—62no—
345C24C2×D1211yescharacteristic
346#41—1232no—
347C2×Q8—64no—
348C2×Q8—122no—
349C2×Q8—122no—
350C2×D4—34no—
351C2×D4—64no—
352C2×D4—64no—
353C2×D4—68no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—122no—
360C2×D4—122no—
361C2×D4—124no—
362C2×D4—124no—
363C2×D4—122no—
364C2×D4—122no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—68no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4C2×D1211yescharacteristic
383C22×C4C2×D1211yescharacteristic
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1211yescharacteristic
386C22×C4C2×D1211yescharacteristic
387C22×C4D3×D414yes—
388C22×C4C23×D311yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C4⋊C4—34no—
396C4⋊C4—32no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—122no—
405C4⋊C4—122no—
406C4⋊C4—122no—
407C4⋊C4—122no—
408C22⋊C4—68no—
409C22⋊C4—64no—
410C22⋊C4—68no—
411C22⋊C4—122no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—64no—
417C42—122no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#52511yescharacteristic
427C22×C3#53611yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3#57511yescharacteristic
432C2×D3—44no—
433C2×D3—44no—
434C2×D3—44no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—82no—
438C2×D3—82no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—41no—
444C3×C4#54712yes—
445C3×C4#54712yes—
446Dic3—42no—
447Dic3—42no—
448Dic3—44no—
449Dic3—81no—
450Dic3—81no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—64no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23#88212yes—
465C23C22×D1211yescharacteristic
466Q8—128no—
467D4—64no—
468D4—64no—
469D4—128no—
470D4—128no—
471D4—1216no—
472D4—122no—
473D4—122no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—64no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—1216no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4#88212yes—
506C2×C4#90412yes—
507C2×C4#90812yes—
508C2×C4#90812yes—
509C2×C4C2×D3×D412yes—
510C2×C4D3×#4111yescharacteristic
511C2×C4D3×#4111yescharacteristic
512C2×C4#101012yes—
513C8—42no—
514C8—42no—
515C2×C3—42no—
516C2×C3#240611yescharacteristic
517C2×C3#300211yescharacteristic
518C2×C3#300311yescharacteristic
519C2×C3#310611yescharacteristic
520C2×C3#311511yescharacteristic
521C2×C3#322811yescharacteristic
522C2×C3#323511yescharacteristic
523D3—84no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—124no—
532C22—124no—
533C22—124no—
534C22—242no—
535C22—242no—
536C22—242no—
537C22—242no—
538C22C2×#88211yescharacteristic
539C22#552411yescharacteristic
540C22#552611yescharacteristic
541C22C2×#90411yescharacteristic
542C22C2×#90811yescharacteristic
543C22C2×#90811yescharacteristic
544C22#574811yescharacteristic
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—124no—
551C4—122no—
552C4—122no—
553C4—241no—
554C4—241no—
555C4#565512yes—
556C4#565512yes—
557C3#2406211yescharacteristic, complemented, Sylow
558C2—42no—
559C2—244no—
560C2#7406311yescharacteristic
561C2#7861011yescharacteristic
562C2#7863711yescharacteristic
563C2#7927011yescharacteristic
564C2#7930111yescharacteristic
565C2#7931911yescharacteristic
566C2#7934711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#1817D3
hypocenterC3#24062
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82069C2semidirect
3. C2×#4737C2semidirect
4. C3×#1817C2semidirect
5. #69933C2semidirect
6. #68671C2semidirect
7. #68670C2semidirect
8. #68633C2semidirect
9. #68632C2semidirect
10. #68629C2semidirect
11. #68629C2semidirect
56. #1817D3semidirect
214. C4⋊Dic3#41semidirect
215. C4⋊Dic3#41semidirect
557. C3#24062semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
#820692
C2×#47372
C3×#18172
#699332
#686712
#686702
#686332
#686322
#686292
#240623