Finite Group Databasebeta

← #10901423

Subgroups — #10901423

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#4763C212yesmaximal, complemented
3#76285C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic, complemented
6#70206C212yesmaximal, complemented
7#70176C212yesmaximal, complemented
8#70154C212yesmaximal, complemented
9#69604C211yesmaximal, characteristic, complemented
10#68662C211yesmaximal, characteristic, complemented
11#68633C211yesmaximal, characteristic, complemented
12#29694—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×#938—21no—
16C2×#938C2212yes—
17C2×#936—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20#5716—28no—
21C2×#890C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C3×#389C2211yescharacteristic
27C2×#832C2212yes—
28C2×#832C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#817C2212yes—
31C2×Dic3⋊C8C2212yes—
32#4989—21no—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4980—21no—
36#4977C2211yescharacteristic
37C2×#807C2212yes—
38#4928C2211yescharacteristic
39#4763—28no—
40#4668—21no—
41#4652—21no—
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#478—32no—
48#2851—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041D311yescharacteristic, complemented
53#2034—32no—
54#2033—32no—
55#1804—32no—
56#1797—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—44no—
64C2×#237—44no—
65C22×C4×D3D411yescharacteristic
66C22×C4×D3D411yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—21no—
69C22×C3×C8C2312yes—
70C3×#127—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77#938—28no—
78#938—24no—
79#938—24no—
80#936—28no—
81D4×Dic3—28no—
82D4×Dic3—24no—
83D4×Dic3D418yes—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97#890—24no—
98C42×D3—24no—
99#869—24no—
100C3×#102—24no—
101C3×#102—42no—
102C3×#95—21no—
103#832—24no—
104#832—42no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110(C2×D3)⋊C8—42no—
111#817—24no—
112Dic3⋊C8—24no—
113#807—24no—
114C22×#41—61no—
115#532—68no—
116C2×C4⋊Q8—31no—
117C2×#127C2×D311yescharacteristic
118C2×#125—31no—
119C2×#124—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C22×C42—31no—
125#478—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×C4⋊C8—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×#102C2×D312yes—
132C2×#102C2×D312yes—
133C2×C22⋊C8—32no—
134#389—31no—
135#389C2×D311yescharacteristic
136#387—61no—
137#387—61no—
138#385—61no—
139#385—61no—
140#382—31no—
141#380—31no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336—31no—
146#336C2×D311yescharacteristic
147C24×C3—22no—
148C23×D3C2×D411yescharacteristic
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4C2×D414yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—44no—
168C2×#67—44no—
169C22×Dic3—22no—
170C22×Dic3—28no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—44no—
174C22×Dic3—44no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178#237—432no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—41no—
189C2×Dic6—24no—
190C2×Dic6—42no—
191C2×Dic6—42no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—42no—
199C3×C42—22no—
200#217—28no—
201#217—28no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—42no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—42no—
210C4⋊Dic3C2×D414yes—
211Dic3⋊C4—42no—
212Dic3⋊C4—42no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C4×Dic3C2×D414yes—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×#41—68no—
220C2×#41—124no—
221C2×#41—124no—
222C22×Q8—61no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—62no—
226C22×D4C22×D312yes—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—62no—
231C22×C8—32no—
232C22×C8C22×D312yes—
233C4⋊Q8—64no—
234#127—24no—
235#125—64no—
236#124—68no—
237#121—64no—
238#121—68no—
239#121—64no—
240C4×D4—38no—
241C4×D4—64no—
242C4×D4—68no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C42—31no—
258C2×C42—31no—
259C2×C42—31no—
260C2×C42—64no—
261C2×C42C22×D311yescharacteristic
262#107—64no—
263#106—64no—
264C4⋊C8—64no—
265#102—24no—
266#102—42no—
267#102—64no—
268#102—122no—
269C22⋊C8—122no—
270#95—21no—
271#95—61no—
272#95—61no—
273#95—61no—
274#95—61no—
275#95—61no—
276C23×C3—24no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3C22×D411yescharacteristic
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—21no—
290C22×D3AGL(1,ℤ/8)12yes—
291C3×D4—24no—
292C3×D4—24no—
293C3×D4—48no—
294C3×D4—48no—
295C3×D4—42no—
296C3×D4—42no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4#13512yes—
308C2×C3×C4AGL(1,ℤ/8)12yes—
309C2×C3×C4C22×D412yes—
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C2×C3×C42−1+412yes—
313#67—416no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—24no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—416no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3#12012yes—
332C2×Dic3#12012yes—
333C4×D3—28no—
334C4×D3—44no—
335C4×D3—42no—
336C4×D3—42no—
337Dic6—48no—
338C3×C8—42no—
339C3⋊C8—42no—
340C24—22no—
341C24—31no—
342#41—1232no—
343C2×Q8—64no—
344C2×Q8—122no—
345C2×Q8—122no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—68no—
358C2×D4—124no—
359C2×D4—124no—
360C2×D4D3×D414yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—68no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—62no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—121no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4D3×D411yescharacteristic
383C22×C4D3×D411yescharacteristic
384C22×C4C23×D311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—34no—
395C4⋊C4—42no—
396C4⋊C4—64no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—122no—
402C4⋊C4—122no—
403C4⋊C4—122no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C42—22no—
409C42—34no—
410C42—62no—
411C42—62no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#53611yescharacteristic
421C22×C3#53711yescharacteristic
422C22×C3C2×#13511yescharacteristic
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
425C22×C3#57511yescharacteristic
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—41no—
432C2×D3—41no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—81no—
436C3×C4—81no—
437C3×C4#53612yes—
438C3×C4#53712yes—
439Dic3—22no—
440Dic3—22no—
441Dic3—41no—
442Dic3—44no—
443Dic3—81no—
444Dic3—81no—
445C23—24no—
446C23—32no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—61no—
454C23—61no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—61no—
459C23C2×D3×D411yescharacteristic
460Q8—128no—
461D4—24no—
462D4—24no—
463D4—48no—
464D4—48no—
465D4—42no—
466D4—42no—
467D4—1216no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—68no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—64no—
488C2×C4—61no—
489C2×C4—124no—
490C2×C4—1216no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4#91112yes—
502C2×C4#91612yes—
503C2×C4C2×D3×D412yes—
504C2×C4#100411yescharacteristic
505C2×C4#101011yescharacteristic
506C2×C4#101012yes—
507C8—42no—
508C8—122no—
509C2×C3—84no—
510C2×C3#261611yescharacteristic
511C2×C3#300211yescharacteristic
512C2×C3#300311yescharacteristic
513C2×C3#322811yescharacteristic
514C2×C3#323411yescharacteristic
515C2×C3#324711yescharacteristic
516C2×C3#325111yescharacteristic
517D3—42no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—121no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22D3×#12011yescharacteristic
533C22#560811yescharacteristic
534C22#567511yescharacteristic
535C22C2×#91111yescharacteristic
536C22C2×#91611yescharacteristic
537C22D3×AGL(1,ℤ/8)11yescharacteristic
538C22#577311yescharacteristic
539C4—22no—
540C4—22no—
541C4—62no—
542C4—62no—
543C4—81no—
544C4—81no—
545C4—124no—
546C4—121no—
547C4—241no—
548C4—241no—
549C4#559712yes—
550C4#564812yes—
551C3#2969411yescharacteristic, complemented, Sylow
552C2—84no—
553C2—122no—
554C2#7697011yescharacteristic
555C2#8000211yescharacteristic
556C2#8014511yescharacteristic
557C2#8034411yescharacteristic
558C2#8040111yescharacteristic
559C2#8157411yescharacteristic
560C2#8172111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2041D3
hypocenterC3#29694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#4763C2semidirect
3. #76285C2semidirect
4. C3×#2041C2semidirect
5. #71762C2semidirect
6. #70206C2semidirect
7. #70176C2semidirect
8. #70154C2semidirect
9. #69604C2semidirect
10. #68662C2semidirect
11. #68633C2semidirect
52. #2041D3semidirect
551. C3#29694semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#47632
#762852
C3×#20412
#717622
#702062
#701762
#701542
#696042
#686622
#686332
#296943