Finite Group Databasebeta

← #10846300

Subgroups — #10846300

Order 768.

1238 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4883C211yesmaximal, characteristic, complemented
2C2×#4883C211yesmaximal, characteristic, complemented
3C2×#4872C211yesmaximal, characteristic, complemented
4C2×#4695C211yesmaximal, characteristic, complemented
5#76220C212yesmaximal, complemented
6C3×#1942C211yesmaximal, characteristic, complemented
7#70628C211yesmaximal, characteristic, complemented
8C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9#69627C212yesmaximal, complemented
10#68897C212yesmaximal, complemented
11#68886C212yesmaximal, complemented
12#25414—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C22×D24C2212yes—
15C22×D24C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17C2×#895—22no—
18C2×#895C2212yes—
19C2×#895C2212yes—
20C2×#893C2212yes—
21C2×#874C2212yes—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#389C2212yes—
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4991—22no—
29#4988C2212yes—
30C2×#809C2212yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#809C2212yes—
34C4×(C2×D3)⋊C4C2211yescharacteristic
35#4883—24no—
36#4883—24no—
37#4872—24no—
38#4695—24no—
39#4671—22no—
40#4665—22no—
41C3×#340C2212yes—
42#4550C2211yescharacteristic
43#4524C2211yescharacteristic
44C2×#493—31no—
45C2×#466—31no—
46C2×#466—31no—
47C2×C4×D8—31no—
48#2428—32no—
49#2041—32no—
50#1999—32no—
51#1970—31no—
52#1942D311yescharacteristic, complemented
53#1879—32no—
54C8×C22⋊C4—31no—
55C22×D12—22no—
56C22×D12—22no—
57C22×D12—22no—
58C22×D12—22no—
59C22×D12C2312yes—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×C4×D3C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4D411yescharacteristic
75C2×(C2×D3)⋊C4D411yescharacteristic
76C2×C4×Dic3C2311yescharacteristic
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79C2×D24—22no—
80C2×D24—22no—
81C2×D24—24no—
82C2×D24—22no—
83C2×D24—22no—
84C2×D24—24no—
85C2×D24—22no—
86C2×D24—42no—
87C2×D24D414yescomplemented
88C2×D3×C8—22no—
89C2×D3×C8—22no—
90C2×D3×C8—22no—
91#895—24no—
92#895—24no—
93#895—42no—
94#895—42no—
95#895—48no—
96#893—24no—
97#893—42no—
98#874—24no—
99#874—42no—
100C3×#107—22no—
101C3×#107D414yescomplemented
102C3×C4×C8—22no—
103C3×#95—22no—
104C3⋊#95D411yescharacteristic
105C3⋊#95D411yescharacteristic
106(C2×D3)⋊C8—22no—
107C8×Dic3—22no—
108#809—24no—
109#809—24no—
110#809—24no—
111#809—42no—
112C22×D8—31no—
113C22×D8—31no—
114C22×D8—32no—
115C23×C8—31no—
116C2×#127—32no—
117C2×#121—32no—
118C2×#121—32no—
119C2×#121—62no—
120C2×C4×D4—32no—
121#493—64no—
122#466—64no—
123#466—64no—
124C4×D8—64no—
125C2×#107C2×D311yescharacteristic
126C2×#107C2×D311yescharacteristic
127C2×#102—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×C22⋊C8—31no—
132C2×C4×C8—31no—
133C2×C4×C8C2×D311yescharacteristic
134#394—62no—
135#392—62no—
136#390—32no—
137#389C2×D312yes—
138#386—62no—
139C4×C22⋊C4—31no—
140#340C2×D312yes—
141#336—31no—
142#336—31no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3C2×D411yescharacteristic
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C22×Dic3C2×D411yescharacteristic
153C22×Dic3C2×D411yescharacteristic
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—44no—
166C2×D12—44no—
167C2×D12—44no—
168C2×D12—44no—
169C2×D12—44no—
170C2×D12—44no—
171C2×D12—44no—
172C2×D12—44no—
173C2×D12—82no—
174C2×D12—82no—
175C2×D12—82no—
176C2×D12—82no—
177C2×D12C2×D414yes—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—82no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8C2×D412yes—
190C2×C3×C8C2×D412yes—
191C2×C3×C8#4112yes—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—44no—
197C3×C4⋊C4C2×D414yes—
198C3×C42—21no—
199C3×C42—21no—
200(C2×D3)⋊C4—22no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—82no—
206(C2×D3)⋊C4D814yescomplemented
207C4×Dic3—21no—
208C4×Dic3—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215D24—28no—
216D24—48no—
217D24—44no—
218D3×C8—44no—
219C22×D4—32no—
220C22×D4—32no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4—62no—
225C23×C4—31no—
226C23×C4—62no—
227C2×D8—34no—
228C2×D8—64no—
229C2×D8—62no—
230C2×D8—62no—
231C2×D8—62no—
232C2×D8—62no—
233C2×D8—64no—
234C2×D8—62no—
235C2×D8—122no—
236C22×C8—31no—
237C22×C8—31no—
238C22×C8—62no—
239C22×C8—62no—
240C22×C8—62no—
241C22×C8C22×D311yescharacteristic
242C22×C8C22×D311yescharacteristic
243#127—64no—
244#127—122no—
245#121—64no—
246#121—64no—
247#121—128no—
248#121—122no—
249#121—122no—
250C4×D4—64no—
251C4×D4—122no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C22×D312yes—
254C2×C4⋊C4C22×D312yes—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—31no—
263C2×C42C22×D311yescharacteristic
264#107—22no—
265#107D1214yescomplemented
266#102—64no—
267#102—64no—
268#102—64no—
269#102—122no—
270C22⋊C8—62no—
271C4×C8—22no—
272C4×C8—62no—
273#95—22no—
274#95—31no—
275#95—31no—
276C23×C3C22×D411yescharacteristic
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—24no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3C2×D812yes—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4C2×D812yes—
309C2×C3×C4#13512yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3C2×D812yes—
325D12—24no—
326D12—24no—
327D12—42no—
328D12—42no—
329D12—42no—
330D12—42no—
331D12—42no—
332D12—42no—
333D12—88no—
334D12—88no—
335D12—88no—
336D12—88no—
337C4×D3—41no—
338C4×D3—41no—
339C4×D3—41no—
340C4×D3—41no—
341C4×D3—88no—
342C3×C8—22no—
343C3×C8#12114yescomplemented
344C3⋊C8—41no—
345C3⋊C8—41no—
346C24—31no—
347C24—62no—
348C24—62no—
349C2×D4—34no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—124no—
362C2×D4—124no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—242no—
370C2×D4—242no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—242no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390D8—68no—
391D8—128no—
392D8—124no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—124no—
402C2×C8C2×D1212yes—
403C2×C8D3×D412yes—
404C2×C8#23912yes—
405C2×C8#23912yes—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4C2×D1214yes—
411C22⋊C4—34no—
412C22⋊C4—62no—
413C22⋊C4—124no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C42—21no—
419C42—21no—
420C42—61no—
421C42—61no—
422C22×C3#53211yescharacteristic
423C22×C3D4211yescharacteristic
424C22×C3D4211yescharacteristic
425C22×C3C22×D811yescharacteristic
426C22×C3C2×#13511yescharacteristic
427C22×C3#57611yescharacteristic
428C22×C3#57611yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—41no—
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—41no—
436C2×D3—44no—
437C2×D3—44no—
438C2×D3—44no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—82no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—44no—
453C3×C4—82no—
454C3×C4C2×#12111yescharacteristic
455C3×C4#54011yescharacteristic
456C3×C4#54611yescharacteristic
457C3×C4#55011yescharacteristic
458Dic3—22no—
459Dic3—41no—
460Dic3—41no—
461C23—32no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—64no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23C2×D3×D411yescharacteristic
483D4—64no—
484D4—64no—
485D4—122no—
486D4—122no—
487D4—122no—
488D4—122no—
489D4—122no—
490D4—122no—
491D4—248no—
492D4—248no—
493D4—248no—
494D4—248no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—32no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—248no—
519C2×C4D3×D812yes—
520C2×C4#91912yes—
521C2×C4C22×D1211yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×#23911yescharacteristic
524C2×C4C2×#23911yescharacteristic
525C2×C4#100911yescharacteristic
526C2×C4#100911yescharacteristic
527C8—22no—
528C8—121no—
529C8—121no—
530C8#89514yescomplemented
531C2×C3#242811yescharacteristic
532C2×C3#316911yescharacteristic
533C2×C3#317611yescharacteristic
534C2×C3D4×D811yescharacteristic
535C2×C3D4×D811yescharacteristic
536C2×C3#328011yescharacteristic
537C2×C3#328711yescharacteristic
538D3—42no—
539D3—84no—
540D3—162no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22—124no—
549C22—124no—
550C22—124no—
551C22—242no—
552C22—242no—
553C22—242no—
554C22—242no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22D4×D1211yescharacteristic
563C22#560911yescharacteristic
564C22#572511yescharacteristic
565C22C2×D3×D811yescharacteristic
566C22C2×#91911yescharacteristic
567C22#576911yescharacteristic
568C22#576911yescharacteristic
569C4—22no—
570C4—22no—
571C4—44no—
572C4—62no—
573C4—82no—
574C4—121no—
575C4—121no—
576C4C2×#89511yescharacteristic
577C4#557711yescharacteristic
578C4#563011yescharacteristic
579C4#569711yescharacteristic
580C3#2541411yescharacteristic, complemented, Sylow
581C2—122no—
582C2—244no—
583C2—482no—
584C2#7622011yescharacteristic
585C2D8×D1211yescharacteristic
586C2#7973411yescharacteristic
587C2#8072911yescharacteristic
588C2#8077311yescharacteristic
589C2#8190911yescharacteristic
590C2#8193311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1942C2
socleC23×C3C22×D4
hypercenter#1942D3
hypocenterC3#25414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
1. C2×#4883C2semidirect
2. C2×#4883C2semidirect
3. C2×#4872C2semidirect
4. C2×#4695C2semidirect
5. #76220C2semidirect
6. C3×#1942C2semidirect
7. #70628C2semidirect
8. C8×(C2×D3)⋊C4C2semidirect
9. #69627C2semidirect
10. #68897C2semidirect
11. #68886C2semidirect
52. #1942D3semidirect
87. C2×D24D4semidirect
101. C3×#107D4semidirect
206. (C2×D3)⋊C4D8semidirect
265. #107D12semidirect
343. C3×C8#121semidirect
530. C8#895semidirect
580. C3#25414semidirect

Maximal subgroups

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

subgroupindex
C2×#48832
C2×#48722
C2×#46952
#762202
C3×#19422
#706282
C8×(C2×D3)⋊C42
#696272
#688972
#688862
#254143