Finite Group Databasebeta

← #10901355

Subgroups — #10901355

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#4890C212yesmaximal, complemented
3#76288C211yesmaximal, characteristic, complemented
4C3×#2057C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70634C212yesmaximal, complemented
7#69627C211yesmaximal, characteristic, complemented
8#68892C212yesmaximal, complemented
9#68822C212yesmaximal, complemented
10#68662C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#29687—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16#5725—28no—
17C2×#896—21no—
18C2×#895—21no—
19C2×#895C2212yes—
20C2×#893—21no—
21C2×#893C2212yes—
22C2×#888C2211yescharacteristic
23C2×#874C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×#107C2212yes—
26C3×#389C2211yescharacteristic
27C2×#821C2212yes—
28C2×#821C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×Dic3⋊C8C2212yes—
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33#4976—21no—
34#4975C2211yescharacteristic
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#807C2212yes—
38#4928C2211yescharacteristic
39#4890—28no—
40#4667—21no—
41#4646—21no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#474—32no—
48#2857—31no—
49#2057—31no—
50#2057D311yescharacteristic, complemented
51#2041—31no—
52#2041—31no—
53#2033—32no—
54#2033—32no—
55#1802—32no—
56#1797—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×Dic6—21no—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C42C2311yescharacteristic
78#947—28no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#896—44no—
94#895—28no—
95#895—24no—
96#895—24no—
97#893—28no—
98#893—24no—
99#893D418yes—
100#888—24no—
101#874—24no—
102C42×D3—24no—
103C3×#107—24no—
104C3×#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#821—24no—
109#821—42no—
110(C2×D3)⋊C8—42no—
111Dic3⋊C8—24no—
112#809—24no—
113#809—42no—
114#807—24no—
115C22×#41—61no—
116#532—68no—
117C2×C4⋊Q8—31no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×#127—31no—
120C2×#124—61no—
121C2×C22⋊Q8—61no—
122C2×#121—32no—
123C2×#121—61no—
124C2×C4×D4—32no—
125C2×C4×D4—61no—
126C22×C42—31no—
127#474—68no—
128C2×#107—32no—
129C2×#107C2×D312yes—
130C2×C4⋊C8—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8—32no—
136#391—61no—
137#389—31no—
138#389—61no—
139#389C2×D311yescharacteristic
140#384—31no—
141#380—31no—
142#380—31no—
143#380—61no—
144#340—32no—
145#340C2×D312yes—
146#336—31no—
147#336C2×D311yescharacteristic
148C23×D3—22no—
149C23×D3C2×D411yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—42no—
152C2×C3×Q8—42no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164#239—432no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—24no—
170C2×D12—24no—
171C2×D12—28no—
172C2×D12—22no—
173C2×D12—42no—
174C2×D12—42no—
175C2×D12—44no—
176C2×D12—44no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12C2×D414yes—
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—28no—
189C2×C4×D3—41no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×C4×D3D814yes—
193C2×Dic6—42no—
194C2×Dic6—42no—
195C2×Dic6—42no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4C2×D414yes—
206C3×C42—22no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—42no—
210(C2×D3)⋊C4—42no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214Dic3⋊C4—42no—
215Dic3⋊C4—42no—
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3C2×D414yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×#41—68no—
225C2×#41—124no—
226C2×#41—124no—
227C22×Q8—21no—
228C22×Q8—61no—
229C22×D4—32no—
230C22×D4—62no—
231C22×D4—61no—
232C22×D4—61no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—62no—
237C22×C8—32no—
238C22×C8C22×D312yes—
239C4⋊Q8—24no—
240C4⋊Q8—64no—
241#127—64no—
242#124—68no—
243C22⋊Q8—124no—
244#121—64no—
245#121—68no—
246#121—64no—
247C4×D4—38no—
248C4×D4—64no—
249C4×D4—68no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—64no—
269C2×C42C22×D311yescharacteristic
270#107—24no—
271#107—64no—
272C4⋊C8—64no—
273#102—64no—
274#102—64no—
275#102—122no—
276#102—122no—
277C22⋊C8—122no—
278#95—21no—
279#95—61no—
280#95—61no—
281#95—61no—
282C23×C3C22×D411yescharacteristic
283C22×D3—21no—
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—24no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3C2×D812yes—
297C3×Q8—48no—
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—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4C2×D812yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×C3×C42−1+412yes—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3#12012yes—
329C2×Dic3#12012yes—
330C2×Dic3C2×D812yes—
331C2×Dic3AGL(1,ℤ/8)12yes—
332D12—24no—
333D12—24no—
334D12—48no—
335D12—48no—
336D12—416no—
337D12—42no—
338D12—42no—
339C4×D3—28no—
340C4×D3—44no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—416no—
344Dic6—84no—
345C3×C8—42no—
346C3⋊C8—42no—
347C24—31no—
348C24—62no—
349#41—1232no—
350C2×Q8—24no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—122no—
354C2×Q8—122no—
355C2×Q8—122no—
356C2×D4—34no—
357C2×D4—64no—
358C2×D4—68no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—122no—
366C2×D4—122no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—122no—
370C2×D4—122no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—34no—
377C22×C4—31no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—68no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—121no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D411yescharacteristic
394C22×C4C23×D311yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—62no—
399C2×C8—62no—
400C2×C8—62no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—122no—
411C4⋊C4—122no—
412C4⋊C4—122no—
413C4⋊C4—122no—
414C4⋊C4D3×D414yes—
415C22⋊C4—68no—
416C22⋊C4—68no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C42—22no—
420C42—34no—
421C42—62no—
422C42—62no—
423C22×C3C2×#12011yescharacteristic
424C22×C3#53611yescharacteristic
425C22×C3#53711yescharacteristic
426C22×C3C22×D811yescharacteristic
427C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
428C22×C3C2×#13711yescharacteristic
429C22×C3#57511yescharacteristic
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—41no—
436C2×D3—41no—
437C2×D3—44no—
438C2×D3—44no—
439C2×D3—44no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—82no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—44no—
447C3×C4—81no—
448C3×C4—81no—
449C3×C4#53612yes—
450C3×C4#53712yes—
451Dic3—22no—
452Dic3—41no—
453Dic3—81no—
454Dic3—81no—
455Dic3#44714yes—
456C23—32no—
457C23—61no—
458C23—61no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—61no—
463C23—64no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23C2×D3×D411yescharacteristic
471Q8—48no—
472Q8—244no—
473D4—64no—
474D4—64no—
475D4—128no—
476D4—128no—
477D4—1216no—
478D4—122no—
479D4—122no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—32no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—68no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—124no—
504C2×C4—1216no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4D3×D812yes—
514C2×C4#91812yes—
515C2×C4C2×D3×D412yes—
516C2×C4#100711yescharacteristic
517C2×C4#100712yes—
518C2×C4#100911yescharacteristic
519C8—42no—
520C8—122no—
521C2×C3#261611yescharacteristic
522C2×C3C2×#44711yescharacteristic
523C2×C3#300711yescharacteristic
524C2×C3#322611yescharacteristic
525C2×C3#323511yescharacteristic
526C2×C3#324511yescharacteristic
527C2×C3#325411yescharacteristic
528D3—42no—
529D3—84no—
530C22—62no—
531C22—62no—
532C22—62no—
533C22—121no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22—124no—
538C22—124no—
539C22—124no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22D3×#12011yescharacteristic
545C22#562911yescharacteristic
546C22#572411yescharacteristic
547C22C2×D3×D811yescharacteristic
548C22C2×#91811yescharacteristic
549C22D3×AGL(1,ℤ/8)11yescharacteristic
550C22#577711yescharacteristic
551C4—22no—
552C4—22no—
553C4—34no—
554C4—44no—
555C4—62no—
556C4—81no—
557C4—81no—
558C4—121no—
559C4—241no—
560C4—241no—
561C4#562912yes—
562C4#572412yes—
563C3#2968711yescharacteristic, complemented, Sylow
564C2—122no—
565C2—244no—
566C2#7720111yescharacteristic
567C2D3×#44711yescharacteristic
568C2#8018411yescharacteristic
569C2#8097511yescharacteristic
570C2#8109811yescharacteristic
571C2#8194111yescharacteristic
572C2#8197911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenter#2057D3
hypocenterC3#29687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
2. C2×#4890C2semidirect
3. #76288C2semidirect
4. C3×#2057C2semidirect
5. #71782C2semidirect
6. #70634C2semidirect
7. #69627C2semidirect
8. #68892C2semidirect
9. #68822C2semidirect
10. #68662C2semidirect
11. #68629C2semidirect
50. #2057D3semidirect
563. C3#29687semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#48902
#762882
C3×#20572
#717822
#706342
#696272
#688922
#688222
#686622
#686292
#296873