Finite Group Databasebeta

← #10846026

Subgroups — #10846026

Order 768.

1070 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
1C3×#2542C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3C2×#5049C212yesmaximal, complemented
4#82417C211yesmaximal, characteristic
5C2×#5012C211yesmaximal, characteristic, complemented
6#71785C211yesmaximal, characteristic, complemented
7#71699C212yesmaximal, complemented
8#71697C212yesmaximal, complemented
9#71695C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#25393—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C22×#215C2212yes—
18C2×#938—21no—
19C2×#930C2212yes—
20C22×#208C2212yes—
21C2×C4×D12C2211yescharacteristic
22C3×#399—22no—
23C3×#390—21no—
24C3×#384—21no—
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31#5083—24no—
32#5049—28no—
33#5012—24no—
34#4987—21no—
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#807C2212yes—
38#4961—22no—
39#4935C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44#4529C2212yes—
45C2×#468—32no—
46C2×#461—31no—
47C2×#436—31no—
48#2542D311yescharacteristic, complemented
49#2537—31no—
50#2036—32no—
51#2000—32no—
52#1956—31no—
53#1947—31no—
54#1894—32no—
55C4×#102—31no—
56C22×C3×Q8C2312yes—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8—42no—
67C3×C22⋊Q8—42no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83C2×#215—24no—
84C2×#215—24no—
85C2×#215—42no—
86#938—44no—
87#930—24no—
88#930—42no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×#208—24no—
96C2×#208—24no—
97C2×#208—42no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C4×D12—22no—
102C4×D12—22no—
103C3×#95—21no—
104#835—24no—
105#835—24no—
106#835—42no—
107C3⋊#95—22no—
108#830—24no—
109#809—22no—
110#809D414yescomplemented
111#807—24no—
112C4⋊(C3⋊C8)D414yescomplemented
113C4×C3⋊C8—22no—
114C22×SD16—32no—
115C22×M16—32no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×C22⋊Q8C2×D312yes—
118C2×C22⋊Q8C2×D312yes—
119C2×#121—32no—
120C2×#121—61no—
121C2×C4×D4—31no—
122C2×C4×D4C2×D311yescharacteristic
123#468—68no—
124#461—64no—
125#436—64no—
126C2×#107—32no—
127C2×C4⋊C8—31no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#102—31no—
132C2×#102—31no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#399—22no—
136#394—62no—
137#390—21no—
138#384—21no—
139#384—31no—
140#380—32no—
141#380—61no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3C2×D412yes—
147C23×D3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C2×#67—44no—
171C2×#67—44no—
172C2×#67—82no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12C2×D412yes—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×C4×D3—42no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D412yes—
192C3×C4⋊C4#4112yes—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C42—42no—
197C3×C42C2×D412yes—
198C3×C42#4112yes—
199#217—44no—
200#215—48no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—42no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—42no—
208C4⋊Dic3—81no—
209#208—48no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C2×D414yes—
218C2×C3⋊C8C2×D412yes—
219C22×Q8C22×D312yes—
220C22×D4—31no—
221C22×D4—62no—
222C22×D4#6711yescharacteristic
223C23×C4—61no—
224C23×C4C22×D312yes—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×M16—64no—
229C2×M16—64no—
230C2×M16—122no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—42no—
239C22⋊Q8—42no—
240#121—64no—
241#121—122no—
242#121—124no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4#6712yes—
264C2×C42—21no—
265C2×C42C22×D311yescharacteristic
266#107—64no—
267C4⋊C8—34no—
268#103—64no—
269#103—64no—
270#103—122no—
271#102—34no—
272#102—62no—
273C22⋊C8—64no—
274C4×C8—62no—
275#95—21no—
276#95—62no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×D4—44no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12112yes—
316C2×C3×C4#12112yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4#13712yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×C3×C4H(5,𝔽2)11yescharacteristic
325#67—88no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332D12—22no—
333D12—22no—
334D12—41no—
335D12—41no—
336C4×D3—84no—
337C3⋊C8—22no—
338C3⋊C8—24no—
339C3⋊C8—41no—
340C24—61no—
341C24C2×#6712yes—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—32no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—124no—
358C2×D4—124no—
359C2×D4—242no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4C2×#6712yes—
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379M16—128no—
380C2×C8—34no—
381C2×C8—32no—
382C2×C8—61no—
383C2×C8—62no—
384C2×C8—61no—
385C2×C8—62no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—61no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—61no—
397C4⋊C4—62no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—122no—
401C4⋊C4—241no—
402C4⋊C4D3×D412yes—
403C4⋊C4#23712yes—
404C22⋊C4—24no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C22⋊C4—122no—
410C42—42no—
411C42D3×D412yes—
412C42#23712yes—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3C2×#12111yescharacteristic
421C22×C3D4211yescharacteristic
422C22×C3#54611yescharacteristic
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3C2×#13711yescharacteristic
425C22×C3#57711yescharacteristic
426C22×C3#57811yescharacteristic
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C3×C4—22no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—82no—
439C3×C4#46814yescomplemented
440C3×C4#53211yescharacteristic
441C3×C4#54011yescharacteristic
442C3×C4D4211yescharacteristic
443C3×C4#55011yescharacteristic
444Dic3—82no—
445Dic3—81no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—62no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23C22×#6711yescharacteristic
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465D4—44no—
466D4—62no—
467D4—62no—
468D4—121no—
469D4—121no—
470D4—248no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—244no—
496C2×C4#93912yes—
497C2×C4#93912yes—
498C2×C4#94212yes—
499C2×C4#94912yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4C2×#23711yescharacteristic
503C2×C4C2×#23711yescharacteristic
504C2×C4#100411yescharacteristic
505C2×C4#100411yescharacteristic
506C8—62no—
507C8—64no—
508C8—121no—
509C2×C3—82no—
510C2×C3#242811yescharacteristic
511C2×C3C2×#46811yescharacteristic
512C2×C3#306311yescharacteristic
513C2×C3#327211yescharacteristic
514C2×C3#327611yescharacteristic
515C2×C3#332911yescharacteristic
516C2×C3#333011yescharacteristic
517D3—82no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—241no—
529C22—241no—
530C22—241no—
531C22—241no—
532C22D4×#6711yescharacteristic
533C22C2×#93911yescharacteristic
534C22#580611yescharacteristic
535C22C2×#94211yescharacteristic
536C22C2×#94911yescharacteristic
537C22#583711yescharacteristic
538C22#583811yescharacteristic
539C4—22no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—241no—
545C4—242no—
546C4#504914yescomplemented
547C4#571611yescharacteristic
548C4#571711yescharacteristic
549C4#571811yescharacteristic
550C4#571911yescharacteristic
551C3#2539311yescharacteristic, complemented, Sylow
552C2—82no—
553C2—242no—
554C2C2×#504911yescharacteristic
555C2#8318611yescharacteristic
556C2#8340011yescharacteristic
557C2#8351411yescharacteristic
558C2#8352311yescharacteristic
559C2#8355511yescharacteristic
560C2#8356011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2542C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenter#2542D3
hypocenterC3#25393
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C3×#2542C2semidirect
2. C2×#5083C2semidirect
3. C2×#5049C2semidirect
5. C2×#5012C2semidirect
6. #71785C2semidirect
7. #71699C2semidirect
8. #71697C2semidirect
9. #71695C2semidirect
10. C4×#809C2semidirect
11. #69466C2semidirect
48. #2542D3semidirect
110. #809D4semidirect
112. C4⋊(C3⋊C8)D4semidirect
439. C3×C4#468semidirect
546. C4#5049semidirect
551. C3#25393semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
C2×#50832
C2×#50492
#824172
C2×#50122
#717852
#716992
#716972
#716952
C4×#8092
#694662
#253933