Finite Group Databasebeta

← #10846048

Subgroups — #10846048

Order 768.

1122 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
1C3×#2541C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic, complemented
3C2×#5042C212yesmaximal, complemented
4#82376C211yesmaximal, characteristic, complemented
5C2×#5012C211yesmaximal, characteristic, complemented
6#71764C211yesmaximal, characteristic, complemented
7#71685C212yesmaximal, complemented
8#71682C212yesmaximal, complemented
9#71679C212yesmaximal, complemented
10C4×#810C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#25395—31nomaximal, Sylow
13C2×C3×#127—21no—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C22×#214C2212yes—
19C2×#924C2212yes—
20C22×#208C2212yes—
21C2×C4×Dic6C2211yescharacteristic
22C3×#392—22no—
23C3×#390—21no—
24C3×C4×C4⋊C4C2211yescharacteristic
25#5221—21no—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31#5075—24no—
32#5042—28no—
33#5012—24no—
34#4972—21no—
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#807C2212yes—
38#4948—22no—
39#4935C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2212yes—
44#4529C2211yescharacteristic
45C2×#468—32no—
46C2×#460—31no—
47C2×#436—31no—
48#2541D311yescharacteristic
49#2538—31no—
50#2033—32no—
51#2000—32no—
52#1957—31no—
53#1947—31no—
54#1895—32no—
55C4×#103—31no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4D411yescharacteristic
61C22×C3×D4C2312yes—
62C23×C3×C4C2312yes—
63C22×Dic6C2311yescharacteristic
64C3×#127—22no—
65C3×#127—22no—
66C3×#127—44no—
67C3×#121—24no—
68C3×#121—24no—
69C3×#121—42no—
70C3×#121—42no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D412yes—
79C2×C3×C42—21no—
80C2×C3×C42C2311yescharacteristic
81C2×#217—22no—
82C2×#217—22no—
83C2×#214—24no—
84C2×#214—24no—
85C2×#214—42no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90#924—24no—
91#924—42no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C2×#208—24no—
96C2×#208—24no—
97C2×#208—42no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×#95—21no—
104#832—24no—
105#832—24no—
106#832—42no—
107C3⋊#95—22no—
108#830—24no—
109#810—22no—
110#810D414yescomplemented
111#807—24no—
112C4⋊(C3⋊C8)D414yescomplemented
113C4×C3⋊C8—22no—
114C22×SD16—32no—
115C22×M16—32no—
116C2×#127—21no—
117C2×#127C2×D311yescharacteristic
118C2×C22⋊Q8—32no—
119C2×#121C2×D312yes—
120C2×#121C2×D312yes—
121C2×C4×Q8—31no—
122C2×C4×D4C2×D311yescharacteristic
123#468—68no—
124#460—64no—
125#436—64no—
126C2×#107—32no—
127C2×C4⋊C8—31no—
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#396—62no—
136#392—22no—
137#390—21no—
138#390—61no—
139#389—61no—
140#384—31no—
141#380—32no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3—22no—
147C24×C3C2×D412yes—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—81no—
166C2×C3×D4—81no—
167C2×C3×D4—82no—
168C2×C3×D4—82no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4#4112yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C42—42no—
199C3×C42C2×D412yes—
200C3×C42#4112yes—
201#217—44no—
202#217—82no—
203#214—48no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—42no—
209C4⋊Dic3—81no—
210Dic3⋊C4—44no—
211Dic3⋊C4—42no—
212C4×Dic3—42no—
213#208—48no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8C2×D414yes—
222C2×C3⋊C8C2×D412yes—
223C22×Q8—31no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4#6711yescharacteristic
229C22×D4C22×D312yes—
230C23×C4C22×D312yes—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×M16—64no—
235C2×M16—64no—
236C2×M16—122no—
237C22×C8—31no—
238C22×C8—32no—
239C22×C8—31no—
240#127—22no—
241#127—22no—
242#127—44no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—24no—
246#121—24no—
247#121—42no—
248#121—42no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4#6711yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4#6712yes—
267C2×C42—21no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#107—64no—
271C4⋊C8—34no—
272#103—34no—
273#103—62no—
274#102—64no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C4×C8—62no—
279#95—21no—
280#95—62no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3C22×D411yescharacteristic
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—44no—
301C3×D4—84no—
302C3×D4—88no—
303C3×D4—88no—
304C3×D4—84no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12112yes—
321C2×C3×C4#12112yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4#13712yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339Dic6—22no—
340Dic6—22no—
341Dic6—41no—
342Dic6—41no—
343C3⋊C8—22no—
344C3⋊C8—24no—
345C3⋊C8—41no—
346C24—22no—
347C24C2×#6712yes—
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—82no—
374C2×D4—82no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391M16—128no—
392C2×C8—34no—
393C2×C8—32no—
394C2×C8—61no—
395C2×C8—62no—
396C2×C8—61no—
397C2×C8—62no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—124no—
409C4⋊C4—122no—
410C4⋊C4—122no—
411C4⋊C4—241no—
412C4⋊C4D3×D412yes—
413C4⋊C4#23712yes—
414C22⋊C4—24no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—42no—
420C42—122no—
421C42D3×D412yes—
422C42#23712yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12111yescharacteristic
438C22×C3D4211yescharacteristic
439C22×C3#54611yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3C2×#13711yescharacteristic
442C22×C3#57611yescharacteristic
443C22×C3#57711yescharacteristic
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4#46814yescomplemented
449C3×C4#53211yescharacteristic
450C3×C4#54011yescharacteristic
451C3×C4D4211yescharacteristic
452C3×C4#55011yescharacteristic
453Dic3—42no—
454Dic3—82no—
455Dic3—81no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23C22×#6711yescharacteristic
471Q8—62no—
472Q8—62no—
473Q8—121no—
474Q8—121no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—44no—
480D4—84no—
481D4—88no—
482D4—88no—
483D4—84no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—62no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4#93212yes—
509C2×C4#93912yes—
510C2×C4#93912yes—
511C2×C4#95112yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23711yescharacteristic
515C2×C4C2×#23711yescharacteristic
516C2×C4#100411yescharacteristic
517C2×C4#100411yescharacteristic
518C8—62no—
519C8—64no—
520C8—121no—
521C2×C3—82no—
522C2×C3—162no—
523C2×C3#242811yescharacteristic
524C2×C3C2×#46811yescharacteristic
525C2×C3#306211yescharacteristic
526C2×C3#327411yescharacteristic
527C2×C3#327811yescharacteristic
528C2×C3#332411yescharacteristic
529C2×C3#333311yescharacteristic
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×#93211yescharacteristic
545C22D4×#6711yescharacteristic
546C22C2×#93911yescharacteristic
547C22#580611yescharacteristic
548C22C2×#95111yescharacteristic
549C22#583511yescharacteristic
550C22#583611yescharacteristic
551C4—22no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—122no—
556C4—242no—
557C4—241no—
558C4#504214yescomplemented
559C4#571611yescharacteristic
560C4#571711yescharacteristic
561C4#571811yescharacteristic
562C4#571911yescharacteristic
563C3#2539511yescharacteristic, complemented, Sylow
564C2—82no—
565C2—162no—
566C2C2×#504211yescharacteristic
567C2#8318611yescharacteristic
568C2#8337911yescharacteristic
569C2#8348611yescharacteristic
570C2#8349211yescharacteristic
571C2#8357511yescharacteristic
572C2#8358411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2541C2
socleC23×C3C22×D4
hypercenter#2541D3
hypocenterC3#25395
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. C2×#5075C2semidirect
3. C2×#5042C2semidirect
4. #82376C2semidirect
5. C2×#5012C2semidirect
6. #71764C2semidirect
7. #71685C2semidirect
8. #71682C2semidirect
9. #71679C2semidirect
10. C4×#810C2semidirect
11. #69466C2semidirect
110. #810D4semidirect
112. C4⋊(C3⋊C8)D4semidirect
448. C3×C4#468semidirect
558. C4#5042semidirect
563. C3#25395semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
C2×#50752
C2×#50422
#823762
C2×#50122
#717642
#716852
#716822
#716792
C4×#8102
#694662
#253953