Finite Group Databasebeta

← #10848699

Subgroups — #10848699

Order 768.

1578 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3C2×#5073C211yesmaximal, characteristic, complemented
4#82081C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#71764C211yesmaximal, characteristic, complemented
7#71684C214yesmaximal, complemented
8#71613C214yesmaximal, complemented
9C4×#830C211yesmaximal, characteristic, complemented
10#25599—31nomaximal, Sylow
11C22×C3×#41—21no—
12C3×#532—24no—
13C3×#532—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×#213C2214yes—
23C2×#930—21no—
24C2×#930—21no—
25C2×#930C2212yes—
26C2×#874C2211yescharacteristic
27C2×#832C2214yes—
28C2×#830C2212yes—
29#5088—24no—
30#5073—24no—
31C2×#809C2214yes—
32#4939—21no—
33#4939—21no—
34#4939C2212yes—
35#4935C2211yescharacteristic
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#532D311yescharacteristic, complemented
41C2×#493—31no—
42C2×#486—31no—
43#2856—31no—
44#1999—34no—
45#1990—31no—
46#1957—31no—
47#1879—34no—
48C4×C22⋊C8—31no—
49C2×C3×#41—28no—
50C2×C3×#41—44no—
51C2×C3×#41—44no—
52C22×C3×Q8—21no—
53C22×C3×D4—22no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C23×C3×C4—22no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yescomplemented
67C3×#127—22no—
68C3×#127D414yescomplemented
69C3×#124—28no—
70C3×#121—28no—
71C3×#121—24no—
72C3×#121—24no—
73C3×C4×D4—28no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#213—28no—
87C2×#213—28no—
88C2×#213—44no—
89#930—44no—
90#930—44no—
91#930—44no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2312yes—
100#874—22no—
101#874—22no—
102#832—44no—
103#830—24no—
104#809—28no—
105C4×C3⋊C8—22no—
106C4×C3⋊C8—22no—
107C22×#41—21no—
108C22×D8—34no—
109#532—24no—
110#532—24no—
111C2×C4⋊Q8C2×D311yescharacteristic
112C2×#127—31no—
113C2×#127C2×D311yescharacteristic
114C2×#124—21no—
115C2×#121—21no—
116C2×#121—32no—
117C2×#121—61no—
118C2×#121—61no—
119C2×#121C2×D312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2×D312yes—
122C22×C42C2×D311yescharacteristic
123#493—64no—
124#486—64no—
125C2×#102—34no—
126C2×#102—34no—
127C2×C22⋊C8—32no—
128C2×C4×C8—31no—
129C2×C4×C8—31no—
130#390—32no—
131#390—61no—
132#390—61no—
133#384—31no—
134#340—34no—
135#336—32no—
136C24×C3—22no—
137C24×C3C2×D411yescharacteristic
138C23×D3—22no—
139C3×#41—432no—
140C2×C3×Q8—24no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×D4—28no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4C2×D414yes—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—28no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4D814yescomplemented
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C22×Dic3—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—24no—
183C2×D12—24no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12—42no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D414yes—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#213—416no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—44no—
205C4⋊Dic3—24no—
206C4⋊Dic3—42no—
207C4⋊Dic3—42no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8#4114yescomplemented
214C2×#41—28no—
215C2×#41—44no—
216C2×#41—44no—
217C22×Q8—21no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—32no—
222C22×D4—62no—
223C22×D4—61no—
224C22×D4—61no—
225C22×D4C22×D312yes—
226C23×C4—22no—
227C23×C4#6711yescharacteristic
228C23×C4#6711yescharacteristic
229C23×C4C22×D311yescharacteristic
230C2×D8—68no—
231C2×D8—68no—
232C2×D8—124no—
233C22×C8—32no—
234C22×C8—32no—
235C4⋊Q8—22no—
236C4⋊Q8#6714yescomplemented
237#127—22no—
238#127—62no—
239#127—62no—
240#127#6714yescomplemented
241#124—28no—
242#121—28no—
243#121—24no—
244#121—24no—
245#121—124no—
246#121—124no—
247#121—124no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42#6711yescharacteristic
264C2×C42#6711yescharacteristic
265C2×C42C22×D311yescharacteristic
266C2×C42C22×D311yescharacteristic
267#102—68no—
268#102—124no—
269C22⋊C8—64no—
270C4×C8—62no—
271C4×C8—62no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C2×D812yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C3×Q8—48no—
295C3×D4—28no—
296C3×D4—416no—
297C3×D4—44no—
298C3×D4—48no—
299C3×D4—48no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—28no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—416no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C2×D812yes—
319C2×C3×C4C2×D812yes—
320C2×C3×C4#13512yes—
321C2×C3×C4AGL(1,ℤ/8)12yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×C3×C4H(5,𝔽2)12yes—
327#67—88no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331D12—44no—
332D12—44no—
333D12—84no—
334D12—84no—
335C3⋊C8—24no—
336C3⋊C8—42no—
337C24—22no—
338C24—62no—
339C24C2×#6711yescharacteristic
340#41—432no—
341C2×Q8—24no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—124no—
361C2×D4—124no—
362C2×D4—124no—
363C2×D4—122no—
364C2×D4—122no—
365C2×D4—122no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4C2×#6714yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—62no—
384C22×C4#21314yescomplemented
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C2×#6711yescharacteristic
390C22×C4C23×D311yescharacteristic
391D8—1216no—
392C2×C8—34no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—64no—
401C4⋊C4—122no—
402C4⋊C4—122no—
403C4⋊C4C2×#6714yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C42—22no—
410C42—22no—
411C42—21no—
412C42—21no—
413C42C2×#6712yes—
414C42C2×#6712yes—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#53211yescharacteristic
431C22×C3#53511yescharacteristic
432C22×C3C22×D811yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
436C2×D3—82no—
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—44no—
446C3×C4—41no—
447C3×C4#44714yes—
448C3×C4#44914yes—
449C3×C4#54612yes—
450C3×C4#55012yes—
451Dic3—82no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23C2×#21312yes—
473C23C22×#6711yescharacteristic
474Q8—48no—
475D4—28no—
476D4—416no—
477D4—48no—
478D4—48no—
479D4—44no—
480D4—124no—
481D4—124no—
482D4—248no—
483D4—244no—
484D4—244no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4C2×#21312yes—
505C2×C4C2×#21312yes—
506C2×C4#93212yes—
507C2×C4#94912yes—
508C2×C4#95012yes—
509C2×C4#95312yes—
510C2×C4#95312yes—
511C2×C4D3×#4112yes—
512C2×C4#100912yes—
513C2×C4C22×#6711yescharacteristic
514C2×C4C22×#6711yescharacteristic
515C8—64no—
516C8—122no—
517C2×C3—42no—
518C2×C3—84no—
519C2×C3#243111yescharacteristic
520C2×C3C2×#44711yescharacteristic
521C2×C3C2×#44911yescharacteristic
522C2×C3#316911yescharacteristic
523C2×C3#318411yescharacteristic
524C2×C3#320511yescharacteristic
525C2×C3#320711yescharacteristic
526D3—162no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22—242no—
545C22—242no—
546C22—242no—
547C22—242no—
548C22C22×#21311yescharacteristic
549C22C2×#93211yescharacteristic
550C22C2×#94911yescharacteristic
551C22C2×#95011yescharacteristic
552C22#583011yescharacteristic
553C22#583211yescharacteristic
554C22C2×#95311yescharacteristic
555C4—22no—
556C4—44no—
557C4—41no—
558C4—41no—
559C4—242no—
560C4#521714yes—
561C4#523814yes—
562C4#582912yes—
563C4#583412yes—
564C3#2559911yescharacteristic, complemented, Sylow
565C2—42no—
566C2—84no—
567C2—482no—
568C2#8329111yescharacteristic
569C2#8329511yescharacteristic
570C2#8330211yescharacteristic
571C2#8330411yescharacteristic
572C2C2×#521711yescharacteristic
573C2C2×#523811yescharacteristic
574C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#25599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5088C2semidirect
3. C2×#5073C2semidirect
4. #82081C2semidirect
5. #71787C2semidirect
6. #71764C2semidirect
7. #71684C2semidirect
8. #71613C2semidirect
9. C4×#830C2semidirect
40. C2×#532D3semidirect
66. C3×C4⋊Q8D4semidirect
68. C3×#127D4semidirect
169. C22×C3×C4D8semidirect
213. C2×C3⋊C8#41semidirect
236. C4⋊Q8#67semidirect
240. #127#67semidirect
384. C22×C4#213semidirect
564. C3#25599semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50882
C2×#50732
#820812
#717872
#717642
#716842
#716132
C4×#8302
#255993