Finite Group Databasebeta

← #10848704

Subgroups — #10848704

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×#5477C211yesmaximal, characteristic, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C2×#5073C211yesmaximal, characteristic, complemented
4C2×#5070C211yesmaximal, characteristic, complemented
5#71684C214yesmaximal, complemented
6C4×#830C211yesmaximal, characteristic, complemented
7#69657C214yesmaximal, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69580C211yesmaximal, characteristic, complemented
10#25599—31nomaximal, Sylow
11C22×#235—21no—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#121—21no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#213C2214yes—
18C2×#930—21no—
19C2×#930C2212yes—
20#5477—24no—
21#5477—24no—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×D12C2212yes—
26C2×#869C2211yescharacteristic
27C3×#390—21no—
28C3×#390—21no—
29C3×#390C2212yes—
30C3×#384C2211yescharacteristic
31C2×#832C2214yes—
32C2×#830C2212yes—
33#5073—24no—
34#5070—24no—
35C2×#809C2214yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#532—31no—
41C2×#493—31no—
42C2×#486—31no—
43#2856D311yescharacteristic, complemented
44#1999—34no—
45#1990—31no—
46#1957—31no—
47#1879—34no—
48C4×C22⋊C8—31no—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57C2×#235—28no—
58C2×#235—44no—
59C2×#235—44no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×Dic6—21no—
65C3×#127—22no—
66C3×#127—22no—
67C3×#121—44no—
68C3×#121—44no—
69C3×#121—44no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#213—28no—
83C2×#213—28no—
84C2×#213—44no—
85#930—28no—
86#930—24no—
87#930—24no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94#875—28no—
95#874—22no—
96#874D414yescomplemented
97C4×D12—28no—
98C4×D12—24no—
99C4×D12—24no—
100#869—22no—
101#869D414yescomplemented
102#832—28no—
103#830—24no—
104#809—44no—
105C4×C3⋊C8—22no—
106C4×C3⋊C8—22no—
107C22×#41—61no—
108C22×D8—34no—
109#532—64no—
110#532—64no—
111C2×C4⋊Q8—31no—
112C2×#127—31no—
113C2×#127C2×D311yescharacteristic
114C2×#124—61no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—32no—
118C2×#121—61no—
119C2×#121C2×D312yes—
120C2×C4×D4—32no—
121C2×C4×D4—61no—
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—21no—
131#390—21no—
132#390C2×D312yes—
133#384C2×D311yescharacteristic
134#340—34no—
135#336—32no—
136C24×C3—22no—
137C24×C3C2×D411yescharacteristic
138C23×D3—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4D814yescomplemented
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—28no—
170C2×#67—44no—
171C2×#67—44no—
172C22×Dic3—22no—
173#235—432no—
174C2×D12—24no—
175C2×D12—24no—
176C2×D12—24no—
177C2×D12—24no—
178C2×D12—22no—
179C2×D12—42no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—42no—
183C2×D12C2×D414yes—
184C2×C4×D3—28no—
185C2×C4×D3—44no—
186C2×C4×D3—44no—
187C2×Dic6—24no—
188C2×Dic6—42no—
189C2×Dic6—42no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#213—416no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—28no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3C2×D414yes—
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—68no—
215C2×#41—124no—
216C2×#41—124no—
217C22×Q8—61no—
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—62no—
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—34no—
236C4⋊Q8—62no—
237#127—22no—
238#127—22no—
239#127—34no—
240#127—62no—
241#124—68no—
242#121—44no—
243#121—44no—
244#121—44no—
245#121—64no—
246#121—64no—
247#121—68no—
248C4×D4—64no—
249C4×D4—64no—
250C4×D4—68no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
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—42no—
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—24no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C3×D4—44no—
295C3×D4—44no—
296C3×D4—88no—
297C3×D4—84no—
298C3×D4—84no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—28no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—44no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4C2×D812yes—
317C2×C3×C4C2×D812yes—
318C2×C3×C4#13512yes—
319C2×C3×C4AGL(1,ℤ/8)12yes—
320C2×C3×C4AGL(1,ℤ/8)12yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4112yes—
324C2×C3×C4H(5,𝔽2)12yes—
325#67—416no—
326C2×Dic3—24no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—28no—
330D12—44no—
331D12—48no—
332D12—48no—
333C4×D3—416no—
334Dic6—48no—
335C3⋊C8—24no—
336C3⋊C8—42no—
337C24—22no—
338C24—62no—
339C24C2×#6711yescharacteristic
340#41—1232no—
341C2×Q8—64no—
342C2×Q8—122no—
343C2×Q8—122no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—34no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—68no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—64no—
363C2×D4—64no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—68no—
381C22×C4—62no—
382C22×C4—124no—
383C22×C4—124no—
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—34no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—64no—
403C4⋊C4—62no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—68no—
408C22⋊C4—68no—
409C42—22no—
410C42—22no—
411C42—21no—
412C42—21no—
413C42D3×D412yes—
414C42D3×D412yes—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
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—44no—
437C2×D3—44no—
438C2×D3—44no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—82no—
447C3×C4#44714yes—
448C3×C4#44914yes—
449C3×C4#54612yes—
450C3×C4#55012yes—
451Dic3—44no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—64no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23C2×#21312yes—
473C23C22×#6711yescharacteristic
474Q8—128no—
475D4—44no—
476D4—44no—
477D4—68no—
478D4—88no—
479D4—84no—
480D4—84no—
481D4—1216no—
482D4—128no—
483D4—128no—
484D4—124no—
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—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—64no—
501C2×C4—1216no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4C2×#21312yes—
505C2×C4C2×#21312yes—
506C2×C4#93212yes—
507C2×C4#93712yes—
508C2×C4#93712yes—
509C2×C4#94912yes—
510C2×C4#95012yes—
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×D3×D411yescharacteristic
513C2×C4#100412yes—
514C2×C4D3×#4112yes—
515C8—64no—
516C8—122no—
517C2×C3—42no—
518C2×C3—162no—
519C2×C3#243111yescharacteristic
520C2×C3C2×#44711yescharacteristic
521C2×C3C2×#44911yescharacteristic
522C2×C3#316911yescharacteristic
523C2×C3#318411yescharacteristic
524C2×C3#320511yescharacteristic
525C2×C3#320711yescharacteristic
526D3—84no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—124no—
542C22—124no—
543C22—124no—
544C22—242no—
545C22—242no—
546C22—242no—
547C22—242no—
548C22C22×#21311yescharacteristic
549C22C2×#93211yescharacteristic
550C22C2×#93711yescharacteristic
551C22#580511yescharacteristic
552C22C2×#94911yescharacteristic
553C22C2×#95011yescharacteristic
554C22#583211yescharacteristic
555C4—22no—
556C4—41no—
557C4—41no—
558C4—82no—
559C4—124no—
560C4#503614yes—
561C4#503714yes—
562C4#559912yes—
563C4#561212yes—
564C3#2559911yescharacteristic, complemented, Sylow
565C2—42no—
566C2—162no—
567C2—244no—
568C2C2×#503611yescharacteristic
569C2C2×#503711yescharacteristic
570C2#8265211yescharacteristic
571C2#8316611yescharacteristic
572C2#8329111yescharacteristic
573C2#8329511yescharacteristic
574C2#8330311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2856C2
socleC23×C3C22×D4
hypercenter#2856D3
hypocenterC3#25599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C3×#2856C2semidirect
3. C2×#5073C2semidirect
4. C2×#5070C2semidirect
5. #71684C2semidirect
6. C4×#830C2semidirect
7. #69657C2semidirect
8. #69582C2semidirect
9. #69580C2semidirect
43. #2856D3semidirect
96. #874D4semidirect
101. #869D4semidirect
162. C22×C3×C4D8semidirect
213. C2×C3⋊C8#41semidirect
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×#54772
C3×#28562
C2×#50732
C2×#50702
#716842
C4×#8302
#696572
#695822
#695802
#255993