Finite Group Databasebeta

← #10848695

Subgroups — #10848695

Order 768.

1298 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C3×#2856C211yesmaximal, characteristic
3C2×#5076C211yesmaximal, characteristic, complemented
4C2×#5070C211yesmaximal, characteristic, complemented
5#71685C214yesmaximal, complemented
6C4×#830C211yesmaximal, characteristic, complemented
7#69660C214yesmaximal, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69579C211yesmaximal, characteristic, complemented
10#25598—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121—21no—
13C2×C3×#121—21no—
14C2×C3×#121C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×#214C2214yes—
17C2×#926—21no—
18C2×#924—21no—
19C2×#924C2212yes—
20#5470—24no—
21#5470—24no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27C3×#390—21no—
28C3×#390—21no—
29C3×#390C2212yes—
30C3×#384C2211yescharacteristic
31C2×#832C2214yes—
32C2×#830C2212yes—
33#5076—24no—
34#5070—24no—
35C2×#810C2214yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#533—31no—
41C2×#492—31no—
42C2×#486—31no—
43#2856D311yescharacteristic
44#2000—34no—
45#1990—31no—
46#1956—31no—
47#1880—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×Dic6—21no—
57C22×Dic6C2312yes—
58C3×#127—22no—
59C3×#127—22no—
60C3×#121—44no—
61C3×#121—44no—
62C3×#121—44no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#214—28no—
77C2×#214—28no—
78C2×#214—44no—
79#926—28no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2312yes—
84#924—28no—
85#924—24no—
86#924—24no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2312yes—
92#870—28no—
93#869—22no—
94#869—22no—
95#869D414yescomplemented
96#869D414yescomplemented
97C4×Dic6—28no—
98C4×Dic6—24no—
99C4×Dic6—24no—
100#832—28no—
101#830—24no—
102#810—44no—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—22no—
105C22×SD16—34no—
106#533—64no—
107#533—64no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8—31no—
110C2×#127C2×D311yescharacteristic
111C2×#125—61no—
112C2×C22⋊Q8—32no—
113C2×C22⋊Q8—61no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2×D312yes—
117C2×C4×Q8—32no—
118C2×C4×Q8—61no—
119C2×#117—61no—
120C22×C42C2×D311yescharacteristic
121#492—64no—
122#486—64no—
123C2×#103—34no—
124C2×#102—34no—
125C2×C22⋊C8—32no—
126C2×C4×C8—31no—
127C2×C4×C8—31no—
128#390—21no—
129#390—21no—
130#390C2×D312yes—
131#384C2×D311yescharacteristic
132#340—34no—
133#336—32no—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—44no—
141C2×C3×D4—44no—
142C2×C3×D4—44no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4SD1614yescomplemented
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—22no—
167C22×Dic3—22no—
168C2×Dic6—24no—
169C2×Dic6—24no—
170C2×Dic6—24no—
171C2×Dic6—22no—
172C2×Dic6—42no—
173C2×Dic6—42no—
174C2×Dic6C2×D414yes—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—28no—
188#214—416no—
189C4⋊Dic3—24no—
190C4⋊Dic3—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3—22no—
193C4⋊Dic3C2×D414yes—
194Dic3⋊C4—28no—
195Dic3⋊C4—28no—
196C4×Dic3—28no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8—32no—
204C22×Q8—61no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C22×D312yes—
209C23×C4#6711yescharacteristic
210C23×C4#6711yescharacteristic
211C23×C4C22×D311yescharacteristic
212C2×SD16—68no—
213C2×SD16—68no—
214C2×SD16—124no—
215C22×C8—32no—
216C22×C8—32no—
217C4⋊Q8—34no—
218C4⋊Q8—34no—
219C4⋊Q8—62no—
220C4⋊Q8—62no—
221#127—22no—
222#127—22no—
223#125—68no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—68no—
227#121—44no—
228#121—44no—
229#121—44no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—68no—
233#117—68no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—62no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—62no—
250C2×C42#6711yescharacteristic
251C2×C42#6711yescharacteristic
252C2×C42C22×D311yescharacteristic
253C2×C42C22×D311yescharacteristic
254#103—124no—
255#102—68no—
256C22⋊C8—64no—
257C4×C8—62no—
258C4×C8—62no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3C2×SD1612yes—
273C23×C3C22×D411yescharacteristic
274C3×D4—44no—
275C3×D4—44no—
276C3×D4—88no—
277C3×D4—84no—
278C3×D4—84no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—28no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12012yes—
296C2×C3×C4C2×SD1612yes—
297C2×C3×C4C2×SD1612yes—
298C2×C3×C4#13512yes—
299C2×C3×C4AGL(1,ℤ/8)12yes—
300C2×C3×C4#13712yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4112yes—
304C2×C3×C4H(5,𝔽2)12yes—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311Dic6—28no—
312Dic6—44no—
313Dic6—48no—
314C3⋊C8—24no—
315C3⋊C8—42no—
316C24—22no—
317C24C2×#6711yescharacteristic
318C2×Q8—34no—
319C2×Q8—64no—
320C2×Q8—62no—
321C2×Q8—64no—
322C2×Q8—64no—
323C2×Q8—122no—
324C2×Q8—122no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4#21414yescomplemented
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—1216no—
358C2×C8—34no—
359C2×C8—62no—
360C2×C8—62no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—24no—
365C4⋊C4—34no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—68no—
369C4⋊C4—68no—
370C4⋊C4—64no—
371C4⋊C4—64no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—68no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—68no—
383C42D3×D412yes—
384C42D3×D412yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53211yescharacteristic
401C22×C3#53511yescharacteristic
402C22×C3C22×SD1611yescharacteristic
403C22×C3C2×#13511yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C3×C4—22no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—82no—
410C3×C4#44814yes—
411C3×C4#44914yes—
412C3×C4#54612yes—
413C3×C4#55012yes—
414Dic3—44no—
415Dic3—44no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×#21412yes—
430C23C22×#6711yescharacteristic
431Q8—68no—
432Q8—128no—
433Q8—124no—
434D4—44no—
435D4—44no—
436D4—88no—
437D4—84no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4#93212yes—
461C2×C4C2×#21412yes—
462C2×C4C2×#21412yes—
463C2×C4#93712yes—
464C2×C4#93712yes—
465C2×C4#95012yes—
466C2×C4#95112yes—
467C2×C4C2×D3×D411yescharacteristic
468C2×C4C2×D3×D411yescharacteristic
469C2×C4#100412yes—
470C2×C4D3×#4112yes—
471C8—64no—
472C8—122no—
473C2×C3—42no—
474C2×C3—162no—
475C2×C3#243111yescharacteristic
476C2×C3C2×#44811yescharacteristic
477C2×C3C2×#44911yescharacteristic
478C2×C3#316711yescharacteristic
479C2×C3#318211yescharacteristic
480C2×C3#320811yescharacteristic
481C2×C3#321011yescharacteristic
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22C2×#93211yescharacteristic
497C22C22×#21411yescharacteristic
498C22C2×#93711yescharacteristic
499C22#580511yescharacteristic
500C22C2×#95011yescharacteristic
501C22C2×#95111yescharacteristic
502C22#583211yescharacteristic
503C4—22no—
504C4—41no—
505C4—41no—
506C4—82no—
507C4—124no—
508C4—124no—
509C4#503714yes—
510C4#504014yes—
511C4#559912yes—
512C4#561212yes—
513C3#2559811yescharacteristic, complemented, Sylow
514C2—42no—
515C2—162no—
516C2C2×#503711yescharacteristic
517C2C2×#504011yescharacteristic
518C2#8265211yescharacteristic
519C2#8316911yescharacteristic
520C2#8329011yescharacteristic
521C2#8329411yescharacteristic
522C2#8331811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2856C2
socleC23×C3C22×D4
hypercenter#2856D3
hypocenterC3#25598
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×#5470C2semidirect
3. C2×#5076C2semidirect
4. C2×#5070C2semidirect
5. #71685C2semidirect
6. C4×#830C2semidirect
7. #69660C2semidirect
8. #69582C2semidirect
9. #69579C2semidirect
95. #869D4semidirect
96. #869D4semidirect
159. C22×C3×C4SD16semidirect
350. C22×C4#214semidirect
513. C3#25598semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#28562
C2×#50762
C2×#50702
#716852
C4×#8302
#696602
#695822
#695792
#255983