Finite Group Databasebeta

← #10848697

Subgroups — #10848697

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×#5728C211yesmaximal, characteristic, complemented
2C2×C3×#486C211yesmaximal, characteristic, complemented
3#83192C211yesmaximal, characteristic
4C2×#5076C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#70213C214yesmaximal, complemented
7#70205C214yesmaximal, complemented
8#69771C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
10#25598—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×#946—21no—
14C2×Q8×Dic3—21no—
15C2×#938—21no—
16C2×#938—21no—
17C2×#938C2212yes—
18C22×#214C2214yes—
19#5728—24no—
20#5728—24no—
21C2×#897C2212yes—
22C2×#892—21no—
23C2×#890—21no—
24C2×#887C2212yes—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C3×#486—24no—
28C2×C3×#102C2214yes—
29C2×C3×C4×C8C2211yescharacteristic
30#5221—21no—
31#5221—21no—
32#5221C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2214yes—
35#5076—24no—
36#4935C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4550C2212yes—
39#4529C2214yes—
40C2×#533—31no—
41C2×#492—31no—
42C2×#486D311yescharacteristic, complemented
43#2856—31no—
44#2000—34no—
45#1990—31no—
46#1956—31no—
47#1880—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C22×#67—22no—
54C22×C4×D3D411yescharacteristic
55C22×C4×D3D411yescharacteristic
56C22×C4×D3C2311yescharacteristic
57C22×Dic6C2312yes—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8D414yescomplemented
61C3×#127—22no—
62C3×#127—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—22no—
69#946—28no—
70Q8×Dic3—28no—
71#938—44no—
72#938—44no—
73#938—44no—
74C2×#214—28no—
75C2×#214—28no—
76C2×#214—44no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2312yes—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90#897—24no—
91#897—24no—
92#892—28no—
93#890—28no—
94#887—24no—
95#887—24no—
96C42×D3—22no—
97C42×D3—22no—
98#869—22no—
99#869D414yescomplemented
100C3×#102—28no—
101C3×C4×C8—22no—
102(C2×D3)⋊C8—24no—
103#816—44no—
104C4×C3⋊C8—22no—
105C22×SD16—34no—
106#533—64no—
107#533—64no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×#127C2×D311yescharacteristic
111C2×#125—61no—
112C2×C22⋊Q8—32no—
113C2×C22⋊Q8—61no—
114C2×#121—32no—
115C2×#121—61no—
116C2×#121—61no—
117C2×C4×Q8—32no—
118C2×C4×Q8—61no—
119C2×#117—61no—
120C22×C42—31no—
121#492—64no—
122#486—24no—
123C2×#103—34no—
124C2×#102C2×D314yes—
125C2×C22⋊C8—32no—
126C2×C4×C8—31no—
127C2×C4×C8C2×D311yescharacteristic
128#390—32no—
129#390—61no—
130#390—61no—
131#384—31no—
132#340—34no—
133#336—32no—
134C24×C3—22no—
135C23×D3C2×D411yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
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×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C2×#67—44no—
154C2×#67—44no—
155C2×#67—44no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3SD1614yescomplemented
171C2×Dic6—24no—
172C2×Dic6—24no—
173C2×Dic6—22no—
174C2×Dic6C2×D414yes—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4C2×D414yes—
181C3×C42C2×D412yes—
182C3×C42C2×D412yes—
183#217—44no—
184#217—44no—
185#217—44no—
186#214—416no—
187(C2×D3)⋊C4—28no—
188C4⋊Dic3—28no—
189C4⋊Dic3—24no—
190C4⋊Dic3—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194Dic3⋊C4—28no—
195C4×Dic3—28no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8—21no—
204C22×Q8—32no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—62no—
208C22×D4C22×D312yes—
209C23×C4—31no—
210C23×C4—31no—
211C23×C4—31no—
212C2×SD16—68no—
213C2×SD16—68no—
214C2×SD16—124no—
215C22×C8—32no—
216C22×C8C22×D312yes—
217C4⋊Q8—22no—
218C4⋊Q8—34no—
219C4⋊Q8—62no—
220C4⋊Q8#6714yescomplemented
221#127—22no—
222#127—22no—
223#125—68no—
224C22⋊Q8—64no—
225C22⋊Q8—68no—
226C22⋊Q8—64no—
227#121—124no—
228#121—124no—
229#121—124no—
230C4×Q8—64no—
231C4×Q8—68no—
232C4×Q8—64no—
233#117—68no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—62no—
251C2×C42—62no—
252C2×C42—62no—
253C2×C42C22×D311yescharacteristic
254#103—124no—
255#102—28no—
256C22⋊C8—64no—
257C4×C8—22no—
258C4×C8—62no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3C22×D411yescharacteristic
267C22×D3—21no—
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3C2×SD1612yes—
274C3×Q8—48no—
275C3×D4—44no—
276C3×D4—44no—
277C3×D4—84no—
278C3×D4—84no—
279C2×C3×C4—24no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4#12012yes—
283C2×C3×C4#12012yes—
284C2×C3×C4C2×SD1612yes—
285C2×C3×C4#13512yes—
286C2×C3×C4AGL(1,ℤ/8)12yes—
287C2×C3×C4#13712yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×C3×C4H(5,𝔽2)12yes—
292#67—88no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—24no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3C2×SD1612yes—
308C4×D3—28no—
309C4×D3—42no—
310C4×D3—42no—
311C4×D3—44no—
312Dic6—28no—
313Dic6—44no—
314C3×C8—42no—
315C3⋊C8—24no—
316C24—22no—
317C24—31no—
318C2×Q8—24no—
319C2×Q8—34no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—62no—
323C2×Q8—64no—
324C2×Q8—64no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—124no—
336C2×D4—124no—
337C2×D4—124no—
338C22×C4—22no—
339C22×C4—31no—
340C22×C4—34no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—1216no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—34no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—68no—
367C4⋊C4—68no—
368C4⋊C4—64no—
369C4⋊C4—64no—
370C4⋊C4—64no—
371C4⋊C4—122no—
372C4⋊C4—122no—
373C4⋊C4C2×#6714yes—
374C22⋊C4—68no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—68no—
379C42—62no—
380C42—62no—
381C42—61no—
382C42—61no—
383C42D3×D412yes—
384C42C2×#6712yes—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3C2×#12011yescharacteristic
393C22×C3#53211yescharacteristic
394C22×C3#53511yescharacteristic
395C22×C3C22×SD1611yescharacteristic
396C22×C3C2×#13511yescharacteristic
397C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C2×#13711yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C3×C4—44no—
407C3×C4#44814yes—
408C3×C4#44914yes—
409C3×C4#54612yes—
410C3×C4#55012yes—
411Dic3—22no—
412Dic3—44no—
413Dic3—41no—
414Dic3—41no—
415Dic3—82no—
416C23—32no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23C2×D3×D411yescharacteristic
431Q8—48no—
432Q8—68no—
433Q8—124no—
434D4—44no—
435D4—44no—
436D4—84no—
437D4—84no—
438D4—248no—
439C2×C4—24no—
440C2×C4—32no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—68no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—64no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—124no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4#91112yes—
462C2×C4#91212yes—
463C2×C4D3×SD1612yes—
464C2×C4#91512yes—
465C2×C4#93712yes—
466C2×C4#93712yes—
467C2×C4C2×D3×D411yescharacteristic
468C2×C4D3×#4112yes—
469C2×C4#100912yes—
470C2×C4C22×#6711yescharacteristic
471C8—42no—
472C8—64no—
473C2×C3—162no—
474C2×C3#243111yescharacteristic
475C2×C3C2×#44811yescharacteristic
476C2×C3C2×#44911yescharacteristic
477C2×C3#316711yescharacteristic
478C2×C3#318211yescharacteristic
479C2×C3#320811yescharacteristic
480C2×C3#321011yescharacteristic
481D3—42no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22#566811yescharacteristic
497C22#567411yescharacteristic
498C22C2×#91111yescharacteristic
499C22C2×#91211yescharacteristic
500C22C2×D3×SD1611yescharacteristic
501C22C2×#91511yescharacteristic
502C22C2×#93711yescharacteristic
503C4—44no—
504C4—62no—
505C4—121no—
506C4—124no—
507C4—121no—
508C4—242no—
509C4#515814yes—
510C4#517014yes—
511C4#567612yes—
512C4#583412yes—
513C3#2559811yescharacteristic, complemented, Sylow
514C2—122no—
515C2—162no—
516C2#8155611yescharacteristic
517C2#8155911yescharacteristic
518C2#8156811yescharacteristic
519C2#8156911yescharacteristic
520C2#8295811yescharacteristic
521C2C2×#515811yescharacteristic
522C2C2×#517011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#486C2
socleC23×C3C22×D4
prime radical#83192C2
hypercenterC2×#486D3
hypocenterC3#25598
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
2. C2×C3×#486C2semidirect
4. C2×#5076C2semidirect
5. #71787C2semidirect
6. #70213C2semidirect
7. #70205C2semidirect
8. #69771C2semidirect
9. C4×(C2×D3)⋊C8C2semidirect
42. C2×#486D3semidirect
60. C3×C4⋊Q8D4semidirect
99. #869D4semidirect
170. C2×C4×D3SD16semidirect
220. C4⋊Q8#67semidirect
513. C3#25598semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×C3×#4862
#831922
C2×#50762
#717872
#702132
#702052
#697712
C4×(C2×D3)⋊C82
#255983