Finite Group Databasebeta

← #10848693

Subgroups — #10848693

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×#5085C211yesmaximal, characteristic, complemented
3C2×#4694C211yesmaximal, characteristic, complemented
4#74375C211yesmaximal, characteristic
5C3×#1956C211yesmaximal, characteristic, complemented
6#70272C214yesmaximal, complemented
7#70240C214yesmaximal, complemented
8C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
9#69582C211yesmaximal, characteristic, complemented
10#25598—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#946C2212yes—
13C2×Q8×Dic3C2212yes—
14C22×#215C2214yes—
15#5728—24no—
16#5728—24no—
17C2×#897—21no—
18C2×#895—21no—
19C2×#895—21no—
20C2×#895C2212yes—
21C2×#892—21no—
22C2×#890—21no—
23C2×#887—21no—
24C2×#874C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×C4×C8C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C2×#821C2214yes—
31C2×(C2×D3)⋊C8C2212yes—
32#5085—24no—
33#4988—21no—
34#4988—21no—
35#4988C2212yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37#4694—24no—
38#4550C2212yes—
39#4529C2214yes—
40C2×#533—31no—
41C2×#492—31no—
42C2×#486—31no—
43#2856—31no—
44#2000—34no—
45#1990—31no—
46#1956D311yescharacteristic, complemented
47#1880—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8C2312yes—
50C22×D12—22no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12C2312yes—
54C22×C4×D3D411yescharacteristic
55C22×C4×D3D411yescharacteristic
56C22×C4×D3C2311yescharacteristic
57C22×Dic6—21no—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8D414yescomplemented
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C42C2311yescharacteristic
67#946—24no—
68#946—24no—
69Q8×Dic3—24no—
70Q8×Dic3—24no—
71C2×#215—28no—
72C2×#215—28no—
73C2×#215—44no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87#897—28no—
88#895—44no—
89#895—44no—
90#895—44no—
91#892—28no—
92#890—28no—
93#887—28no—
94#874—22no—
95#874—22no—
96C42×D3—22no—
97C42×D3—22no—
98#869—22no—
99#869D414yescomplemented
100C3×#103—44no—
101C3×C4×C8—22no—
102#821—28no—
103(C2×D3)⋊C8—24no—
104C4×C3⋊C8—22no—
105C22×SD16—34no—
106#533—64no—
107#533—64no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×#127—31no—
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—64no—
123C2×#103C2×D314yes—
124C2×#102—34no—
125C2×C22⋊C8—32no—
126C2×C4×C8—31no—
127C2×C4×C8C2×D311yescharacteristic
128#390—32no—
129#390—61no—
130#390—61no—
131#384C2×D311yescharacteristic
132#340—34no—
133#336—32no—
134C23×D3—22no—
135C23×D3C2×D411yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8C2×D414yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—24no—
152C2×D12—24no—
153C2×D12—44no—
154C2×D12—44no—
155C2×D12—44no—
156C2×D12—42no—
157C2×D12—42no—
158C2×D12—42no—
159C2×D12—42no—
160C2×D12—42no—
161C2×D12—42no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3SD1614yescomplemented
172C2×Dic6—24no—
173C2×Dic6—42no—
174C2×Dic6—42no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C42C2×D412yes—
184C3×C42C2×D412yes—
185#215—416no—
186(C2×D3)⋊C4—28no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—44no—
190C4⋊Dic3—28no—
191C4⋊Dic3—24no—
192C4⋊Dic3—22no—
193C4⋊Dic3C2×D414yes—
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—61no—
204C22×Q8C22×D312yes—
205C22×D4—32no—
206C22×D4—62no—
207C22×D4—61no—
208C22×D4—61no—
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⋊Q8D1214yescomplemented
221#127—62no—
222#127—62no—
223#125—68no—
224C22⋊Q8—68no—
225C22⋊Q8—64no—
226C22⋊Q8—64no—
227#121—124no—
228#121—124no—
229#121—124no—
230C4×Q8—68no—
231C4×Q8—64no—
232C4×Q8—64no—
233#117—68no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
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—44no—
255#102—68no—
256C22⋊C8—64no—
257C4×C8—22no—
258C4×C8—62no—
259C23×C3C22×D411yescharacteristic
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3C2×SD1612yes—
274C3×Q8—28no—
275C3×Q8—44no—
276C2×C3×C4—24no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
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—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—24no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3C2×SD1612yes—
304D12—44no—
305D12—44no—
306D12—88no—
307D12—84no—
308D12—84no—
309C4×D3—28no—
310C4×D3—42no—
311C4×D3—42no—
312C4×D3—44no—
313Dic6—48no—
314C3×C8—42no—
315C3⋊C8—24no—
316C24—31no—
317C24—62no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—64no—
322C2×Q8—122no—
323C2×Q8—122no—
324C2×Q8C2×D1214yes—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—124no—
332C2×D4—122no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—122no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—31no—
341C22×C4—34no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—62no—
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—24no—
366C4⋊C4—24no—
367C4⋊C4—34no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C22⋊C4—68no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—68no—
379C42—62no—
380C42—62no—
381C42—61no—
382C42—61no—
383C42C2×D1212yes—
384C42D3×D412yes—
385C22×C3C2×#12011yescharacteristic
386C22×C3#53211yescharacteristic
387C22×C3#53511yescharacteristic
388C22×C3C22×SD1611yescharacteristic
389C22×C3C2×#13511yescharacteristic
390C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C22×C3C2×#13711yescharacteristic
392C2×D3—22no—
393C2×D3—22no—
394C2×D3—22no—
395C2×D3—41no—
396C2×D3—41no—
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C3×C4—44no—
407C3×C4—82no—
408C3×C4#44814yes—
409C3×C4#44914yes—
410C3×C4#54612yes—
411C3×C4#55012yes—
412Dic3—22no—
413Dic3—41no—
414Dic3—44no—
415Dic3—41no—
416C23—32no—
417C23—61no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—122no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23C2×D3×D411yescharacteristic
431Q8—28no—
432Q8—44no—
433Q8—128no—
434D4—124no—
435D4—124no—
436D4—248no—
437D4—244no—
438D4—244no—
439C2×C4—24no—
440C2×C4—32no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—68no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—64no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—124no—
460C2×C4—122no—
461C2×C4#88212yes—
462C2×C4#88212yes—
463C2×C4D3×SD1612yes—
464C2×C4#91412yes—
465C2×C4#91812yes—
466C2×C4#91912yes—
467C2×C4C22×D1211yescharacteristic
468C2×C4C2×D3×D411yescharacteristic
469C2×C4#100412yes—
470C2×C4D3×#4112yes—
471C8—42no—
472C8—64no—
473C2×C3#243111yescharacteristic
474C2×C3C2×#44811yescharacteristic
475C2×C3C2×#44911yescharacteristic
476C2×C3#316711yescharacteristic
477C2×C3#318211yescharacteristic
478C2×C3#320811yescharacteristic
479C2×C3#321011yescharacteristic
480D3—42no—
481D3—162no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—242no—
490C22—242no—
491C22—242no—
492C22—242no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22C2×#88211yescharacteristic
497C22#566811yescharacteristic
498C22#571511yescharacteristic
499C22C2×D3×SD1611yescharacteristic
500C22C2×#91411yescharacteristic
501C22C2×#91811yescharacteristic
502C22C2×#91911yescharacteristic
503C4—44no—
504C4—62no—
505C4—82no—
506C4—121no—
507C4—121no—
508C4—124no—
509C4#480614yes—
510C4#481014yes—
511C4#555412yes—
512C4#559912yes—
513C3#2559811yescharacteristic, complemented, Sylow
514C2—122no—
515C2—482no—
516C2#7631311yescharacteristic
517C2C2×#480611yescharacteristic
518C2C2×#481011yescharacteristic
519C2#8159211yescharacteristic
520C2#8159511yescharacteristic
521C2#8176611yescharacteristic
522C2#8180311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radical#74375C2
hypercenter#1956D3
hypocenterC3#25598
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
2. C2×#5085C2semidirect
3. C2×#4694C2semidirect
5. C3×#1956C2semidirect
6. #70272C2semidirect
7. #70240C2semidirect
8. C4×(C2×D3)⋊C8C2semidirect
9. #69582C2semidirect
46. #1956D3semidirect
60. C3×C4⋊Q8D4semidirect
99. #869D4semidirect
171. C2×C4×D3SD16semidirect
220. C4⋊Q8D12semidirect
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×#50852
C2×#46942
#743752
C3×#19562
#702722
#702402
C4×(C2×D3)⋊C82
#695822
#255983