Finite Group Databasebeta

← #10848690

Subgroups — #10848690

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×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic, complemented
3C2×#5088C211yesmaximal, characteristic, complemented
4#82081C211yesmaximal, characteristic
5#71787C211yesmaximal, characteristic, complemented
6#71785C211yesmaximal, characteristic, complemented
7#71699C214yesmaximal, complemented
8#71633C214yesmaximal, complemented
9C4×#830C211yesmaximal, characteristic, complemented
10#25598—31nomaximal, Sylow
11C3×#533—24no—
12C3×#533—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×#117—21no—
21C22×C3×C42C2211yescharacteristic
22C22×#215C2214yes—
23C2×#930—21no—
24C2×#930—21no—
25C2×#930C2212yes—
26C2×#874C2211yescharacteristic
27C2×#835C2214yes—
28C2×#830C2212yes—
29#5088—24no—
30#5085—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×#533D311yescharacteristic, complemented
41C2×#492—31no—
42C2×#486—31no—
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×Q8C2312yes—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×D12—21no—
56C22×D12—21no—
57C22×D12C2312yes—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8D414yescomplemented
61C3×C4⋊Q8D414yescomplemented
62C3×#125—28no—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—24no—
66C3×C4×Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×#117—28no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#215—28no—
85C2×#215—28no—
86C2×#215—44no—
87#930—44no—
88#930—44no—
89#930—44no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2312yes—
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2312yes—
98#874—22no—
99#874—22no—
100#835—44no—
101#830—24no—
102#809—28no—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—22no—
105C22×SD16—34no—
106#533—24no—
107#533—24no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×#127—31no—
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2×D312yes—
114C2×#121—32no—
115C2×#121—61no—
116C2×#121—61no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2×D312yes—
119C2×#117—21no—
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—32no—
129#390—61no—
130#390—61no—
131#384—31no—
132#340—34no—
133#336—32no—
134C24×C3C2×D411yescharacteristic
135C23×D3—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×Q8C2×D414yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4SD1614yescomplemented
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C2×#67—44no—
162C2×#67—44no—
163C2×#67—44no—
164C22×Dic3—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—24no—
168C2×D12—24no—
169C2×D12—42no—
170C2×D12—42no—
171C2×D12—42no—
172C2×D12—42no—
173C2×D12—42no—
174C2×D12—42no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C42—28no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#215—416no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194C4⋊Dic3—24no—
195C4⋊Dic3—42no—
196C4⋊Dic3—42no—
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—21no—
204C22×Q8C22×D312yes—
205C22×D4—32no—
206C22×D4—62no—
207C22×D4—61no—
208C22×D4—61no—
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—22no—
218C4⋊Q8—22no—
219C4⋊Q8#6714yescomplemented
220C4⋊Q8#6714yescomplemented
221#127—62no—
222#127—62no—
223#125—28no—
224C22⋊Q8—28no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227#121—124no—
228#121—124no—
229#121—124no—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233#117—28no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
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×C3C2×SD1612yes—
266C23×C3C22×D411yescharacteristic
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C3×Q8—28no—
275C3×Q8—44no—
276C3×Q8—48no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—28no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
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—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—44no—
295C2×C3×C4#12012yes—
296C2×C3×C4#12012yes—
297C2×C3×C4C2×SD1612yes—
298C2×C3×C4C2×SD1612yes—
299C2×C3×C4#13512yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4#13712yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4112yes—
305C2×C3×C4H(5,𝔽2)12yes—
306#67—88no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310D12—44no—
311D12—44no—
312D12—84no—
313D12—84no—
314C3⋊C8—24no—
315C3⋊C8—42no—
316C24—62no—
317C24C2×#6711yescharacteristic
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8C2×#6714yes—
325C2×D4—64no—
326C2×D4—64no—
327C2×D4—62no—
328C2×D4—62no—
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—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—62no—
350C22×C4#21514yescomplemented
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—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—64no—
371C4⋊C4—122no—
372C4⋊C4—122no—
373C4⋊C4C2×#6714yes—
374C22⋊C4—28no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—28no—
379C42—22no—
380C42—22no—
381C42—21no—
382C42—21no—
383C42C2×#6712yes—
384C42C2×#6712yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
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—82no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C3×C4—22no—
407C3×C4—41no—
408C3×C4—44no—
409C3×C4—41no—
410C3×C4—44no—
411C3×C4#44814yes—
412C3×C4#44914yes—
413C3×C4#54612yes—
414C3×C4#55012yes—
415Dic3—82no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—122no—
423C23—122no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23C2×#21512yes—
430C23C22×#6711yescharacteristic
431Q8—28no—
432Q8—48no—
433Q8—44no—
434D4—124no—
435D4—124no—
436D4—248no—
437D4—244no—
438D4—244no—
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—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4C2×#21512yes—
461C2×C4C2×#21512yes—
462C2×C4#94212yes—
463C2×C4#94912yes—
464C2×C4#95012yes—
465C2×C4#95312yes—
466C2×C4#95312yes—
467C2×C4D3×#4112yes—
468C2×C4#100912yes—
469C2×C4C22×#6711yescharacteristic
470C2×C4C22×#6711yescharacteristic
471C8—64no—
472C8—122no—
473C2×C3—42no—
474C2×C3#243111yescharacteristic
475C2×C3C2×#44811yescharacteristic
476C2×C3C2×#44911yescharacteristic
477C2×C3#316711yescharacteristic
478C2×C3#318211yescharacteristic
479C2×C3#320811yescharacteristic
480C2×C3#321011yescharacteristic
481D3—162no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—242no—
490C22—242no—
491C22—242no—
492C22—242no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22C22×#21511yescharacteristic
497C22C2×#94211yescharacteristic
498C22C2×#94911yescharacteristic
499C22C2×#95011yescharacteristic
500C22#583011yescharacteristic
501C22#583211yescharacteristic
502C22C2×#95311yescharacteristic
503C4—22no—
504C4—41no—
505C4—44no—
506C4—41no—
507C4—44no—
508C4—242no—
509C4#522714yes—
510C4#523814yes—
511C4#582912yes—
512C4#583412yes—
513C3#2559811yescharacteristic, complemented, Sylow
514C2—42no—
515C2—482no—
516C2#8334311yescharacteristic
517C2#8334711yescharacteristic
518C2#8335511yescharacteristic
519C2#8335611yescharacteristic
520C2C2×#522711yescharacteristic
521C2C2×#523811yescharacteristic
522C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radical#82081C2
hypercenterC2×#533D3
hypocenterC3#25598
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×#5085C2semidirect
3. C2×#5088C2semidirect
5. #71787C2semidirect
6. #71785C2semidirect
7. #71699C2semidirect
8. #71633C2semidirect
9. C4×#830C2semidirect
40. C2×#533D3semidirect
60. C3×C4⋊Q8D4semidirect
61. C3×C4⋊Q8D4semidirect
154. C22×C3×C4SD16semidirect
219. C4⋊Q8#67semidirect
220. C4⋊Q8#67semidirect
350. C22×C4#215semidirect
513. C3#25598semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50852
C2×#50882
#820812
#717872
#717852
#716992
#716332
C4×#8302
#255983