Finite Group Databasebeta

← #10877220

Subgroups — #10877220

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic
2D3×#384C211yesmaximal, characteristic
3C3×#1803C211yesmaximal, characteristic, complemented
4#71626C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic
6#70454C211yesmaximal, characteristic, complemented
7#70268C212yesmaximal
8#70250C212yesmaximal, complemented
9#69568C211yesmaximal, characteristic, complemented
10#68875C212yesmaximal
11#68810C212yesmaximal, complemented
12#27680—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2211yescharacteristic
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4—21no—
17C2×D3×C4⋊C4C2212yes—
18C2×C42×D3C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#103C2212yes—
21C3×#384C2211yescharacteristic
22#5229C2211yescharacteristic
23C2×#835C2212yes—
24C2×(C2×D3)⋊C8C2212yes—
25C2×#817C2212yes—
26#4989—21no—
27#4987C2211yescharacteristic
28#4978—21no—
29#4975—21no—
30#4975—21no—
31#4975C2212yes—
32#4972C2211yescharacteristic
33C2×#808C2212yes—
34C4×(C2×D3)⋊C4—21no—
35#4935C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C4×Dic3⋊C4—21no—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×C22⋊Q8—31no—
45C2×#384—31no—
46#2099—32no—
47#2096—32no—
48#1948—31no—
49#1948—31no—
50#1872—32no—
51#1853—32no—
52#1803—31no—
53#1803D311yescharacteristic, complemented
54#1797—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C4×D3—22no—
57C22×C4×D3D411yescharacteristic
58C22×C4×D3D411yescharacteristic
59C22×C4×D3C2311yescharacteristic
60C22×C3×C8C2312yes—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70#946—22no—
71#946Q814yes—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89D3×C4⋊C4—42no—
90D3×C4⋊C4—42no—
91D3×C4⋊C4—44no—
92D3×C4⋊C4—44no—
93C42×D3—22no—
94C42×D3—22no—
95C3×C4⋊C8—22no—
96C3×#103—42no—
97#835—42no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103(C2×D3)⋊C8—24no—
104#817—42no—
105#808—42no—
106C4⋊(C3⋊C8)—22no—
107C2×C22⋊Q8—31no—
108C2×C4×Q8C2×D311yescharacteristic
109C22×C4⋊C4—32no—
110C22×C4⋊C4—61no—
111C22×C4⋊C4—61no—
112C22×C42—31no—
113C2×#106—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×C4⋊C8C2×D311yescharacteristic
117C2×#103—32no—
118C2×#103C2×D312yes—
119C2×C22⋊C8—32no—
120#391—31no—
121#389—31no—
122#387—61no—
123#385—61no—
124#384—32no—
125#384—31no—
126#384—61no—
127#384—61no—
128#384C2×D311yescharacteristic
129#380—31no—
130C4×C4⋊C4—31no—
131C4×C4⋊C4—61no—
132C4×C22⋊C4—61no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340C2×D312yes—
137#336—32no—
138C23×D3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8C2×Q812yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3C2×D411yescharacteristic
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×C4×D3—22no—
166C2×C4×D3—44no—
167C2×C4×D3—44no—
168C2×C4×D3—44no—
169C2×C4×D3SD1614yes—
170C2×C4×D3C2×Q812yes—
171C2×C4×D3#4112yes—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C42—42no—
183C3×C42#4112yes—
184C3×C42#4112yes—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3C2×Q812yes—
195Dic3⋊C4—24no—
196Dic3⋊C4—44no—
197Dic3⋊C4—44no—
198Dic3⋊C4—44no—
199Dic3⋊C4—42no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—21no—
203C4×Dic3—21no—
204C4×Dic3—42no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8C22×D311yescharacteristic
209C23×C4—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—62no—
213C22×C8—32no—
214C22×C8C22×D312yes—
215C22⋊Q8—34no—
216C22⋊Q8—62no—
217C4×Q8—22no—
218C4×Q8—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—122no—
233C2×C4⋊C4—122no—
234C2×C4⋊C4—124no—
235C2×C4⋊C4—124no—
236C2×C4⋊C4C22×D312yes—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42—21no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—61no—
247C2×C42C22×D311yescharacteristic
248#106—122no—
249#106—122no—
250C4⋊C8—22no—
251C4⋊C8—62no—
252#103—42no—
253#103—122no—
254C22⋊C8—64no—
255#95—61no—
256#95—61no—
257#95—61no—
258#95—61no—
259#95—61no—
260C23×C3C22×D411yescharacteristic
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3C22⋊Q812yes—
266C22×D3C2×SD1612yes—
267C22×D3#13712yes—
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—41no—
271C3×Q8—41no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4#12312yes—
284C2×C3×C4#12312yes—
285C2×C3×C4C2×SD1612yes—
286C2×C3×C4#13712yes—
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C2×C3×C4H(5,𝔽2)11yescharacteristic
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—22no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3C22⋊Q812yes—
309C2×Dic3C2×SD1612yes—
310C2×Dic3AGL(1,ℤ/8)12yes—
311C4×D3—28no—
312C4×D3—22no—
313C4×D3—22no—
314C4×D3—22no—
315C4×D3—22no—
316C4×D3—28no—
317C4×D3—88no—
318C3×C8—42no—
319C3⋊C8—42no—
320C24—31no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8D3×Q812yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—34no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—124no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4C23×D311yescharacteristic
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—62no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—32no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—64no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—61no—
374C4⋊C4—122no—
375C4⋊C4—124no—
376C4⋊C4—124no—
377C4⋊C4—122no—
378C4⋊C4—124no—
379C4⋊C4—122no—
380C22⋊C4—64no—
381C22⋊C4—122no—
382C42—42no—
383C42—62no—
384C42—62no—
385C42—61no—
386C42—61no—
387C42—122no—
388C42#23512yes—
389C42#23712yes—
390C22×C3C2×C22⋊Q811yescharacteristic
391C22×C3C2×#12311yescharacteristic
392C22×C3#53511yescharacteristic
393C22×C3C22×SD1611yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×#13711yescharacteristic
396C22×C3C2×#13711yescharacteristic
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—22no—
403C2×D3#47514yes—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—81no—
407C3×C4—82no—
408C3×C4#48314yes—
409C3×C4#55311yescharacteristic
410C3×C4#55311yescharacteristic
411C3×C4#55411yescharacteristic
412C3×C4#55411yescharacteristic
413Dic3—22no—
414Dic3—22no—
415Dic3—22no—
416Dic3—42no—
417Dic3—82no—
418Dic3—81no—
419C23—32no—
420C23—32no—
421C23—32no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23C2×D3×D411yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—68no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—68no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—248no—
467C2×C4#88312yes—
468C2×C4#88312yes—
469C2×C4D3×SD1612yes—
470C2×C4#91812yes—
471C2×C4C2×#23511yescharacteristic
472C2×C4C2×#23711yescharacteristic
473C2×C4#100411yescharacteristic
474C2×C4C2×D3×Q811yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C8—42no—
478C8—122no—
479C2×C3#248711yescharacteristic
480C2×C3C2×#47511yescharacteristic
481C2×C3#307911yescharacteristic
482C2×C3C2×#48311yescharacteristic
483C2×C3#309211yescharacteristic
484C2×C3#320811yescharacteristic
485C2×C3#321811yescharacteristic
486D3—24no—
487C22—34no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22C2×#88311yescharacteristic
495C22#560711yescharacteristic
496C22D3×C22⋊Q811yescharacteristic
497C22C2×D3×SD1611yescharacteristic
498C22C2×#91811yescharacteristic
499C22D3×AGL(1,ℤ/8)11yescharacteristic
500C22D3×#13711yescharacteristic
501C4—22no—
502C4—42no—
503C4—62no—
504C4—62no—
505C4—62no—
506C4—82no—
507C4—81no—
508C4—122no—
509C4—242no—
510C4—241no—
511C4#480514yes—
512C4#551111yescharacteristic
513C4#553011yescharacteristic
514C4#561411yescharacteristic
515C4#563511yescharacteristic
516C3#2768011yescharacteristic, complemented, Sylow
517C2—64no—
518C2#7391211yescharacteristic
519C2C2×#480511yescharacteristic
520C2#7889211yescharacteristic
521C2#8038711yescharacteristic
522C2#8048311yescharacteristic
523C2D3×#47511yescharacteristic
524C2#8110911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1803D3
hypocenterC3#27680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. C3×#1803C2semidirect
4. #71626C2semidirect
6. #70454C2semidirect
8. #70250C2semidirect
9. #69568C2semidirect
11. #68810C2semidirect
53. #1803D3semidirect
516. C3#27680semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
D3×#3842
C3×#18032
#716262
#704752
#704542
#702682
#702502
#695682
#688752
#688102
#276803