Finite Group Databasebeta

← #10877221

Subgroups — #10877221

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C22⋊Q8C211yesmaximal, characteristic
2C2×#4975C211yesmaximal, characteristic
3C3×#1872C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71556C211yesmaximal, characteristic
6#70250C212yesmaximal, complemented
7#70221C212yesmaximal, complemented
8#69928C211yesmaximal, characteristic
9#69908C211yesmaximal, characteristic
10#68842C212yesmaximal, complemented
11#68796C212yesmaximal, complemented
12#27680—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4—21no—
18C22×Dic3⋊C4—21no—
19C22×C4×Dic3C2211yescharacteristic
20C2×C3×#103C2212yes—
21C2×C3×C22⋊C8C2211yescharacteristic
22#5229C2211yescharacteristic
23C2×#835C2212yes—
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#817C2212yes—
27C2×Dic3⋊C8C2212yes—
28#4984C2211yescharacteristic
29#4980—21no—
30#4975—21no—
31#4975—21no—
32#4975C2212yes—
33#4975C2212yes—
34Dic3×C4⋊C4—21no—
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#808C2212yes—
37#4950—21no—
38Dic3×C22⋊C4—21no—
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—32no—
49#1872—31no—
50#1872D311yescharacteristic
51#1853—31no—
52#1853—31no—
53#1803—32no—
54#1797—31no—
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C23×Dic3D411yescharacteristic
61C22×C3×C8C2312yes—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8Q814yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71Q8×Dic3—24no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—42no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—44no—
84C2×Dic3⋊C4—44no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—21no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C3×#103—42no—
94C3×C22⋊C8—22no—
95#835—42no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101#830—22no—
102#817—42no—
103Dic3⋊C8—24no—
104#808—42no—
105C2×C22⋊Q8C2×D311yescharacteristic
106C2×C4×Q8—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—61no—
109C22×C4⋊C4—61no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42—31no—
112C2×#106—32no—
113C2×#106—32no—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103C2×D312yes—
117C2×C22⋊C8—31no—
118C2×C22⋊C8C2×D311yescharacteristic
119#391—31no—
120#389—31no—
121#387—61no—
122#385—61no—
123#384—32no—
124#384—32no—
125#384—61no—
126#384—61no—
127#380—31no—
128C4×C4⋊C4—31no—
129C4×C4⋊C4—61no—
130C4×C22⋊C4—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D312yes—
135#336—32no—
136C24×C3C2×D411yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×Q812yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×Q812yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3SD1614yes—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×Q812yes—
178C3×C22⋊C4—24no—
179#217—42no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—42no—
185C4⋊Dic3—42no—
186C4⋊Dic3—42no—
187Dic3⋊C4—44no—
188Dic3⋊C4—44no—
189Dic3⋊C4—44no—
190Dic3⋊C4—44no—
191C4×Dic3—22no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195C4×Dic3—42no—
196C4×Dic3#4114yes—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C22×Q8C22×D311yescharacteristic
201C23×C4—21no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4—61no—
205C23×C4C22×D311yescharacteristic
206C22×C8—32no—
207C22×C8C22×D312yes—
208C22⋊Q8—22no—
209C22⋊Q8Dic614yes—
210C4×Q8—64no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—122no—
225C2×C4⋊C4—124no—
226C2×C4⋊C4—124no—
227C2×C4⋊C4C22×D312yes—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—61no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—64no—
236C2×C42—61no—
237C2×C42—61no—
238#106—122no—
239#106—122no—
240C4⋊C8—64no—
241#103—42no—
242#103—122no—
243C22⋊C8—22no—
244C22⋊C8—62no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—61no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C22⋊Q812yes—
255C23×C3C2×SD1612yes—
256C23×C3#13712yes—
257C23×C3C22×D411yescharacteristic
258C3×Q8—24no—
259C3×Q8—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—84no—
274C2×C3×C4C22⋊Q812yes—
275C2×C3×C4C22×Q811yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4112yes—
278C2×C3×C4H(5,𝔽2)12yes—
279C2×Dic3—28no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—28no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—84no—
297C2×Dic3#12312yes—
298C2×Dic3#12312yes—
299C2×Dic3C2×SD1612yes—
300C2×Dic3C2×SD1612yes—
301C2×Dic3AGL(1,ℤ/8)12yes—
302C2×Dic3#13712yes—
303C3×C8—42no—
304C3⋊C8—42no—
305C24D3×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8C2×Dic612yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—31no—
316C22×C4—34no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—62no—
331C22×C4—122no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4C2×Dic612yes—
335C22×C4D3×D411yescharacteristic
336C22×C4#23712yes—
337C22×C4C23×D311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8—62no—
344C4⋊C4—24no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—62no—
353C4⋊C4—122no—
354C4⋊C4—124no—
355C4⋊C4—124no—
356C4⋊C4—124no—
357C4⋊C4—122no—
358C4⋊C4—122no—
359C4⋊C4—124no—
360C4⋊C4C2×Dic612yes—
361C22⋊C4—24no—
362C22⋊C4—122no—
363C42—34no—
364C42—62no—
365C42—62no—
366C42—62no—
367C42—122no—
368C42—122no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3#47514yes—
376C22×C3C2×C22⋊Q811yescharacteristic
377C22×C3C2×#12311yescharacteristic
378C22×C3#53511yescharacteristic
379C22×C3C22×SD1611yescharacteristic
380C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C3×C4—22no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—82no—
387C3×C4#55312yes—
388C3×C4#55412yes—
389Dic3—22no—
390Dic3—22no—
391Dic3—22no—
392Dic3—81no—
393Dic3—82no—
394Dic3—81no—
395Dic3#48314yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23#87812yes—
401C23D3×SD1612yes—
402C23#91812yes—
403C23C2×D3×D411yescharacteristic
404Q8—24no—
405Q8—42no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—68no—
426C2×C4—68no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—84no—
434C2×C4—121no—
435C2×C4—122no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—244no—
444C2×C4#87812yes—
445C2×C4C22×Dic611yescharacteristic
446C2×C4C2×#23711yescharacteristic
447C2×C4#100412yes—
448C2×C4D3×#4112yes—
449C8—42no—
450C8—122no—
451C2×C3—24no—
452C2×C3#248711yescharacteristic
453C2×C3C2×#47511yescharacteristic
454C2×C3#307911yescharacteristic
455C2×C3C2×#48311yescharacteristic
456C2×C3#309211yescharacteristic
457C2×C3#320811yescharacteristic
458C2×C3#321811yescharacteristic
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22#479114yes—
466C22C2×#87811yescharacteristic
467C22#559211yescharacteristic
468C22D3×#12311yescharacteristic
469C22C2×D3×SD1611yescharacteristic
470C22C2×#91811yescharacteristic
471C22D3×AGL(1,ℤ/8)11yescharacteristic
472C22D3×#13711yescharacteristic
473C4—22no—
474C4—34no—
475C4—42no—
476C4—42no—
477C4—62no—
478C4—62no—
479C4—62no—
480C4—82no—
481C4—242no—
482C4—241no—
483C4—241no—
484C4#553912yes—
485C4#560112yes—
486C3#2768011yescharacteristic, complemented, Sylow
487C2—24no—
488C2#7473811yescharacteristic
489C2C2×#479111yescharacteristic
490C2#7887011yescharacteristic
491C2#8013811yescharacteristic
492C2#8017911yescharacteristic
493C2D3×#48311yescharacteristic
494C2#8146811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1872D3
hypocenterC3#27680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
6. #70250C2semidirect
7. #70221C2semidirect
10. #68842C2semidirect
11. #68796C2semidirect
486. C3#27680semidirect

Maximal subgroups

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

subgroupindex
Dic3×C22⋊Q82
C2×#49752
C3×#18722
#716922
#715562
#702502
#702212
#699282
#699082
#688422
#687962
#276803