Finite Group Databasebeta

← #10740326

Subgroups — #10740326

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4984C212yesmaximal
2C2×#4661C212yesmaximal
3C2×#4653C212yesmaximal
4C3×#1444C211yesmaximal, characteristic
5#71566C212yesmaximal
6#71565C212yesmaximal
7#69415C211yesmaximal, characteristic
8#69415C211yesmaximal, characteristic
9#69402C212yesmaximal
10#15778—31nomaximal, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4⋊Dic3C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32#4984—28no—
33#4661—28no—
34#4653—28no—
35C2×#398—32no—
36C2×#389—32no—
37C2×#387—32no—
38#1444D311yescharacteristic
39#1442—31no—
40#1442—31no—
41#1441—32no—
42#1441—32no—
43#1434—32no—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—41no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—41no—
65C2×C4⋊Dic3—41no—
66C2×C4⋊Dic3—42no—
67C2×C4⋊Dic3D418yes—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—24no—
72C2×C4×Dic3—24no—
73C2×C4×Dic3—24no—
74C3×#95—24no—
75C3×#95—41no—
76C3×#95D414yes—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—22no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—24no—
86C3⋊#95Q814yes—
87C22×C4⋊C4—32no—
88C22×C4⋊C4—32no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4—31no—
92C22×C4⋊C4C2×D312yes—
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C42—32no—
95#398—68no—
96#389—68no—
97#387—68no—
98C2×#95—31no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—31no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95C2×D312yes—
109C2×#95C2×D311yescharacteristic
110C2×#95C2×D311yescharacteristic
111C24×C3C2411yescharacteristic
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D414yes—
136C22×C3×C4C2×D414yes—
137C22×C3×C4C2×Q812yes—
138C22×C3×C4#4112yes—
139C22×C3×C4#4114yes—
140C22×C3×C4#4114yes—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3C2×D414yes—
167C22×Dic3C2×D414yes—
168C22×Dic3C2×Q814yes—
169C3×C4⋊C4—216no—
170C3×C4⋊C4—44no—
171C4⋊Dic3—216no—
172C4⋊Dic3—44no—
173C4⋊Dic3—48no—
174C4⋊Dic3—44no—
175Dic3⋊C4—48no—
176C4×Dic3—48no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D312yes—
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D311yescharacteristic
185C23×C4C22×D312yes—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—38no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—121no—
202C2×C4⋊C4—121no—
203C2×C4⋊C4—122no—
204C2×C42—64no—
205C2×C42—64no—
206C2×C42—64no—
207#95—24no—
208#95—34no—
209#95—41no—
210#95—64no—
211#95—64no—
212#95—64no—
213#95—64no—
214#95—64no—
215#95—64no—
216#95—64no—
217#95—64no—
218#95—62no—
219#95D1214yes—
220C23×C3C22×D411yescharacteristic
221C23×C3C22×D412yes—
222C23×C3C22×Q811yescharacteristic
223C23×C3C2×#4111yescharacteristic
224C23×C3C2×#4112yes—
225C23×C3C2×#4112yes—
226C23×C3H(5,𝔽2)11yescharacteristic
227C23×C32−1+411yescharacteristic
228C23×C32−1+411yescharacteristic
229C23×C32−1+412yes—
230C23×C32−1+411yescharacteristic
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4#12118yes—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—24no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C24C23×D311yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—34no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4C2×D1212yes—
340C22×C4C2×D1214yes—
341C22×C4D3×D414yes—
342C22×C4#23712yes—
343C22×C4#23714yes—
344C22×C4D3×Q812yes—
345C22×C4#23914yes—
346C4⋊C4—216no—
347C4⋊C4—44no—
348C4⋊C4—616no—
349C4⋊C4—124no—
350C4⋊C4—128no—
351C4⋊C4—124no—
352C4⋊C4—128no—
353C42—128no—
354C22×C3C2×#12112yes—
355C22×C3#52512yes—
356C22×C3#53312yes—
357C22×C3#53612yes—
358C22×C3#53711yescharacteristic
359C22×C3#53911yescharacteristic
360C22×C3#54112yes—
361C22×C3#54112yes—
362C22×C3#54411yescharacteristic
363C22×C3D4211yescharacteristic
364C22×C3#54712yes—
365C22×C3#54711yescharacteristic
366C22×C3#54712yes—
367C22×C3#54812yes—
368C22×C3D4×Q812yes—
369C22×C3D4×Q811yescharacteristic
370C22×C3#55111yescharacteristic
371C22×C3#55212yes—
372C22×C3#55212yes—
373C22×C3#55511yescharacteristic
374C22×C3#55711yescharacteristic
375C22×C3#56311yescharacteristic
376C22×C3#56411yescharacteristic
377C3×C4—28no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—44no—
381C3×C4—42no—
382Dic3—44no—
383Dic3—44no—
384Dic3—44no—
385Dic3—82no—
386C23C22×D1211yescharacteristic
387C23C2×D3×D412yes—
388C23C2×#23711yescharacteristic
389C23C2×#23712yes—
390C23#100411yescharacteristic
391C23C2×D3×Q811yescharacteristic
392C23C2×#23912yes—
393C23#100711yescharacteristic
394C23#101012yes—
395C23#101011yescharacteristic
396C23#101011yescharacteristic
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4#89518yes—
454C2×C3#241312yes—
455C2×C3#242912yes—
456C2×C3#246212yes—
457C2×C3#246311yescharacteristic
458C2×C3#253711yescharacteristic
459C2×C3#261211yescharacteristic
460C2×C3#267711yescharacteristic
461C2×C3#271912yes—
462C2×C3#272211yescharacteristic
463C2×C3#272611yescharacteristic
464C2×C3#276911yescharacteristic
465C22C2×#89512yes—
466C22#551011yescharacteristic
467C22#555512yes—
468C22#555511yescharacteristic
469C22Q8×D1211yescharacteristic
470C22#558612yes—
471C22#558711yescharacteristic
472C22#559812yes—
473C22#560211yescharacteristic
474C22#562311yescharacteristic
475C22#562512yes—
476C22#563212yes—
477C22#564212yes—
478C22#564212yes—
479C22#564812yes—
480C22#565111yescharacteristic
481C22C3⋊#56411yescharacteristic
482C22#569812yes—
483C22#571711yescharacteristic
484C22#572111yescharacteristic
485C22#572612yes—
486C22#572812yes—
487C22#573411yescharacteristic
488C4—28no—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—42no—
493C4—124no—
494C4—124no—
495C4—124no—
496C4—242no—
497C3#1577811yescharacteristic, complemented, Sylow
498C2#7436811yescharacteristic
499C2#7440712yes—
500C2#7453611yescharacteristic
501C2#7612711yescharacteristic
502C2#7622112yes—
503C2#7668111yescharacteristic
504C2#7717812yes—
505C2#7720011yescharacteristic
506C2#7786012yes—
507C2#7789211yescharacteristic
508C2#7806111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1444C2
hypercenter#1444D3
hypocenterC3#15778
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
497. C3#15778semidirect

Maximal subgroups

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

subgroupindex
C2×#49842
C2×#46612
C2×#46532
C3×#14442
#715662
#715652
#694152
#694022
#157783