Finite Group Databasebeta

← #10740466

Subgroups — #10740466

Order 768.

1170 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#384C211yesmaximal, characteristic
2C2×#4981C211yesmaximal, characteristic
3#71567C212yesmaximal
4#71567C212yesmaximal
5#71566C212yesmaximal
6#71565C212yesmaximal
7#71564C212yesmaximal
8#71564C211yesmaximal, characteristic
9#71536C212yesmaximal
10#15790—31nomaximal, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C3×#384—24no—
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33#4981—24no—
34C2×#398—31no—
35C2×#384D311yescharacteristic
36#1442—32no—
37#1442—32no—
38#1441—32no—
39#1441—32no—
40#1436—32no—
41#1436—32no—
42#1436—31no—
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4—41no—
59C2×C3×C4⋊C4—42no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C4⋊C4—41no—
63C2×C3×C4⋊C4D414yes—
64C2×C3×C42—22no—
65C2×C3×C42—24no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—41no—
68C2×C4⋊Dic3—41no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—24no—
72C2×C4×Dic3—41no—
73C2×C4×Dic3—42no—
74C3⋊#95—22no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—42no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—41no—
85C3⋊#95—42no—
86C3⋊#95Q818yes—
87C22×C4⋊C4—32no—
88C22×C4⋊C4—31no—
89C22×C4⋊C4C2×D312yes—
90C22×C4⋊C4C2×D312yes—
91C22×C4⋊C4C2×D311yescharacteristic
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C42—31no—
94C22×C42C2×D311yescharacteristic
95#398—64no—
96#384—24no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—31no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—31no—
108C2×#95—32no—
109C2×#95—32no—
110C24×C3C2411yescharacteristic
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×Q814yes—
137C22×C3×C4#4114yes—
138C22×C3×C4#4114yes—
139C22×C3×C4#4114yes—
140C22×Dic3—22no—
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×Q814yes—
167C22×Dic3C2×Q814yes—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—48no—
170C3×C4⋊C4—48no—
171C3×C4⋊C4—44no—
172C3×C42—28no—
173C4⋊Dic3—44no—
174Dic3⋊C4—48no—
175C4×Dic3—44no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4C22×D312yes—
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D312yes—
184C23×C4C22×D311yescharacteristic
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—121no—
201C2×C4⋊C4—121no—
202C2×C4⋊C4#6714yes—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—121no—
206C2×C42—122no—
207#95—38no—
208#95—64no—
209#95—64no—
210#95—64no—
211#95—64no—
212#95—62no—
213#95—64no—
214#95—122no—
215#95—121no—
216#95—122no—
217#95—122no—
218#95—122no—
219#95—122no—
220C23×C3C22×D411yescharacteristic
221C23×C3C22×Q812yes—
222C23×C3C2×#4112yes—
223C23×C3C2×#4112yes—
224C23×C3C2×#4112yes—
225C23×C3H(5,𝔽2)11yescharacteristic
226C23×C3H(5,𝔽2)11yescharacteristic
227C23×C3H(5,𝔽2)12yes—
228C23×C32−1+412yes—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
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—24no—
242C2×C3×C4—24no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4#12414yes—
257C2×C3×C4#12518yes—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
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—
286C24C23×D311yescharacteristic
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—34no—
310C22×C4—34no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
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#23514yes—
338C22×C4#23714yes—
339C22×C4D3×Q814yes—
340C22×C4#23914yes—
341C22×C4C2×#6712yes—
342C22×C4C2×#6712yes—
343C22×C4C2×#6712yes—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—124no—
349C4⋊C4—128no—
350C42—28no—
351C42—124no—
352C22×C3#52511yescharacteristic
353C22×C3C2×#12411yescharacteristic
354C22×C3C2×#12512yes—
355C22×C3#53312yes—
356C22×C3#53411yescharacteristic
357C22×C3#53912yes—
358C22×C3#54012yes—
359C22×C3#54212yes—
360C22×C3#54412yes—
361C22×C3#54612yes—
362C22×C3D4×Q812yes—
363C22×C3#55212yes—
364C22×C3#55312yes—
365C22×C3#55312yes—
366C22×C3#55412yes—
367C22×C3#55412yes—
368C22×C3#55612yes—
369C22×C3Q8211yescharacteristic
370C22×C3#55911yescharacteristic
371C22×C3#56011yescharacteristic
372C22×C3#56311yescharacteristic
373C3×C4—24no—
374C3×C4—28no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—82no—
378Dic3—44no—
379Dic3—44no—
380Dic3—82no—
381Dic3—82no—
382C23C2×#23512yes—
383C23C2×#23712yes—
384C23#100411yescharacteristic
385C23#100411yescharacteristic
386C23#100412yes—
387C23C2×D3×Q812yes—
388C23C2×#23912yes—
389C23#100712yes—
390C23C22×#6711yescharacteristic
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—64no—
419C2×C4—64no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—64no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
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#88918yes—
447C2×C4#94714yes—
448C2×C3#247312yes—
449C2×C3#249212yes—
450C2×C3#250712yes—
451C2×C3#257012yes—
452C2×C3#259111yescharacteristic
453C2×C3#270911yescharacteristic
454C2×C3#273411yescharacteristic
455C2×C3#279012yes—
456C2×C3#279312yes—
457C22C2×#88912yes—
458C22#551112yes—
459C22#551312yes—
460C22#551412yes—
461C22#551412yes—
462C22#551512yes—
463C22#571311yescharacteristic
464C22#571912yes—
465C22#572011yescharacteristic
466C22#572211yescharacteristic
467C22#572812yes—
468C22#572912yes—
469C22#573012yes—
470C22#573212yes—
471C22#573312yes—
472C22#573511yescharacteristic
473C22#579111yescharacteristic
474C22#580411yescharacteristic
475C22#580612yes—
476C22C2×#94711yescharacteristic
477C22Q8×#6712yes—
478C4—24no—
479C4—28no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—124no—
484C4—124no—
485C4—242no—
486C4—242no—
487C3#1579011yescharacteristic, complemented, Sylow
488C2#7433112yes—
489C2#7434112yes—
490C2#7435112yes—
491C2#7437312yes—
492C2#8217412yes—
493C2#8218211yescharacteristic
494C2#8318311yescharacteristic
495C2#8325011yescharacteristic
496C2#8325812yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#384C2
hypercenterC2×#384D3
hypocenterC3#15790
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
487. C3#15790semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3842
C2×#49812
#715672
#715662
#715652
#715642
#715362
#157903