Finite Group Databasebeta

← #10720459

Subgroups — #10720459

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4977C211yesmaximal, characteristic
2C2×#4648C212yesmaximal
3C3×#1441C211yesmaximal, characteristic
4#71565C211yesmaximal, characteristic
5#71564C212yesmaximal
6#69411C212yesmaximal
7#69407C212yesmaximal
8#69406C212yesmaximal
9#69405C212yesmaximal
10#14168—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×C4×Dic3C2212yes—
17C22×C4×Dic3C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
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⋊#95C2212yes—
32#4977C418yescomplemented
33#4648—28no—
34C2×#382—31no—
35C2×#376—32no—
36#1441—32no—
37#1441—31no—
38#1441D311yescharacteristic
39#1436—32no—
40#1436—32no—
41#1436—32no—
42#1436—32no—
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—41no—
55C2×C3×C4⋊C4—41no—
56C2×C3×C4⋊C4C2×C414yes—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—41no—
59C2×C4⋊Dic3—41no—
60C2×C4⋊Dic3—41no—
61C2×C4⋊Dic3C2×C414yes—
62C2×Dic3⋊C4—24no—
63C2×Dic3⋊C4—24no—
64C2×Dic3⋊C4—42no—
65C2×C4×Dic3—24no—
66C2×C4×Dic3—24no—
67C2×C4×Dic3—24no—
68C2×C4×Dic3—24no—
69C2×C4×Dic3—24no—
70C2×C4×Dic3—24no—
71C2×C4×Dic3C2×C414yes—
72C2×C4×Dic3D414yes—
73C2×C4×Dic3Q814yes—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—41no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—24no—
83C3⋊#95—41no—
84C3⋊#95—42no—
85C3⋊#95C2×C418yes—
86C3⋊#95C2×C418yes—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—32no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4C2×D311yescharacteristic
91C22×C4⋊C4C2×D311yescharacteristic
92C22×C42—31no—
93C22×C42—32no—
94C22×C42—32no—
95#382—38no—
96#376—68no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—31no—
107C2×#95C2×D312yes—
108C2×#95C2×D312yes—
109C2×#95C2×D311yescharacteristic
110C24×C3C2411yescharacteristic
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
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—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4C22×C412yes—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C22×C412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×Q812yes—
140C22×C3×C4#4114yes—
141C22×C3×C4#4114yes—
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×Dic3C22×C414yes—
165C22×Dic3C22×C414yes—
166C22×Dic3C2×D414yes—
167C22×Dic3C2×Q814yes—
168C22×Dic3#4114yes—
169C22×Dic3#4114yes—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—44no—
172C4⋊Dic3—44no—
173C4⋊Dic3—44no—
174Dic3⋊C4—48no—
175C4×Dic3—216no—
176C4×Dic3—216no—
177C4×Dic3C4⋊C4116yescomplemented
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D312yes—
185C23×C4C22×D312yes—
186C23×C4C22×D311yescharacteristic
187C2×C4⋊C4—24no—
188C2×C4⋊C4—34no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—122no—
196C2×C4⋊C4—121no—
197C2×C4⋊C4—121no—
198C2×C4⋊C4—121no—
199C2×C4⋊C4C4×D314yes—
200C2×C42—34no—
201C2×C42—34no—
202C2×C42—34no—
203C2×C42—64no—
204C2×C42—64no—
205C2×C42—64no—
206C2×C42—64no—
207C2×C42—64no—
208C2×C42—64no—
209#95—24no—
210#95—24no—
211#95—38no—
212#95—38no—
213#95—41no—
214#95—64no—
215#95—64no—
216#95—64no—
217#95—64no—
218#95—64no—
219#95—64no—
220#95—121no—
221#95—122no—
222C23×C3C23×C411yescharacteristic
223C23×C3C22×D411yescharacteristic
224C23×C3C22×Q811yescharacteristic
225C23×C3C2×#4112yes—
226C23×C3C2×#4112yes—
227C23×C3C2×#4112yes—
228C23×C3C2×#4112yes—
229C23×C3H(5,𝔽2)11yescharacteristic
230C23×C3H(5,𝔽2)11yescharacteristic
231C23×C32−1+411yescharacteristic
232C23×C32−1+411yescharacteristic
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4C2×C4⋊C414yes—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—24no—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—24no—
278C2×Dic3—24no—
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—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3C2×C4⋊C418yes—
292C24C23×D311yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—34no—
318C22×C4—34no—
319C22×C4—34no—
320C22×C4—34no—
321C22×C4—34no—
322C22×C4—34no—
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×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4C2×C4×D312yes—
346C22×C4C2×C4×D312yes—
347C22×C4C2×C4×D312yes—
348C22×C4D3×D412yes—
349C22×C4#23714yes—
350C22×C4D3×Q812yes—
351C22×C4#23914yes—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—124no—
355C4⋊C4—128no—
356C4⋊C4—124no—
357C42—316no—
358C42—616no—
359C42—616no—
360C22×C3C22×C4⋊C411yescharacteristic
361C22×C3C4×#4112yes—
362C22×C3C4×#4112yes—
363C22×C3#51811yescharacteristic
364C22×C3#51911yescharacteristic
365C22×C3#52512yes—
366C22×C3#52512yes—
367C22×C3#53312yes—
368C22×C3#53312yes—
369C22×C3#53711yescharacteristic
370C22×C3#53711yescharacteristic
371C22×C3#54411yescharacteristic
372C22×C3#54411yescharacteristic
373C22×C3#55112yes—
374C22×C3#55111yescharacteristic
375C22×C3#55111yescharacteristic
376C22×C3#55212yes—
377C22×C3#55212yes—
378C22×C3#55212yes—
379C22×C3#55212yes—
380C22×C3#55511yescharacteristic
381C22×C3#55511yescharacteristic
382C22×C3#55512yes—
383C3×C4—24no—
384C3×C4—44no—
385C3×C4—44no—
386C3×C4—42no—
387C3×C4—42no—
388Dic3—28no—
389Dic3—44no—
390Dic3—44no—
391Dic3—44no—
392C23C22×C4×D311yescharacteristic
393C23C2×D3×D411yescharacteristic
394C23C2×#23712yes—
395C23#100411yescharacteristic
396C23#100411yescharacteristic
397C23C2×D3×Q811yescharacteristic
398C23C2×#23912yes—
399C23D3×#4112yes—
400C23D3×#4112yes—
401C23#101011yescharacteristic
402C23#101011yescharacteristic
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—38no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—64no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4D3×C4⋊C414yes—
462C2×C3#229612yes—
463C2×C3#231611yescharacteristic
464C2×C3#237112yes—
465C2×C3#237211yescharacteristic
466C2×C3#237211yescharacteristic
467C2×C3#260912yes—
468C2×C3#260911yescharacteristic
469C2×C3#260911yescharacteristic
470C2×C3#261211yescharacteristic
471C2×C3#261211yescharacteristic
472C2×C3#261212yes—
473C22#548611yescharacteristic
474C22C2×D3×C4⋊C411yescharacteristic
475C22#551611yescharacteristic
476C22C4×#23712yes—
477C22C4×#23912yes—
478C22#558711yescharacteristic
479C22#559711yescharacteristic
480C22#559711yescharacteristic
481C22#560612yes—
482C22#561612yes—
483C22#562311yescharacteristic
484C22#562311yescharacteristic
485C22#562712yes—
486C22#563712yes—
487C22#564212yes—
488C22#565012yes—
489C22#565911yescharacteristic
490C22#567011yescharacteristic
491C22#568812yes—
492C22#569412yes—
493C22#570812yes—
494C22#572812yes—
495C22#573411yescharacteristic
496C4—24no—
497C4—44no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—68no—
502C4—124no—
503C4—124no—
504C4—124no—
505C3#1416811yescharacteristic, complemented, Sylow
506C2#7386911yescharacteristic
507C2#7468011yescharacteristic
508C2#7497112yes—
509C2#7596312yes—
510C2#7649511yescharacteristic
511C2#7696811yescharacteristic
512C2#7720011yescharacteristic
513C2#7733012yes—
514C2#7781511yescharacteristic
515C2#7781712yes—
516C2#7820711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1441C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1441D3
hypocenterC3#14168
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
32. #4977C4semidirect
177. C4×Dic3C4⋊C4semidirect
505. C3#14168semidirect

Maximal subgroups

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

subgroupindex
C2×#49772
C2×#46482
C3×#14412
#715652
#715642
#694112
#694072
#694062
#694052
#141683