Finite Group Databasebeta

← #10726329

Subgroups — #10726329

Order 768.

1182 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×#4975C211yesmaximal, characteristic
2C2×#4651C212yesmaximal
3C3×#1441C211yesmaximal, characteristic
4#71566C212yesmaximal
5#71565C211yesmaximal, characteristic
6#69411C212yesmaximal
7#69410C212yesmaximal
8#69409C212yesmaximal
9#69409C212yesmaximal
10#14676—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C2×C3×#95C2212yes—
20C2×C3×#95C2211yescharacteristic
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4975C418yescomplemented
33#4651—28no—
34C2×#384—31no—
35C2×#383—32no—
36#1441—32no—
37#1441—32no—
38#1441—32no—
39#1441—32no—
40#1441—31no—
41#1441—32no—
42#1441D311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C2×C3×C4⋊C4—41no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—42no—
55C2×C3×C4⋊C4—41no—
56C2×C3×C4⋊C4C2×C414yes—
57C2×C4⋊Dic3—41no—
58C2×C4⋊Dic3—41no—
59C2×C4⋊Dic3—42no—
60C2×C4⋊Dic3—41no—
61C2×C4⋊Dic3C2×C414yes—
62C2×Dic3⋊C4—42no—
63C2×Dic3⋊C4—42no—
64C2×Dic3⋊C4—42no—
65C2×Dic3⋊C4—42no—
66C2×Dic3⋊C4—42no—
67C2×Dic3⋊C4—42no—
68C2×Dic3⋊C4—42no—
69C2×Dic3⋊C4C2×C418yes—
70C2×Dic3⋊C4C2×C418yes—
71C2×C4×Dic3C2×C414yes—
72C2×C4×Dic3D414yes—
73C2×C4×Dic3Q814yes—
74C3×#95—24no—
75C3×#95—42no—
76C3×#95—41no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—42no—
85C3⋊#95—41no—
86C3⋊#95—42no—
87C22×C4⋊C4—32no—
88C22×C4⋊C4—32no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4—31no—
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C42—31no—
95#384—38no—
96#383—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—31no—
106C2×#95—32no—
107C2×#95C2×D312yes—
108C2×#95C2×D311yescharacteristic
109C2×#95C2×D312yes—
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—21no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
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×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C22×C412yes—
138C22×C3×C4C22×C412yes—
139C22×C3×C4C22×C412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×Q812yes—
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×Dic3C22×C414yes—
167C22×Dic3C22×C414yes—
168C22×Dic3C2×D414yes—
169C22×Dic3C2×Q814yes—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—44no—
172C4⋊Dic3—44no—
173C4⋊Dic3—44no—
174Dic3⋊C4—48no—
175Dic3⋊C4—48no—
176Dic3⋊C4—48no—
177C4×Dic3C4⋊C4116yescomplemented
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4C22×D312yes—
183C23×C4C22×D312yes—
184C23×C4C22×D311yescharacteristic
185C23×C4C22×D311yescharacteristic
186C23×C4C22×D311yescharacteristic
187C2×C4⋊C4—38no—
188C2×C4⋊C4—38no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—122no—
195C2×C4⋊C4—122no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4—122no—
198C2×C4⋊C4—122no—
199C2×C4⋊C4—122no—
200C2×C4⋊C4—122no—
201C2×C4⋊C4—121no—
202C2×C4⋊C4—121no—
203C2×C4⋊C4—121no—
204C2×C4⋊C4—122no—
205C2×C4⋊C4C4×D314yes—
206C2×C42—34no—
207C2×C42—34no—
208C2×C42—34no—
209#95—24no—
210#95—41no—
211#95—42no—
212#95—64no—
213#95—64no—
214#95—64no—
215#95—64no—
216#95—64no—
217#95—121no—
218#95—122no—
219#95—122no—
220#95—122no—
221#95—122no—
222C23×C3C23×C411yescharacteristic
223C23×C3C22×D411yescharacteristic
224C23×C3C22×Q811yescharacteristic
225C23×C3H(5,𝔽2)11yescharacteristic
226C23×C3H(5,𝔽2)12yes—
227C23×C3H(5,𝔽2)11yescharacteristic
228C23×C3H(5,𝔽2)12yes—
229C23×C32−1+412yes—
230C23×C32−1+411yescharacteristic
231C23×C32−1+412yes—
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—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
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—42no—
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—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—
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—21no—
298C22×C4—22no—
299C22×C4—21no—
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—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
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×C4—62no—
346C22×C4—62no—
347C22×C4C2×C4×D312yes—
348C22×C4C2×C4×D312yes—
349C22×C4C2×C4×D312yes—
350C22×C4D3×D412yes—
351C22×C4D3×Q812yes—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—128no—
355C4⋊C4—128no—
356C4⋊C4—124no—
357C4⋊C4—128no—
358C4⋊C4—124no—
359C42—316no—
360C22×C3C22×C4⋊C411yescharacteristic
361C22×C3#51811yescharacteristic
362C22×C3#51812yes—
363C22×C3#51911yescharacteristic
364C22×C3#51912yes—
365C22×C3#53711yescharacteristic
366C22×C3#53711yescharacteristic
367C22×C3#53712yes—
368C22×C3#53712yes—
369C22×C3#54412yes—
370C22×C3#54412yes—
371C22×C3#54411yescharacteristic
372C22×C3#54411yescharacteristic
373C22×C3#56011yescharacteristic
374C22×C3#56011yescharacteristic
375C22×C3#56012yes—
376C22×C3#56312yes—
377C22×C3#56312yes—
378C22×C3#56312yes—
379C22×C3#56311yescharacteristic
380C22×C3#56312yes—
381C22×C3#56312yes—
382C22×C3#56311yescharacteristic
383C3×C4—24no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—82no—
387C3×C4—82no—
388Dic3—28no—
389Dic3—44no—
390Dic3—82no—
391Dic3—82no—
392C23C22×C4×D311yescharacteristic
393C23C2×D3×D411yescharacteristic
394C23#100411yescharacteristic
395C23#100412yes—
396C23#100411yescharacteristic
397C23C2×D3×Q811yescharacteristic
398C23#100712yes—
399C23#100912yes—
400C23#101011yescharacteristic
401C23#101012yes—
402C23#101011yescharacteristic
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—38no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
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—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—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#231611yescharacteristic
463C2×C3#231612yes—
464C2×C3#238011yescharacteristic
465C2×C3#238111yescharacteristic
466C2×C3#238112yes—
467C2×C3#282311yescharacteristic
468C2×C3#282312yes—
469C2×C3#282311yescharacteristic
470C2×C3#284311yescharacteristic
471C2×C3#284312yes—
472C2×C3#284311yescharacteristic
473C22#548611yescharacteristic
474C22C2×D3×C4⋊C411yescharacteristic
475C22#551611yescharacteristic
476C22#554512yes—
477C22#557212yes—
478C22#559511yescharacteristic
479C22#559711yescharacteristic
480C22#559711yescharacteristic
481C22#560812yes—
482C22#562012yes—
483C22#562311yescharacteristic
484C22#562311yescharacteristic
485C22#562912yes—
486C22#564612yes—
487C22#564612yes—
488C22#565112yes—
489C22#566611yescharacteristic
490C22#567211yescharacteristic
491C22#568912yes—
492C22#569912yes—
493C22#571212yes—
494C22#572912yes—
495C22#573511yescharacteristic
496C4—24no—
497C4—42no—
498C4—42no—
499C4—68no—
500C4—82no—
501C4—82no—
502C4—124no—
503C4—242no—
504C4—242no—
505C3#1467611yescharacteristic, complemented, Sylow
506C2#7386911yescharacteristic
507C2#7468511yescharacteristic
508C2#7497712yes—
509C2#7619512yes—
510C2#7649711yescharacteristic
511C2#7708911yescharacteristic
512C2#7724611yescharacteristic
513C2#7745012yes—
514C2#7794312yes—
515C2#7800811yescharacteristic
516C2#7833811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1441C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1441D3
hypocenterC3#14676
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
32. #4975C4semidirect
177. C4×Dic3C4⋊C4semidirect
505. C3#14676semidirect

Maximal subgroups

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

subgroupindex
C2×#49752
C2×#46512
C3×#14412
#715662
#715652
#694112
#694102
#694092
#146763