Finite Group Databasebeta

← #10725753

Subgroups — #10725753

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4986C211yesmaximal, characteristic
2C3×#1445C211yesmaximal, characteristic
3#71564C211yesmaximal, characteristic
4#69418C212yesmaximal
5#69414C212yesmaximal
6#69407C212yesmaximal
7#69406C212yesmaximal
8#69405C212yesmaximal
9#69401C212yesmaximal
10#14614—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×Dic3⋊C4C2212yes—
14C22×C4×Dic3C2211yescharacteristic
15C22×C4×Dic3C2212yes—
16C2×C3×#95C2212yes—
17C2×C3×#95C2212yes—
18C2×C3×#95C2211yescharacteristic
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95C2212yes—
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⋊#95C2211yescharacteristic
32#4986C418yescomplemented
33C2×#400—31no—
34#1445D311yescharacteristic
35#1443—32no—
36#1442—32no—
37#1436—32no—
38#1436—32no—
39#1436—32no—
40#1436—31no—
41#1434—32no—
42C23×C3×C4C2312yes—
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C4⋊C4—41no—
52C2×C3×C4⋊C4—41no—
53C2×C3×C4⋊C4C2×C414yes—
54C2×C4⋊Dic3—41no—
55C2×C4⋊Dic3—41no—
56C2×C4⋊Dic3C2×C414yes—
57C2×Dic3⋊C4—24no—
58C2×Dic3⋊C4—24no—
59C2×Dic3⋊C4—42no—
60C2×C4×Dic3—24no—
61C2×C4×Dic3—41no—
62C2×C4×Dic3—42no—
63C2×C4×Dic3—42no—
64C2×C4×Dic3—42no—
65C3×#95—24no—
66C3×#95—24no—
67C3×#95—41no—
68C3×#95—41no—
69C3⋊#95—24no—
70C3⋊#95—24no—
71C3⋊#95—24no—
72C3⋊#95—24no—
73C3⋊#95—42no—
74C3⋊#95—42no—
75C3⋊#95—41no—
76C3⋊#95—42no—
77C3⋊#95—42no—
78C3⋊#95C2×C414yes—
79C3⋊#95C2×C418yes—
80C3⋊#95C2×C418yes—
81C22×C4⋊C4—31no—
82C22×C4⋊C4—32no—
83C22×C4⋊C4C2×D311yescharacteristic
84C22×C42—31no—
85C22×C42—32no—
86#400—38no—
87C2×#95—32no—
88C2×#95—32no—
89C2×#95—31no—
90C2×#95—32no—
91C2×#95—32no—
92C2×#95—32no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95—31no—
98C2×#95—32no—
99C2×#95C2×D312yes—
100C2×#95C2×D312yes—
101C2×#95C2×D311yescharacteristic
102C2×#95C2×D311yescharacteristic
103C24×C3C2411yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
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—22no—
128C22×C3×C4—22no—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C22×C412yes—
131C22×C3×C4C22×C412yes—
132C22×C3×C4#4114yes—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
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×Dic3C22×C414yes—
158C22×Dic3C22×C414yes—
159C22×Dic3#4114yes—
160C22×Dic3#4114yes—
161C3×C4⋊C4—44no—
162C4⋊Dic3—44no—
163Dic3⋊C4—48no—
164C4×Dic3—44no—
165C4×Dic3—48no—
166C23×C4—32no—
167C23×C4—32no—
168C23×C4—32no—
169C23×C4—32no—
170C23×C4C22×D312yes—
171C23×C4C22×D311yescharacteristic
172C23×C4C22×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C23×C4C22×D312yes—
175C2×C4⋊C4—34no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—64no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—122no—
181C2×C4⋊C4—121no—
182C2×C4⋊C4—121no—
183C2×C4⋊C4C4×D314yes—
184C2×C42—64no—
185C2×C42—121no—
186C2×C42—122no—
187C2×C42—122no—
188C2×C42—122no—
189#95—24no—
190#95—24no—
191#95—34no—
192#95—38no—
193#95—38no—
194#95—41no—
195#95—41no—
196#95—64no—
197#95—64no—
198#95—64no—
199#95—64no—
200#95—121no—
201#95—122no—
202#95—122no—
203#95—122no—
204#95—122no—
205C23×C3C23×C411yescharacteristic
206C23×C3C2×#4112yes—
207C23×C3C2×#4112yes—
208C23×C3C2×#4112yes—
209C23×C3H(5,𝔽2)11yescharacteristic
210C23×C3H(5,𝔽2)11yescharacteristic
211C23×C3H(5,𝔽2)12yes—
212C23×C32−1+412yes—
213C23×C32−1+412yes—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—24no—
254C2×Dic3—24no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C24C23×D311yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
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—34no—
298C22×C4—34no—
299C22×C4—34no—
300C22×C4—34no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
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×C4C2×C4×D312yes—
326C22×C4C2×C4×D312yes—
327C22×C4C2×C4×D312yes—
328C22×C4#23714yes—
329C4⋊C4—44no—
330C4⋊C4—124no—
331C4⋊C4—128no—
332C42—124no—
333C42—128no—
334C22×C3C4×#4111yescharacteristic
335C22×C3C4×#4112yes—
336C22×C3#51811yescharacteristic
337C22×C3#51911yescharacteristic
338C22×C3#52012yes—
339C22×C3#53912yes—
340C22×C3#53912yes—
341C22×C3#53912yes—
342C22×C3#54112yes—
343C22×C3#54112yes—
344C22×C3#54212yes—
345C22×C3#55112yes—
346C22×C3#55111yescharacteristic
347C22×C3#55212yes—
348C22×C3#55312yes—
349C22×C3#55312yes—
350C22×C3#55612yes—
351C22×C3#55911yescharacteristic
352C22×C3H(3,𝔽4)11yescharacteristic
353C22×C3#56212yes—
354C22×C3#56311yescharacteristic
355C3×C4—42no—
356C3×C4—42no—
357C3×C4—42no—
358C3×C4—44no—
359C3×C4—82no—
360Dic3—44no—
361Dic3—44no—
362Dic3—44no—
363Dic3—44no—
364C23C22×C4×D311yescharacteristic
365C23C2×#23712yes—
366C23#100411yescharacteristic
367C23#100411yescharacteristic
368C23#100712yes—
369C23D3×#4112yes—
370C23D3×#4112yes—
371C23#100912yes—
372C23#101012yes—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—64no—
413C2×C4—64no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C3#233211yescharacteristic
431C2×C3#233612yes—
432C2×C3#237312yes—
433C2×C3#237411yescharacteristic
434C2×C3#237911yescharacteristic
435C2×C3#273912yes—
436C2×C3#277612yes—
437C2×C3#278212yes—
438C2×C3#279812yes—
439C22#548611yescharacteristic
440C22#551611yescharacteristic
441C22#552211yescharacteristic
442C22C4×#23712yes—
443C22#557412yes—
444C22#560112yes—
445C22#562512yes—
446C22#564112yes—
447C22#564712yes—
448C22#565812yes—
449C22#566012yes—
450C22#566212yes—
451C22#566512yes—
452C22#567911yescharacteristic
453C22#568311yescharacteristic
454C22#569312yes—
455C22#569612yes—
456C22#570712yes—
457C22#570912yes—
458C22#571311yescharacteristic
459C22#573511yescharacteristic
460C4—42no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—124no—
466C4—124no—
467C4—124no—
468C4—124no—
469C3#1461411yescharacteristic, complemented, Sylow
470C2#7405711yescharacteristic
471C2#7495912yes—
472C2#7497312yes—
473C2#7630811yescharacteristic
474C2#7649611yescharacteristic
475C2#7762912yes—
476C2#7789912yes—
477C2#7808612yes—
478C2#7837712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1445C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1445D3
hypocenterC3#14614
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
32. #4986C4semidirect
469. C3#14614semidirect

Maximal subgroups

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

subgroupindex
C2×#49862
C3×#14452
#715642
#694182
#694142
#694072
#694062
#694052
#694012
#146143