Finite Group Databasebeta

← #10720414

Subgroups — #10720414

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5229C211yesmaximal, characteristic
2C3×#1446C211yesmaximal, characteristic
3#72134C211yesmaximal, characteristic
4#69411C212yesmaximal
5#69406C212yesmaximal
6#69405C212yesmaximal
7#69403C212yesmaximal
8#69402C212yesmaximal
9Dic3×#95C212yesmaximal
10#14162—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×Dic3⋊C4C2212yes—
14C22×C4×Dic3C2212yes—
15C22×C4×Dic3C2212yes—
16C22×C4×Dic3C2211yescharacteristic
17C2×C3×#95C2212yes—
18C2×C3×#95C2212yes—
19C2×C3×#95C2211yescharacteristic
20C2×C3×#95C2211yescharacteristic
21#5229C418yescomplemented
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×#391—31no—
34#1446—31no—
35#1446D311yescharacteristic
36#1441—32no—
37#1436—32no—
38#1436—32no—
39#1434—32no—
40#1434—32no—
41C4×#95—32no—
42C22×C3×Q8—24no—
43C22×C3×Q8—42no—
44C22×C3×Q8—41no—
45C22×C3×Q8—41no—
46C22×C3×Q8C2×C414yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C2×C4⋊Dic3—41no—
57C2×C4⋊Dic3—41no—
58C2×C4⋊Dic3C2×C414yes—
59C2×Dic3⋊C4—24no—
60C2×Dic3⋊C4—24no—
61C2×Dic3⋊C4—42no—
62C2×C4×Dic3—24no—
63C2×C4×Dic3—24no—
64C2×C4×Dic3—24no—
65C2×C4×Dic3—24no—
66C2×C4×Dic3—42no—
67C2×C4×Dic3—42no—
68C2×C4×Dic3—42no—
69C2×C4×Dic3C2×C414yes—
70C3×#95—24no—
71C3×#95—41no—
72C3×#95—42no—
73C3×#95—41no—
74C3⋊#95—24no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—41no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95—41no—
82C3⋊#95C2×C418yes—
83C3⋊#95C2×C418yes—
84C3⋊#95Q818yescomplemented
85C23×Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87C22×C4⋊C4—32no—
88C22×C42—31no—
89C22×C42—32no—
90C22×C42—32no—
91#391—38no—
92C2×#95—32no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—31no—
101C2×#95—32no—
102C2×#95—31no—
103C2×#95C2×D312yes—
104C2×#95C2×D311yescharacteristic
105C2×#95C2×D312yes—
106C2×#95C2×D311yescharacteristic
107C24×C3C2411yescharacteristic
108C2×C3×Q8—28no—
109C2×C3×Q8—44no—
110C2×C3×Q8—44no—
111C2×C3×Q8—48no—
112C2×C3×Q8—48no—
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—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—21no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C22×C412yes—
138C22×C3×C4C22×C412yes—
139C22×C3×C4C2×Q814yes—
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×Q814yes—
167C22×Dic3C2×Q814yes—
168C22×Dic3#4114yes—
169C22×Dic3#4114yes—
170C4⋊Dic3—44no—
171Dic3⋊C4—48no—
172C4×Dic3—44no—
173C4×Dic3—48no—
174C4×Dic3—48no—
175C22×Q8—24no—
176C22×Q8—42no—
177C22×Q8—41no—
178C22×Q8—41no—
179C22×Q8C4×D314yes—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4—32no—
183C23×C4—32no—
184C23×C4C22×D311yescharacteristic
185C23×C4C22×D311yescharacteristic
186C23×C4C22×D312yes—
187C23×C4C22×D312yes—
188C23×C4C22×D311yescharacteristic
189C2×C4⋊C4—34no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—64no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4—121no—
194C2×C4⋊C4—121no—
195C2×C42—34no—
196C2×C42—64no—
197C2×C42—64no—
198C2×C42—64no—
199C2×C42—64no—
200C2×C42—122no—
201C2×C42—122no—
202C2×C42—122no—
203#95—24no—
204#95—38no—
205#95—38no—
206#95—38no—
207#95—41no—
208#95—42no—
209#95—41no—
210#95—64no—
211#95—64no—
212#95—64no—
213#95—64no—
214#95—122no—
215#95—121no—
216#95—122no—
217#95—121no—
218C23×C3C23×C411yescharacteristic
219C23×C3C22×Q812yes—
220C23×C3C2×#4112yes—
221C23×C3C2×#4112yes—
222C23×C3C2×#4112yes—
223C23×C3C2×#4112yes—
224C23×C3H(5,𝔽2)12yes—
225C23×C3H(5,𝔽2)11yescharacteristic
226C23×C3H(5,𝔽2)11yescharacteristic
227C3×Q8—416no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
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#11714yes—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—24no—
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—24no—
270C2×Dic3—24no—
271C2×Dic3—24no—
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×Dic3C4×Q818yes—
285C24C23×D311yescharacteristic
286C2×Q8—28no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—48no—
290C2×Q8—48no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—34no—
315C22×C4—34no—
316C22×C4—34no—
317C22×C4—34no—
318C22×C4—34no—
319C22×C4—34no—
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×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4C2×C4×D312yes—
343C22×C4C2×C4×D312yes—
344C22×C4C2×C4×D312yes—
345C22×C4#23714yes—
346C22×C4D3×Q814yes—
347C22×C4#23914yes—
348C4⋊C4—124no—
349C4⋊C4—128no—
350C42—124no—
351C42—128no—
352C42—128no—
353C22×C3C2×#11711yescharacteristic
354C22×C3C2×C4×Q812yes—
355C22×C3C4×#4112yes—
356C22×C3#51811yescharacteristic
357C22×C3#51811yescharacteristic
358C22×C3#52912yes—
359C22×C3#52912yes—
360C22×C3#53312yes—
361C22×C3#53312yes—
362C22×C3#53812yes—
363C22×C3#53812yes—
364C22×C3#55112yes—
365C22×C3#55112yes—
366C22×C3#55112yes—
367C22×C3#55412yes—
368C22×C3#55412yes—
369C22×C3#55412yes—
370C22×C3#55511yescharacteristic
371C22×C3#55511yescharacteristic
372C22×C3#55511yescharacteristic
373C22×C3Q8211yescharacteristic
374C3×C4—24no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—44no—
378C3×C4—44no—
379Dic3—28no—
380Dic3—44no—
381Dic3—44no—
382Dic3—44no—
383C23C22×C4×D311yescharacteristic
384C23C2×#23712yes—
385C23#100411yescharacteristic
386C23#100412yes—
387C23#100411yescharacteristic
388C23C2×D3×Q812yes—
389C23C2×#23912yes—
390C23D3×#4112yes—
391C23D3×#4112yes—
392Q8—416no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—38no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—64no—
423C2×C4—64no—
424C2×C4—64no—
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—64no—
435C2×C4—64no—
436C2×C4—64no—
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#87714yes—
450C2×C3#229512yes—
451C2×C3#231211yescharacteristic
452C2×C3#236812yes—
453C2×C3#236911yescharacteristic
454C2×C3#237211yescharacteristic
455C2×C3#257212yes—
456C2×C3#257412yes—
457C2×C3#257412yes—
458C2×C3#258612yes—
459C22C2×#87711yescharacteristic
460C22#548611yescharacteristic
461C22#548611yescharacteristic
462C22C4×D3×Q812yes—
463C22C4×#23912yes—
464C22#559312yes—
465C22#559312yes—
466C22#562212yes—
467C22#562412yes—
468C22#563612yes—
469C22#563612yes—
470C22#563712yes—
471C22#563712yes—
472C22#567011yescharacteristic
473C22#567011yescharacteristic
474C22#567912yes—
475C22#568012yes—
476C22#572211yescharacteristic
477C22#572812yes—
478C22#573012yes—
479C22#573411yescharacteristic
480C4—24no—
481C4—44no—
482C4—44no—
483C4—42no—
484C4—42no—
485C4—68no—
486C4—124no—
487C4—124no—
488C4—124no—
489C3#1416211yescharacteristic, complemented, Sylow
490C2#7382011yescharacteristic
491C2#7647712yes—
492C2#7649411yescharacteristic
493C2#7649511yescharacteristic
494C2#7661612yes—
495C2#7680012yes—
496C2#7680112yes—
497C2#7719012yes—
498C2#7719312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1446C2
socle, prime radical, cosocleC24×C3C24
hypercenter#1446D3
hypocenterC3#14162
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
21. #5229C4semidirect
84. C3⋊#95Q8semidirect
489. C3#14162semidirect

Maximal subgroups

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

subgroupindex
C2×#52292
C3×#14462
#721342
#694112
#694062
#694052
#694032
#694022
Dic3×#952
#141623