Finite Group Databasebeta

← #10795128

Subgroups — #10795128

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic
2#73820C212yesmaximal, complemented
3C3×#1781C211yesmaximal, characteristic
4#72130C211yesmaximal, characteristic
5#70028C214yesmaximal, complemented
6Dic3×#99C214yesmaximal, complemented
7#69495C212yesmaximal, complemented
8#20724—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#217C412yescomplemented
11C22×#217C2211yescharacteristic
12C22×#217C2211yescharacteristic
13#5798—24no—
14#5798—24no—
15C2×D4×Dic3C412yescomplemented
16C2×D4×Dic3C412yescomplemented
17C2×D4×Dic3C2212yes—
18C2×D4×Dic3C2212yes—
19C2×#926C412yescomplemented
20C2×C3×#99C2214yes—
21C3×#379C2212yes—
22#5247C2212yes—
23C2×#834C2214yes—
24#4945—24no—
25#4944—24no—
26#4942C2214yescomplemented
27Dic3×C22⋊C4C2214yescomplemented
28#4527C2218yescomplemented
29C2×#518—31no—
30#2312—32no—
31#1819—32no—
32#1799—34no—
33#1781—31no—
34#1781D311yescharacteristic
35C4×#99—34no—
36C25×C3C2×C411yescharacteristic
37C25×C3C2311yescharacteristic
38C22×C3×D4—22no—
39C22×C3×D4—22no—
40C22×C3×D4—28no—
41C22×C3×D4C2×C412yes—
42C22×C3×D4C2312yes—
43C23×C3×C4C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C23×Dic3C2×C412yes—
47C23×Dic3C2×C412yes—
48C23×Dic3C2311yescharacteristic
49C23×Dic3C2311yescharacteristic
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4C2314yes—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—24no—
56C2×#217—24no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217C2×C412yes—
60C2×#217C2×C412yes—
61C2×#217C2×C412yes—
62C2×#217C2×C412yes—
63C2×#217C2×C412yes—
64C2×#217D414yes—
65C2×#217D414yes—
66C2×#217C2314yes—
67C2×#217C2312yes—
68D4×Dic3—28no—
69D4×Dic3—28no—
70D4×Dic3—28no—
71D4×Dic3—28no—
72#926—24no—
73#926—24no—
74C2×C4⋊Dic3C2×C412yes—
75C2×C4⋊Dic3C2×C412yes—
76C2×Dic3⋊C4—24no—
77C2×C4×Dic3—24no—
78C2×C4×Dic3C2×C412yes—
79C2×C4×Dic3C2×C412yes—
80C3×#99—28no—
81#834—28no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C23×D4C2×D311yescharacteristic
85#518—64no—
86#518—64no—
87C2×C4×D4—32no—
88C2×C4×D4—32no—
89C2×C4×D4—32no—
90C2×C4×D4—32no—
91C2×#117—32no—
92C22×C22⋊C4—31no—
93C22×C22⋊C4—32no—
94C22×C22⋊C4—31no—
95C2×#99—34no—
96C2×#99C2×D314yes—
97#388—64no—
98#388—64no—
99#380—34no—
100#379—32no—
101#379C2×D312yes—
102C4×C22⋊C4—34no—
103#342—38no—
104C24×C3—21no—
105C24×C3—21no—
106C24×C3—24no—
107C24×C3—24no—
108C24×C3—81no—
109C24×C3—81no—
110C24×C3—81no—
111C24×C3—81no—
112C24×C3C22×C411yescharacteristic
113C24×C3C22×C411yescharacteristic
114C24×C3C22×C412yes—
115C24×C3C2×D412yes—
116C24×C3C2×D412yes—
117C24×C3C2411yescharacteristic
118C2×C3×D4—216no—
119C2×C3×D4—216no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4#4118yes—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4C22×C412yes—
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—24no—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—81no—
137C22×Dic3—81no—
138C22×Dic3—82no—
139C22×Dic3—82no—
140C22×Dic3—81no—
141C22×Dic3—81no—
142C22×Dic3C22×C412yes—
143C22×Dic3C22×C412yes—
144C22×Dic3C22×C412yes—
145C22×Dic3C22×C412yes—
146C22×Dic3C2×D414yes—
147C22×Dic3C2×D414yes—
148C3×C22⋊C4—28no—
149C3×C22⋊C4—82no—
150C3×C22⋊C4—82no—
151#217—22no—
152#217—22no—
153#217—24no—
154#217—24no—
155#217—24no—
156#217—28no—
157#217—82no—
158#217—82no—
159#217C22⋊C414yes—
160#217C22⋊C414yes—
161#217C22⋊C418yes—
162C4⋊Dic3—24no—
163C4⋊Dic3—24no—
164Dic3⋊C4—84no—
165C4×Dic3—24no—
166C4×Dic3—24no—
167C4×Dic3—84no—
168C25C4×D311yescharacteristic
169C25C22×D311yescharacteristic
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—28no—
173C22×D4C4×D312yes—
174C22×D4C22×D312yes—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—31no—
180C23×C4—31no—
181C23×C4C4×D311yescharacteristic
182C4×D4—68no—
183C4×D4—68no—
184C4×D4—68no—
185C4×D4—68no—
186#117—64no—
187#117—64no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—64no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—34no—
196C2×C22⋊C4—34no—
197C2×C22⋊C4—32no—
198C2×C22⋊C4—32no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—32no—
201C2×C22⋊C4—34no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4C22×D314yes—
209C2×C42—32no—
210C2×C42—32no—
211C2×C42—64no—
212#99—28no—
213#99—68no—
214#95—64no—
215#95—64no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—81no—
237C23×C3—81no—
238C23×C3—81no—
239C23×C3—81no—
240C23×C3—88no—
241C23×C3—88no—
242C23×C3C2×C22⋊C411yescharacteristic
243C23×C3C2×C22⋊C411yescharacteristic
244C23×C3C2×C22⋊C412yes—
245C23×C3#11714yes—
246C23×C3C23×C411yescharacteristic
247C23×C3C22×D412yes—
248C23×C3C2×#4114yes—
249C3×D4—216no—
250C3×D4—216no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4—44no—
258C2×C3×C4#11714yes—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—44no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—84no—
271C2×Dic3—84no—
272C2×Dic3—82no—
273C2×Dic3—82no—
274C2×Dic3C2×C22⋊C414yes—
275C2×Dic3C2×C22⋊C414yes—
276C24—24no—
277C24—21no—
278C24—21no—
279C24—24no—
280C24—81no—
281C24—81no—
282C24—81no—
283C24—81no—
284C24C2×C4×D312yes—
285C24C2×C4×D311yescharacteristic
286C24C2×C4×D311yescharacteristic
287C24D3×D412yes—
288C24D3×D412yes—
289C24C23×D311yescharacteristic
290C2×D4—216no—
291C2×D4—216no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4#23718yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—34no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—64no—
309C22×C4—64no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—64no—
313C22×C4—241no—
314C22×C4—241no—
315C22×C4—242no—
316C22×C4—242no—
317C22×C4—241no—
318C22×C4—241no—
319C22×C4C2×C4×D312yes—
320C4⋊C4—64no—
321C4⋊C4—64no—
322C4⋊C4—244no—
323C22⋊C4—28no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—38no—
327C22⋊C4—64no—
328C22⋊C4—64no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—64no—
332C22⋊C4—68no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—242no—
336C22⋊C4—242no—
337C42—64no—
338C42—64no—
339C42—244no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—24no—
354C22×C3—44no—
355C22×C3—44no—
356C22×C3—44no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—84no—
362C22×C3—84no—
363C22×C3—84no—
364C22×C3—84no—
365C22×C3C22×C22⋊C411yescharacteristic
366C22×C3C2×#11712yes—
367C22×C3#52514yes—
368C3×C4—24no—
369C3×C4—84no—
370Dic3—24no—
371Dic3—24no—
372Dic3—42no—
373Dic3—42no—
374Dic3—84no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—88no—
400C23—88no—
401C23#87714yes—
402C23D3×C22⋊C411yescharacteristic
403C23D3×C22⋊C411yescharacteristic
404C23D3×C22⋊C412yes—
405C23C22×C4×D311yescharacteristic
406C23C2×D3×D412yes—
407C23C2×#23714yes—
408D4—216no—
409D4—216no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—34no—
415C2×C4—34no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4—244no—
431C2×C4—244no—
432C2×C4—242no—
433C2×C4—242no—
434C2×C4#87714yes—
435C2×C3—24no—
436C2×C3—22no—
437C2×C3—42no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—44no—
441C2×C3#228011yescharacteristic
442C2×C3#288311yescharacteristic
443C2×C3#288311yescharacteristic
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22C2×#87712yes—
470C22C2×D3×C22⋊C411yescharacteristic
471C22#558614yes—
472C4—24no—
473C4—64no—
474C4—64no—
475C4—84no—
476C4—122no—
477C4—122no—
478C4—244no—
479C3#2072411yescharacteristic, complemented, Sylow
480C2—24no—
481C2—22no—
482C2—41no—
483C2—41no—
484C2—42no—
485C2—44no—
486C2#7381711yescharacteristic
487C2#7868811yescharacteristic
488C2#7868811yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1781C2
socleC22×C3C22×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenter#1781D3
hypocenterC3#20724
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #73820C2semidirect
5. #70028C2semidirect
6. Dic3×#99C2semidirect
7. #69495C2semidirect
10. C22×#217C4semidirect
15. C2×D4×Dic3C4semidirect
16. C2×D4×Dic3C4semidirect
19. C2×#926C4semidirect
26. #4942C22semidirect
27. Dic3×C22⋊C4C22semidirect
28. #4527C22semidirect
479. C3#20724semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
#738202
C3×#17812
#721302
#700282
Dic3×#992
#694952
#207243