Finite Group Databasebeta

← #10795129

Subgroups — #10795129

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5486C211yesmaximal, characteristic
2C3×#2312C211yesmaximal, characteristic, complemented
3#82093C211yesmaximal, characteristic, complemented
4#69530C214yesmaximal, complemented
5#69498C212yesmaximal
6#69495C212yesmaximal, complemented
7C4×#806C214yesmaximal, complemented
8#20724—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217C2211yescharacteristic
12C2×C4×#67C2212yes—
13C2×C4×#67C2212yes—
14#5486—24no—
15#5486—24no—
16C2×#67⋊C4C412yescomplemented
17C2×#67⋊C4C412yescomplemented
18C2×D3×C22⋊C4C412yescomplemented
19C2×#877C412yescomplemented
20C3×#388—24no—
21C3×#380C2212yescomplemented
22C3×#379C2212yes—
23C3×C4×C22⋊C4C2212yescomplemented
24#5247C2212yes—
25#5212C2212yescomplemented
26C4×#217C2212yescomplemented
27C2×#806C2218yes—
28#4951—24no—
29#4527C2218yescomplemented
30C2×#518—31no—
31#2312—31no—
32#2312D311yescharacteristic, complemented
33#1819—32no—
34#1799—34no—
35#1781—32no—
36C4×#99—34no—
37C25×C3C2311yescharacteristic
38C24×D3C2×C411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—22no—
42C22×#67—22no—
43C22×#67—28no—
44C22×#67C2×C412yes—
45C22×#67C2312yes—
46C23×Dic3C2×C411yescharacteristic
47C23×Dic3C2×C411yescharacteristic
48C23×Dic3C2×C411yescharacteristic
49C22×C4×D3C2×C412yes—
50C22×C4×D3C2×C412yes—
51C2×C3×C4⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4C2×C412yes—
55C2×C3×C22⋊C4C2×C412yes—
56C2×C3×C22⋊C4D414yes—
57C2×C3×C22⋊C4D414yes—
58C2×C3×C22⋊C4C2314yes—
59C2×C3×C42—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217C2×C412yes—
65C2×#217C2314yes—
66C2×#217C2312yes—
67C4×#67—24no—
68C4×#67—24no—
69C4×#67—24no—
70C4×#67—24no—
71C2×(C2×D3)⋊C4C2×C412yes—
72C2×(C2×D3)⋊C4C2×C412yes—
73C2×C4⋊Dic3—22no—
74C2×Dic3⋊C4C2×C412yes—
75C2×Dic3⋊C4C2×C412yes—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2×C412yes—
78C2×C4×Dic3C2×C412yes—
79#67⋊C4—28no—
80#67⋊C4—28no—
81D3×C22⋊C4—28no—
82#877—28no—
83C3×#95—22no—
84C3×#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#806—216no—
88C23×D4—31no—
89#518—64no—
90#518—64no—
91C2×C4×D4—32no—
92C2×C4×D4—32no—
93C2×C4×D4—32no—
94C2×C4×D4—32no—
95C2×#117—32no—
96C22×C22⋊C4—31no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4C2×D311yescharacteristic
99C2×#99—38no—
100#388—24no—
101#388—64no—
102#380—32no—
103#380C2×D312yescomplemented
104#379—32no—
105#379C2×D312yes—
106C4×C22⋊C4—32no—
107C4×C22⋊C4C2×D312yescomplemented
108#342—38no—
109C24×C3—24no—
110C24×C3—81no—
111C24×C3—81no—
112C24×C3C22×C411yescharacteristic
113C24×C3C22×C411yescharacteristic
114C24×C3C2×D412yes—
115C24×C3C2×D412yes—
116C24×C3C2411yescharacteristic
117C23×D3—22no—
118C23×D3—24no—
119C23×D3—81no—
120C23×D3—81no—
121C23×D3C22×C412yes—
122C22×C3×C4—24no—
123C22×C3×C4—81no—
124C22×C3×C4—81no—
125C22×C3×C4C22×C412yes—
126C22×C3×C4C22×C412yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C2×#67—216no—
132C2×#67—216no—
133C2×#67—24no—
134C2×#67—22no—
135C2×#67—22no—
136C2×#67—24no—
137C2×#67—24no—
138C2×#67—24no—
139C2×#67#4118yes—
140C22×Dic3—22no—
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—24no—
145C22×Dic3—81no—
146C22×Dic3—81no—
147C22×Dic3C22×C412yes—
148C22×Dic3C22×C412yes—
149C22×Dic3C22×C412yes—
150C2×C4×D3—24no—
151C2×C4×D3—24no—
152C2×C4×D3—82no—
153C2×C4×D3—82no—
154C3×C4⋊C4—82no—
155C3×C22⋊C4—28no—
156C3×C22⋊C4—82no—
157C3×C22⋊C4—82no—
158C3×C22⋊C4C22⋊C418yes—
159C3×C22⋊C4C22⋊C418yes—
160C3×C42—82no—
161#217—24no—
162#217—22no—
163#217—22no—
164#217—28no—
165#217—82no—
166#217—82no—
167(C2×D3)⋊C4—24no—
168(C2×D3)⋊C4—24no—
169C4⋊Dic3—82no—
170Dic3⋊C4—24no—
171Dic3⋊C4—24no—
172C4×Dic3—24no—
173C4×Dic3—24no—
174C4×Dic3—82no—
175C25—31no—
176C25C22×D311yescharacteristic
177C22×D4—32no—
178C22×D4—32no—
179C22×D4—68no—
180C22×D4—62no—
181C22×D4—62no—
182C23×C4—31no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—32no—
186C23×C4—32no—
187C23×C4C22×D311yescharacteristic
188C23×C4C22×D311yescharacteristic
189C4×D4—64no—
190C4×D4—64no—
191C4×D4—64no—
192C4×D4—64no—
193C4×D4—68no—
194C4×D4—68no—
195#117—68no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—62no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—32no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—32no—
205C2×C22⋊C4—32no—
206C2×C22⋊C4—34no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—68no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4C4×D312yes—
213C2×C22⋊C4C4×D312yes—
214C2×C22⋊C4D1214yes—
215C2×C22⋊C4#6714yes—
216C2×C22⋊C4C22×D314yes—
217C2×C42—22no—
218C2×C42—32no—
219C2×C42—32no—
220C2×C42—62no—
221#99—616no—
222#95—22no—
223#95—22no—
224#95—62no—
225#95—62no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—24no—
229C23×C3—24no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—88no—
241C23×C3C2×C22⋊C412yes—
242C23×C3C2×C22⋊C412yes—
243C23×C3C23×C411yescharacteristic
244C23×C3C22×D411yescharacteristic
245C23×C3C22×D411yescharacteristic
246C23×C3C2×#4114yes—
247C22×D3—22no—
248C22×D3—22no—
249C22×D3—28no—
250C22×D3—42no—
251C22×D3—42no—
252C22×D3—82no—
253C22×D3—82no—
254C22×D3—88no—
255C22×D3#11714yes—
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—44no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4C2×C22⋊C414yes—
266C2×C3×C4C2×C22⋊C414yes—
267#67—216no—
268#67—28no—
269#67—28no—
270C2×Dic3—28no—
271C2×Dic3—22no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—44no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—82no—
281C2×Dic3—82no—
282C2×Dic3#11714yes—
283C4×D3—84no—
284C4×D3—84no—
285C24—24no—
286C24—32no—
287C24—64no—
288C24—62no—
289C24—81no—
290C24—81no—
291C24—241no—
292C24—241no—
293C24C2×C4×D311yescharacteristic
294C24C2×C4×D311yescharacteristic
295C24C2×D1212yes—
296C24C2×#6712yes—
297C24C23×D311yescharacteristic
298C2×D4—38no—
299C2×D4—64no—
300C2×D4—64no—
301C2×D4—616no—
302C2×D4—616no—
303C2×D4—64no—
304C2×D4—62no—
305C2×D4—64no—
306C2×D4—62no—
307C22×C4—24no—
308C22×C4—32no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—64no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—64no—
316C22×C4—64no—
317C22×C4—64no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—242no—
321C22×C4—242no—
322C22×C4—241no—
323C22×C4—241no—
324C22×C4C2×C4×D312yes—
325C22×C4C2×C4×D312yes—
326C22×C4C2×D1212yes—
327C22×C4C2×D1212yes—
328C22×C4C2×#6712yes—
329C22×C4C2×#6712yes—
330C4⋊C4—64no—
331C4⋊C4—64no—
332C4⋊C4—82no—
333C4⋊C4—242no—
334C22⋊C4—28no—
335C22⋊C4—64no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—64no—
339C22⋊C4—64no—
340C22⋊C4—68no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—242no—
344C22⋊C4—242no—
345C22⋊C4(C2×D3)⋊C418yes—
346C22⋊C4(C2×D3)⋊C418yes—
347C42—64no—
348C42—64no—
349C42—82no—
350C42—242no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—84no—
367C22×C3—84no—
368C22×C3C22×C22⋊C411yescharacteristic
369C22×C3C2×#11712yes—
370C22×C3#52512yes—
371C22×C3#52512yes—
372C2×D3—24no—
373C2×D3—44no—
374C2×D3—42no—
375C2×D3—42no—
376C2×D3—44no—
377C2×D3—88no—
378C3×C4—24no—
379C3×C4—24no—
380C3×C4—84no—
381Dic3—24no—
382Dic3—42no—
383Dic3—42no—
384Dic3—84no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—34no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—68no—
401C23—62no—
402C23—62no—
403C23—88no—
404C23—122no—
405C23—122no—
406C23—242no—
407C23—242no—
408C23—248no—
409C23C2×(C2×D3)⋊C412yes—
410C23C2×(C2×D3)⋊C412yes—
411C23C22×C4×D311yescharacteristic
412C23C22×D1211yescharacteristic
413C23D3×#4114yes—
414C23C22×#6711yescharacteristic
415D4—616no—
416D4—68no—
417D4—68no—
418C2×C4—34no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—68no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—244no—
439C2×C4—244no—
440C2×C4—242no—
441C2×C4—242no—
442C2×C4C2×(C2×D3)⋊C414yes—
443C2×C4C2×(C2×D3)⋊C414yes—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3#228011yescharacteristic
450C2×C3#288311yescharacteristic
451C2×C3#288311yescharacteristic
452D3—44no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—64no—
469C22—84no—
470C22—84no—
471C22—124no—
472C22—124no—
473C22—122no—
474C22—122no—
475C22—248no—
476C22D3×#11712yes—
477C22#552412yes—
478C22C22×(C2×D3)⋊C411yescharacteristic
479C22#582812yes—
480C4—24no—
481C4—24no—
482C4—64no—
483C4—84no—
484C4—122no—
485C4—122no—
486C4—244no—
487C3#2072411yescharacteristic, complemented, Sylow
488C2—22no—
489C2—22no—
490C2—22no—
491C2—42no—
492C2—42no—
493C2—124no—
494C2#8226711yescharacteristic
495C2#8254611yescharacteristic
496C2#8254611yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2312C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenter#2312D3
hypocenterC3#20724
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. C3×#2312C2semidirect
3. #82093C2semidirect
4. #69530C2semidirect
6. #69495C2semidirect
7. C4×#806C2semidirect
16. C2×#67⋊C4C4semidirect
17. C2×#67⋊C4C4semidirect
18. C2×D3×C22⋊C4C4semidirect
19. C2×#877C4semidirect
21. C3×#380C22semidirect
23. C3×C4×C22⋊C4C22semidirect
25. #5212C22semidirect
26. C4×#217C22semidirect
29. #4527C22semidirect
32. #2312D3semidirect
103. #380C2×D3semidirect
107. C4×C22⋊C4C2×D3semidirect
487. C3#20724semidirect

Maximal subgroups

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

subgroupindex
C2×#54862
C3×#23122
#820932
#695302
#694982
#694952
C4×#8062
#207243