Finite Group Databasebeta

← #10824667

Subgroups — #10824667

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic, complemented
2#82396C212yesmaximal, complemented
3C2×#4690C211yesmaximal, characteristic, complemented
4#72043C211yesmaximal, characteristic
5#72040C211yesmaximal, characteristic, complemented
6#71609C212yesmaximal, complemented
7#70636C211yesmaximal, characteristic, complemented
8#70621C211yesmaximal, characteristic, complemented
9#70179C212yesmaximal, complemented
10#70176C212yesmaximal, complemented
11C4×#817C211yesmaximal, characteristic, complemented
12#23702—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#938C2212yes—
16C2×#872C2211yescharacteristic
17C3×C4×D8—24no—
18C2×C3×#107C2211yescharacteristic
19C2×C3×#102C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21#5221C2212yes—
22C2×#832C2212yes—
23C2×#832C2212yes—
24#5213—22no—
25#5210—22no—
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2212yes—
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4987C2212yes—
32#4972C2212yes—
33#4957—22no—
34#4945—22no—
35C4×C4⋊Dic3C2212yes—
36C2×#803C2211yescharacteristic
37#4690—24no—
38#4551C2211yescharacteristic
39#4551C2211yescharacteristic
40#4529C2212yes—
41#4529C2212yes—
42C2×C4×D8D311yescharacteristic, complemented
43C2×#433—31no—
44#2324—32no—
45#2086—31no—
46#2075—31no—
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106—31no—
52#1762—32no—
53C22×C3×D4C2312yes—
54C23×C3×C4—22no—
55C22×#67—22no—
56C22×C4×D3C2311yescharacteristic
57C2×C3×D8—24no—
58C2×C3×D8—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4×D4—24no—
62C3×C4×D4—24no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C22⋊C4—22no—
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—22no—
69#938—24no—
70#938—24no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4D411yescharacteristic
73C2×(C2×D3)⋊C4D411yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4D411yescharacteristic
78C2×Dic3⋊C4D411yescharacteristic
79C2×C4×Dic3—22no—
80C2×C4×Dic3C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83#872—22no—
84#872—22no—
85C3×#107—22no—
86C3×#102—24no—
87C3×C4×C8D414yescomplemented
88#832—24no—
89#832—24no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93(C2×D3)⋊C8—22no—
94(C2×D3)⋊C8—22no—
95#817—24no—
96Dic3⋊C8—22no—
97Dic3⋊C8—22no—
98#803—22no—
99C22×D8C2×D311yescharacteristic
100C2×#121—32no—
101C2×C4×D4C2×D312yes—
102C2×#117—31no—
103C4×D8—24no—
104#433—64no—
105C2×#107C2×D311yescharacteristic
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#102C2×D312yes—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×#97—31no—
115C2×C4×C8C2×D311yescharacteristic
116#390—32no—
117#390—62no—
118#389—32no—
119#388—62no—
120#386—62no—
121#385—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—31no—
126#340—32no—
127#340—32no—
128C24×C3—22no—
129C23×D3C2×D411yescharacteristic
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4#4114yes—
136C22×C3×C4—22no—
137C22×C3×C4—44no—
138C22×C3×C4—82no—
139C22×C3×C4—82no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C2×#67—28no—
144C2×#67—82no—
145C2×#67—82no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C3×D8—28no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D412yes—
159C2×C3×C8C2×D412yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C22⋊C4—44no—
163C3×C42C2×D412yes—
164C3×C42#4112yes—
165#217—28no—
166#217—82no—
167#217—82no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170(C2×D3)⋊C4—82no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3#4114yes—
175Dic3⋊C4—22no—
176Dic3⋊C4—22no—
177Dic3⋊C4—82no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C4×Dic3—82no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C22×D4—62no—
186C22×D4C22×D312yes—
187C23×C4—22no—
188C23×C4—31no—
189C2×D8—24no—
190C2×D8—22no—
191C22×C8—31no—
192C22×C8—31no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195#121—64no—
196#121—64no—
197C4×D4—24no—
198C4×D4—24no—
199#117—62no—
200#117—62no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—32no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4C22×D312yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C42—31no—
215C2×C42—62no—
216C2×C42C22×D311yescharacteristic
217#107—22no—
218#106—64no—
219C4⋊C8—62no—
220C4⋊C8—62no—
221#102—24no—
222#102—64no—
223#102—64no—
224C22⋊C8—62no—
225C22⋊C8—62no—
226#97—62no—
227C4×C8#6714yescomplemented
228#95—62no—
229#95—62no—
230#95—62no—
231C23×C3—24no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3C22×D411yescharacteristic
239C22×D3—21no—
240C22×D3—22no—
241C22×D3—21no—
242C22×D3—21no—
243C22×D3—22no—
244C3×D4—24no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4—42no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—88no—
255C2×C3×C4#12112yes—
256C2×C3×C4#12112yes—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4112yes—
261#67—88no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—24no—
266C2×Dic3—22no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—22no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C4×D3—41no—
276C4×D3—42no—
277C4×D3—41no—
278C3×C8—22no—
279C3×C8—22no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C24—22no—
283C24—31no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—68no—
290C2×D4—242no—
291C2×D4—242no—
292C2×D4#23514yes—
293C22×C4—22no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—44no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4—61no—
303C22×C4—62no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4C23×D311yescharacteristic
309D8—28no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8C2×#6712yes—
317C2×C8C2×#6712yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—34no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—62no—
326C4⋊C4—242no—
327C22⋊C4—44no—
328C22⋊C4—68no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—242no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—61no—
335C42—61no—
336C42—242no—
337C42#23712yes—
338C42C2×#6712yes—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3—82no—
346C22×C3C2×#12111yescharacteristic
347C22×C3#52512yes—
348C22×C3#57611yescharacteristic
349C22×C3#57611yescharacteristic
350C22×C3#57712yes—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—82no—
359C3×C4#52511yescharacteristic
360C3×C4#52912yes—
361C3×C4#53211yescharacteristic
362Dic3—44no—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—61no—
375C23—61no—
376C23—62no—
377C23—61no—
378C23—62no—
379C23C2×D3×D411yescharacteristic
380D4—24no—
381D4—24no—
382D4—42no—
383D4—42no—
384D4—248no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—62no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4—64no—
399C2×C4—88no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—121no—
403C2×C4—122no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4#93812yes—
409C2×C4#93812yes—
410C2×C4C2×#23512yes—
411C2×C4C2×#23711yescharacteristic
412C2×C4D3×#4112yes—
413C2×C4C22×#6711yescharacteristic
414C8—22no—
415C8—22no—
416C8—121no—
417C8—121no—
418C2×C3—84no—
419C2×C3#239211yescharacteristic
420C2×C3#305811yescharacteristic
421C2×C3#306011yescharacteristic
422C2×C3#311812yes—
423C2×C3#311912yes—
424D3—81no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22—122no—
436C22—122no—
437C22#554712yes—
438C22#575711yescharacteristic
439C22#575711yescharacteristic
440C22#577012yes—
441C22C2×#93811yescharacteristic
442C4—22no—
443C4—22no—
444C4—82no—
445C4—121no—
446C4—121no—
447C4—121no—
448C4—124no—
449C4—242no—
450C4#553812yes—
451C4#579111yescharacteristic
452C4#583211yescharacteristic
453C3#2370211yescharacteristic, complemented, Sylow
454C2—84no—
455C2—241no—
456C2#7959412yes—
457C2#7959512yes—
458C2#8240111yescharacteristic
459C2#8383611yescharacteristic
460C2#8391611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radical#72043C2
hypercenterC2×C4×D8D3
hypocenterC3#23702
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D8C2semidirect
2. #82396C2semidirect
3. C2×#4690C2semidirect
5. #72040C2semidirect
6. #71609C2semidirect
7. #70636C2semidirect
8. #70621C2semidirect
9. #70179C2semidirect
10. #70176C2semidirect
11. C4×#817C2semidirect
42. C2×C4×D8D3semidirect
87. C3×C4×C8D4semidirect
227. C4×C8#67semidirect
453. C3#23702semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
#823962
C2×#46902
#720432
#720402
#716092
#706362
#706212
#701792
#701762
C4×#8172
#237023