Finite Group Databasebeta

← #10824604

Subgroups — #10824604

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic, complemented
2C4×#938C212yesmaximal, complemented
3C2×#4685C211yesmaximal, characteristic, complemented
4#72043C211yesmaximal, characteristic
5#72040C211yesmaximal, characteristic, complemented
6C4×#832C212yesmaximal, complemented
7#70637C211yesmaximal, characteristic, complemented
8#70622C211yesmaximal, characteristic, complemented
9#70179C212yesmaximal, complemented
10#70176C212yesmaximal, complemented
11C4×#817C211yesmaximal, characteristic, complemented
12#23697—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#938C2212yes—
16C2×C4×#67—22no—
17C2×C42×D3C2211yescharacteristic
18C3×C4×D8—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#102C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22#5221C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25C4×#217—22no—
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2212yes—
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4987C2212yes—
32#4972C2212yes—
33#4956—22no—
34#4944—22no—
35C4×C4⋊Dic3C2212yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37#4685—24no—
38#4551C2211yescharacteristic
39#4551C2211yescharacteristic
40#4529C2212yes—
41#4529C2212yes—
42C2×C4×D8D311yescharacteristic, complemented
43C2×#431—31no—
44C4×#121—32no—
45#2087—31no—
46#2076—31no—
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106—31no—
52C4×#102—32no—
53C22×C3×D4C2312yes—
54C23×C3×C4—22no—
55C22×#67—22no—
56C22×C4×D3D411yescharacteristic
57C22×C4×D3D411yescharacteristic
58C22×C4×D3C2311yescharacteristic
59C2×C3×D8—24no—
60C2×C3×D8—22no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—24no—
64C3×C4×D4—24no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71#938—24no—
72#938—24no—
73C4×#67—48no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C42×D3—22no—
85C42×D3—22no—
86C3×#107—22no—
87C3×#102—24no—
88C3×C4×C8D414yescomplemented
89#832—24no—
90#832—24no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#817—24no—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99C4×C3⋊C8—22no—
100C22×D8C2×D311yescharacteristic
101C2×#121—32no—
102C2×C4×D4—62no—
103C2×C4×D4C2×D312yes—
104C22×C42—31no—
105C4×D8—24no—
106#431—64no—
107C2×#107C2×D311yescharacteristic
108C2×#106—32no—
109C2×C4⋊C8—31no—
110C2×C4⋊C8—31no—
111C2×#102—32no—
112C2×#102—32no—
113C2×#102C2×D312yes—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C4×C8—31no—
117C2×C4×C8C2×D311yescharacteristic
118#390—32no—
119#389—32no—
120#388—62no—
121#386—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124C4×C22⋊C4—62no—
125#340—31no—
126#340—31no—
127#340—32no—
128#340—32no—
129C24×C3—22no—
130C23×D3C2×D411yescharacteristic
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4#4114yes—
137C22×C3×C4—22no—
138C22×C3×C4—44no—
139C22×C3×C4—44no—
140C22×C3×C4—44no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2411yescharacteristic
144C2×#67—28no—
145C2×#67—44no—
146C2×#67—44no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—24no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C3×D8—28no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8C2×D412yes—
165C2×C3×C8C2×D412yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C22⋊C4—44no—
169C3×C42C2×D412yes—
170C3×C42#4112yes—
171#217—28no—
172#217—44no—
173#217—44no—
174(C2×D3)⋊C4—44no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3#4114yes—
179Dic3⋊C4—44no—
180C4×Dic3—22no—
181C4×Dic3—22no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—44no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C22×D4—62no—
190C22×D4C22×D312yes—
191C23×C4—22no—
192C23×C4—31no—
193C23×C4—31no—
194C23×C4—31no—
195C2×D8—24no—
196C2×D8—22no—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C22×C8C22×D311yescharacteristic
201#121—64no—
202#121—64no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—128no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—62no—
219C2×C42—62no—
220C2×C42—62no—
221C2×C42C22×D311yescharacteristic
222#107—22no—
223#106—64no—
224C4⋊C8—62no—
225C4⋊C8—62no—
226#102—24no—
227#102—64no—
228#102—64no—
229C22⋊C8—62no—
230C22⋊C8—62no—
231C4×C8—62no—
232C4×C8#6714yescomplemented
233#95—62no—
234#95—62no—
235#95—62no—
236C23×C3—24no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3C22×D411yescharacteristic
244C22×D3—21no—
245C22×D3—22no—
246C22×D3—21no—
247C22×D3—22no—
248C22×D3#13512yes—
249C3×D4—24no—
250C3×D4—24no—
251C3×D4—42no—
252C3×D4—42no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—88no—
259C2×C3×C4#12112yes—
260C2×C3×C4#12112yes—
261C2×C3×C4AGL(1,ℤ/8)14yes—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4112yes—
266#67—416no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—24no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3#13512yes—
280C4×D3—28no—
281C4×D3—41no—
282C4×D3—42no—
283C4×D3—41no—
284C4×D3—44no—
285C3×C8—22no—
286C3×C8—22no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C24—22no—
290C24—31no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—68no—
297C2×D4—124no—
298C2×D4—124no—
299C2×D4#23514yes—
300C22×C4—22no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—64no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—61no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—62no—
318C22×C4D3×D411yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4C23×D311yescharacteristic
321D8—28no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8C2×#6712yes—
329C2×C8C2×#6712yes—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—34no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—124no—
337C22⋊C4—44no—
338C22⋊C4—68no—
339C22⋊C4—124no—
340C22⋊C4—124no—
341C22⋊C4—124no—
342C42—62no—
343C42—62no—
344C42—61no—
345C42—61no—
346C42—124no—
347C42#23712yes—
348C42C2×#6712yes—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3C2×#12111yescharacteristic
357C22×C3#52512yes—
358C22×C3C2×#13511yescharacteristic
359C22×C3C2×#13511yescharacteristic
360C22×C3C2×AGL(1,ℤ/8)12yes—
361C2×D3—24no—
362C2×D3—22no—
363C2×D3—42no—
364C2×D3—41no—
365C2×D3—41no—
366C3×C4—22no—
367C3×C4—82no—
368C3×C4#46914yes—
369C3×C4#52511yescharacteristic
370C3×C4#52912yes—
371C3×C4#53211yescharacteristic
372Dic3—22no—
373Dic3—44no—
374Dic3—41no—
375Dic3—41no—
376Dic3—82no—
377C23—24no—
378C23—32no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—62no—
386C23—61no—
387C23—61no—
388C23—62no—
389C23C2×D3×D411yescharacteristic
390D4—24no—
391D4—24no—
392D4—42no—
393D4—42no—
394D4—1216no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—32no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—68no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—62no—
408C2×C4—64no—
409C2×C4—88no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—124no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4#91114yes—
420C2×C4#93812yes—
421C2×C4#93812yes—
422C2×C4C2×#23512yes—
423C2×C4C2×#23711yescharacteristic
424C2×C4D3×#4112yes—
425C2×C4C22×#6711yescharacteristic
426C8—22no—
427C8—22no—
428C8—121no—
429C8—121no—
430C2×C3—84no—
431C2×C3#239211yescharacteristic
432C2×C3#305011yescharacteristic
433C2×C3C2×#46911yescharacteristic
434C2×C3#310512yes—
435C2×C3#310612yes—
436D3—42no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—62no—
441C22—64no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—121no—
447C22—121no—
448C22—122no—
449C22#554712yes—
450C22C2×#91112yes—
451C22D3×#13511yescharacteristic
452C22D3×#13511yescharacteristic
453C22C2×#93811yescharacteristic
454C4—22no—
455C4—62no—
456C4—82no—
457C4—121no—
458C4—121no—
459C4—124no—
460C4—242no—
461C4#515914yes—
462C4#553812yes—
463C4#579111yescharacteristic
464C4#583211yescharacteristic
465C3#2369711yescharacteristic, complemented, Sylow
466C2—84no—
467C2—122no—
468C2#7959212yes—
469C2#7959312yes—
470C2#8240111yescharacteristic
471C2C2×#515911yescharacteristic
472C2#8391411yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radical#72043C2
hypercenterC2×C4×D8D3
hypocenterC3#23697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ 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. C4×#938C2semidirect
3. C2×#4685C2semidirect
5. #72040C2semidirect
6. C4×#832C2semidirect
7. #70637C2semidirect
8. #70622C2semidirect
9. #70179C2semidirect
10. #70176C2semidirect
11. C4×#817C2semidirect
42. C2×C4×D8D3semidirect
88. C3×C4×C8D4semidirect
232. C4×C8#67semidirect
465. C3#23697semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
C4×#9382
C2×#46852
#720432
#720402
C4×#8322
#706372
#706222
#701792
#701762
C4×#8172
#236973