Finite Group Databasebeta

← #10684916

Subgroups — #10684916

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#407C211yesmaximal, characteristic
2#84027C211yesmaximal, characteristic
3C2×#5211C211yesmaximal, characteristic
4#67682C214yesmaximal, complemented
5#7943—31nomaximal, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—21no—
8C22×#217—21no—
9C2×#926C412yescomplemented
10C3×#407—21no—
11C3×#407—21no—
12C3×#407C412yescomplemented
13C3×#407C2212yes—
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8C412yescomplemented
16#5247—21no—
17#5212—21no—
18#5212—21no—
19#5211—21no—
20#5211—21no—
21#5211—21no—
22#5211C412yescomplemented
23#5211C2212yes—
24C4×#217C412yescomplemented
25C4×#217C2212yes—
26C2×C3⋊#95—21no—
27C2×C3⋊#95C412yescomplemented
28C2×#407D311yescharacteristic
29#2280—31no—
30C2×#381—31no—
31#1284—34no—
32C25×C3D411yescharacteristic
33C23×C3×C4—44no—
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4Q811yescharacteristic
46C23×C3×C4Q811yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—22no—
49C2×C3×M16—22no—
50C2×C3×M16—22no—
51C2×C3×M16—44no—
52C22×C3×C8—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217—44no—
59#926—44no—
60C2×C4⋊Dic3—22no—
61C2×C4×Dic3C2×C414yescomplemented
62C3×C22⋊C8—22no—
63C3×C22⋊C8—22no—
64C3×C22⋊C8—22no—
65C3×C22⋊C8C2×C414yescomplemented
66C3⋊#95—22no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95C2×C414yes—
72C24×C4C2×D311yescharacteristic
73C22×M16—21no—
74C2×#117—32no—
75C22×C22⋊C4—61no—
76#407—21no—
77#407—21no—
78#407Dic312yescomplemented
79#407C2×D312yes—
80C2×C22⋊C8—21no—
81C2×C22⋊C8Dic312yescomplemented
82#381—32no—
83#381—32no—
84#381—61no—
85#381—61no—
86#381—61no—
87#380—61no—
88#380—61no—
89#379—61no—
90C4×C22⋊C4—32no—
91C4×C22⋊C4—32no—
92C2×#95—32no—
93C2×#95—61no—
94C24×C3—24no—
95C24×C3—42no—
96C24×C3—42no—
97C24×C3C22⋊C411yescharacteristic
98C24×C3C22⋊C411yescharacteristic
99C24×C3C22⋊C411yescharacteristic
100C24×C3C22⋊C411yescharacteristic
101C24×C3C4⋊C411yescharacteristic
102C24×C3C4⋊C411yescharacteristic
103C24×C3C2×D411yescharacteristic
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4C4211yescharacteristic
119C22×C3×C4C4211yescharacteristic
120C22×C3×C4C4211yescharacteristic
121C22×C3×C4C4211yescharacteristic
122C22×C3×C4C22⋊C411yescharacteristic
123C22×C3×C4C22⋊C411yescharacteristic
124C22×C3×C4C22⋊C411yescharacteristic
125C22×C3×C4C22⋊C411yescharacteristic
126C22×C3×C4C22⋊C411yescharacteristic
127C22×C3×C4C22⋊C411yescharacteristic
128C22×C3×C4C22⋊C411yescharacteristic
129C22×C3×C4C22⋊C411yescharacteristic
130C22×C3×C4C4⋊C411yescharacteristic
131C22×C3×C4C4⋊C411yescharacteristic
132C22×C3×C4C4⋊C411yescharacteristic
133C22×C3×C4C4⋊C411yescharacteristic
134C22×C3×C4C4⋊C411yescharacteristic
135C22×C3×C4C4⋊C411yescharacteristic
136C22×C3×C4C4⋊C411yescharacteristic
137C22×C3×C4C4⋊C411yescharacteristic
138C22×C3×C4C4⋊C411yescharacteristic
139C22×C3×C4C4⋊C411yescharacteristic
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×Q811yescharacteristic
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—44no—
149C22×Dic3—44no—
150C22×Dic3—44no—
151C3×M16—44no—
152C3×M16—44no—
153C2×C3×C8—24no—
154C2×C3×C8—44no—
155#217—44no—
156#217—44no—
157#217—44no—
158#217—44no—
159#217—44no—
160C4⋊Dic3—44no—
161C4×Dic3—44no—
162C25D1211yescharacteristic
163C23×C4—44no—
164C23×C4—62no—
165C23×C4Dic611yescharacteristic
166C23×C4Dic611yescharacteristic
167C23×C4C4×D311yescharacteristic
168C23×C4C4×D311yescharacteristic
169C23×C4C4×D311yescharacteristic
170C23×C4C4×D311yescharacteristic
171C23×C4D1211yescharacteristic
172C23×C4C2×Dic311yescharacteristic
173C23×C4C2×Dic311yescharacteristic
174C23×C4#6711yescharacteristic
175C23×C4#6711yescharacteristic
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—44no—
182C22×C8—22no—
183#117—124no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—124no—
191C2×C42—34no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8C2×Dic314yescomplemented
196#95—34no—
197#95—62no—
198#95—62no—
199#95—62no—
200#95—62no—
201#95—62no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—416no—
223C23×C3#9511yescharacteristic
224C23×C3#9511yescharacteristic
225C23×C3#9511yescharacteristic
226C23×C3#9511yescharacteristic
227C23×C3#9914yes—
228C23×C3C2×C22⋊C411yescharacteristic
229C23×C3C2×C22⋊C411yescharacteristic
230C23×C3C2×C4⋊C411yescharacteristic
231C2×C3×C4—22no—
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—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4#9511yescharacteristic
260C2×C3×C4#9511yescharacteristic
261C2×C3×C4#9511yescharacteristic
262C2×C3×C4#9511yescharacteristic
263C2×C3×C4C2×C4211yescharacteristic
264C2×C3×C4C2×C22⋊C411yescharacteristic
265C2×C3×C4C2×C4⋊C411yescharacteristic
266C2×C3×C4C2×C4⋊C411yescharacteristic
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—88no—
274C3×C8—44no—
275C24—24no—
276C24—42no—
277C24—42no—
278C24C4⋊Dic311yescharacteristic
279C24C4⋊Dic311yescharacteristic
280C24(C2×D3)⋊C411yescharacteristic
281C24(C2×D3)⋊C411yescharacteristic
282C24(C2×D3)⋊C411yescharacteristic
283C24(C2×D3)⋊C411yescharacteristic
284C24C2×D1211yescharacteristic
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—64no—
300C22×C4—64no—
301C22×C4—124no—
302C22×C4—124no—
303C22×C4—124no—
304C22×C4C4×Dic311yescharacteristic
305C22×C4C4×Dic311yescharacteristic
306C22×C4C4×Dic311yescharacteristic
307C22×C4C4×Dic311yescharacteristic
308C22×C4Dic3⋊C411yescharacteristic
309C22×C4Dic3⋊C411yescharacteristic
310C22×C4Dic3⋊C411yescharacteristic
311C22×C4Dic3⋊C411yescharacteristic
312C22×C4Dic3⋊C411yescharacteristic
313C22×C4Dic3⋊C411yescharacteristic
314C22×C4Dic3⋊C411yescharacteristic
315C22×C4Dic3⋊C411yescharacteristic
316C22×C4C4⋊Dic311yescharacteristic
317C22×C4C4⋊Dic311yescharacteristic
318C22×C4(C2×D3)⋊C411yescharacteristic
319C22×C4(C2×D3)⋊C411yescharacteristic
320C22×C4(C2×D3)⋊C411yescharacteristic
321C22×C4(C2×D3)⋊C411yescharacteristic
322C22×C4#21711yescharacteristic
323C22×C4#21711yescharacteristic
324C22×C4#21711yescharacteristic
325C22×C4#21711yescharacteristic
326C22×C4C2×Dic611yescharacteristic
327C22×C4C2×C4×D311yescharacteristic
328C22×C4C2×C4×D311yescharacteristic
329C22×C4C22×Dic311yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332M16—44no—
333M16—44no—
334C2×C8—24no—
335C2×C8—44no—
336C4⋊C4—124no—
337C22⋊C4—124no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22⋊C4—124no—
341C22⋊C4—124no—
342C42—124no—
343C22×C3—28no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—44no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3H(3,ℤ/4)12yes—
367C22×C3#34212yes—
368C22×C3#34212yes—
369C22×C3C2×#9511yescharacteristic
370C22×C3C2×#9912yes—
371C22×C3#42112yes—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380Dic3—84no—
381Dic3—84no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—416no—
403C23#80614yes—
404C23C3⋊#9511yescharacteristic
405C23C3⋊#9511yescharacteristic
406C23C3⋊#9511yescharacteristic
407C23C3⋊#9511yescharacteristic
408C23C2×C4⋊Dic311yescharacteristic
409C23C2×(C2×D3)⋊C411yescharacteristic
410C23C2×(C2×D3)⋊C411yescharacteristic
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—248no—
446C2×C4C3⋊#9511yescharacteristic
447C2×C4C3⋊#9511yescharacteristic
448C2×C4C3⋊#9511yescharacteristic
449C2×C4C3⋊#9511yescharacteristic
450C2×C4C2×C4×Dic311yescharacteristic
451C2×C4C2×Dic3⋊C411yescharacteristic
452C2×C4C2×Dic3⋊C411yescharacteristic
453C2×C4C2×#21711yescharacteristic
454C8—44no—
455C2×C3—22no—
456C2×C3—22no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—22no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3C2×H(3,ℤ/4)11yescharacteristic
463C2×C3#173311yescharacteristic
464C2×C3C2×#34211yescharacteristic
465C22—28no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22#452712yes—
489C22#452712yes—
490C22#455712yes—
491C22C2×#80612yes—
492C22#513912yes—
493C22C2×C3⋊#9511yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—244no—
503C4—244no—
504C3#794311yescharacteristic, complemented, Sylow
505C2—22no—
506C2—22no—
507C2—22no—
508C2—22no—
509C2—22no—
510C2—41no—
511C2—41no—
512C2C2×#452711yescharacteristic
513C2#7198211yescharacteristic
514C2C2×#455711yescharacteristic
subgroupG/H
centerC22C2×C3⋊#95
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#407C2
socleC22×C3C2×#95
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#407D3
hypocenterC3#7943
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
4. #67682C2semidirect
9. C2×#926C4semidirect
12. C3×#407C4semidirect
15. C2×C3×C22⋊C8C4semidirect
22. #5211C4semidirect
24. C4×#217C4semidirect
27. C2×C3⋊#95C4semidirect
61. C2×C4×Dic3C2×C4semidirect
65. C3×C22⋊C8C2×C4semidirect
78. #407Dic3semidirect
81. C2×C22⋊C8Dic3semidirect
195. C22⋊C8C2×Dic3semidirect
504. C3#7943semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4072
#840272
C2×#52112
#676822
#79433