Finite Group Databasebeta

← #10685562

Subgroups — #10685562

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#407C211yesmaximal, characteristic
2#84027C211yesmaximal, characteristic
3C2×#5207C211yesmaximal, characteristic
4#67686C214yesmaximal, complemented
5#8115—31nomaximal, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—21no—
8C2×#926C412yes—
9C22×#208—21no—
10C3×#407—21no—
11C3×#407—21no—
12C3×#407C412yescomplemented
13C3×#407C2212yes—
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8C412yescomplemented
16#5247C412yes—
17#5212—21no—
18#5212—21no—
19#5211C2212yes—
20C4×#217—21no—
21C4×#217—21no—
22#5207—21no—
23#5207—21no—
24#5207C412yescomplemented
25#5207C2212yes—
26C2×#830—21no—
27C2×#830C412yescomplemented
28C2×#407—31no—
29C2×#407D311yescharacteristic
30#2280—31no—
31#1306—34no—
32C25×C3C2×C411yescharacteristic
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×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4Q811yescharacteristic
46C23×C3×C4Q811yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C2×C3×M16—22no—
49C2×C3×M16—22no—
50C2×C3×M16—44no—
51C22×C3×C8—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217C2×C414yes—
55#926—44no—
56C2×C4⋊Dic3—22no—
57C2×C4×Dic3—22no—
58C2×#208—22no—
59C2×#208—22no—
60C2×#208—44no—
61C22×C3⋊C8—22no—
62C3×C22⋊C8—22no—
63C3×C22⋊C8—22no—
64C3×C22⋊C8—22no—
65C3×C22⋊C8C2×C414yes—
66C3⋊#95—22no—
67C3⋊#95—22no—
68#830—22no—
69#830—22no—
70#830—22no—
71#830C2×C414yes—
72C24×C4C2×D311yescharacteristic
73C22×M16—21no—
74C22×M16—61no—
75C2×#117—32no—
76#407—21no—
77#407—21no—
78#407—32no—
79#407—32no—
80#407—61no—
81#407—61no—
82#407Dic312yescomplemented
83#407C2×D312yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—32no—
86C2×C22⋊C8—61no—
87C2×C22⋊C8Dic312yescomplemented
88#381—32no—
89#380—61no—
90#380—61no—
91#379—32no—
92C4×C22⋊C4—61no—
93C4×C22⋊C4—61no—
94C24×C3—42no—
95C24×C3—42no—
96C24×C3—42no—
97C24×C3C4211yescharacteristic
98C24×C3C4211yescharacteristic
99C24×C3C22⋊C411yescharacteristic
100C24×C3C22⋊C411yescharacteristic
101C24×C3C4⋊C411yescharacteristic
102C24×C3C4⋊C411yescharacteristic
103C24×C3C22×C411yescharacteristic
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
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×C4C22⋊C411yescharacteristic
121C22×C3×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×Q811yescharacteristic
146C22×Dic3—24no—
147C3×M16—44no—
148C3×M16—44no—
149C2×C3×C8—24no—
150C2×C3×C8—44no—
151#217—44no—
152#217—44no—
153#217—44no—
154C4⋊Dic3—44no—
155C4×Dic3—44no—
156#208—44no—
157#208—44no—
158C2×C3⋊C8—24no—
159C2×C3⋊C8—44no—
160C25C4×D311yescharacteristic
161C23×C4—44no—
162C23×C4Dic611yescharacteristic
163C23×C4Dic611yescharacteristic
164C23×C4C4×D311yescharacteristic
165C23×C4C4×D311yescharacteristic
166C23×C4C4×D311yescharacteristic
167C23×C4D1211yescharacteristic
168C23×C4D1211yescharacteristic
169C23×C4C2×Dic311yescharacteristic
170C23×C4C2×Dic311yescharacteristic
171C23×C4#6711yescharacteristic
172C23×C4#6711yescharacteristic
173C23×C4#6711yescharacteristic
174C23×C4#6711yescharacteristic
175C23×C4C22×D311yescharacteristic
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—44no—
179C2×M16—62no—
180C2×M16—62no—
181C2×M16—124no—
182C22×C8—22no—
183C22×C8—62no—
184#117—124no—
185C2×C4⋊C4—62no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C42—62no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—34no—
194C22⋊C8—62no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197C22⋊C8C2×Dic314yes—
198#95—62no—
199#95—62no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—416no—
222C23×C3#9511yescharacteristic
223C23×C3#9511yescharacteristic
224C23×C3#9511yescharacteristic
225C23×C3#9511yescharacteristic
226C23×C3C2×C4211yescharacteristic
227C23×C3C2×C22⋊C411yescharacteristic
228C23×C3C2×C4⋊C411yescharacteristic
229C2×C3×C4—22no—
230C2×C3×C4—22no—
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—22no—
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#9511yescharacteristic
257C2×C3×C4#9511yescharacteristic
258C2×C3×C4#9511yescharacteristic
259C2×C3×C4#9511yescharacteristic
260C2×C3×C4#9914yes—
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C2×C4⋊C411yescharacteristic
264C2×C3×C4C2×C4⋊C411yescharacteristic
265C2×Dic3—44no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C3×C8—44no—
269C3⋊C8—44no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C4×Dic311yescharacteristic
274C24C4×Dic311yescharacteristic
275C24Dic3⋊C411yescharacteristic
276C24Dic3⋊C411yescharacteristic
277C24(C2×D3)⋊C411yescharacteristic
278C24(C2×D3)⋊C411yescharacteristic
279C24C2×C4×D311yescharacteristic
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—64no—
295C22×C4C4×Dic311yescharacteristic
296C22×C4C4×Dic311yescharacteristic
297C22×C4Dic3⋊C411yescharacteristic
298C22×C4Dic3⋊C411yescharacteristic
299C22×C4Dic3⋊C411yescharacteristic
300C22×C4Dic3⋊C411yescharacteristic
301C22×C4Dic3⋊C411yescharacteristic
302C22×C4Dic3⋊C411yescharacteristic
303C22×C4C4⋊Dic311yescharacteristic
304C22×C4C4⋊Dic311yescharacteristic
305C22×C4C4⋊Dic311yescharacteristic
306C22×C4C4⋊Dic311yescharacteristic
307C22×C4(C2×D3)⋊C411yescharacteristic
308C22×C4(C2×D3)⋊C411yescharacteristic
309C22×C4(C2×D3)⋊C411yescharacteristic
310C22×C4(C2×D3)⋊C411yescharacteristic
311C22×C4(C2×D3)⋊C411yescharacteristic
312C22×C4(C2×D3)⋊C411yescharacteristic
313C22×C4#21711yescharacteristic
314C22×C4#21711yescharacteristic
315C22×C4#21711yescharacteristic
316C22×C4#21711yescharacteristic
317C22×C4C2×Dic611yescharacteristic
318C22×C4C2×C4×D311yescharacteristic
319C22×C4C2×D1211yescharacteristic
320C22×C4C22×Dic311yescharacteristic
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323M16—44no—
324M16—44no—
325M16—124no—
326M16—124no—
327C2×C8—24no—
328C2×C8—44no—
329C2×C8—64no—
330C2×C8—124no—
331C4⋊C4—124no—
332C22⋊C4—124no—
333C22⋊C4—124no—
334C22⋊C4—124no—
335C42—124no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
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—44no—
355C22×C3—44no—
356C22×C3—44no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3#34412yes—
362C22×C3#34412yes—
363C22×C3C2×#9511yescharacteristic
364C22×C3C2×#9912yes—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4#34212yes—
372C3×C4#34212yes—
373Dic3—84no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—416no—
396C23C3⋊#9511yescharacteristic
397C23C3⋊#9511yescharacteristic
398C23C3⋊#9511yescharacteristic
399C23C3⋊#9511yescharacteristic
400C23C2×C4×Dic311yescharacteristic
401C23C2×Dic3⋊C411yescharacteristic
402C23C2×(C2×D3)⋊C411yescharacteristic
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—124no—
431C2×C4—124no—
432C2×C4—124no—
433C2×C4#80614yes—
434C2×C4C3⋊#9511yescharacteristic
435C2×C4C3⋊#9511yescharacteristic
436C2×C4C3⋊#9511yescharacteristic
437C2×C4C3⋊#9511yescharacteristic
438C2×C4C2×Dic3⋊C411yescharacteristic
439C2×C4C2×C4⋊Dic311yescharacteristic
440C2×C4C2×(C2×D3)⋊C411yescharacteristic
441C2×C4C2×#21711yescharacteristic
442C8—44no—
443C8—124no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3C2×#34211yescharacteristic
452C2×C3C2×#34411yescharacteristic
453C2×C3C2×#34411yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22#455612yes—
480C22#455912yes—
481C22C2×#80612yes—
482C22C2×C3⋊#9511yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—244no—
490C4#452712yes—
491C4#452712yes—
492C3#811511yescharacteristic, complemented, Sylow
493C2—22no—
494C2—22no—
495C2—22no—
496C2—41no—
497C2—41no—
498C2—41no—
499C2—41no—
500C2C2×#452711yescharacteristic
501C2C2×#455611yescharacteristic
502C2C2×#455911yescharacteristic
subgroupG/H
centerC22C2×C3⋊#95
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#407C2
socleC22×C3C2×#95
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×#407D3
hypocenterC3#8115
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
4. #67686C2semidirect
12. C3×#407C4semidirect
15. C2×C3×C22⋊C8C4semidirect
24. #5207C4semidirect
27. C2×#830C4semidirect
82. #407Dic3semidirect
87. C2×C22⋊C8Dic3semidirect
492. C3#8115semidirect

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×#52072
#676862
#81153