Finite Group Databasebeta

← #10690903

Subgroups — #10690903

Order 768.

1270 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2848C211yesmaximal, characteristic
2C22×C3⋊#95C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4538C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9143—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C24×Dic3—21no—
9C2×C3×C22⋊Q8C412yescomplemented
10C2×C3×C22⋊Q8C2211yescharacteristic
11C3×#385—21no—
12C3×#385—22no—
13C3×#380C412yescomplemented
14C3×#380C2211yescharacteristic
15C3×#379—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95—24no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4538—24no—
22#2848D311yescharacteristic
23C22×#95—31no—
24#1779—31no—
25C2×#328—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic, complemented
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C412yes—
33C23×C3×C4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—28no—
37C23×Dic3—24no—
38C23×Dic3—22no—
39C23×Dic3—44no—
40C23×Dic3—42no—
41C23×Dic3—42no—
42C3×C22⋊Q8—22no—
43C3×C22⋊Q8—22no—
44C3×C22⋊Q8—44no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
48C2×C3×C4⋊C4C2×C412yescomplemented
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3×#95—21no—
56C3⋊#95—24no—
57C3⋊#95—22no—
58C3⋊#95D414yes—
59#830—22no—
60#830—22no—
61#830—44no—
62C24×C4—61no—
63C24×C4C2×D311yescharacteristic
64C2×C22⋊Q8Dic312yescomplemented
65C2×C22⋊Q8C2×D311yescharacteristic
66C2×C22⋊C8—32no—
67C2×C22⋊C8—62no—
68#385—21no—
69#385—22no—
70#380Dic312yescomplemented
71#380C2×D311yescharacteristic
72#379—21no—
73C2×#95—32no—
74C2×#95—64no—
75C2×#95—61no—
76#328—64no—
77C24×C3—24no—
78C24×C3—24no—
79C24×C3—21no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3C22⋊C412yes—
84C24×C3SD1612yes—
85C24×C3Q1612yes—
86C24×C3C2×D411yescharacteristic
87C2×C3×Q8—22no—
88C2×C3×Q8—42no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—48no—
103C22×C3×C4—48no—
104C22×C3×C4C22⋊C412yes—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4112yes—
109C22×Dic3—216no—
110C22×Dic3—28no—
111C22×Dic3—24no—
112C22×Dic3—48no—
113C22×Dic3—416no—
114C22×Dic3—416no—
115C22×Dic3—44no—
116C22×Dic3—48no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—44no—
122C3×C4⋊C4—42no—
123C3×C22⋊C4—44no—
124C3×C22⋊C4—44no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C2×C3⋊C8—44no—
130C2×C3⋊C8—42no—
131C25#6711yescharacteristic
132C22×Q8C2×Dic311yescharacteristic, complemented
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—64no—
137C23×C4—68no—
138C23×C4—62no—
139C23×C4—124no—
140C23×C4—122no—
141C23×C4—122no—
142C23×C4C2×Dic312yes—
143C23×C4#6712yes—
144C23×C4#6711yescharacteristic
145C23×C4C22×D311yescharacteristic
146C22×C8—62no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4C2×Dic311yescharacteristic, complemented
153C2×C4⋊C4C2×Dic312yescomplemented
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C22⋊C8—62no—
159C22⋊C8—62no—
160C22⋊C8—124no—
161#95—22no—
162#95—21no—
163#95—34no—
164#95—64no—
165#95—62no—
166C23×C3—24no—
167C23×C3—24no—
168C23×C3—28no—
169C23×C3—24no—
170C23×C3—24no—
171C23×C3—28no—
172C23×C3—28no—
173C23×C3—28no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3#9914yes—
185C23×C3#10314yes—
186C23×C3#10414yes—
187C23×C3C2×C22⋊C411yescharacteristic
188C23×C3#12011yescharacteristic
189C23×C3#12311yescharacteristic
190C23×C3C2×SD1611yescharacteristic
191C23×C3C2×Q1611yescharacteristic
192C23×C3AGL(1,ℤ/8)11yescharacteristic
193C23×C3#13711yescharacteristic
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—48no—
203C2×C3×C4—48no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4C4×D412yes—
214C2×C3×C4#12112yes—
215C2×Dic3—28no—
216C2×Dic3—432no—
217C2×Dic3—44no—
218C2×Dic3—48no—
219C2×Dic3—48no—
220C2×Dic3—44no—
221C3⋊C8—82no—
222C24—24no—
223C24—24no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24#21512yes—
229C24#21612yes—
230C24#21712yes—
231C24C2×#6711yescharacteristic
232C2×Q8—22no—
233C2×Q8—42no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—616no—
250C22×C4—68no—
251C22×C4—64no—
252C22×C4—128no—
253C22×C4—1216no—
254C22×C4—124no—
255C22×C4—1216no—
256C22×C4—128no—
257C22×C4—122no—
258C22×C4—122no—
259C22×C4#21712yes—
260C22×C4D3×D412yes—
261C22×C4#23712yes—
262C22×C4C22×Dic311yescharacteristic
263C22×C4C2×#6711yescharacteristic
264C2×C8—124no—
265C2×C8—122no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22×C3—28no—
277C22×C3—28no—
278C22×C3—28no—
279C22×C3—28no—
280C22×C3—24no—
281C22×C3—21no—
282C22×C3—24no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—24no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—24no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—24no—
292C22×C3—24no—
293C22×C3—22no—
294C22×C3#32818yescomplemented
295C22×C3#38611yescharacteristic
296C22×C3C2×#9912yes—
297C22×C3C2×#10311yescharacteristic
298C22×C3#41711yescharacteristic
299C22×C3C2×#10412yes—
300C22×C3#45011yescharacteristic
301C22×C3#45111yescharacteristic
302C22×C3#48311yescharacteristic
303C22×C3#48411yescharacteristic
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—42no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4—82no—
310Dic3—48no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—28no—
315C23—28no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—28no—
327C23—28no—
328C23—22no—
329C23#83414yes—
330C23#83514yes—
331C23#83714yes—
332C23#93511yescharacteristic
333C23#93711yescharacteristic
334C23C2×#21511yescharacteristic
335C23C2×#21611yescharacteristic
336C23#94911yescharacteristic
337C23#95111yescharacteristic
338C23C2×#21711yescharacteristic
339Q8—82no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—68no—
359C2×C4—1232no—
360C2×C4—128no—
361C2×C4—128no—
362C2×C4—124no—
363C2×C4—124no—
364C2×C4D4×Dic312yes—
365C2×C4#93912yes—
366C8—242no—
367C2×C3—21no—
368C2×C3—28no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3C2×#32812yes—
373C2×C3#150412yes—
374C2×C3#187211yescharacteristic
375C2×C3#189611yescharacteristic
376C2×C3#189811yescharacteristic
377C22—28no—
378C22—28no—
379C22—28no—
380C22—28no—
381C22—22no—
382C22—24no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—24no—
387C22—24no—
388C22—22no—
389C22—21no—
390C22—24no—
391C22—21no—
392C22—21no—
393C22—24no—
394C22—21no—
395C22#453818yescomplemented
396C22#504311yescharacteristic
397C22#504411yescharacteristic
398C22#504611yescharacteristic
399C22#505011yescharacteristic
400C22C2×#83412yes—
401C22#522011yescharacteristic
402C22C2×#83511yescharacteristic
403C22#523211yescharacteristic
404C22C2×#83712yes—
405C4—22no—
406C4—22no—
407C4—42no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—128no—
412C3#914311yescharacteristic, complemented, Sylow
413C2—21no—
414C2—21no—
415C2—21no—
416C2—28no—
417C2—21no—
418C2#7167011yescharacteristic
419C2C2×#453812yes—
420C2#7169211yescharacteristic
421C2#7170812yes—
422C2#7171111yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2848C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2848D3
hypocenterC3#9143
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#4538C2semidirect
5. #67624C2semidirect
9. C2×C3×C22⋊Q8C4semidirect
13. C3×#380C4semidirect
28. C22×C3×Q8C2×C4semidirect
47. C2×C3×C4⋊C4C2×C4semidirect
48. C2×C3×C4⋊C4C2×C4semidirect
64. C2×C22⋊Q8Dic3semidirect
70. #380Dic3semidirect
132. C22×Q8C2×Dic3semidirect
152. C2×C4⋊C4C2×Dic3semidirect
153. C2×C4⋊C4C2×Dic3semidirect
294. C22×C3#328semidirect
395. C22#4538semidirect
412. C3#9143semidirect

Maximal subgroups

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

subgroupindex
C3×#28482
C22×C3⋊#952
#721262
C2×#45382
#676242
#91433