Finite Group Databasebeta

← #10690908

Subgroups — #10690908

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2855C211yesmaximal, characteristic
2C2×C4×C4⋊Dic3C211yesmaximal, characteristic
3C2×#4538C212yesmaximal
4#71531C211yesmaximal, characteristic
5#67624C212yesmaximal
6#9144—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C22×C4⋊Dic3—21no—
10C22×C4×Dic3—21no—
11C3×#391C412yescomplemented
12C3×#389C412yescomplemented
13C3×#387—22no—
14C3×#385—21no—
15C3×#381—21no—
16C3×#380C2211yescharacteristic
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19C4×C4⋊Dic3—24no—
20#4538—24no—
21#4524—22no—
22#2855D311yescharacteristic
23C2×C4×C4⋊C4—31no—
24#1817—31no—
25C2×#328—32no—
26#1294—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C23×Dic3—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C42—22no—
41C2×C3×C42—22no—
42C2×C3×C42C2×C412yes—
43C2×C3×C42D412yes—
44C2×C4⋊Dic3—24no—
45C2×C4⋊Dic3—28no—
46C2×C4×Dic3—24no—
47C2×C4×Dic3—42no—
48C2×C4×Dic3—42no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95—21no—
54C3⋊#95—22no—
55C3⋊#95D414yes—
56#830—22no—
57#830—22no—
58C2×C22⋊Q8C2×D311yescharacteristic
59C22×C4⋊C4—61no—
60C22×C42—61no—
61C22×C42C2×D311yescharacteristic
62C2×C22⋊C8—32no—
63#391Dic312yescomplemented
64#389Dic312yescomplemented
65#387—22no—
66#385—21no—
67#381—21no—
68#380C2×D311yescharacteristic
69C4×C4⋊C4—64no—
70C2×#95—32no—
71#336—62no—
72#328—64no—
73C24×C3C2×D411yescharacteristic
74C2×C3×Q8—22no—
75C2×C3×Q8—42no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—24no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4Q1612yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4112yes—
95C22×Dic3—28no—
96C22×Dic3—24no—
97C22×Dic3—44no—
98C22×Dic3—42no—
99C22×Dic3—42no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—44no—
103C3×C4⋊C4—42no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C42—24no—
109C3×C42—24no—
110C4⋊Dic3—216no—
111C4×Dic3—48no—
112C2×C3⋊C8—44no—
113C2×C3⋊C8—42no—
114C22×Q8C2×Dic311yescharacteristic
115C23×C4—62no—
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—64no—
125C2×C4⋊C4—68no—
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C4⋊C4C2×Dic312yes—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—64no—
133C2×C42—122no—
134C2×C42—122no—
135C2×C42C2×Dic312yes—
136C2×C42#6712yes—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—21no—
143#95—34no—
144#95—62no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3#12011yescharacteristic
149C23×C3#12311yescharacteristic
150C23×C3C2×SD1611yescharacteristic
151C23×C3C2×Q1611yescharacteristic
152C23×C3AGL(1,ℤ/8)11yescharacteristic
153C23×C3#13711yescharacteristic
154C3×Q8—82no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4#10314yes—
177C2×C3×C4#10414yes—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12112yes—
180C2×Dic3—28no—
181C2×Dic3—44no—
182C2×Dic3—48no—
183C2×Dic3—48no—
184C2×Dic3—44no—
185C3⋊C8—82no—
186C24C2×#6711yescharacteristic
187C2×Q8—22no—
188C2×Q8—42no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—68no—
201C22×C4—64no—
202C22×C4—124no—
203C22×C4—122no—
204C22×C4—122no—
205C22×C4#21512yes—
206C22×C4#21612yes—
207C22×C4#21712yes—
208C22×C4#21712yes—
209C22×C4D3×D412yes—
210C22×C4#23712yes—
211C22×C4C22×Dic311yescharacteristic
212C22×C4C2×#6711yescharacteristic
213C2×C8—124no—
214C2×C8—122no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—616no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C42—24no—
225C42—24no—
226C42—128no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3#38611yescharacteristic
236C22×C3#41012yes—
237C22×C3C2×#10311yescharacteristic
238C22×C3#41711yescharacteristic
239C22×C3C2×#10412yes—
240C22×C3#45011yescharacteristic
241C22×C3#45111yescharacteristic
242C22×C3#48311yescharacteristic
243C22×C3#48411yescharacteristic
244C3×C4—24no—
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—82no—
251Dic3—48no—
252C23—22no—
253C23—22no—
254C23#93511yescharacteristic
255C23#93711yescharacteristic
256C23C2×#21511yescharacteristic
257C23C2×#21611yescharacteristic
258C23#94911yescharacteristic
259C23#95111yescharacteristic
260C23C2×#21711yescharacteristic
261Q8—82no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—68no—
284C2×C4—124no—
285C2×C4—128no—
286C2×C4—128no—
287C2×C4—124no—
288C2×C4#83514yes—
289C2×C4#83714yes—
290C2×C4D4×Dic312yes—
291C2×C4#93912yes—
292C8—242no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3#150312yes—
298C2×C3#150912yes—
299C2×C3#187211yescharacteristic
300C2×C3#189611yescharacteristic
301C2×C3#189911yescharacteristic
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22#504311yescharacteristic
311C22#504411yescharacteristic
312C22#504611yescharacteristic
313C22#505011yescharacteristic
314C22#522011yescharacteristic
315C22C2×#83511yescharacteristic
316C22#523211yescharacteristic
317C22C2×#83712yes—
318C22#524012yes—
319C4—24no—
320C4—22no—
321C4—22no—
322C4—24no—
323C4—81no—
324C4—81no—
325C4—82no—
326C4—128no—
327C3#914411yescharacteristic, complemented, Sylow
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#7169211yescharacteristic
333C2#7170511yescharacteristic
334C2#7171111yescharacteristic
335C2#7175512yes—
336C2#7183512yes—
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2855C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2855D3
hypocenterC3#9144
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
11. C3×#391C4semidirect
12. C3×#389C4semidirect
63. #391Dic3semidirect
64. #389Dic3semidirect
327. C3#9144semidirect

Maximal subgroups

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

subgroupindex
C3×#28552
C2×C4×C4⋊Dic32
C2×#45382
#715312
#676242
#91443