Finite Group Databasebeta

← #10691201

Subgroups — #10691201

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2628C211yesmaximal, characteristic
2#82204C211yesmaximal, characteristic
3#71775C212yesmaximal, complemented
4#71571C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9193—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#400C412yes—
11C3×#399C412yes—
12C3×#395—21no—
13C3×#393—21no—
14C3×#389—21no—
15C3×#382C2211yescharacteristic
16C3×#380—21no—
17#4999—24no—
18#4925C2212yes—
19#4924—21no—
20#4924C2212yes—
21#4547—24no—
22#4528—21no—
23#4525—21no—
24#2905—31no—
25#2628D311yescharacteristic
26#1549—32no—
27#1458—31no—
28#1286—32no—
29C22×C3×Q8—21no—
30C23×C3×C4D411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×#117—22no—
34C3×#117—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C412yes—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×#208—22no—
47C2×#208—22no—
48C2×#208—42no—
49C2×#208—42no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95—21no—
54C3×#95C2×C412yes—
55#830—22no—
56C4⋊(C3⋊C8)—24no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)—22no—
59C4⋊(C3⋊C8)D414yes—
60C4×C3⋊C8—22no—
61C22×M16—61no—
62C2×C4⋊Q8C2×D311yescharacteristic
63C2×#117C2×D311yescharacteristic
64#431—32no—
65#424—64no—
66#423—32no—
67#423—61no—
68#400Dic312yes—
69#399Dic312yes—
70#395—21no—
71#393—21no—
72#389—21no—
73#382C2×D311yescharacteristic
74#380—21no—
75#332—64no—
76#324—61no—
77#323—61no—
78C24×C3C22⋊C411yescharacteristic
79C2×C3×Q8—42no—
80C2×C3×Q8—82no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4SD1612yes—
91C22×C3×C4Q1612yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—82no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—82no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42C2×D412yes—
109C3×C42#4112yes—
110#208—44no—
111#208—42no—
112#208—42no—
113C2×C3⋊C8—24no—
114C2×C3⋊C8—44no—
115C2×C3⋊C8—42no—
116C22×Q8—21no—
117C23×C4#6711yescharacteristic
118C2×M16—62no—
119C2×M16—62no—
120C2×M16—122no—
121C2×M16—122no—
122C22×C8—62no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4C2×Dic312yes—
131C2×C4⋊C4C2×Dic311yescharacteristic
132C2×C4⋊C4#6711yescharacteristic
133C2×C22⋊C4—22no—
134C2×C22⋊C4C2×Dic311yescharacteristic
135C2×C22⋊C4#6711yescharacteristic
136C2×C42#6711yescharacteristic
137C2×C42C22×D311yescharacteristic
138C4⋊C8—34no—
139C4⋊C8—64no—
140C4⋊C8—62no—
141C4⋊C8—62no—
142C22⋊C8—62no—
143C4×C8—62no—
144#95—22no—
145#95—22no—
146#95—21no—
147#95C2×Dic312yes—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3#10312yes—
153C23×C3C2×C22⋊C411yescharacteristic
154C3×Q8—84no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4#10312yes—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12011yescharacteristic
176C2×C3×C4#12112yes—
177C2×C3×C4#12311yescharacteristic
178C2×C3×C4C2×SD1611yescharacteristic
179C2×C3×C4C2×Q1611yescharacteristic
180C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C3⋊C8—44no—
183C3⋊C8—82no—
184C24#21711yescharacteristic
185C2×Q8—42no—
186C2×Q8—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4#21512yes—
194C22×C4#21612yes—
195C22×C4#21711yescharacteristic
196C22×C4#21711yescharacteristic
197C22×C4#21711yescharacteristic
198C22×C4C22×Dic311yescharacteristic
199C22×C4C2×#6711yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—124no—
202M16—122no—
203M16—122no—
204C2×C8—64no—
205C2×C8—124no—
206C2×C8—122no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—82no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—82no—
218C42—21no—
219C42—21no—
220C42D3×D412yes—
221C42#23712yes—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10311yescharacteristic
231C22×C3#41711yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—82no—
238C3×C4—82no—
239C3×C4#45011yescharacteristic
240C3×C4#45111yescharacteristic
241C3×C4#48311yescharacteristic
242C3×C4#48411yescharacteristic
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#83512yes—
248C23C2×#21711yescharacteristic
249Q8—84no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4#83512yes—
269C2×C4D4×Dic312yes—
270C2×C4#93511yescharacteristic
271C2×C4#93711yescharacteristic
272C2×C4#93912yes—
273C2×C4C2×#21511yescharacteristic
274C2×C4C2×#21611yescharacteristic
275C2×C4#94911yescharacteristic
276C2×C4#95111yescharacteristic
277C8—124no—
278C8—242no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3#187211yescharacteristic
283C2×C3#191011yescharacteristic
284C2×C3#191511yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22#522011yescharacteristic
293C22C2×#83511yescharacteristic
294C22#523211yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4#504311yescharacteristic
303C4#504411yescharacteristic
304C4#504611yescharacteristic
305C4#505011yescharacteristic
306C3#919311yescharacteristic, complemented, Sylow
307C2—21no—
308C2—21no—
309C2—42no—
310C2#7169211yescharacteristic
311C2#7185411yescharacteristic
312C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2628C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2628D3
hypocenterC3#9193
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #71775C2semidirect
5. #67631C2semidirect
306. C3#9193semidirect

Maximal subgroups

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

subgroupindex
C3×#26282
#822042
#717752
#715712
#676312
#91933