Finite Group Databasebeta

← #10692882

Subgroups — #10692882

Order 768.

448 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2567C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71776C212yesmaximal
4#71625C211yesmaximal, characteristic
5#67621C212yesmaximal
6#9454—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#395—21no—
10C3×#395—21no—
11C3×#391C412yes—
12C3×#389—21no—
13C3×#389—21no—
14C3×#384C412yes—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5024—24no—
17#4926—21no—
18#4926C2212yes—
19#4926C2212yes—
20#4925—21no—
21#4547—24no—
22#4528—21no—
23#4528—21no—
24#2983—31no—
25#2567D311yescharacteristic
26#1556—32no—
27#1467—31no—
28#1285—32no—
29C22×C3×Q8—21no—
30C22×C3×Q8C2×C411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C42C2×C412yes—
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—21no—
50#830—22no—
51#830—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yes—
56#803—22no—
57#803—22no—
58#803—22no—
59C4×C3⋊C8—22no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C4×Q8C2×D311yescharacteristic
62#446—64no—
63#432—32no—
64#432—32no—
65#432—61no—
66#431—61no—
67#395—21no—
68#395—21no—
69#391Dic312yes—
70#389—21no—
71#389—21no—
72#384Dic312yes—
73C4×C4⋊C4C2×D311yescharacteristic
74#332—64no—
75#324—61no—
76#324—61no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8—42no—
81C2×C3×Q8—82no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C3×C4⋊C4—24no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—24no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—82no—
99C3×C4⋊C4—82no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42—21no—
104C3×C42—42no—
105C3×C42C2×D412yes—
106C3×C42#4112yes—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C22×Q8—21no—
110C22×Q8C2×Dic311yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4C2×Dic312yes—
120C2×C4⋊C4C2×Dic311yescharacteristic
121C2×C4⋊C4#6711yescharacteristic
122C2×C4⋊C4#6711yescharacteristic
123C2×C42C2×Dic312yes—
124C2×C42#6711yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—34no—
128C4⋊C8—62no—
129C4⋊C8—62no—
130C4⋊C8—62no—
131C22⋊C8—62no—
132C22⋊C8—62no—
133#97—62no—
134#97—62no—
135#97—62no—
136C4×C8—62no—
137#95—22no—
138#95—22no—
139#95—21no—
140C23×C3C2×C22⋊C411yescharacteristic
141C3×Q8—44no—
142C3×Q8—84no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4#10112yes—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12112yes—
166C2×C3×C4#12311yescharacteristic
167C3⋊C8—44no—
168C3⋊C8—82no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—42no—
173C2×Q8—82no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4#21711yescharacteristic
177C22×C4#21711yescharacteristic
178C22×C4#21711yescharacteristic
179C22×C4#21711yescharacteristic
180C22×C4C22×Dic311yescharacteristic
181C22×C4C2×#6711yescharacteristic
182C22×C4C2×#6711yescharacteristic
183C2×C8—64no—
184C2×C8—122no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—24no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—42no—
199C42D3×D412yes—
200C42#23712yes—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3#38611yescharacteristic
204C22×C3C2×#10111yescharacteristic
205C22×C3#41311yescharacteristic
206C3×C4—21no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—44no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4—41no—
214C3×C4—82no—
215C3×C4#45612yes—
216C23C2×#21711yescharacteristic
217Q8—44no—
218Q8—84no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4#83612yes—
239C2×C4D4×Dic312yes—
240C2×C4#93511yescharacteristic
241C2×C4#93711yescharacteristic
242C2×C4#93912yes—
243C8—124no—
244C8—242no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3#188611yescharacteristic
248C2×C3#190111yescharacteristic
249C2×C3#191011yescharacteristic
250C22—21no—
251C22—21no—
252C22#522011yescharacteristic
253C22C2×#83611yescharacteristic
254C22#523711yescharacteristic
255C4—22no—
256C4—22no—
257C4—21no—
258C4—22no—
259C4—44no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—82no—
264C4#509512yes—
265C3#945411yescharacteristic, complemented, Sylow
266C2—21no—
267C2—21no—
268C2#7169811yescharacteristic
269C2#7178411yescharacteristic
270C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2567C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2567D3
hypocenterC3#9454
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
265. C3#9454semidirect

Maximal subgroups

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

subgroupindex
C3×#25672
#824362
#717762
#716252
#676212
#94543