Finite Group Databasebeta

← #10690708

Subgroups — #10690708

Order 768.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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×#2327C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71624C211yesmaximal, characteristic
4#71621C212yesmaximal
5#67614C212yesmaximal
6#9104—31nomaximal, Sylow
7C2×C3×C4⋊Q8C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#391C411yescharacteristic
11C3×#391C411yescharacteristic
12C3×#387—22no—
13C3×#387—22no—
14C3×#382C411yescharacteristic
15C3×#376C2211yescharacteristic
16#5024—24no—
17#4926—22no—
18#4925C2212yes—
19#4925C2212yes—
20#4535—24no—
21#4528—22no—
22#2983—31no—
23#2327D311yescharacteristic
24#1479—32no—
25#1461—31no—
26#1274—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D412yes—
40C2×C3×C42—21no—
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42D412yes—
44C2×C3×C42C2311yescharacteristic
45C3×#95—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—21no—
49C3×#95—21no—
50#830—22no—
51#830—22no—
52C4⋊(C3⋊C8)—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#803—24no—
56C4×C3⋊C8—22no—
57C4×C3⋊C8D414yes—
58C2×C4⋊Q8Dic311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×Q8C2×D311yescharacteristic
61#446—64no—
62#432—62no—
63#431—32no—
64#431—32no—
65#391Dic311yescharacteristic
66#391Dic311yescharacteristic
67#387—22no—
68#387—22no—
69#382Dic311yescharacteristic
70#376C2×D311yescharacteristic
71Q8⋊C8—64no—
72#324—62no—
73C2×C3×Q8—22no—
74C2×C3×Q8—21no—
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8M1612yes—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C3×C4⋊C4—28no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C42—22no—
97C3×C42—22no—
98C3×C42—21no—
99C3×C42—22no—
100C3×C42—42no—
101C3×C42M1612yes—
102C3×C42C2×D412yes—
103C3×C42#4112yes—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—42no—
106C22×Q8C2×Dic311yescharacteristic
107C22×Q8C2×Dic311yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×Dic311yescharacteristic
117C2×C4⋊C4C2×Dic311yescharacteristic
118C2×C4⋊C4#6712yes—
119C2×C42—21no—
120C2×C42C2×Dic311yescharacteristic
121C2×C42C2×Dic311yescharacteristic
122C2×C42#6712yes—
123C2×C42C22×D311yescharacteristic
124C4⋊C8—64no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127C22⋊C8—62no—
128C22⋊C8—62no—
129#97—64no—
130C4×C8—34no—
131C4×C8—62no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—21no—
137C23×C3C2×C22⋊C411yescharacteristic
138C3×Q8—24no—
139C3×Q8—42no—
140C3×Q8—42no—
141C3×Q8—42no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4C4×D412yes—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12112yes—
163C2×C3×C4#12311yescharacteristic
164C2×C3×C4C2×M1611yescharacteristic
165C2×C3×C4C2×M1611yescharacteristic
166C2×C3×C4#13112yes—
167C3⋊C8—44no—
168C3⋊C8—82no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8#20812yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4#21712yes—
182C22×C4#21712yes—
183C22×C4C22×Dic311yescharacteristic
184C22×C4C2×#6711yescharacteristic
185C22×C4C2×#6711yescharacteristic
186C2×C8—64no—
187C2×C8—122no—
188C4⋊C4—28no—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C42—21no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—42no—
199C42#20812yes—
200C42D3×D412yes—
201C42#23712yes—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3#38611yescharacteristic
205C22×C3#40711yescharacteristic
206C22×C3#40811yescharacteristic
207C3×C4—22no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4—42no—
212C3×C4—42no—
213C3×C4—42no—
214C3×C4—81no—
215C3×C4—81no—
216C3×C4#43512yes—
217C3×C4#43612yes—
218C23C2×#21711yescharacteristic
219Q8—24no—
220Q8—42no—
221Q8—42no—
222Q8—42no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4C2×#20811yescharacteristic
242C2×C4C2×#20811yescharacteristic
243C2×C4D4×Dic312yes—
244C2×C4#93511yescharacteristic
245C2×C4#93711yescharacteristic
246C2×C4#93912yes—
247C2×C4#94812yes—
248C8—124no—
249C8—242no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3#187011yescharacteristic
253C2×C3#190911yescharacteristic
254C2×C3#191211yescharacteristic
255C22—21no—
256C22—21no—
257C22#520711yescharacteristic
258C22#522011yescharacteristic
259C22#523411yescharacteristic
260C4—21no—
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—81no—
268C4—81no—
269C4#501112yes—
270C4#501212yes—
271C3#910411yescharacteristic, complemented, Sylow
272C2—21no—
273C2—21no—
274C2#7161111yescharacteristic
275C2#7186411yescharacteristic
276C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2327C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2327D3
hypocenterC3#9104
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
271. C3#9104semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#824362
#716242
#716212
#676142
#91043