Finite Group Databasebeta

← #10692618

Subgroups — #10692618

Order 768.

554 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2548C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71759C212yesmaximal
4#71637C211yesmaximal, characteristic, complemented
5#67620C212yesmaximal, complemented
6#9406—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×C3×#124—21no—
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#394C414yes—
11C3×#390—21no—
12C3×#390—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#836—21no—
15#5024—24no—
16#4926—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4544—24no—
20#4530—21no—
21#2983—31no—
22#2548D311yescharacteristic
23#1801—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C3×#127—22no—
31C3×#127—22no—
32C3×#124—44no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C412yes—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4—24no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C42—21no—
41C2×C3×C42D412yes—
42C2×C3×C42C2311yescharacteristic
43C2×#208—22no—
44C3×#95C2×C412yes—
45#836—42no—
46#830—22no—
47C4⋊(C3⋊C8)—24no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51#803—24no—
52C4×C3⋊C8—22no—
53C2×#127C2×D311yescharacteristic
54C2×#124—21no—
55C2×C4×Q8C2×D311yescharacteristic
56#446—64no—
57#432—62no—
58#431—32no—
59#423—32no—
60C2×#101—61no—
61#394Dic314yes—
62#390—21no—
63#390—22no—
64C4×C4⋊C4C2×D311yescharacteristic
65#343—61no—
66#331—64no—
67C24×C3—22no—
68C2×C3×Q8—22no—
69C2×C3×Q8—21no—
70C2×C3×Q8—22no—
71C2×C3×Q8—21no—
72C2×C3×D4—24no—
73C2×C3×D4—44no—
74C2×C3×D4—41no—
75C2×C3×D4—42no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—82no—
81C22×C3×C4—22no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C3×C4⋊C4—24no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—42no—
90C3×C22⋊C4—48no—
91C3×C22⋊C4—84no—
92C3×C42—22no—
93C3×C42—22no—
94C3×C42—42no—
95C3×C42C2×D412yes—
96C3×C42#4112yes—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—24no—
100C2×C3⋊C8—42no—
101C22×Q8#6711yescharacteristic
102C22×D4—21no—
103C22×D4—21no—
104C22×D4C2×Dic312yes—
105C2×M16—62no—
106#127—22no—
107#127—22no—
108#124—44no—
109C4×Q8—22no—
110C4×Q8—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2×Dic312yes—
113C2×C4⋊C4#6711yescharacteristic
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C42—21no—
117C2×C42#6712yes—
118C2×C42C22×D311yescharacteristic
119C4⋊C8—34no—
120C4⋊C8—64no—
121C4⋊C8—62no—
122C4⋊C8—62no—
123#101—122no—
124C22⋊C8—62no—
125#97—64no—
126C4×C8—62no—
127#95C2×Dic312yes—
128C23×C3—44no—
129C23×C3—44no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3C2×C22⋊C411yescharacteristic
134C3×Q8—42no—
135C3×Q8—42no—
136C3×D4—84no—
137C3×D4—82no—
138C3×D4—82no—
139C3×D4—82no—
140C3×D4—82no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—41no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4C4×D412yes—
152C2×C3×C4#12011yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4#12311yescharacteristic
155C2×C3×C4AGL(1,ℤ/8)12yes—
156C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
157C2×C3×C4#13711yescharacteristic
158C3⋊C8—44no—
159C3⋊C8—82no—
160C24—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×D4—24no—
166C2×D4—44no—
167C2×D4—42no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—82no—
174C22×C4—22no—
175C22×C4#21712yes—
176C22×C4#21712yes—
177C22×C4C22×Dic311yescharacteristic
178C22×C4C2×#6711yescharacteristic
179C22×C4C2×#6711yescharacteristic
180M16—122no—
181M16—122no—
182C2×C8—64no—
183C2×C8—122no—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C22⋊C4—48no—
188C22⋊C4—84no—
189C42—22no—
190C42—22no—
191C42—42no—
192C42D3×D412yes—
193C42#23712yes—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—84no—
197C22×C3—84no—
198C22×C3—82no—
199C22×C3—82no—
200C22×C3—82no—
201C22×C3#38611yescharacteristic
202C22×C3#41711yescharacteristic
203C22×C3#41811yescharacteristic
204C3×C4—21no—
205C3×C4—22no—
206C3×C4—42no—
207C3×C4—42no—
208C3×C4—82no—
209C3×C4#45412yes—
210C3×C4#46812yes—
211C3×C4#46912yes—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23C2×#21711yescharacteristic
218Q8—42no—
219Q8—42no—
220D4—84no—
221D4—82no—
222D4—82no—
223D4—82no—
224D4—82no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4#93212yes—
236C2×C4D4×Dic312yes—
237C2×C4#93511yescharacteristic
238C2×C4#93711yescharacteristic
239C2×C4#93912yes—
240C2×C4#94911yescharacteristic
241C2×C4#95111yescharacteristic
242C8—124no—
243C8—242no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—162no—
247C2×C3#188811yescharacteristic
248C2×C3#189511yescharacteristic
249C2×C3#191411yescharacteristic
250C22—21no—
251C22—21no—
252C22—84no—
253C22—84no—
254C22—82no—
255C22—82no—
256C22—82no—
257C22#521511yescharacteristic
258C22#522011yescharacteristic
259C22#523211yescharacteristic
260C4—21no—
261C4—22no—
262C4—42no—
263C4—42no—
264C4—82no—
265C4#503912yes—
266C4#504212yes—
267C4#506212yes—
268C3#940611yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—162no—
272C2#7168211yescharacteristic
273C2#7172411yescharacteristic
274C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2548C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#2548D3
hypocenterC3#9406
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
4. #71637C2semidirect
5. #67620C2semidirect
268. C3#9406semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#824362
#717592
#716372
#676202
#94063