Finite Group Databasebeta

← #10692615

Subgroups — #10692615

Order 768.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#82476C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3C3×#1801C211yesmaximal, characteristic, complemented
4#70502C212yesmaximal
5#67715C212yesmaximal, complemented
6#9406—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#947—21no—
9C2×#874C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×#101—21no—
12#5024—24no—
13#4991C414yes—
14#4988—21no—
15#4939—22no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4926—22no—
18#4925C2212yes—
19C3×#343—21no—
20#4485—24no—
21#2983—31no—
22#2548—31no—
23#1801D311yescharacteristic, complemented
24#1545—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×D12—21no—
28C22×D12—21no—
29C22×D12C2×C412yes—
30C2×C3×M16—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4C2×C412yes—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C42D412yes—
36C2×C3×C42C2311yescharacteristic
37#947—44no—
38C2×(C2×D3)⋊C4—24no—
39C2×(C2×D3)⋊C4—24no—
40C2×C4⋊Dic3—21no—
41C2×C4×Dic3—21no—
42#874—22no—
43#874—22no—
44C3×C4⋊C8—22no—
45C3×C4⋊C8—22no—
46C3×#101—42no—
47C3⋊#95C2×C412yes—
48#830—22no—
49C4⋊(C3⋊C8)—24no—
50C4⋊(C3⋊C8)D414yes—
51#803—24no—
52C4×C3⋊C8—22no—
53C2×#127—31no—
54C2×#124—61no—
55C2×C4×Q8C2×D311yescharacteristic
56#446—64no—
57#432—62no—
58#431—32no—
59#423C2×D312yes—
60C2×#101—21no—
61#394—34no—
62#390—61no—
63#390—62no—
64C4×C4⋊C4—31no—
65#343—21no—
66#331—64no—
67C23×D3—22no—
68C2×C3×Q8—22no—
69C2×C3×Q8—21no—
70C2×C3×Q8—22no—
71C2×C3×Q8—21no—
72C22×C3×C4C22⋊C412yes—
73C22×C3×C4C22⋊C412yes—
74C22×C3×C4C22×C411yescharacteristic
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—22no—
78C2×D12—24no—
79C2×D12—44no—
80C2×D12—41no—
81C2×D12—42no—
82C2×D12—41no—
83C2×D12—42no—
84C2×D12—41no—
85C2×D12—41no—
86C2×D12—82no—
87C3×M16—42no—
88C3×M16—42no—
89C2×C3×C8—42no—
90C3×C4⋊C4—24no—
91C3×C4⋊C4—22no—
92C3×C42—22no—
93C3×C42—22no—
94C3×C42C2×D412yes—
95C3×C42#4112yes—
96(C2×D3)⋊C4—48no—
97(C2×D3)⋊C4—84no—
98C4⋊Dic3—42no—
99C4×Dic3—42no—
100C2×C3⋊C8—24no—
101C22×Q8#6711yescharacteristic
102C22×D4—32no—
103C22×D4—61no—
104C22×D4—61no—
105C2×M16—22no—
106#127—62no—
107#127—62no—
108#124—124no—
109C4×Q8—22no—
110C4×Q8—22no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4C4×D312yes—
113C2×C4⋊C4#6711yescharacteristic
114C2×C22⋊C4—64no—
115C2×C22⋊C4—64no—
116C2×C42—61no—
117C2×C42D1212yes—
118C2×C42C22×D311yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—34no—
122C4⋊C8—64no—
123#101—42no—
124C22⋊C8—62no—
125#97—64no—
126C4×C8—62no—
127#95—32no—
128C23×C3C2×C22⋊C411yescharacteristic
129C22×D3—44no—
130C22×D3—44no—
131C22×D3—42no—
132C22×D3—42no—
133C22×D3—42no—
134C3×Q8—42no—
135C3×Q8—42no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—41no—
142C2×C3×C4—42no—
143C2×C3×C4C4×D412yes—
144C2×C3×C4#12011yescharacteristic
145C2×C3×C4#12112yes—
146C2×C3×C4#12311yescharacteristic
147C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
148C2×C3×C4AGL(1,ℤ/8)12yes—
149C2×C3×C4#13711yescharacteristic
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153D12—84no—
154D12—82no—
155D12—82no—
156D12—82no—
157D12—82no—
158C3×C8—82no—
159C3⋊C8—44no—
160C24—62no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—21no—
164C2×Q8—22no—
165C2×D4—64no—
166C2×D4—124no—
167C2×D4—121no—
168C2×D4—121no—
169C2×D4—121no—
170C2×D4—122no—
171C2×D4—121no—
172C2×D4—122no—
173C2×D4—242no—
174C22×C4—62no—
175C22×C4(C2×D3)⋊C412yes—
176C22×C4(C2×D3)⋊C412yes—
177C22×C4C2×C4×D311yescharacteristic
178C22×C4C2×D1211yescharacteristic
179C22×C4C2×#6711yescharacteristic
180M16—42no—
181M16—42no—
182C2×C8—42no—
183C2×C8—64no—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—122no—
187C22⋊C4—128no—
188C22⋊C4—244no—
189C42—22no—
190C42—22no—
191C42—122no—
192C42D3×D412yes—
193C42#23712yes—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#38611yescharacteristic
197C22×C3#41711yescharacteristic
198C22×C3#41811yescharacteristic
199C2×D3—84no—
200C2×D3—84no—
201C2×D3—82no—
202C2×D3—82no—
203C2×D3—82no—
204C3×C4—21no—
205C3×C4—22no—
206C3×C4—42no—
207C3×C4—42no—
208C3×C4#45412yes—
209C3×C4#46812yes—
210C3×C4#46912yes—
211Dic3—82no—
212C23—124no—
213C23—124no—
214C23—122no—
215C23—122no—
216C23—122no—
217C23C2×(C2×D3)⋊C411yescharacteristic
218Q8—42no—
219Q8—42no—
220D4—244no—
221D4—242no—
222D4—242no—
223D4—242no—
224D4—242no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4#67⋊C412yes—
236C2×C4#88211yescharacteristic
237C2×C4#88611yescharacteristic
238C2×C4#93211yescharacteristic
239C2×C4#93912yes—
240C2×C4#94211yescharacteristic
241C2×C4#94912yes—
242C8—82no—
243C8—124no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3#188811yescharacteristic
247C2×C3#189511yescharacteristic
248C2×C3#191411yescharacteristic
249D3—162no—
250C22—21no—
251C22—21no—
252C22—244no—
253C22—244no—
254C22—242no—
255C22—242no—
256C22—242no—
257C22#495811yescharacteristic
258C22#496611yescharacteristic
259C22#500111yescharacteristic
260C4—21no—
261C4—22no—
262C4—42no—
263C4—42no—
264C4—242no—
265C4#475312yes—
266C4#503912yes—
267C4#504912yes—
268C3#940611yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—482no—
272C2#6951711yescharacteristic
273C2#7010411yescharacteristic
274C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1801C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1801D3
hypocenterC3#9406
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. C3×#1801C2semidirect
5. #67715C2semidirect
23. #1801D3semidirect
268. C3#9406semidirect

Maximal subgroups

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

subgroupindex
#824762
#824362
C3×#18012
#705022
#677152
#94063