Finite Group Databasebeta

← #11785498

Subgroups — #11785498

Order 1152.

653 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 653 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C2×#64116C211yesmaximal, characteristic
2#10591232C212yesmaximal
3#10591138C212yesmaximal, complemented
4C3×#4557C212yesmaximal, complemented
5#68710—32nomaximal
6C2×#3828C2211yescharacteristic
7C2×C32×M16C2211yescharacteristic
8C22×C3⋊(C3⋊C8)C2211yescharacteristic
9#64116—22no—
10#64116—22no—
11C2×#3743C412yescomplemented
12C2×C8×C3⋊D3C412yescomplemented
13C3×#926C2214yes—
14#63581—24no—
15#63576—22no—
16#63574—22no—
17C2×#907—32no—
18C2×#907—61no—
19#5006—34no—
20#4998—34no—
21C3×H(3,ℤ/4)—32no—
22#4557D312yescomplemented
23C22×C32×C4C2311yescharacteristic
24C2×#638D412yes—
25#3828—22no—
26#3828—22no—
27C2×C4⋊(C3⋊D3)Q811yescharacteristic
28C2×C4×C3⋊D3C2×C412yes—
29C2×#634Q811yescharacteristic
30C32×M16—22no—
31C2×C32×C8C2×C412yes—
32C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
33C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
34C2×C3⋊(C3⋊C8)D412yes—
35C2×C3⋊(C3⋊C8)D412yes—
36#3743—24no—
37C8×C3⋊D3—22no—
38C8×C3⋊D3—22no—
39C3×#217—24no—
40C3×C4⋊Dic3—24no—
41C3×C4×Dic3—24no—
42#1812—91noSylow
43C2×#235—32no—
44C2×#235—61no—
45C2×C3×M16—21no—
46C2×C3×M16C2×D312yes—
47C3×#117—34no—
48#926C2×D314yes—
49C22×C3⋊C8—32no—
50C22×C3⋊C8—61no—
51#907—64no—
52#907—64no—
53#907—64no—
54#907—64no—
55C2×ΓL(1,𝔽25)—34no—
56C2×ΓL(1,𝔽25)—62no—
57C2×D3×C8—34no—
58C2×D3×C8—62no—
59#808—64no—
60#807—64no—
61#806—68no—
62C23×C32C2×D411yescharacteristic
63C22×C3⋊D3C4⋊C412yes—
64C2×C32×C4C22×C411yescharacteristic
65C2×C32×C4C2×D412yes—
66C2×C32×C4C2×Q811yescharacteristic
67C2×C32×C4#4112yes—
68#638—24no—
69C2×C3⋊Dic3C4⋊C412yes—
70C4⋊(C3⋊D3)—22no—
71C4×C3⋊D3—22no—
72C4×C3⋊D3—22no—
73#634—22no—
74C2×C3×Dic3—44no—
75C32×C8—22no—
76C3⋊(C3⋊C8)C22⋊C411yescharacteristic
77C3⋊(C3⋊C8)C22⋊C411yescharacteristic
78C3⋊(C3⋊C8)C22⋊C412yes—
79C2×#131—91no—
80#427—92no—
81#410—92no—
82H(3,ℤ/4)—32no—
83C22×C3×C4—21no—
84C22×C3×C4C22×D312yes—
85C2×#67—34no—
86C2×#67—62no—
87#235—64no—
88#235—64no—
89#235—64no—
90#235—64no—
91C2×D12—32no—
92C2×D12—61no—
93C2×C4×D3—34no—
94C2×C4×D3—62no—
95C2×Dic6—32no—
96C2×Dic6—61no—
97C3×M16—24no—
98C3×M16—24no—
99C2×C3×C8—22no—
100C2×C3×C8C4×D314yes—
101C3×C4⋊C4—64no—
102C3×C22⋊C4—64no—
103C3×C42—64no—
104#217—24no—
105C4⋊Dic3—24no—
106C4×Dic3—24no—
107C2×C3⋊C8—32no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—34no—
110C2×C3⋊C8—34no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—61no—
113C2×C3⋊C8—62no—
114C2×C3⋊C8—62no—
115ΓL(1,𝔽25)—68no—
116ΓL(1,𝔽25)—68no—
117D3×C8—64no—
118D3×C8—64no—
119D3×C8—64no—
120D3×C8—64no—
121C22×C32—41no—
122C22×C32C2×C4⋊C411yescharacteristic
123C22×C32#12112yes—
124C2×C3⋊D3—22no—
125C2×C3⋊D3—42no—
126C32×C4C2×C22⋊C411yescharacteristic
127C32×C4#11711yescharacteristic
128C32×C4C22⋊Q812yes—
129C3⋊Dic3—22no—
130C3×Dic3—84no—
131C2×#41—91no—
132#131—182no—
133#131—182no—
134C2×M16—92no—
135C2×M16D3211yescharacteristic
136C22×C8—91no—
137C22×C8—92no—
138#117—34no—
139#107—182no—
140#106—182no—
141#99—184no—
142C23×C3—21no—
143C23×C3D3×D412yes—
144C22×D3—34no—
145C22×D3—62no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—124no—
151C2×C3×C4C2×C4×D312yes—
152C2×C3×C4#23514yes—
153C2×C3×C4D3×D414yes—
154C2×C3×C4D3×Q812yes—
155#67—68no—
156#67—68no—
157C2×Dic3—34no—
158C2×Dic3—44no—
159C2×Dic3—62no—
160D12—64no—
161D12—64no—
162C4×D3—64no—
163C4×D3—64no—
164C4×D3—64no—
165C4×D3—64no—
166Dic6—64no—
167Dic6—64no—
168C3×C8—24no—
169C3×C8—24no—
170C3⋊C8—32no—
171C3⋊C8—32no—
172C3⋊C8—34no—
173C3⋊C8—61no—
174C3⋊C8—61no—
175C3⋊C8—62no—
176C2×C32—21no—
177C2×C32—22no—
178C2×C32#38011yescharacteristic
179C3⋊D3—42no—
180#41—182no—
181#41—182no—
182C2×Q8—91no—
183C2×D4—92no—
184C2×D4—91no—
185C22×C4—92no—
186C22×C4C2×D3211yescharacteristic
187M16—22no—
188M16—184no—
189C2×C8—91no—
190C2×C8—91no—
191C2×C8—92no—
192C2×C8—92no—
193C2×C8—182no—
194C2×C8—182no—
195C2×C8#62412yes—
196C4⋊C4—64no—
197C22⋊C4—64no—
198C42—64no—
199C22×C3—21no—
200C22×C3—22no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3#88414yes—
204C22×C3D3×C4⋊C412yes—
205C2×D3—64no—
206C2×D3—64no—
207C2×D3—124no—
208C2×D3—124no—
209C3×C4—22no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—244no—
213C3×C4#87212yes—
214C3×C4D3×C22⋊C412yes—
215C3×C4#89614yes—
216Dic3—64no—
217Dic3—64no—
218Dic3—84no—
219C32#181211yescharacteristic, complemented, Sylow
220C23—92no—
221C23#380211yescharacteristic
222Q8—182no—
223D4—184no—
224D4—182no—
225C2×C4—92no—
226C2×C4—124no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4#379711yescharacteristic
230C2×C4#379812yes—
231C2×C4C2×#62411yescharacteristic
232C2×C4#381112yes—
233C8—22no—
234C8—91no—
235C8—91no—
236C8—92no—
237C2×C3—22no—
238C2×C3—24no—
239C2×C3—22no—
240C2×C3—21no—
241C2×C3—42no—
242C2×C3#466712yes—
243D3—124no—
244D3—124no—
245C22—41no—
246C22—182no—
247C22—362no—
248C22#6390011yescharacteristic
249C22#6391212yes—
250C4—182no—
251C4—244no—
252C4#6386512yes—
253C4#6387411yescharacteristic
254C4#6397411yescharacteristic
255C3—21no—
256C3#6871012yescomplemented
257C2—22no—
258C2—21no—
259C2—362no—
260C2#1059135511yescharacteristic
subgroupG/H
centerC4#63874
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#380
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1812
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. #10591138C2semidirect
4. C3×#4557C2semidirect
11. C2×#3743C4semidirect
12. C2×C8×C3⋊D3C4semidirect
22. #4557D3semidirect
219. C32#1812semidirect
256. C3#68710semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105912322
#105911382
C3×#45572
#687103