Finite Group Databasebeta

← #11800021

Subgroups — #11800021

Order 1152.

303 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 76 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#10592836C211yesmaximal, characteristic
2C3×#5125C211yesmaximal, characteristic
3C3×#4553C211yesmaximal, characteristic
4#10589137C212yesmaximal, complemented
5#10589131C212yesmaximal, complemented
6#70724—31nomaximal
7#69117—31nomaximal
8C2×C32×M16C2211yescharacteristic
9#64133—21no—
10C2×#3746C2211yescharacteristic
11C2×C3×#208—21no—
12#63644C2211yescharacteristic
13#63643—21no—
14C3×#819—21no—
15C3×#819C2212yes—
16C3×#812C2212yes—
17C3×#430—31no—
18#5137—31no—
19#5137—31no—
20#5137—61no—
21#5125D311yescharacteristic
22C3×#349—31no—
23#4553D311yescharacteristic
24#4517—32no—
25#4513—32no—
26#4496—32no—
27#4494—32no—
28C22×C32×C4D411yescharacteristic
29C2×#634—21no—
30C32×M16—22no—
31C2×C32×C8D411yescharacteristic
32C2×C32×C8C2311yescharacteristic
33#3757—41no—
34C4⋊(C3⋊Dic3)—21no—
35#3750—41no—
36#3746—22no—
37C3×#208—22no—
38C3×#208—41no—
39C2×C3×C3⋊C8—41no—
40C3×C3⋊C16—22no—
41#2239—91noSylow
42C2×C3×M16—21no—
43C2×C3×M16—61no—
44C2×C3×M16C2×D311yescharacteristic
45C2×C3×M16C2×D311yescharacteristic
46#924—62no—
47#924—61no—
48#924—61no—
49C2×#208—21no—
50C2×Dic12—31no—
51C2×Dic12—31no—
52C2×Dic12—61no—
53C3×M32—32no—
54C3×#108—32no—
55C3×#108—61no—
56#819—21no—
57#819C2×D312yes—
58#817—31no—
59#817—31no—
60#817—61no—
61#816—62no—
62#816—61no—
63#816—61no—
64#812C2×D312yes—
65C23×C32SD1611yescharacteristic
66C2×C32×C4—21no—
67C2×C32×C4—21no—
68C2×C32×C4SD1611yescharacteristic
69C2×C32×C4C2×D411yescharacteristic
70C2×C3⋊Dic3—41no—
71C2×C3⋊Dic3—41no—
72#634—41no—
73#634—41no—
74C32×C8—21no—
75C32×C8#4112yes—
76C3×C3⋊C8—42no—
77#470—91no—
78#430—31no—
79#365—92no—
80#362—92no—
81#349—31no—
82C22×C3×C4—21no—
83C22×C3×C4D1211yescharacteristic
84C22×C3×C4#6711yescharacteristic
85C2×Dic6—62no—
86C2×Dic6—61no—
87C2×Dic6—61no—
88C3×M16—22no—
89C3×M16—22no—
90C3×M16—42no—
91C3×M16—62no—
92C3×M16—121no—
93C2×C3×C8—21no—
94C2×C3×C8—21no—
95C2×C3×C8—121no—
96C2×C3×C8D1211yescharacteristic
97C2×C3×C8#6711yescharacteristic
98C2×C3×C8C22×D311yescharacteristic
99C2×C3×C8C22×D311yescharacteristic
100#217—122no—
101#217—121no—
102#217—121no—
103C4⋊Dic3—62no—
104C4⋊Dic3—61no—
105C4⋊Dic3—61no—
106Dic3⋊C4—122no—
107Dic3⋊C4—121no—
108Dic3⋊C4—121no—
109#208—22no—
110#208—41no—
111C2×C3⋊C8—41no—
112Dic12—62no—
113Dic12—62no—
114Dic12—122no—
115C3×C16—62no—
116C3⋊C16—22no—
117C22×C32—21no—
118C22×C32—41no—
119C22×C32C2×SD1611yescharacteristic
120C32×C4—21no—
121C32×C4#12311yescharacteristic
122C32×C4AGL(1,ℤ/8)11yescharacteristic
123C3⋊Dic3—81no—
124C3⋊Dic3—81no—
125C2×Q16—91no—
126C2×M16—61no—
127C2×M16D3211yescharacteristic
128C22⋊Q8—181no—
129M32—32no—
130#108—32no—
131#108—61no—
132#106—91no—
133#103—181no—
134C23×C3—21no—
135C23×C3#20411yescharacteristic
136C23×C3#21511yescharacteristic
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4#20411yescharacteristic
146C2×C3×C4#21511yescharacteristic
147C2×C3×C4C2×D1211yescharacteristic
148C2×C3×C4C2×#6711yescharacteristic
149C2×Dic3—122no—
150C2×Dic3—122no—
151C2×Dic3—121no—
152C2×Dic3—121no—
153C2×Dic3—121no—
154C2×Dic3—121no—
155Dic6—122no—
156Dic6—122no—
157Dic6—121no—
158Dic6—121no—
159Dic6—121no—
160Dic6—121no—
161C3×C8—21no—
162C3×C8—21no—
163C3×C8—22no—
164C3×C8—41no—
165C3×C8—122no—
166C3×C8#23512yes—
167C3×C8#23712yes—
168C3⋊C8—42no—
169C2×C32—21no—
170C2×C32—41no—
171C2×C32#48111yescharacteristic
172C2×Q8—181no—
173C22×C4#62611yescharacteristic
174Q16—182no—
175M16—22no—
176M16—62no—
177M16—121no—
178C2×C8—121no—
179C2×C8#62611yescharacteristic
180C2×C8C2×D3211yescharacteristic
181C4⋊C4—181no—
182C4⋊C4—361no—
183C22⋊C4—361no—
184C16—62no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3C2×#20411yescharacteristic
193C22×C3C2×#21511yescharacteristic
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—41no—
199C3×C4#88611yescharacteristic
200C3×C4#90811yescharacteristic
201C3×C4#92911yescharacteristic
202C3×C4#93211yescharacteristic
203Dic3—242no—
204Dic3—242no—
205Dic3—241no—
206Dic3—241no—
207Dic3—241no—
208Dic3—241no—
209C32#223911yescharacteristic, complemented, Sylow
210C23#371711yescharacteristic
211Q8—361no—
212Q8—361no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—361no—
216C2×C4—361no—
217C2×C4#371711yescharacteristic
218C2×C4C2×#62611yescharacteristic
219C8—21no—
220C8—122no—
221C8#380412yes—
222C2×C3—22no—
223C2×C3—21no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3—81no—
229C2×C3#473511yescharacteristic
230C2×C3#496811yescharacteristic
231C22—21no—
232C22—41no—
233C22C2×#371711yescharacteristic
234C4—21no—
235C4—721no—
236C4—721no—
237C4#6383011yescharacteristic
238C4#6396611yescharacteristic
239C3—21no—
240C3#6911711yescharacteristic, complemented
241C3#7072411yescharacteristic, complemented
242C2—21no—
243C2—41no—
244C2#1059112511yescharacteristic
subgroupG/H
centerC2#10591125
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×M16C22
socleC2×C32#481
prime radicalC23×C32SD16
hypercenterC2×M16D32
hypocenterC32#2239
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #10589137C2semidirect
5. #10589131C2semidirect
209. C32#2239semidirect
240. C3#69117semidirect
241. C3#70724semidirect

Maximal subgroups

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

subgroupindex
#105928362
C3×#51252
C3×#45532
#105891372
#105891312
#707243
#691173