Finite Group Databasebeta

← #11800022

Subgroups — #11800022

Order 1152.

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

normal subgroups only

This group has 299 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#10592824C211yesmaximal, characteristic
2C3×#5137C211yesmaximal, characteristic
3C3×#4553C211yesmaximal, characteristic
4#10589136C212yesmaximal, complemented
5#10589130C212yesmaximal, complemented
6#71090—31nomaximal
7#69117—31nomaximal
8C2×C32×M16C2211yescharacteristic
9C2×#3748—21no—
10C3×#924—21no—
11C2×C3×Dic12C2211yescharacteristic
12#63646—21no—
13#63646C2212yes—
14C3×#817C2211yescharacteristic
15C3×#816—21no—
16C3×#812C2212yes—
17C3×#470—31no—
18#5137D311yescharacteristic
19#5125—31no—
20#5125—31no—
21#5125—61no—
22C3×#349—31no—
23#4565—32no—
24#4553D311yescharacteristic
25#4517—32no—
26#4513—32no—
27#4487—32no—
28C22×C32×C4D411yescharacteristic
29C2×C3×Dic6—21no—
30C32×M16—22no—
31C2×C32×C8D411yescharacteristic
32C2×C32×C8C2311yescharacteristic
33#3748—22no—
34#3748—41no—
35C2×C3⋊(C3⋊C8)—41no—
36C3×#217—41no—
37C3×C4⋊Dic3—21no—
38C3×Dic3⋊C4—41no—
39C3×Dic12—22no—
40C3×C3⋊C16—22no—
41#2239—91noSylow
42C2×C3×Q16—31no—
43C2×C3×M16—21no—
44C2×C3×M16C2×D311yescharacteristic
45C2×C3×M16C2×D311yescharacteristic
46C3×C22⋊Q8—61no—
47#924—21no—
48C2×#208—62no—
49C2×#208—61no—
50C2×#208—61no—
51C2×Dic12C2×D311yescharacteristic
52C3×M32—32no—
53C3×#106—31no—
54C3×#103—61no—
55#819—32no—
56#819—32no—
57#819—62no—
58#819—62no—
59#819—61no—
60#819—61no—
61#817C2×D311yescharacteristic
62#816—21no—
63#812C2×D312yes—
64C23×C32SD1611yescharacteristic
65C2×C32×C4—21no—
66C2×C32×C4—21no—
67C2×C32×C4SD1611yescharacteristic
68C2×C32×C4C2×D411yescharacteristic
69C2×C3×Dic3—41no—
70C2×C3×Dic3—41no—
71C3×Dic6—41no—
72C3×Dic6—41no—
73C32×C8—21no—
74C32×C8#4112yes—
75C3⋊(C3⋊C8)—42no—
76#470—31no—
77#430—91no—
78#365—92no—
79#362—92no—
80#349—31no—
81C2×C3×Q8—61no—
82C22×C3×C4—21no—
83C22×C3×C4D1211yescharacteristic
84C22×C3×C4#6711yescharacteristic
85C2×Dic6—21no—
86C3×Q16—62no—
87C3×M16—24no—
88C3×M16—22no—
89C3×M16—22no—
90C2×C3×C8—21no—
91C2×C3×C8—21no—
92C2×C3×C8D1211yescharacteristic
93C2×C3×C8#6711yescharacteristic
94C2×C3×C8C22×D311yescharacteristic
95C2×C3×C8C22×D311yescharacteristic
96C3×C4⋊C4—61no—
97C3×C4⋊C4—121no—
98C3×C22⋊C4—121no—
99#217—41no—
100C4⋊Dic3—21no—
101Dic3⋊C4—41no—
102#208—64no—
103#208—62no—
104#208—62no—
105#208—122no—
106#208—121no—
107#208—121no—
108C2×C3⋊C8—122no—
109C2×C3⋊C8—121no—
110C2×C3⋊C8—121no—
111Dic12—22no—
112C3×C16—62no—
113C3⋊C16—22no—
114C22×C32—21no—
115C22×C32—41no—
116C22×C32C2×SD1611yescharacteristic
117C32×C4—21no—
118C32×C4#12311yescharacteristic
119C32×C4AGL(1,ℤ/8)11yescharacteristic
120C3×Dic3—81no—
121C3×Dic3—81no—
122C2×Q16—31no—
123C2×M16—181no—
124C2×M16D3211yescharacteristic
125C22⋊Q8—61no—
126M32—32no—
127#108—92no—
128#108—181no—
129#106—31no—
130#103—61no—
131C23×C3—21no—
132C23×C3#20411yescharacteristic
133C23×C3#21411yescharacteristic
134C3×Q8—121no—
135C3×Q8—121no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—121no—
145C2×C3×C4—121no—
146C2×C3×C4#20411yescharacteristic
147C2×C3×C4#21411yescharacteristic
148C2×C3×C4C2×D1211yescharacteristic
149C2×C3×C4C2×#6711yescharacteristic
150C2×Dic3—41no—
151C2×Dic3—41no—
152Dic6—41no—
153Dic6—41no—
154C3×C8—22no—
155C3×C8—21no—
156C3×C8—21no—
157C3×C8—22no—
158C3×C8#23712yes—
159C3×C8#23712yes—
160C3⋊C8—124no—
161C3⋊C8—122no—
162C3⋊C8—122no—
163C2×C32—21no—
164C2×C32—41no—
165C2×C32#48111yescharacteristic
166C2×Q8—61no—
167C22×C4#62611yescharacteristic
168Q16—62no—
169M16—22no—
170M16—182no—
171M16—361no—
172C2×C8—361no—
173C2×C8#62611yescharacteristic
174C2×C8C2×D3211yescharacteristic
175C4⋊C4—61no—
176C4⋊C4—121no—
177C22⋊C4—121no—
178C16—62no—
179C22×C3—22no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—81no—
186C22×C3C2×#20411yescharacteristic
187C22×C3C2×#21411yescharacteristic
188C3×C4—22no—
189C3×C4—21no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—241no—
194C3×C4—241no—
195C3×C4#88611yescharacteristic
196C3×C4#90811yescharacteristic
197C3×C4#93511yescharacteristic
198C3×C4#94911yescharacteristic
199Dic3—81no—
200Dic3—81no—
201C32#223911yescharacteristic, complemented, Sylow
202C23#371611yescharacteristic
203Q8—121no—
204Q8—121no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—121no—
208C2×C4—121no—
209C2×C4#371611yescharacteristic
210C2×C4C2×#62611yescharacteristic
211C8—21no—
212C8—362no—
213C8#380512yes—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—41no—
218C2×C3—42no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3#473511yescharacteristic
222C2×C3#503411yescharacteristic
223C22—21no—
224C22—41no—
225C22C2×#371611yescharacteristic
226C4—21no—
227C4—241no—
228C4—241no—
229C4#6382511yescharacteristic
230C4#6396211yescharacteristic
231C3—21no—
232C3#6911711yescharacteristic, complemented
233C3#7109011yescharacteristic, complemented
234C2—21no—
235C2—41no—
236C2#1059109211yescharacteristic
subgroupG/H
centerC2#10591092
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. #10589136C2semidirect
5. #10589130C2semidirect
201. C32#2239semidirect
232. C3#69117semidirect
233. C3#71090semidirect

Maximal subgroups

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

subgroupindex
#105928242
C3×#51372
C3×#45532
#105891362
#105891302
#710903
#691173