Finite Group Databasebeta

← #11786077

Subgroups — #11786077

Order 1152.

521 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 156 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×#63642C211yesmaximal, characteristic
2#10591359C211yesmaximal, characteristic
3#10591230C211yesmaximal, characteristic
4C3×#4551C212yesmaximal
5#10589163C212yesmaximal
6#68849—32nomaximal
7C22×C32×C8C2211yescharacteristic
8C2×C4×C3⋊Dic3C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×C3×C4⋊Dic3C2212yes—
11C2×C3×C4⋊Dic3C2212yes—
12C2×#3724—22no—
13#63642C414yescomplemented
14#63579—22no—
15C2×#815—32no—
16C2×#815—61no—
17#4977—32no—
18#4975—32no—
19C3×#340—32no—
20#4551D312yes—
21#4529—32no—
22#4529—32no—
23C22×C32×C4C2311yescharacteristic
24C22×C3⋊Dic3D412yes—
25C22×C3×Dic3—22no—
26C22×C3×Dic3—22no—
27C2×C32×C8—21no—
28C2×C32×C8—21no—
29C2×C32×C8C2×C412yes—
30C4×C3⋊Dic3C2×C412yes—
31C4×C3⋊Dic3Q812yes—
32C2×C3⋊(C3⋊C8)—21no—
33C2×C3⋊(C3⋊C8)—21no—
34C2×C3⋊(C3⋊C8)C2×C412yes—
35C3×C4⋊Dic3—22no—
36C3×C4⋊Dic3—22no—
37C3×C4⋊Dic3—22no—
38C3×C4⋊Dic3—22no—
39#3724—44no—
40#1836—91noSylow
41C22×C3×C8—21no—
42C22×C3×C8C2×D312yes—
43C2×C3×C4⋊C4—32no—
44C2×C3×C4⋊C4—32no—
45C2×C4⋊Dic3C2×D312yes—
46C2×C4⋊Dic3C2×D312yes—
47C2×Dic3⋊C4—62no—
48C2×Dic3⋊C4—62no—
49C2×C4×Dic3—32no—
50C2×C4×Dic3—61no—
51C22×C3⋊C8—32no—
52C22×C3⋊C8—61no—
53C3⋊#95—62no—
54C3⋊#95—62no—
55#815—38no—
56#815—64no—
57C23×C32C2×D411yescharacteristic
58C2×C32×C4C22×C411yescharacteristic
59C2×C32×C4C2×Q811yescharacteristic
60C2×C32×C4#4111yescharacteristic
61C2×C32×C4#4111yescharacteristic
62C2×C32×C4#4111yescharacteristic
63C2×C32×C4#4111yescharacteristic
64C2×C3⋊Dic3—22no—
65C2×C3⋊Dic3—22no—
66C2×C3⋊Dic3C4⋊C414yes—
67C2×C3×Dic3—42no—
68C2×C3×Dic3—42no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3—42no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—42no—
73C32×C8—22no—
74C3⋊(C3⋊C8)—22no—
75C2×#97—91no—
76#384—91no—
77#382—91no—
78#340—32no—
79#340—92no—
80C22×C3×C4—21no—
81C22×C3×C4—62no—
82C22×C3×C4—62no—
83C22×C3×C4C22×D312yes—
84C22×Dic3—22no—
85C22×Dic3—22no—
86C22×Dic3—32no—
87C22×Dic3—32no—
88C22×Dic3—62no—
89C2×C3×C8—22no—
90C2×C3×C8—22no—
91C2×C3×C8—22no—
92C2×C3×C8—22no—
93C2×C3×C8—22no—
94C2×C3×C8C4×D314yes—
95C3×C4⋊C4—62no—
96C3×C4⋊C4—62no—
97C3×C4⋊C4—62no—
98C3×C4⋊C4—62no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102C4⋊Dic3—22no—
103Dic3⋊C4—124no—
104Dic3⋊C4—124no—
105C4×Dic3—34no—
106C4×Dic3—34no—
107C4×Dic3—62no—
108C4×Dic3—62no—
109C2×C3⋊C8—34no—
110C2×C3⋊C8—62no—
111C2×C3⋊C8—62no—
112C2×C3⋊C8—62no—
113C2×C3⋊C8—62no—
114C2×C3⋊C8—62no—
115C22×C32C2×C4⋊C411yescharacteristic
116C22×C32#12411yescharacteristic
117C22×C32#12411yescharacteristic
118C22×C32AGL(1,ℤ/8)12yes—
119C22×C32#13712yes—
120C32×C4#11711yescharacteristic
121C32×C4#11711yescharacteristic
122C32×C4#12511yescharacteristic
123C32×C4#12511yescharacteristic
124C3⋊Dic3—24no—
125C3×Dic3—82no—
126C3×Dic3—82no—
127C22×C8—91no—
128C22×C8D3211yescharacteristic
129C2×C4⋊C4—32no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—182no—
132C2×C42—91no—
133#97—94no—
134#95—182no—
135C23×C3—21no—
136C23×C3D3×D412yes—
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—122no—
144C2×C3×C4—122no—
145C2×C3×C4—122no—
146C2×C3×C4—122no—
147C2×C3×C4—122no—
148C2×C3×C4—122no—
149C2×C3×C4C2×C4×D312yes—
150C2×C3×C4#23512yes—
151C2×C3×C4#23512yes—
152C2×C3×C4#23712yes—
153C2×C3×C4#23712yes—
154C2×C3×C4D3×Q812yes—
155C2×Dic3—34no—
156C2×Dic3—34no—
157C2×Dic3—42no—
158C2×Dic3—42no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—64no—
164C2×Dic3—62no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3—122no—
169C2×Dic3—122no—
170C3×C8—24no—
171C3×C8—24no—
172C3⋊C8—64no—
173C3⋊C8—64no—
174C2×C32#38211yescharacteristic
175C2×C32#42812yes—
176C2×C32#48912yes—
177C2×C32#49011yescharacteristic
178C2×C32#49111yescharacteristic
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—92no—
182C22×C4C2×D3211yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—92no—
186C2×C8—181no—
187C2×C8—181no—
188C2×C8#62412yes—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—364no—
194C42—92no—
195C42—92no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3#88512yes—
202C22×C3#88512yes—
203C22×C3D3×C4⋊C412yes—
204C22×C3#91112yes—
205C22×C3#91412yes—
206C22×C3#91512yes—
207C22×C3#91812yes—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—242no—
213C3×C4—242no—
214C3×C4#87212yes—
215C3×C4#87712yes—
216C3×C4#88912yes—
217C3×C4#88912yes—
218Dic3—64no—
219Dic3—64no—
220Dic3—82no—
221Dic3—82no—
222Dic3—124no—
223C32#183611yescharacteristic, complemented, Sylow
224C23#380211yescharacteristic
225C2×C4—94no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4#379711yescharacteristic
235C2×C4#379811yescharacteristic
236C2×C4#379811yescharacteristic
237C2×C4#380511yescharacteristic
238C2×C4#380511yescharacteristic
239C2×C4C2×#62411yescharacteristic
240C8—22no—
241C8—182no—
242C2×C3—21no—
243C2×C3—22no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—22no—
247C2×C3#465012yes—
248C2×C3#476412yes—
249C2×C3#476512yes—
250C2×C3#479412yes—
251C2×C3#479712yes—
252C2×C3#486112yes—
253C2×C3#488112yes—
254C22#6379812yes—
255C22#6381112yes—
256C22#6390011yescharacteristic
257C22#6391511yescharacteristic
258C22#6391511yescharacteristic
259C4—184no—
260C4—242no—
261C4—242no—
262C4#6387011yescharacteristic
263C4#6387011yescharacteristic
264C4#6387411yescharacteristic
265C4#6395711yescharacteristic
266C3—21no—
267C3#6884912yescomplemented
268C2#1059068812yes—
269C2#1059086612yes—
270C2#1059088811yescharacteristic
271C2#1059090411yescharacteristic
272C2#1059136011yescharacteristic
subgroupG/H
centerC23#3802
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socle, prime radicalC23×C32C2×D4
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1836
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
13. #63642C4semidirect
223. C32#1836semidirect
267. C3#68849semidirect

Maximal subgroups

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

subgroupindex
C2×#636422
#105913592
#105912302
C3×#45512
#105891632
#688493