Finite Group Databasebeta

← #11922832

Subgroups — #11922832

Order 1152.

770 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 131 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×#64259C212yesmaximal
2C3×C42×A4C211yesmaximal, characteristic
3C4⋊(C3⋊SL(2,ℤ/4))C214yesmaximal
4#85283—33nomaximal
5#82076—31nomaximal
6C2×C3×C4×A4C2211yescharacteristic
7C2×C3×C4×A4C2212yes—
8C2×C3⋊SL(2,ℤ/4)C2214yes—
9#64259—24no—
10#64078—41nomaximal
11C2×A4⋊Q8—36no—
12C22×C3×C42D311yescharacteristic
13C2×#924—32no—
14C2×#869—31no—
15C42×A4D313yes—
16C4⋊SL(2,ℤ/4)—312no—
17#5212—34no—
18#4935—31no—
19#4929—32no—
20#4928—34no—
21C22×C3×A4C2311yescharacteristic
22C2×#634—42no—
23C3×C4×A4D412yes—
24C3×C4×A4Q814yescomplemented
25C3⋊SL(2,ℤ/4)—24no—
26C32×C42—41no—
27C4⋊(C3⋊Dic3)—44no—
28#2857—91noSylow
29C23×C3×C4C2×D312yes—
30C23×C3×C4C2×D311yescharacteristic
31C22×Dic6—32no—
32C2×C4×A4C2×D316yes—
33C2×C4×A4C2×D313yes—
34C2×SL(2,ℤ/4)—312no—
35A4⋊Q8—612no—
36C2×C3×C42—31no—
37C2×C3×C42—32no—
38C2×C3×C42—31no—
39C2×#217—34no—
40C2×C4⋊Dic3—34no—
41C2×C4⋊Dic3—34no—
42C2×C4⋊Dic3—32no—
43#924—64no—
44#924—64no—
45C2×Dic3⋊C4—34no—
46#869—61no—
47#869—62no—
48#869—61no—
49#869—123no—
50C3⋊#95—38no—
51C2×C3×A4C2×D411yescharacteristic
52C2×C3×A4C2×Q812yes—
53C2×C32×C4—42no—
54C2×C32×C4—41no—
55C2×C3⋊Dic3—44no—
56#634—84no—
57C2×C4⋊Q8—91no—
58C2×C22⋊Q8—92no—
59C22×C42C3⋊D311yescharacteristic
60#391—92no—
61#389—94no—
62#384—91no—
63#380—94no—
64C24×C3C22×D311yescharacteristic
65C22×A4C22×D313yes—
66C22×C3×C4—32no—
67C22×C3×C4—31no—
68C22×C3×C4—32no—
69C22×C3×C4—31no—
70C22×C3×C4—32no—
71C22×C3×C4—34no—
72C22×C3×C4Dic614yescomplemented
73C22×C3×C4D1212yes—
74C22×Dic3—34no—
75C22×Dic3—34no—
76C2×Dic6—62no—
77C2×Dic6—64no—
78C2×Dic6—64no—
79C2×Dic6—62no—
80C2×Dic6—126no—
81C4×A4Dic6112yescomplemented
82C4×A4D1216yes—
83SL(2,ℤ/4)—612no—
84C3×C42—31no—
85C3×C42—32no—
86C3×C42—43no—
87C3×C42—61no—
88C3×C42S411yescharacteristic
89#217—64no—
90#217—64no—
91C4⋊Dic3—64no—
92C4⋊Dic3—64no—
93C4⋊Dic3—68no—
94C4⋊Dic3—64no—
95C4⋊Dic3—1212no—
96Dic3⋊C4—68no—
97C22×C32—41no—
98C3×A4C4⋊Q811yescharacteristic, complemented
99C32×C4—44no—
100C32×C4—42no—
101C3⋊Dic3—84no—
102C22×Q8—92no—
103C23×C4C2×C3⋊D312yes—
104C23×C4C2×C3⋊D311yescharacteristic
105C4⋊Q8—181no—
106C4⋊Q8—182no—
107C4⋊Q8—181no—
108C22⋊Q8—184no—
109C22⋊Q8—184no—
110C2×C4⋊C4—92no—
111C2×C4⋊C4—94no—
112C2×C4⋊C4—94no—
113C2×C4⋊C4—94no—
114C2×C22⋊C4—94no—
115C2×C42—31no—
116C2×C42—32no—
117C2×C42—31no—
118#95—98no—
119C23×C3—31no—
120C23×C3—31no—
121C23×C3—32no—
122C23×C3C2×Dic612yes—
123C23×C3C2×D1211yescharacteristic
124C2×A4C2×Dic616yes—
125C2×A4C2×D1213yes—
126C2×C3×C4—32no—
127C2×C3×C4—31no—
128C2×C3×C4—34no—
129C2×C3×C4—34no—
130C2×C3×C4—32no—
131C2×C3×C4—32no—
132C2×C3×C4—32no—
133C2×C3×C4—34no—
134C2×C3×C4—46no—
135C2×C3×C4—43no—
136C2×C3×C4—62no—
137C2×C3×C4—61no—
138C2×C3×C4C2×S412yes—
139C2×C3×C4C2×S411yescharacteristic
140C2×Dic3—64no—
141C2×Dic3—64no—
142C2×Dic3—68no—
143C2×Dic3—64no—
144C2×Dic3—64no—
145C2×Dic3—1212no—
146Dic6—124no—
147Dic6—124no—
148Dic6—2412no—
149C2×C32—42no—
150C2×C32—41no—
151C24C22×C3⋊D311yescharacteristic
152C2×Q8—184no—
153C2×Q8—182no—
154C2×Q8—184no—
155C2×Q8—182no—
156C22×C4—32no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—34no—
162C22×C4—94no—
163C22×C4—94no—
164C22×C4#63414yescomplemented
165C22×C4C4⋊(C3⋊D3)12yes—
166C4⋊C4—188no—
167C4⋊C4—184no—
168C4⋊C4—184no—
169C4⋊C4—188no—
170C4⋊C4—184no—
171C22⋊C4—184no—
172C22⋊C4—184no—
173C42—31no—
174C42—32no—
175C42—61no—
176C42C3⋊S411yescharacteristic
177C22×C3—32no—
178C22×C3—31no—
179C22×C3—32no—
180C22×C3—32no—
181C22×C3—31no—
182C22×C3—31no—
183C22×C3—43no—
184C22×C3—61no—
185C22×C3#86911yescharacteristic, complemented
186C22×C3C22×S411yescharacteristic
187A4#86913yescomplemented
188C3×C4—34no—
189C3×C4—32no—
190C3×C4—412no—
191C3×C4—46no—
192C3×C4A4⋊Q814yescomplemented
193C3×C4C4⋊S412yes—
194Dic3—124no—
195Dic3—124no—
196Dic3—2412no—
197C32—41noSylow
198C23—31no—
199C23—31no—
200C23—32no—
201C23C2×#63412yes—
202C23C2×C4⋊(C3⋊D3)11yescharacteristic
203Q8—364no—
204Q8—364no—
205C2×C4—32no—
206C2×C4—31no—
207C2×C4—34no—
208C2×C4—34no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—34no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—184no—
216C2×C4—184no—
217C2×C4—188no—
218C2×C4—184no—
219C2×C4—184no—
220C2×C4C2×C3⋊S412yes—
221C2×C4C2×C3⋊S411yescharacteristic
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—32no—
225C2×C3—46no—
226C2×C3—43no—
227C2×C3C2×A4⋊Q812yes—
228C2×C3C2×C4⋊S411yescharacteristic
229C22—32no—
230C22—31no—
231C22—32no—
232C22—32no—
233C22—31no—
234C22—31no—
235C22—61no—
236C22#6407811yescharacteristic, complemented
237C22C22×C3⋊S411yescharacteristic
238C4—34no—
239C4—32no—
240C4—364no—
241C4—364no—
242C4#6425914yescomplemented
243C4C4⋊(C3⋊S4)12yes—
244C3—43no—
245C3#8528311yescharacteristic, complemented
246C2—31no—
247C2—31no—
248C2—32no—
249C2C2×#6425912yes—
250C2C2×C4⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC22×C3×A4C23
FittingC22×C3×C42D3
socleC24×C3C22×D3
hypercenterC42C3⋊S4
hypocenterC3×A4C4⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
24. C3×C4×A4Q8semidirect
72. C22×C3×C4Dic6semidirect
81. C4×A4Dic6semidirect
98. C3×A4C4⋊Q8semidirect
164. C22×C4#634semidirect
185. C22×C3#869semidirect
187. A4#869semidirect
192. C3×C4A4⋊Q8semidirect
236. C22#64078semidirect
242. C4#64259semidirect
245. C3#85283semidirect

Maximal subgroups

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

subgroupindex
C2×#642592
C3×C42×A42
C4⋊(C3⋊SL(2,ℤ/4))2
#852833
#820763
#640784