Finite Group Databasebeta

← #11921104

Subgroups — #11921104

Order 1152.

696 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C212yesmaximal, complemented
3#73148—33nomaximal
4#72132—31nomaximal
5C22×C3×SL(2,𝔽3)—22no—
6C22×C3×SL(2,𝔽3)C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×#3757—41nomaximal
9C3⋊#859—22no—
10C23×C3×Q8D311yescharacteristic
11C23×SL(2,𝔽3)D313yes—
12C2×#859—36no—
13C2×#835—32no—
14C2×#835—32no—
15#5212—31no—
16C2×#830—31no—
17C24×C32—41no—
18C22×C3⋊Dic3—42no—
19C2×C3×SL(2,𝔽3)—22no—
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21C2×C3×SL(2,𝔽3)D412yes—
22#3757—44no—
23#1787—91noSylow
24C22×C3×Q8—22no—
25C22×C3×Q8—31no—
26C22×C3×Q8—32no—
27C22×C3×Q8—62no—
28C22×C3×Q8Dic312yescomplemented
29C22×C3×Q8C2×D311yescharacteristic
30C23×C3×C4—31no—
31C22×SL(2,𝔽3)—26no—
32C22×SL(2,𝔽3)Dic316yescomplemented
33C22×SL(2,𝔽3)C2×D313yes—
34C2×#217—61no—
35C2×#217—123no—
36C2×C4⋊Dic3—32no—
37C22×C3⋊C8—32no—
38#859—66no—
39#835—62no—
40#835—64no—
41#835—62no—
42C3⋊#95—61no—
43#830—62no—
44C23×C32—42no—
45C23×C32—42no—
46C23×C32—42no—
47C23×C32—41no—
48C23×C32—82no—
49C23×C32—82no—
50C2×C3⋊Dic3—44no—
51C2×C3⋊Dic3—82no—
52C2×C3⋊Dic3—82no—
53C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
54C23×Q8C3⋊D311yescharacteristic
55C2×#103—92no—
56C2×#103—92no—
57C2×C22⋊C8—91no—
58#380—91no—
59C24×C3—43no—
60C24×C3S411yescharacteristic
61C2×C3×Q8—22no—
62C2×C3×Q8—32no—
63C2×C3×Q8—32no—
64C2×C3×Q8—31no—
65C2×C3×Q8—34no—
66C2×C3×Q8—61no—
67C2×C3×Q8—62no—
68C2×C3×Q8—64no—
69C2×C3×Q8—62no—
70C2×C3×Q8—64no—
71C2×C3×Q8C2×Dic311yescharacteristic
72C2×C3×Q8#6712yes—
73C22×C3×C4—31no—
74C22×C3×C4—32no—
75C22×C3×C4—32no—
76C22×C3×C4—31no—
77C22×C3×C4—62no—
78C22×C3×C4—62no—
79C22×Dic3—62no—
80C22×Dic3—126no—
81C2×SL(2,𝔽3)—26no—
82C2×SL(2,𝔽3)C2×Dic313yes—
83C2×SL(2,𝔽3)#6716yes—
84#217—1212no—
85#217—122no—
86C4⋊Dic3—62no—
87C4⋊Dic3—62no—
88C2×C3⋊C8—62no—
89C2×C3⋊C8—62no—
90C2×C3⋊C8—62no—
91C22×C32—44no—
92C22×C32—41no—
93C22×C32—41no—
94C22×C32—42no—
95C22×C32—41no—
96C22×C32—42no—
97C22×C32—84no—
98C22×C32—82no—
99C22×C32—84no—
100C22×C32—82no—
101C3⋊Dic3—84no—
102C22×Q8—22no—
103C22×Q8—31no—
104C22×Q8—32no—
105C22×Q8—62no—
106C22×Q8C3⋊Dic312yescomplemented
107C22×Q8C2×C3⋊D311yescharacteristic
108C23×C4—31no—
109C22×C8—92no—
110C2×C4⋊C4—92no—
111C2×C22⋊C4—181no—
112#103—182no—
113#103—184no—
114#103—182no—
115C22⋊C8—182no—
116#95—181no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—43no—
120C23×C3—46no—
121C23×C3—46no—
122C23×C3—46no—
123C23×C3—86no—
124C23×C3—86no—
125C23×C3#22612yes—
126C23×C3GL(2,𝔽3)12yes—
127C23×C3SL(2,ℤ/4)12yes—
128C23×C3C2×S411yescharacteristic
129C3×Q8—31no—
130C3×Q8—32no—
131C3×Q8—34no—
132C3×Q8—61no—
133C3×Q8—62no—
134C3×Q8—64no—
135C3×Q8#21711yescharacteristic, complemented
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—62no—
142C2×C3×C4—64no—
143C2×C3×C4—62no—
144C2×C3×C4—62no—
145C2×Dic3—1212no—
146C2×Dic3—122no—
147C2×Dic3—122no—
148C2×Dic3—122no—
149C2×Dic3—246no—
150C2×Dic3—246no—
151SL(2,𝔽3)#21713yescomplemented
152C3⋊C8—122no—
153C2×C32—41no—
154C2×C32—42no—
155C2×C32—41no—
156C2×C32—42no—
157C2×C32—41no—
158C2×C32—84no—
159C24C3⋊S411yescharacteristic
160C2×Q8—22no—
161C2×Q8—32no—
162C2×Q8—32no—
163C2×Q8—31no—
164C2×Q8—34no—
165C2×Q8—61no—
166C2×Q8—62no—
167C2×Q8—64no—
168C2×Q8—62no—
169C2×Q8—64no—
170C2×Q8C2×C3⋊Dic311yescharacteristic
171C2×Q8#63812yes—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—62no—
177C22×C4—62no—
178C22×C4—182no—
179C2×C8—182no—
180C2×C8—182no—
181C2×C8—182no—
182C4⋊C4—182no—
183C4⋊C4—182no—
184C22⋊C4—362no—
185C22×C3—24no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—22no—
189C22×C3—43no—
190C22×C3—43no—
191C22×C3—46no—
192C22×C3—412no—
193C22×C3—43no—
194C22×C3—46no—
195C22×C3—812no—
196C22×C3—86no—
197C22×C3—812no—
198C22×C3—86no—
199C22×C3#85914yescomplemented
200C22×C3C2×#22611yescharacteristic
201C22×C3C2×GL(2,𝔽3)11yescharacteristic
202C22×C3#98312yes—
203C22×C3C2×SL(2,ℤ/4)11yescharacteristic
204C22×C3GL(2,ℤ/4)12yes—
205C3×C4—32no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—62no—
209Dic3—2412no—
210Dic3—242no—
211C32—41noSylow
212C23—22no—
213C23—22no—
214C23C3⋊#22612yes—
215C23C3⋊GL(2,𝔽3)12yes—
216C23C3⋊SL(2,ℤ/4)12yes—
217C23C2×C3⋊S411yescharacteristic
218Q8—31no—
219Q8—32no—
220Q8—34no—
221Q8—61no—
222Q8—62no—
223Q8—64no—
224Q8#375711yescharacteristic, complemented
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—62no—
231C2×C4—64no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—362no—
235C2×C4—362no—
236C2×C4—362no—
237C8—362no—
238C2×C3—24no—
239C2×C3—43no—
240C2×C3—46no—
241C2×C3—43no—
242C2×C3—46no—
243C2×C3—43no—
244C2×C3—812no—
245C2×C3C2×#85911yescharacteristic
246C2×C3#541912yes—
247C2×C3#542011yescharacteristic
248C2×C3#542112yes—
249C2×C3#543211yescharacteristic
250C22—24no—
251C22—22no—
252C22—24no—
253C22—22no—
254C22C3⋊#85914yescomplemented
255C22C2×C3⋊#22611yescharacteristic
256C22C2×C3⋊GL(2,𝔽3)11yescharacteristic
257C22#6426412yes—
258C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
259C22#6426912yes—
260C4—32no—
261C4—31no—
262C4—31no—
263C4—62no—
264C4—722no—
265C3—43no—
266C3#7314811yescharacteristic, complemented
267C2—24no—
268C2C2×C3⋊#85911yescharacteristic
269C2#1059365312yes—
270C2#1059365411yescharacteristic
271C2#1059365512yes—
272C2#1059366611yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC23×C3×Q8D3
socleC23×C3C2×S4
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC24C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#859C2semidirect
6. C22×C3×SL(2,𝔽3)C4semidirect
28. C22×C3×Q8Dic3semidirect
32. C22×SL(2,𝔽3)Dic3semidirect
53. C3×SL(2,𝔽3)C22⋊C4semidirect
106. C22×Q8C3⋊Dic3semidirect
135. C3×Q8#217semidirect
151. SL(2,𝔽3)#217semidirect
199. C22×C3#859semidirect
224. Q8#3757semidirect
254. C22C3⋊#859semidirect
266. C3#73148semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3⋊#8592
#731483
#721323
C2×#37574