Finite Group Databasebeta

← #11921107

Subgroups — #11921107

Order 1152.

564 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊#859C212yesmaximal, complemented
3#73151—33nomaximal
4#72135—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—22no—
9#63658—41nomaximal
10C22×C3×#41D311yescharacteristic
11C22×#41⋊C3D313yes—
12C2×#859—36no—
13C2×#835—32no—
14C2×#832—32no—
15#5212—31no—
16C2×#830—31no—
17C22×C32×C4—41no—
18C22×C3⋊Dic3—42no—
19C3×#41⋊C3—22no—
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21C2×C3×SL(2,𝔽3)D412yes—
22#1788—91noSylow
23C2×C3×#41—22no—
24C2×C3×#41—32no—
25C2×C3×#41—62no—
26C2×C3×#41Dic312yescomplemented
27C22×C3×Q8C2×D311yescharacteristic
28C22×C3×D4—31no—
29C23×C3×C4—31no—
30C2×#41⋊C3—26no—
31C2×#41⋊C3Dic316yescomplemented
32C22×SL(2,𝔽3)C2×D313yes—
33C2×#217—61no—
34C2×C4⋊Dic3—32no—
35C22×C3⋊C8—32no—
36#859—66no—
37#835—62no—
38#835—62no—
39#832—64no—
40C3⋊#95—61no—
41C3⋊#95—123no—
42#830—62no—
43C23×C32—41no—
44C2×C32×C4—42no—
45C2×C32×C4—82no—
46C2×C3⋊Dic3—44no—
47C2×C3⋊Dic3—82no—
48C2×C3⋊Dic3—82no—
49C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
50C22×#41C3⋊D311yescharacteristic
51C2×#103—92no—
52C2×#102—92no—
53C2×C22⋊C8—91no—
54#380—91no—
55C24×C3—31no—
56C3×#41—22no—
57C3×#41—64no—
58C3×#41—62no—
59C3×#41—64no—
60C2×C3×Q8—32no—
61C2×C3×Q8—31no—
62C2×C3×Q8C2×Dic311yescharacteristic
63C2×C3×Q8#6712yes—
64C2×C3×D4—32no—
65C2×C3×D4—34no—
66C2×C3×D4—61no—
67C2×C3×D4—62no—
68C22×C3×C4—31no—
69C22×C3×C4—32no—
70C22×C3×C4—32no—
71C22×C3×C4—43no—
72C22×C3×C4—62no—
73C22×C3×C4—62no—
74C22×C3×C4S411yescharacteristic
75C22×Dic3—62no—
76C22×Dic3—126no—
77#41⋊C3—26no—
78C2×SL(2,𝔽3)C2×Dic313yes—
79C2×SL(2,𝔽3)#6716yes—
80#217—122no—
81C4⋊Dic3—62no—
82C4⋊Dic3—62no—
83C2×C3⋊C8—62no—
84C2×C3⋊C8—62no—
85C2×C3⋊C8—62no—
86C22×C32—41no—
87C22×C32—42no—
88C22×C32—41no—
89C22×C32—41no—
90C22×C32—42no—
91C32×C4—82no—
92C3⋊Dic3—84no—
93C2×#41—22no—
94C2×#41—32no—
95C2×#41—62no—
96C2×#41C3⋊Dic312yescomplemented
97C22×Q8C2×C3⋊D311yescharacteristic
98C22×D4—31no—
99C23×C4—31no—
100C22×C8—92no—
101C2×C4⋊C4—92no—
102C2×C22⋊C4—181no—
103#103—182no—
104#103—182no—
105#102—184no—
106C22⋊C8—182no—
107#95—181no—
108C23×C3—32no—
109C23×C3—43no—
110C23×C3—61no—
111C23×C3—62no—
112C23×C3—61no—
113C23×C3—62no—
114C23×C3C2×S411yescharacteristic
115C3×Q8—31no—
116C3×Q8—32no—
117C3×Q8—61no—
118C3×Q8#21711yescharacteristic
119C3×D4—34no—
120C3×D4—62no—
121C3×D4—64no—
122C2×C3×C4—22no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—46no—
128C2×C3×C4—62no—
129C2×C3×C4—64no—
130C2×C3×C4—62no—
131C2×C3×C4—86no—
132C2×C3×C4SL(2,ℤ/4)12yes—
133C2×Dic3—1212no—
134C2×Dic3—122no—
135C2×Dic3—122no—
136C2×Dic3—122no—
137C2×Dic3—246no—
138C2×Dic3—246no—
139SL(2,𝔽3)#21713yescomplemented
140C3⋊C8—122no—
141C2×C32—41no—
142C2×C32—42no—
143C2×C32—41no—
144C2×C32—42no—
145C2×C32—41no—
146C24—31no—
147#41—22no—
148#41—64no—
149#41—62no—
150#41—64no—
151C2×Q8—32no—
152C2×Q8—31no—
153C2×Q8C2×C3⋊Dic311yescharacteristic
154C2×Q8#63812yes—
155C2×D4—32no—
156C2×D4—34no—
157C2×D4—61no—
158C2×D4—62no—
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—182no—
165C22×C4C3⋊S411yescharacteristic
166C2×C8—182no—
167C2×C8—182no—
168C2×C8—182no—
169C4⋊C4—182no—
170C4⋊C4—182no—
171C22⋊C4—362no—
172C22×C3—43no—
173C22×C3—46no—
174C22×C3—43no—
175C22×C3—43no—
176C22×C3—46no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3—122no—
181C22×C3—122no—
182C22×C3#98411yescharacteristic
183C22×C3#98512yes—
184C22×C3#98611yescharacteristic
185C22×C3C2×SL(2,ℤ/4)11yescharacteristic
186C22×C3GL(2,ℤ/4)12yes—
187C3×C4—22no—
188C3×C4—32no—
189C3×C4—31no—
190C3×C4—31no—
191C3×C4—86no—
192Dic3—2412no—
193Dic3—242no—
194C32—41noSylow
195C23—32no—
196C23—61no—
197C23—62no—
198C23—61no—
199C23—62no—
200C23C2×C3⋊S411yescharacteristic
201Q8—31no—
202Q8—32no—
203Q8—61no—
204Q8#375711yescharacteristic
205D4—34no—
206D4—62no—
207D4—64no—
208C2×C4—22no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—62no—
214C2×C4—64no—
215C2×C4—62no—
216C2×C4—362no—
217C2×C4—362no—
218C2×C4—362no—
219C2×C4C3⋊SL(2,ℤ/4)12yes—
220C8—362no—
221C2×C3—43no—
222C2×C3—46no—
223C2×C3—43no—
224C2×C3—46no—
225C2×C3—43no—
226C2×C3—122no—
227C2×C3#542411yescharacteristic
228C2×C3#542512yes—
229C2×C3#542611yescharacteristic
230C2×C3#542712yes—
231C2×C3#543211yescharacteristic
232C22—62no—
233C22—62no—
234C22—62no—
235C22—122no—
236C22—122no—
237C22#6426511yescharacteristic
238C22#6426612yes—
239C22#6426711yescharacteristic
240C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
241C22#6426912yes—
242C4—22no—
243C4—32no—
244C4—31no—
245C4—31no—
246C4—722no—
247C3—43no—
248C3#7315111yescharacteristic, complemented
249C2—122no—
250C2#1059365811yescharacteristic
251C2#1059365912yes—
252C2#1059366011yescharacteristic
253C2#1059366112yes—
254C2#1059366611yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC22×C3×#41D3
socleC23×C3C2×S4
prime radicalC22×C3×#41⋊C3C2
cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#859C2semidirect
6. C2×C3×#41⋊C3C4semidirect
26. C2×C3×#41Dic3semidirect
31. C2×#41⋊C3Dic3semidirect
49. C3×SL(2,𝔽3)C22⋊C4semidirect
96. C2×#41C3⋊Dic3semidirect
139. SL(2,𝔽3)#217semidirect
248. C3#73151semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3⋊#8592
#731513
#721353
#636584