Finite Group Databasebeta

← #11921112

Subgroups — #11921112

Order 1152.

472 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 69 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2#10593654C212yesmaximal, complemented
3#73156—33nomaximal
4#72144—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C3⋊#859—22no—
10#63661—41nomaximal
11C2×C3×2−1+4D311yescharacteristic
12C2×2−1+4⋊C3D313yes—
13#5420—36no—
14#5240—31no—
15#5237—31no—
16#5236—32no—
17#5225—32no—
18C2×C32×D4—41no—
19C3×#41⋊C3—21no—
20C2×C3×SL(2,𝔽3)—21no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×SL(2,𝔽3)D412yes—
23#3757—42no—
24#1796—91noSylow
25C3×2−1+4—22no—
26C3×2−1+4—62no—
27C2×C3×#41—31no—
28C2×C3×#41—31no—
29C2×C3×#41—31no—
30C2×C3×#41Dic311yescharacteristic, complemented
31C22×C3×Q8—31no—
32C22×C3×Q8Dic311yescharacteristic, complemented
33C22×C3×Q8C2×D311yescharacteristic
342−1+4⋊C3—26no—
35C2×#41⋊C3Dic313yescomplemented
36C22×SL(2,𝔽3)Dic313yescomplemented
37C22×SL(2,𝔽3)C2×D313yes—
38#926—32no—
39C2×#208—32no—
40#859—66no—
41#837—64no—
42#836—61no—
43#835—62no—
44#835—62no—
45#834—61no—
46#834—61no—
47#834—123no—
48#833—61no—
49C23×C32—41no—
50C23×C32—41no—
51C32×D4—82no—
52C2×C32×C4—41no—
53C2×C3⋊Dic3—82no—
54C3×SL(2,𝔽3)C22⋊C411yescharacteristic
55C2×2−1+4C3⋊D311yescharacteristic
56#421—92no—
57#419—92no—
58#413—91no—
59#410—91no—
60C3×#41—21no—
61C3×#41—34no—
62C3×#41—62no—
63C3×#41—62no—
64C3×#41—62no—
65C3×#41—62no—
66C3×#41—62no—
67C3×#41—61no—
68C2×C3×Q8—21no—
69C2×C3×Q8—31no—
70C2×C3×Q8—32no—
71C2×C3×Q8—31no—
72C2×C3×Q8—31no—
73C2×C3×Q8—62no—
74C2×C3×Q8—62no—
75C2×C3×Q8—61no—
76C2×C3×Q8C2×Dic311yescharacteristic
77C2×C3×Q8#6712yes—
78C2×C3×D4—31no—
79C2×C3×D4—31no—
80C2×C3×D4—31no—
81C2×C3×D4—43no—
82C2×C3×D4S411yescharacteristic
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87#41⋊C3—23no—
88C2×SL(2,𝔽3)—23no—
89C2×SL(2,𝔽3)C2×Dic313yes—
90C2×SL(2,𝔽3)#6716yes—
91#217—62no—
92#217—126no—
93C4⋊Dic3—62no—
94C4×Dic3—62no—
95#208—62no—
96#208—62no—
97C2×C3⋊C8—62no—
98C22×C32—41no—
99C22×C32—42no—
100C22×C32—81no—
101C22×C32—161no—
102C22×C32—161no—
103C32×C4—81no—
104C3⋊Dic3—162no—
1052−1+4—22no—
1062−1+4—62no—
107C2×#41—31no—
108C2×#41—31no—
109C2×#41—31no—
110C2×#41C3⋊Dic311yescharacteristic, complemented
111C22×Q8—31no—
112C22×Q8C3⋊Dic311yescharacteristic, complemented
113C22×Q8C2×C3⋊D311yescharacteristic
114C2×M16—92no—
115#117—92no—
116#104—184no—
117#103—182no—
118#103—182no—
119#101—181no—
120#100—181no—
121#99—181no—
122#99—181no—
123C23×C3—31no—
124C23×C3—43no—
125C23×C3—43no—
126C23×C3SL(2,ℤ/4)11yescharacteristic
127C23×C3C2×S411yescharacteristic
128C3×Q8—31no—
129C3×Q8—32no—
130C3×Q8—32no—
131C3×Q8—62no—
132C3×Q8—61no—
133C3×Q8—61no—
134C3×Q8#21711yescharacteristic
135C3×D4—22no—
136C3×D4—32no—
137C3×D4—62no—
138C3×D4—61no—
139C3×D4—62no—
140C3×D4—86no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—31no—
146C2×C3×C4—43no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—64no—
152C2×C3×C4SL(2,ℤ/4)11yescharacteristic
153C2×Dic3—122no—
154C2×Dic3—246no—
155SL(2,𝔽3)#21713yes—
156C3⋊C8—122no—
157C2×C32—41no—
158C2×C32—82no—
159C2×C32—81no—
160C2×C32—161no—
161#41—21no—
162#41—34no—
163#41—62no—
164#41—62no—
165#41—62no—
166#41—62no—
167#41—62no—
168#41—61no—
169C2×Q8—21no—
170C2×Q8—31no—
171C2×Q8—32no—
172C2×Q8—31no—
173C2×Q8—31no—
174C2×Q8—62no—
175C2×Q8—62no—
176C2×Q8—61no—
177C2×Q8C2×C3⋊Dic311yescharacteristic
178C2×Q8#63812yes—
179C2×D4—31no—
180C2×D4—31no—
181C2×D4—31no—
182C2×D4C3⋊S411yescharacteristic
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—61no—
187M16—182no—
188M16—182no—
189C2×C8—182no—
190C4⋊C4—182no—
191C22⋊C4—182no—
192C42—182no—
193C22×C3—21no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—43no—
197C22×C3—46no—
198C22×C3—61no—
199C22×C3—83no—
200C22×C3—121no—
201C22×C3—163no—
202C22×C3—163no—
203C22×C3C2×SL(2,ℤ/4)11yescharacteristic
204C22×C3GL(2,ℤ/4)12yes—
205C3×C4—21no—
206C3×C4—32no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—61no—
210C3×C4—83no—
211Dic3—242no—
212Dic3—486no—
213C32—41noSylow
214C23—31no—
215C23C3⋊SL(2,ℤ/4)11yescharacteristic
216C23C2×C3⋊S411yescharacteristic
217Q8—31no—
218Q8—32no—
219Q8—32no—
220Q8—62no—
221Q8—61no—
222Q8—61no—
223Q8#375711yescharacteristic
224D4—22no—
225D4—32no—
226D4—62no—
227D4—61no—
228D4—62no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—31no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—64no—
239C2×C4—362no—
240C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
241C8—362no—
242C2×C3—22no—
243C2×C3—21no—
244C2×C3—41no—
245C2×C3—43no—
246C2×C3—86no—
247C2×C3—83no—
248C2×C3—121no—
249C2×C3—163no—
250C2×C3#543211yescharacteristic
251C22—21no—
252C22—41no—
253C22—41no—
254C22—61no—
255C22—121no—
256C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
257C22#6426912yes—
258C4—21no—
259C4—32no—
260C4—31no—
261C4—31no—
262C4—61no—
263C4—722no—
264C3—43no—
265C3#7315611yescharacteristic, complemented
266C2—22no—
267C2—21no—
268C2—41no—
269C2—121no—
270C2#1059366611yescharacteristic
subgroupG/H
centerC2#10593666
commutatorC2×C3×SL(2,𝔽3)C2×C4
FrattiniC23C2×C3⋊S4
FittingC2×C3×2−1+4D3
socleC2×C3#5432
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593654C2semidirect
6. C2×C3×#41⋊C3C4semidirect
7. C22×C3×SL(2,𝔽3)C4semidirect
30. C2×C3×#41Dic3semidirect
32. C22×C3×Q8Dic3semidirect
35. C2×#41⋊C3Dic3semidirect
36. C22×SL(2,𝔽3)Dic3semidirect
110. C2×#41C3⋊Dic3semidirect
112. C22×Q8C3⋊Dic3semidirect
265. C3#73156semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
#105936542
#731563
#721443
#636614