Finite Group Databasebeta

← #11921087

Subgroups — #11921087

Order 1152.

410 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 55 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×#64267C211yesmaximal, characteristic
2C3×#568⋊C3C211yesmaximal, characteristic, complemented
3#10593657C211yesmaximal, characteristic, complemented
4#73131—33nomaximal
5#72023—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64267—21no—
8#64267—21no—
9C2×C3⋊GL(2,𝔽3)C412yescomplemented
10C3×#131⋊C3—21no—
11C3⋊GU(2,𝔽3)—21no—
12#63650—41nomaximal
13C2×#986—33no—
14C3×#568D311yescharacteristic, complemented
15C2×#949—31no—
16#568⋊C3D313yescomplemented
17#5423—33no—
18#5236—31no—
19#5131—31no—
20C2×#823—31no—
21C3⋊#338—31no—
22#4559—31no—
23C2×C4⋊(C3⋊D3)—41no—
24C3×#41⋊C3D411yescharacteristic
25C3×#41⋊C3D411yescharacteristic
26C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
27C3⋊GL(2,𝔽3)—22no—
28C32×M16—41no—
29#3748—41no—
30#1903—91noSylow
31C2×C3×#41C2×D311yescharacteristic
32C22×D12—31no—
33C2×#41⋊C3C2×D313yes—
34#986—63no—
35#986—63no—
36C2×GL(2,𝔽3)—36no—
37C3×#131—21no—
38C3×#131—61no—
39C2×C3×M16—31no—
40C2×C3×M16—31no—
41#949—61no—
42#949—61no—
43#949—61no—
44#949—61no—
45C2×#215—32no—
46C2×#213—32no—
47#926—31no—
48C2×#208—31no—
49C2×#208—31no—
50#131⋊C3—23no—
51GU(2,𝔽3)—63no—
52#837—61no—
53#837—61no—
54#823—61no—
55#823—61no—
56#823—123no—
57#821—62no—
58C22×C3⋊D3—42no—
59C2×C32×C4—41no—
60C4⋊(C3⋊D3)—81no—
61C4⋊(C3⋊D3)—81no—
62C3×SL(2,𝔽3)C22⋊C411yescharacteristic
63C32×C8—81no—
64C3⋊(C3⋊C8)—81no—
65C2×AGL(1,ℤ/8)—91no—
66#568C3⋊D311yescharacteristic, complemented
67#421—91no—
68#418—91no—
69C2×#100—91no—
70#344—91no—
71#338—91no—
72C23×D3—61no—
73C3×#41—31no—
74C3×#41—31no—
75C3×#41D1211yescharacteristic
76C3×#41#6711yescharacteristic
77C2×C3×Q8C4×D311yescharacteristic
78C2×C3×D4—31no—
79C22×C3×C4—31no—
80C2×D12—32no—
81C2×D12—61no—
82C2×D12—61no—
83C2×D12—61no—
84C2×D12—61no—
85C2×D12—61no—
86C2×D12—123no—
87#41⋊C3D1213yes—
88#41⋊C3#6713yes—
89C2×SL(2,𝔽3)C4×D313yes—
90GL(2,𝔽3)—66no—
91C3×M16—31no—
92C3×M16—43no—
93C3×M16—61no—
94C3×M16—61no—
95C3×M16S411yescharacteristic
96C2×C3×C8—32no—
97C2×C3×C8—61no—
98#217—61no—
99#215—62no—
100#215—62no—
101#213—62no—
102#213—62no—
103C4⋊Dic3—61no—
104C4×Dic3—61no—
105#208—61no—
106#208—61no—
107#208—61no—
108#208—61no—
109#208—123no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—61no—
112C22×C32—41no—
113C2×C3⋊D3—82no—
114C2×C3⋊D3—162no—
115C32×C4—41no—
116C32×C4—41no—
117C2×#41C2×C3⋊D311yescharacteristic
118C22×D4—91no—
119AGL(1,ℤ/8)—181no—
120AGL(1,ℤ/8)—181no—
121AGL(1,ℤ/8)—181no—
122AGL(1,ℤ/8)—181no—
123C2×SD16—92no—
124C2×D8—92no—
125#131—21no—
126#131—61no—
127C2×M16—31no—
128C2×M16—31no—
129C2×M16—91no—
130C2×M16—91no—
131#117—91no—
132#104—181no—
133#104—181no—
134#102—182no—
135#100—181no—
136#100—181no—
137C23×C3—31no—
138C22×D3—62no—
139C22×D3—126no—
140C22×D3—121no—
141C22×D3—121no—
142C22×D3—242no—
143C3×Q8—31no—
144C3×Q8(C2×D3)⋊C411yescharacteristic
145C3×D4—31no—
146C3×D4—31no—
147C3×D4—61no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—43no—
154C2×C3×C4C2×S411yescharacteristic
155C2×Dic3—121no—
156D12—62no—
157D12—62no—
158D12—121no—
159D12—121no—
160D12—243no—
161D12—243no—
162SL(2,𝔽3)(C2×D3)⋊C413yes—
163C3×C8—21no—
164C3×C8—61no—
165C3×C8—83no—
166C3⋊C8—62no—
167C3⋊C8—121no—
168C3⋊C8—243no—
169C2×C32—41no—
170C2×C32—81no—
171C3⋊D3—162no—
172C24—181no—
173#41—31no—
174#41—31no—
175#41C4⋊(C3⋊D3)11yescharacteristic
176#41#63811yescharacteristic
177C2×Q8C4×C3⋊D311yescharacteristic
178C2×D4—31no—
179C2×D4—92no—
180C2×D4—181no—
181C2×D4—181no—
182C2×D4—181no—
183C2×D4—181no—
184C2×D4—181no—
185C22×C4—31no—
186SD16—182no—
187SD16—182no—
188D8—182no—
189D8—182no—
190M16—31no—
191M16—61no—
192M16—61no—
193M16—181no—
194M16—181no—
195M16—181no—
196M16—181no—
197M16C3⋊S411yescharacteristic
198C2×C8—32no—
199C2×C8—61no—
200C2×C8—92no—
201C2×C8—181no—
202C4⋊C4—181no—
203C22⋊C4—181no—
204C42—181no—
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—43no—
208C22×C3—121no—
209C22×C3C4×S411yescharacteristic
210C2×D3—122no—
211C2×D3—246no—
212C2×D3—242no—
213C2×D3—242no—
214C2×D3—242no—
215C2×D3—486no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—43no—
219C3×C4—43no—
220C3×C4C4⋊S411yescharacteristic
221C3×C4GL(2,ℤ/4)11yescharacteristic
222Dic3—241no—
223C32—41noSylow
224C23—31no—
225C23—182no—
226C23—361no—
227C23—361no—
228C23—722no—
229Q8—31no—
230Q8#375211yescharacteristic
231D4—31no—
232D4—31no—
233D4—61no—
234D4—182no—
235D4—182no—
236D4—361no—
237D4—361no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—31no—
243C2×C4—361no—
244C2×C4C2×C3⋊S411yescharacteristic
245C8—21no—
246C8—61no—
247C8—182no—
248C8—361no—
249C2×C3—21no—
250C2×C3—43no—
251C2×C3—61no—
252C2×C3—61no—
253C2×C3—83no—
254C2×C3(C2×S4)⋊C411yescharacteristic
255D3—242no—
256D3—486no—
257C22—31no—
258C22—31no—
259C22—121no—
260C22—362no—
261C22—722no—
262C22—722no—
263C22—722no—
264C22C4×C3⋊S411yescharacteristic
265C4—31no—
266C4—31no—
267C4—721no—
268C4C4⋊(C3⋊S4)11yescharacteristic
269C4#6426911yescharacteristic
270C3—43no—
271C3#7313111yescharacteristic, complemented
272C2—21no—
273C2—61no—
274C2—61no—
275C2—722no—
276C2#1059364711yescharacteristic
subgroupG/H
centerC2#10593647
commutatorC2×C3×SL(2,𝔽3)C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC3×#568D3
socleC2×C3(C2×S4)⋊C4
prime radicalC2×#64267C2
cosocleC2×C3×#41⋊C3C22
hypercenterM16C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×#568⋊C3C2semidirect
3. #10593657C2semidirect
9. C2×C3⋊GL(2,𝔽3)C4semidirect
14. C3×#568D3semidirect
16. #568⋊C3D3semidirect
66. #568C3⋊D3semidirect
271. C3#73131semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C3×#568⋊C32
#105936572
#731313
#720233
#636504