Finite Group Databasebeta

← #11920712

Subgroups — #11920712

Order 1152.

373 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 41 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
2#5819⋊C3C211yesmaximal, characteristic, complemented
3C3×#5423C211yesmaximal, characteristic, complemented
4#73131—31nomaximal
5#70400—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#63559—41nomaximal
13C2×#986—31no—
14C2×#986—61no—
15C2×#949—31no—
16#5819D311yescharacteristic, complemented
17#568⋊C3—31no—
18#5423D311yescharacteristic, complemented
19C3×#421—31no—
20C2×#823—31no—
21#5001—31no—
22C3⋊#338—31no—
23#4533—31no—
24C2×C4⋊(C3⋊D3)—41no—
25C3×#41⋊C3D411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C3⋊GL(2,𝔽3)—22no—
29C3×#208—41no—
30C3×#208—41no—
31#1903—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33C22×D12—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3C2×D311yescharacteristic
36#986—62no—
37#986—62no—
38#986—61no—
39#986—61no—
40C2×GL(2,𝔽3)—32no—
41C2×GL(2,𝔽3)—62no—
42C2×C3×M16—31no—
43C3×#117—31no—
44#949—61no—
45#949—61no—
46#949—61no—
47#949—61no—
48#948—21no—
49#948—61no—
50C2×#215—32no—
51C2×#213—32no—
52C2×#208—31no—
53C2×#208—31no—
54C2×#208—31no—
55#131⋊C3—61no—
56GU(2,𝔽3)—21no—
57C3×#104—61no—
58C3×#104—61no—
59#823—61no—
60#823—61no—
61#823—121no—
62#809—62no—
63C22×C3⋊D3—42no—
64C2×C32×C4—41no—
65C4⋊(C3⋊D3)—81no—
66C4⋊(C3⋊D3)—81no—
67C3×SL(2,𝔽3)C22⋊C411yescharacteristic
68C3×C3⋊C8—81no—
69C3×C3⋊C8—81no—
70C2×AGL(1,ℤ/8)—91no—
71#568—31no—
72#421—31no—
73#418—91no—
74C2×#100—91no—
75#344—91no—
76#338—91no—
77C23×D3—61no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41D1211yescharacteristic
81C3×#41#6711yescharacteristic
82C2×C3×Q8C4×D311yescharacteristic
83C2×C3×D4—31no—
84C22×C3×C4—31no—
85C2×D12—32no—
86C2×D12—61no—
87C2×D12—61no—
88C2×D12—61no—
89C2×D12—61no—
90C2×D12—61no—
91C2×D12—121no—
92C2×D12—241no—
93#41⋊C3—21no—
94#41⋊C3—21no—
95#41⋊C3D1211yescharacteristic
96#41⋊C3#6711yescharacteristic
97C2×SL(2,𝔽3)—21no—
98C2×SL(2,𝔽3)C4×D311yescharacteristic
99GL(2,𝔽3)—64no—
100GL(2,𝔽3)—62no—
101C3×M16—61no—
102C3×M16—61no—
103C3×M16—121no—
104C2×C3×C8—61no—
105C3×C4⋊C4—61no—
106C3×C22⋊C4—61no—
107C3×C42—61no—
108#215—62no—
109#215—62no—
110#213—62no—
111#213—62no—
112#208—31no—
113#208—41no—
114#208—61no—
115#208—61no—
116#208—61no—
117#208—61no—
118#208S411yescharacteristic
119C2×C3⋊C8—32no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—61no—
122C22×C32—41no—
123C2×C3⋊D3—82no—
124C2×C3⋊D3—162no—
125C32×C4—41no—
126C32×C4—41no—
127C2×#41D3211yescharacteristic
128C22×D4—91no—
129AGL(1,ℤ/8)—181no—
130AGL(1,ℤ/8)—181no—
131AGL(1,ℤ/8)—181no—
132AGL(1,ℤ/8)—181no—
133C2×SD16—92no—
134C2×D8—92no—
135#131—61no—
136#131—181no—
137C2×M16—31no—
138C2×M16—91no—
139C2×M16—91no—
140C2×M16—91no—
141#117—31no—
142#104—61no—
143#104—61no—
144#102—182no—
145#100—181no—
146#100—181no—
147C23×C3—31no—
148C22×D3—62no—
149C22×D3—122no—
150C22×D3—121no—
151C22×D3—121no—
152C22×D3—242no—
153C22×D3—242no—
154C3×Q8—31no—
155C3×Q8(C2×D3)⋊C411yescharacteristic
156C3×D4—31no—
157C3×D4—31no—
158C3×D4—61no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—41no—
165C2×C3×C4—81no—
166C2×C3×C4—121no—
167C2×C3×C4C2×S411yescharacteristic
168D12—62no—
169D12—62no—
170D12—121no—
171D12—121no—
172D12—241no—
173D12—241no—
174D12—242no—
175D12—242no—
176SL(2,𝔽3)—21no—
177SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
178C3×C8—121no—
179C3×C8—241no—
180C3⋊C8—21no—
181C3⋊C8—62no—
182C3⋊C8—61no—
183C3⋊C8—81no—
184C2×C32—41no—
185C2×C32—81no—
186C3⋊D3—162no—
187C24—181no—
188#41—31no—
189#41—31no—
190#41#62611yescharacteristic
191#41#62611yescharacteristic
192C2×Q8#62411yescharacteristic
193C2×D4—31no—
194C2×D4—92no—
195C2×D4—181no—
196C2×D4—181no—
197C2×D4—181no—
198C2×D4—181no—
199C2×D4—181no—
200C22×C4—31no—
201SD16—182no—
202SD16—182no—
203D8—182no—
204D8—182no—
205M16—31no—
206M16—61no—
207M16—61no—
208M16—91no—
209M16—181no—
210M16—181no—
211M16—181no—
212M16—181no—
213C2×C8—61no—
214C2×C8—92no—
215C2×C8—92no—
216C2×C8—181no—
217C4⋊C4—61no—
218C22⋊C4—61no—
219C42—61no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—41no—
223C22×C3—81no—
224C22×C3—121no—
225C22×C3C4×S411yescharacteristic
226C2×D3—122no—
227C2×D3—242no—
228C2×D3—244no—
229C2×D3—242no—
230C2×D3—242no—
231C2×D3—242no—
232C2×D3—484no—
233C2×D3—482no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—81no—
239C3×C4—81no—
240C3×C4—241no—
241C3×C4C4⋊S411yescharacteristic
242C3×C4GL(2,ℤ/4)11yescharacteristic
243C32—41noSylow
244C23—31no—
245C23—182no—
246C23—361no—
247C23—361no—
248C23—722no—
249Q8—31no—
250Q8#372211yescharacteristic
251D4—31no—
252D4—31no—
253D4—61no—
254D4—182no—
255D4—182no—
256D4—361no—
257D4—361no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—121no—
264C2×C4D3×S411yescharacteristic
265C8—61no—
266C8—121no—
267C8—182no—
268C8—181no—
269C2×C3—21no—
270C2×C3—41no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—81no—
274C2×C3—82no—
275C2×C3—81no—
276C2×C3(C2×S4)⋊C411yescharacteristic
277D3—242no—
278D3—482no—
279D3—484no—
280C22—31no—
281C22—31no—
282C22—121no—
283C22—362no—
284C22—722no—
285C22—722no—
286C22—722no—
287C22(C3⋊S4)⋊C411yescharacteristic
288C4—31no—
289C4—31no—
290C4—241no—
291C4#6420711yescharacteristic
292C4#6421011yescharacteristic
293C3—41no—
294C3—81no—
295C3#7313111yescharacteristic, complemented
296C2—21no—
297C2—61no—
298C2—61no—
299C2—722no—
300C2#1059324211yescharacteristic
subgroupG/H
centerC2#10593242
commutatorC2×C3×SL(2,𝔽3)C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC2×C3(C2×S4)⋊C4
prime radicalC2×#64267C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5819⋊C3C2semidirect
3. C3×#5423C2semidirect
9. C2×C3⋊GL(2,𝔽3)C4semidirect
16. #5819D3semidirect
18. #5423D3semidirect
295. C3#73131semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
#5819⋊C32
C3×#54232
#731313
#704003
#635594