Finite Group Databasebeta

← #11920711

Subgroups — #11920711

Order 1152.

348 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 43 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×#986C211yesmaximal, characteristic
2#5819⋊C3C211yesmaximal, characteristic, complemented
3#10593657C211yesmaximal, characteristic, complemented
4#73131—31nomaximal
5#72090—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#986—21no—
8C3×#986—21no—
9C2×C3×GL(2,𝔽3)C412yescomplemented
10C3⋊(#131⋊C3)—21no—
11C3⋊GU(2,𝔽3)—21no—
12#63558—41nomaximal
13C2×#986D311yescharacteristic
14C2×C3×AGL(1,ℤ/8)—31no—
15#5819D311yescharacteristic, complemented
16#568⋊C3—31no—
17#5423—31no—
18#5423—61no—
19#5236—31no—
20C2×#833—31no—
21#5215—31no—
22C3⋊#338—31no—
23#4556—31no—
24C2×C3×D12—41no—
25C3×#41⋊C3D411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C3×GL(2,𝔽3)—22no—
29#3748—41no—
30C3×#208—41no—
31#1903—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33C22×C3×D4—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3C2×D311yescharacteristic
36#986—21no—
37#986—21no—
38C2×GL(2,𝔽3)Dic312yescomplemented
39C3×AGL(1,ℤ/8)—61no—
40C3×AGL(1,ℤ/8)—61no—
41C3×AGL(1,ℤ/8)—61no—
42C3×AGL(1,ℤ/8)—61no—
43C2×C3×SD16—32no—
44C2×C3×D8—32no—
45C2×C3×M16—31no—
46#948—21no—
47#948—61no—
48#926—31no—
49C2×#208—31no—
50C2×#208—31no—
51C2×#208—31no—
52#131⋊C3—61no—
53GU(2,𝔽3)—62no—
54GU(2,𝔽3)—61no—
55#837—61no—
56#837—61no—
57#833—61no—
58#833—61no—
59#832—62no—
60#823—121no—
61C22×C3×D3—42no—
62C2×C32×C4—41no—
63C3×D12—81no—
64C3×D12—81no—
65C3×SL(2,𝔽3)C22⋊C411yescharacteristic
66C3⋊(C3⋊C8)—81no—
67C3×C3⋊C8—81no—
68C2×AGL(1,ℤ/8)—31no—
69#568—31no—
70#421—91no—
71#418—91no—
72C2×#100—91no—
73#344—91no—
74#338—91no—
75C24×C3—61no—
76C3×#41—31no—
77C3×#41—31no—
78C3×#41D1211yescharacteristic
79C3×#41#6711yescharacteristic
80C2×C3×Q8C4×D311yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—32no—
83C2×C3×D4—61no—
84C2×C3×D4—61no—
85C2×C3×D4—61no—
86C2×C3×D4—61no—
87C2×C3×D4—61no—
88C22×C3×C4—31no—
89C2×D12—41no—
90#41⋊C3—21no—
91#41⋊C3—21no—
92#41⋊C3#6711yescharacteristic
93#41⋊C3#6711yescharacteristic
94C2×SL(2,𝔽3)—21no—
95C2×SL(2,𝔽3)C2×Dic311yescharacteristic
96GL(2,𝔽3)—22no—
97C3×SD16—62no—
98C3×SD16—62no—
99C3×D8—62no—
100C3×D8—62no—
101C3×M16—61no—
102C3×M16—61no—
103C3×M16—121no—
104C2×C3×C8—32no—
105#217—61no—
106C4⋊Dic3—61no—
107C4×Dic3—61no—
108#208—31no—
109#208—61no—
110#208—61no—
111#208—61no—
112#208—61no—
113#208—121no—
114#208—241no—
115#208S411yescharacteristic
116C2×C3⋊C8—32no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—61no—
119C22×C32—41no—
120C2×C3×D3—82no—
121C2×C3×D3—162no—
122C32×C4—41no—
123C32×C4—41no—
124C2×#41D3211yescharacteristic
125C22×D4—31no—
126AGL(1,ℤ/8)—61no—
127AGL(1,ℤ/8)—61no—
128AGL(1,ℤ/8)—61no—
129AGL(1,ℤ/8)—61no—
130C2×SD16—32no—
131C2×D8—32no—
132#131—61no—
133#131—181no—
134C2×M16—31no—
135C2×M16—91no—
136C2×M16—91no—
137C2×M16—91no—
138#117—91no—
139#104—181no—
140#104—181no—
141#102—182no—
142#100—181no—
143#100—181no—
144C23×C3—31no—
145C23×C3—62no—
146C23×C3—121no—
147C23×C3—121no—
148C23×C3—242no—
149C22×D3—42no—
150C3×Q8—31no—
151C3×Q8(C2×D3)⋊C411yescharacteristic
152C3×D4—31no—
153C3×D4—31no—
154C3×D4—61no—
155C3×D4—62no—
156C3×D4—62no—
157C3×D4—121no—
158C3×D4—121no—
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×C4C2×S411yescharacteristic
167C2×Dic3—121no—
168D12—81no—
169D12—81no—
170SL(2,𝔽3)—21no—
171SL(2,𝔽3)#21711yescharacteristic
172C3×C8—62no—
173C3×C8—241no—
174C3⋊C8—21no—
175C3⋊C8—61no—
176C3⋊C8—121no—
177C3⋊C8—241no—
178C3⋊C8—242no—
179C2×C32—41no—
180C2×C32—81no—
181C3×D3—162no—
182C24—61no—
183#41—31no—
184#41—31no—
185#41#62511yescharacteristic
186#41#62611yescharacteristic
187C2×Q8D3×Dic311yescharacteristic
188C2×D4—31no—
189C2×D4—32no—
190C2×D4—61no—
191C2×D4—61no—
192C2×D4—61no—
193C2×D4—61no—
194C2×D4—61no—
195C22×C4—31no—
196SD16—62no—
197SD16—62no—
198D8—62no—
199D8—62no—
200M16—31no—
201M16—61no—
202M16—61no—
203M16—91no—
204M16—181no—
205M16—181no—
206M16—181no—
207M16—181no—
208C2×C8—32no—
209C2×C8—92no—
210C2×C8—181no—
211C2×C8—181no—
212C4⋊C4—181no—
213C22⋊C4—181no—
214C42—181no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—41no—
218C22×C3—81no—
219C22×C3—121no—
220C22×C3—122no—
221C22×C3—242no—
222C22×C3—242no—
223C22×C3—242no—
224C22×C3C4×S411yescharacteristic
225C2×D3—82no—
226C2×D3—162no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—81no—
232C3×C4—81no—
233C3×C4C4⋊S411yescharacteristic
234C3×C4GL(2,ℤ/4)11yescharacteristic
235Dic3—241no—
236C32—41noSylow
237C23—31no—
238C23—62no—
239C23—121no—
240C23—121no—
241C23—242no—
242Q8—31no—
243Q8(C2×D3)⋊Dic311yescharacteristic
244D4—31no—
245D4—31no—
246D4—61no—
247D4—62no—
248D4—62no—
249D4—121no—
250D4—121no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—361no—
257C2×C4D3×S411yescharacteristic
258C8—62no—
259C8—61no—
260C8—181no—
261C8—361no—
262C2×C3—21no—
263C2×C3—41no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—81no—
267C2×C3—81no—
268C2×C3—161no—
269C2×C3—242no—
270C2×C3(C2×S4)⋊C411yescharacteristic
271D3—162no—
272C22—31no—
273C22—31no—
274C22—121no—
275C22—122no—
276C22—242no—
277C22—242no—
278C22—242no—
279C22Dic3×S411yescharacteristic
280C4—31no—
281C4—31no—
282C4—721no—
283C4#6420711yescharacteristic
284C4#6420911yescharacteristic
285C3—41no—
286C3—81no—
287C3#7313111yescharacteristic, complemented
288C2—21no—
289C2—61no—
290C2—61no—
291C2—242no—
292C2(C2×S4)⋊Dic311yescharacteristic
subgroupG/H
centerC2(C2×S4)⋊Dic3
commutatorC2×C3×SL(2,𝔽3)C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC2×C3(C2×S4)⋊C4
prime radicalC2×C3×#986C2
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. #10593657C2semidirect
9. C2×C3×GL(2,𝔽3)C4semidirect
15. #5819D3semidirect
38. C2×GL(2,𝔽3)Dic3semidirect
287. C3#73131semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9862
#5819⋊C32
#105936572
#731313
#720903
#635584