Finite Group Databasebeta

← #11921034

Subgroups — #11921034

Order 1152.

454 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 59 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×#520⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73078—33nomaximal
5#71667—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64267—21no—
8#64267C412yescomplemented
9C2×C3⋊GL(2,𝔽3)—21no—
10C3×C4×SL(2,𝔽3)—21no—
11C3⋊GU(2,𝔽3)—21no—
12#63636—41nomaximal
13C2×#986—33no—
14C3×#520D311yescharacteristic, complemented
15C2×#949—31no—
16#520⋊C3D313yescomplemented
17C2×GU(2,𝔽3)—33no—
18C2×#837—31no—
19#5001—31no—
20#4988—31no—
21#4532—31no—
22#4530—31no—
23C2×C4⋊(C3⋊D3)—41no—
24C3×#41⋊C3C2×C411yescharacteristic
25C3×#41⋊C3D411yescharacteristic
26C2×C3×SL(2,𝔽3)D411yescharacteristic
27C3⋊GL(2,𝔽3)—22no—
28C32×C4⋊C4—41no—
29C2×C3⋊(C3⋊C8)—41no—
30#1883—91noSylow
31C2×C3×#41C2×D311yescharacteristic
32C22×D12—31no—
33C2×#41⋊C3C2×D313yes—
34#986—36no—
35#986—63no—
36C2×GL(2,𝔽3)—63no—
37C3×C4×Q8—21no—
38C3×C4×D4—61no—
39C3×#117—31no—
40C2×C3×C4⋊C4—31no—
41#949—32no—
42#949—32no—
43#949—61no—
44#949—61no—
45C2×#215—61no—
46C2×#213—61no—
47C2×(C2×D3)⋊C4—62no—
48C2×C4×Dic3—31no—
49C2×#208—31no—
50C2×#208—31no—
51C4×SL(2,𝔽3)—23no—
52GU(2,𝔽3)—63no—
53#837—61no—
54#837—61no—
55#809—61no—
56#809—61no—
57#809—123no—
58C22×C3⋊D3—81no—
59C2×C32×C4—41no—
60C2×C32×C4—81no—
61C4⋊(C3⋊D3)—42no—
62C4⋊(C3⋊D3)—81no—
63C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
64C3⋊(C3⋊C8)—81no—
65C2×AGL(1,ℤ/8)—91no—
66#520C3⋊D311yescharacteristic, complemented
67C2×#104—91no—
68#418—91no—
69#390—91no—
70#343—91no—
71#339—91no—
72C23×D3—61no—
73C3×#41—31no—
74C3×#41—31no—
75C3×#41C4×D311yescharacteristic
76C3×#41D1211yescharacteristic
77C2×C3×Q8#6711yescharacteristic
78C2×C3×D4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—32no—
81C22×Dic3—61no—
82C2×D12—32no—
83C2×D12—61no—
84C2×D12—61no—
85C2×D12—61no—
86C2×D12—61no—
87C2×D12—61no—
88C2×D12—123no—
89#41⋊C3C4×D313yes—
90#41⋊C3D1213yes—
91C2×SL(2,𝔽3)#6713yes—
92GL(2,𝔽3)—66no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4—43no—
95C3×C4⋊C4—61no—
96C3×C4⋊C4S411yescharacteristic
97C3×C22⋊C4—61no—
98C3×C42—61no—
99#215—62no—
100#215—62no—
101#213—62no—
102#213—62no—
103(C2×D3)⋊C4—124no—
104C4×Dic3—61no—
105C4×Dic3—61no—
106#208—32no—
107#208—61no—
108#208—61no—
109#208—61no—
110C2×C3⋊C8—61no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—123no—
113C22×C32—41no—
114C2×C3⋊D3—82no—
115C2×C3⋊D3—161no—
116C2×C3⋊D3—161no—
117C32×C4—41no—
118C32×C4—41no—
119C32×C4—161no—
120C2×#41C2×C3⋊D311yescharacteristic
121C22×D4—91no—
122AGL(1,ℤ/8)—92no—
123AGL(1,ℤ/8)—92no—
124AGL(1,ℤ/8)—181no—
125AGL(1,ℤ/8)—181no—
126C2×SD16—181no—
127C2×D8—181no—
128C2×M16—91no—
129C2×M16—91no—
130C4×Q8—21no—
131C4×D4—61no—
132#117—31no—
133C2×C4⋊C4—31no—
134C2×C22⋊C4—182no—
135C2×C42—91no—
136#104—181no—
137#104—181no—
138#102—181no—
139#102—181no—
140C23×C3—31no—
141C22×D3—62no—
142C22×D3—121no—
143C22×D3—122no—
144C22×D3—122no—
145C22×D3—121no—
146C22×D3—243no—
147C3×Q8—31no—
148C3×Q8(C2×D3)⋊C411yescharacteristic
149C3×D4—31no—
150C3×D4—31no—
151C3×D4—61no—
152C2×C3×C4—21no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—43no—
159C2×C3×C4—61no—
160C2×C3×C4—83no—
161C2×C3×C4—122no—
162C2×C3×C4C2×S411yescharacteristic
163C2×Dic3—62no—
164C2×Dic3—121no—
165C2×Dic3—121no—
166D12—62no—
167D12—62no—
168D12—126no—
169D12—121no—
170D12—121no—
171D12—243no—
172SL(2,𝔽3)(C2×D3)⋊C413yescomplemented
173C3⋊C8—62no—
174C3⋊C8—121no—
175C3⋊C8—243no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C3⋊D3—162no—
180C24—181no—
181#41—31no—
182#41—31no—
183#41C4×C3⋊D311yescharacteristic
184#41C4⋊(C3⋊D3)11yescharacteristic
185C2×Q8#63811yescharacteristic
186C2×D4—31no—
187C2×D4—92no—
188C2×D4—181no—
189C2×D4—181no—
190C2×D4—181no—
191C2×D4—181no—
192C2×D4—181no—
193C22×C4—31no—
194C22×C4—32no—
195C22×C4—181no—
196SD16—182no—
197SD16—182no—
198D8—182no—
199D8—182no—
200M16—92no—
201M16—181no—
202M16—181no—
203M16—181no—
204C2×C8—181no—
205C2×C8—181no—
206C4⋊C4—31no—
207C4⋊C4—61no—
208C4⋊C4C3⋊S411yescharacteristic
209C22⋊C4—61no—
210C22⋊C4—364no—
211C42—61no—
212C42—181no—
213C42—181no—
214C22×C3—31no—
215C22×C3—31no—
216C22×C3—43no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3GL(2,ℤ/4)11yescharacteristic
220C2×D3—124no—
221C2×D3—122no—
222C2×D3—246no—
223C2×D3—242no—
224C2×D3—241no—
225C2×D3—241no—
226C2×D3—483no—
227C2×D3—483no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—41no—
231C3×C4—43no—
232C3×C4—43no—
233C3×C4—121no—
234C3×C4—163no—
235C3×C4C4×S411yescharacteristic
236C3×C4C4⋊S411yescharacteristic
237Dic3—122no—
238C32—41noSylow
239C23—31no—
240C23—182no—
241C23—361no—
242C23—362no—
243C23—362no—
244C23—361no—
245Q8—31no—
246Q8#375211yescharacteristic
247D4—31no—
248D4—31no—
249D4—61no—
250D4—182no—
251D4—182no—
252D4—361no—
253D4—361no—
254C2×C4—21no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—182no—
263C2×C4—361no—
264C2×C4—361no—
265C2×C4C2×C3⋊S411yescharacteristic
266C8—182no—
267C8—361no—
268C2×C3—43no—
269C2×C3—43no—
270C2×C3—43no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3(C2×S4)⋊C411yescharacteristic
274C2×C3#542911yescharacteristic
275C2×C3#543111yescharacteristic
276D3—242no—
277D3—486no—
278C22—31no—
279C22—31no—
280C22—61no—
281C22—61no—
282C22—364no—
283C22—362no—
284C22—722no—
285C22—721no—
286C22—721no—
287C22#6426911yescharacteristic
288C4—31no—
289C4—31no—
290C4—41no—
291C4—121no—
292C4—362no—
293C4C4×C3⋊S411yescharacteristic
294C4C4⋊(C3⋊S4)11yescharacteristic
295C3—43no—
296C3#7307811yescharacteristic, complemented
297C2—61no—
298C2—61no—
299C2—722no—
300C2#1059364711yescharacteristic
301C2#1059366311yescharacteristic
302C2#1059366511yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC3×#520D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×#64267C2
cosocleC2×C3×#41⋊C3C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C2×C3⋊GU(2,𝔽3)C2semidirect
8. #64267C4semidirect
14. C3×#520D3semidirect
16. #520⋊C3D3semidirect
63. C3×SL(2,𝔽3)C22⋊C4semidirect
66. #520C3⋊D3semidirect
172. SL(2,𝔽3)(C2×D3)⋊C4semidirect
296. C3#73078semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C3×#520⋊C32
C2×C3⋊GU(2,𝔽3)2
#730783
#716673
#636364