Finite Group Databasebeta

← #11922916

Subgroups — #11922916

Order 1152.

718 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 89 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
3#10593658C211yesmaximal, characteristic, complemented
4#10593628C212yesmaximal, complemented
5C4×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
6#85366—33nomaximal
7#82564—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2212yes—
11#64102—41nomaximal
12C3×C4×SL(2,𝔽3)C2212yescomplemented
13C3⋊#859C2211yescharacteristic, complemented
14C3⋊#859C2211yescharacteristic, complemented
15C2×#986—33no—
16C3×#520D311yescharacteristic, complemented
17C2×#949—31no—
18#5521—31no—
19#520⋊C3D313yescomplemented
20#5424—33no—
21#5394—36no—
22C4×GL(2,𝔽3)—36no—
23#5233—31no—
24#5026—32no—
25C4×#215—32no—
26#5014—32no—
27C4×#213—32no—
28#4997—31no—
29C2×#809—31no—
30#4964—31no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C4×C3⋊D3—42no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊GL(2,𝔽3)C2×C414yescomplemented
36C32×C4⋊C4—41no—
37#3752—42no—
38C4×C3⋊Dic3—41no—
39#2946—91noSylow
40C2×C3×#41C2×D311yescharacteristic
41C22×D12—31no—
42C2×#41⋊C3C2×D313yes—
43#986—312no—
44C2×GL(2,𝔽3)—36no—
45C3×C4×Q8C2×D312yes—
46C3×C4×D4—32no—
47C3×#117—31no—
48C2×C3×C4⋊C4—31no—
49#949—34no—
50#949—34no—
51C2×#215—32no—
52C2×#213—32no—
53#926—31no—
54C2×#208—31no—
55C22×C3⋊C8—31no—
56#893—61no—
57#893—61no—
58#893—123no—
59D3×C22⋊C4—62no—
60C4×D12—32no—
61C4×D12—32no—
62C4×SL(2,𝔽3)C2×D316yescomplemented
63#859—33no—
64#859—33no—
65#835—31no—
66#835—31no—
67#832—31no—
68#832—31no—
69#809—32no—
70#809—32no—
71#808—32no—
72#807—32no—
73#803—32no—
74C4×C3⋊C8—32no—
75C22×C3⋊D3—42no—
76C2×C32×C4—41no—
77C2×C32×C4—42no—
78C2×C3⋊Dic3—41no—
79C2×C3⋊Dic3—41no—
80C4⋊(C3⋊D3)—44no—
81C4×C3⋊D3—84no—
82C3×SL(2,𝔽3)C22×C411yescharacteristic
83C2×AGL(1,ℤ/8)—91no—
84#520C3⋊D311yescharacteristic, complemented
85#518—91no—
86#442—92no—
87#440—92no—
88C4×SD16—92no—
89C4×D8—92no—
90#428—91no—
91#416—91no—
92C2×#102—91no—
93#405—91no—
94C23×D3—61no—
95C3×#41—32no—
96C3×#41C4×D312yescomplemented
97C2×C3×Q8C22×D311yescharacteristic
98C2×C3×D4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C2×D12—32no—
102C2×D12—34no—
103C2×D12—62no—
104C2×D12—61no—
105C2×D12—123no—
106C2×C4×D3—62no—
107C2×C4×D3—62no—
108C2×C4×D3—126no—
109#41⋊C3C4×D316yescomplemented
110C2×SL(2,𝔽3)C22×D313yes—
111GL(2,𝔽3)—312no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—43no—
115C3×C4⋊C4S411yescharacteristic
116C3×C22⋊C4—32no—
117C3×C42—32no—
118#217—61no—
119#215—34no—
120#215—34no—
121#213—38no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—126no—
125C4⋊Dic3—31no—
126C4⋊Dic3—31no—
127C4×Dic3—61no—
128C4×Dic3—123no—
129#208—34no—
130C2×C3⋊C8—32no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—31no—
133C2×C3⋊C8—62no—
134C22×C32—41no—
135C2×C3⋊D3—44no—
136C2×C3⋊D3—82no—
137C2×C3⋊D3—82no—
138C32×C4—42no—
139C32×C4—82no—
140C3⋊Dic3—42no—
141C3⋊Dic3—81no—
142C2×#41C2×C3⋊D311yescharacteristic
143C22×D4—91no—
144AGL(1,ℤ/8)—94no—
145AGL(1,ℤ/8)—94no—
146C2×SD16—92no—
147C2×D8—92no—
148C2×M16—91no—
149C22×C8—91no—
150C4×Q8C2×C3⋊D312yes—
151C4×D4—32no—
152C4×D4—92no—
153C4×D4—92no—
154C4×D4—181no—
155C4×D4—181no—
156#117—31no—
157#117—91no—
158C2×C4⋊C4—31no—
159C2×C22⋊C4—182no—
160#107—92no—
161#106—92no—
162#103—91no—
163#103—91no—
164#102—92no—
165#102—92no—
166#102—91no—
167#102—91no—
168#97—92no—
169C4×C8—92no—
170C23×C3—31no—
171C22×D3—62no—
172C22×D3—64no—
173C22×D3—126no—
174C22×D3—241no—
175C22×D3—241no—
176C3×Q8—31no—
177C3×Q8C2×C4×D311yescharacteristic
178C3×D4—32no—
179C3×D4—61no—
180C2×C3×C4—31no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—43no—
185C2×C3×C4—46no—
186C2×C3×C4—241no—
187C2×C3×C4C2×S412yes—
188C2×C3×C4C2×S411yescharacteristic
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—123no—
192C2×Dic3—123no—
193D12—34no—
194D12—62no—
195D12—64no—
196D12—1212no—
197C4×D3—242no—
198C4×D3—242no—
199C4×D3—2412no—
200SL(2,𝔽3)C2×C4×D313yes—
201C3⋊C8—34no—
202C3⋊C8—61no—
203C3⋊C8—61no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C3⋊D3—84no—
208C24—181no—
209#41—32no—
210#41C4×C3⋊D312yescomplemented
211C2×Q8C22×C3⋊D311yescharacteristic
212C2×D4—31no—
213C2×D4—92no—
214C2×D4—94no—
215C2×D4—182no—
216C2×D4—181no—
217C22×C4—31no—
218C22×C4—61no—
219C22×C4—182no—
220C22×C4—182no—
221SD16—94no—
222SD16—94no—
223D8—98no—
224M16—94no—
225C2×C8—92no—
226C2×C8—91no—
227C2×C8—91no—
228C2×C8—182no—
229C4⋊C4—32no—
230C4⋊C4—31no—
231C4⋊C4—91no—
232C4⋊C4—91no—
233C4⋊C4C3⋊S411yescharacteristic
234C22⋊C4—32no—
235C22⋊C4—182no—
236C22⋊C4—181no—
237C22⋊C4—182no—
238C42—32no—
239C42—181no—
240C22×C3—32no—
241C22×C3—43no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3C22×S411yescharacteristic
245C2×D3—64no—
246C2×D3—1212no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—246no—
250C2×D3—246no—
251C2×D3—244no—
252C3×C4—22no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—46no—
256C3×C4—62no—
257C3×C4—86no—
258C3×C4C4×S412yes—
259Dic3—126no—
260Dic3—121no—
261Dic3—121no—
262Dic3—243no—
263C32—41noSylow
264C23—31no—
265C23—184no—
266C23—182no—
267C23—721no—
268C23—721no—
269Q8—31no—
270Q8C2×C4×C3⋊D311yescharacteristic
271D4—32no—
272D4—61no—
273D4—94no—
274D4—182no—
275D4—184no—
276C2×C4—31no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—241no—
283C2×C4—722no—
284C2×C4—722no—
285C2×C4C2×C3⋊S412yes—
286C2×C4C2×C3⋊S411yescharacteristic
287C8—94no—
288C8—181no—
289C8—181no—
290C2×C3—43no—
291C2×C3—43no—
292C2×C3—43no—
293C2×C3—62no—
294C2×C3C2×C4×S411yescharacteristic
295C2×C3#592511yescharacteristic
296C2×C3#592611yescharacteristic
297D3—124no—
298D3—2412no—
299C22—32no—
300C22—61no—
301C22—61no—
302C22—184no—
303C22—362no—
304C22—362no—
305C22—724no—
306C22C22×C3⋊S411yescharacteristic
307C4—22no—
308C4—31no—
309C4—31no—
310C4—62no—
311C4—361no—
312C4—361no—
313C4C4×C3⋊S412yes—
314C3—43no—
315C3#8536611yescharacteristic, complemented
316C2—62no—
317C2—364no—
318C2C2×C4×C3⋊S411yescharacteristic
319C2#1059658811yescharacteristic
320C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC3×#520D3
socleC22×C3C22×S4
prime radicalC2×#64267C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC4⋊C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. #10593658C2semidirect
4. #10593628C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
9. #64267C4semidirect
12. C3×C4×SL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C3⋊#859C22semidirect
16. C3×#520D3semidirect
19. #520⋊C3D3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. C3⋊GL(2,𝔽3)C2×C4semidirect
62. C4×SL(2,𝔽3)C2×D3semidirect
84. #520C3⋊D3semidirect
96. C3×#41C4×D3semidirect
109. #41⋊C3C4×D3semidirect
210. #41C4×C3⋊D3semidirect
315. C3#85366semidirect

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
#105936582
#105936282
C4×C3⋊GL(2,𝔽3)2
#853663
#825643
#641024