Finite Group Databasebeta

← #11923068

Subgroups — #11923068

Order 1152.

764 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2#10593660C212yesmaximal, complemented
3#10593658C212yesmaximal, complemented
4#10593654C212yesmaximal, complemented
5#85519—33nomaximal
6#84072—31nomaximal
7C2×C3×#41⋊C3C414yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#64135—41nomaximal
12C3⋊#859C2214yescomplemented
13C22×C3×#41D311yescharacteristic
14C22×#41⋊C3D313yes—
15C2×#926—31no—
16C22×#208—31no—
17#5426—36no—
18#5424—36no—
19#5420—36no—
20#5233—32no—
21#5233—32no—
22#5232—32no—
23#5232—32no—
24#5225—32no—
25#5215—32no—
26C22×C32×C4—41no—
27C3×#41⋊C3C2×C412yes—
28C3×#41⋊C3C2×C412yescomplemented
29C2×C3×SL(2,𝔽3)C2×C412yescomplemented
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31#3757—42no—
32C4⋊(C3⋊Dic3)—42no—
33C4×C3⋊Dic3—42no—
34#2896—91noSylow
35C2×C3×#41—34no—
36C2×C3×#41—31no—
37C2×C3×#41—31no—
38C2×C3×#41Dic314yescomplemented
39C2×C3×#41C2×D311yescharacteristic
40C2×C3×#41C2×D311yescharacteristic
41C22×C3×Q8C2×D311yescharacteristic
42C22×C3×D4—31no—
43C23×C3×C4—31no—
44C2×#41⋊C3Dic3112yescomplemented
45C2×#41⋊C3C2×D313yes—
46C2×#41⋊C3C2×D313yes—
47C22×SL(2,𝔽3)C2×D313yes—
48C2×#217—61no—
49#926—32no—
50#926—32no—
51#926—61no—
52#926—61no—
53#926—123no—
54C2×C4⋊Dic3—32no—
55C2×C4×Dic3—61no—
56C2×#208—32no—
57C2×#208—32no—
58C2×#208—64no—
59C22×C3⋊C8—32no—
60#859—312no—
61#835—34no—
62#835—34no—
63#832—34no—
64#832—34no—
65C23×C32—41no—
66C2×C32×C4—44no—
67C2×C32×C4—41no—
68C2×C32×C4—41no—
69C2×C3⋊Dic3—44no—
70C3×SL(2,𝔽3)C22×C411yescharacteristic
71C22×#41C3⋊D311yescharacteristic
72C22×M16—91no—
73C2×#117—91no—
74#419—92no—
75#418—92no—
76#417—92no—
77#417—92no—
78#416—92no—
79#416—92no—
80C24×C3—31no—
81C3×#41—34no—
82C3×#41—34no—
83C3×#41—32no—
84C3×#41—32no—
85C3×#41—62no—
86C3×#41—62no—
87C3×#41C2×Dic312yes—
88C3×#41C2×Dic312yes—
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8C2×Dic312yes—
92C2×C3×Q8C22×D311yescharacteristic
93C2×C3×D4—31no—
94C2×C3×D4—31no—
95C2×C3×D4—34no—
96C2×C3×D4—62no—
97C2×C3×D4—61no—
98C22×C3×C4—31no—
99C22×C3×C4—34no—
100C22×C3×C4—34no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—43no—
106C22×C3×C4S411yescharacteristic
107C22×Dic3—62no—
108#41⋊C3C2×Dic316yes—
109#41⋊C3C2×Dic316yescomplemented
110C2×SL(2,𝔽3)C2×Dic316yescomplemented
111C2×SL(2,𝔽3)C22×D313yes—
112#217—62no—
113#217—62no—
114#217—126no—
115C4⋊Dic3—34no—
116C4⋊Dic3—62no—
117C4⋊Dic3—126no—
118C4×Dic3—62no—
119C4×Dic3—62no—
120C4×Dic3—126no—
121#208—64no—
122#208—64no—
123C2×C3⋊C8—34no—
124C2×C3⋊C8—64no—
125C22×C32—42no—
126C22×C32—41no—
127C22×C32—82no—
128C32×C4—42no—
129C32×C4—42no—
130C3⋊Dic3—84no—
131C2×#41—31no—
132C2×#41—31no—
133C2×#41—34no—
134C2×#41C3⋊Dic314yescomplemented
135C2×#41C2×C3⋊D311yescharacteristic
136C2×#41C2×C3⋊D311yescharacteristic
137C22×Q8C2×C3⋊D311yescharacteristic
138C22×D4—31no—
139C23×C4—31no—
140C2×M16—92no—
141C2×M16—92no—
142C2×M16—184no—
143C22×C8—92no—
144#117—92no—
145#117—92no—
146#117—181no—
147#117—181no—
148C2×C4⋊C4—92no—
149C2×C22⋊C4—181no—
150C2×C42—181no—
151#103—94no—
152#103—94no—
153#102—94no—
154#102—94no—
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—34no—
158C23×C3—43no—
159C23×C3—122no—
160C23×C3C2×S411yescharacteristic
161C3×Q8—32no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8C22×Dic311yescharacteristic
165C3×D4—32no—
166C3×D4—32no—
167C3×D4—61no—
168C3×D4—61no—
169C3×D4—64no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—34no—
179C2×C3×C4—34no—
180C2×C3×C4—43no—
181C2×C3×C4—43no—
182C2×C3×C4—412no—
183C2×C3×C4SL(2,ℤ/4)14yes—
184C2×C3×C4C2×S411yescharacteristic
185C2×C3×C4C2×S411yescharacteristic
186C2×Dic3—64no—
187C2×Dic3—1212no—
188C2×Dic3—124no—
189SL(2,𝔽3)C22×Dic313yes—
190C3⋊C8—64no—
191C2×C32—42no—
192C2×C32—41no—
193C2×C32—82no—
194C24—31no—
195#41—34no—
196#41—34no—
197#41—32no—
198#41—32no—
199#41—62no—
200#41—62no—
201#41C2×C3⋊Dic312yes—
202#41C2×C3⋊Dic312yes—
203C2×Q8—31no—
204C2×Q8—32no—
205C2×Q8C2×C3⋊Dic312yes—
206C2×Q8C22×C3⋊D311yescharacteristic
207C2×D4—31no—
208C2×D4—31no—
209C2×D4—34no—
210C2×D4—62no—
211C2×D4—61no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—34no—
218C22×C4—34no—
219C22×C4—182no—
220C22×C4C3⋊S411yescharacteristic
221M16—184no—
222M16—184no—
223C2×C8—94no—
224C2×C8—184no—
225C4⋊C4—94no—
226C4⋊C4—182no—
227C22⋊C4—182no—
228C22⋊C4—182no—
229C42—182no—
230C42—182no—
231C22×C3—22no—
232C22×C3—32no—
233C22×C3—32no—
234C22×C3—46no—
235C22×C3—43no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3—86no—
239C22×C3—124no—
240C22×C3C2×SL(2,ℤ/4)12yes—
241C22×C3C22×S411yescharacteristic
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—32no—
245C3×C4—46no—
246C3×C4—46no—
247C3×C4C2×SL(2,ℤ/4)12yes—
248C3×C4C2×SL(2,ℤ/4)12yes—
249Dic3—124no—
250Dic3—2412no—
251C32—41noSylow
252C23—31no—
253C23—31no—
254C23—34no—
255C23—122no—
256C23C2×C3⋊S411yescharacteristic
257Q8—32no—
258Q8—31no—
259Q8—61no—
260Q8C22×C3⋊Dic311yescharacteristic
261D4—32no—
262D4—32no—
263D4—61no—
264D4—61no—
265D4—64no—
266C2×C4—32no—
267C2×C4—32no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—34no—
275C2×C4—34no—
276C2×C4—184no—
277C2×C4—364no—
278C2×C4C3⋊SL(2,ℤ/4)14yes—
279C2×C4C2×C3⋊S411yescharacteristic
280C2×C4C2×C3⋊S411yescharacteristic
281C8—184no—
282C2×C3—22no—
283C2×C3—46no—
284C2×C3—43no—
285C2×C3—62no—
286C2×C3—62no—
287C2×C3—86no—
288C2×C3#592312yes—
289C2×C3C22×SL(2,ℤ/4)11yescharacteristic
290C22—22no—
291C22—32no—
292C22—32no—
293C22—62no—
294C22—62no—
295C22—124no—
296C22C2×C3⋊SL(2,ℤ/4)12yes—
297C22C22×C3⋊S411yescharacteristic
298C4—31no—
299C4—31no—
300C4—32no—
301C4—364no—
302C4C2×C3⋊SL(2,ℤ/4)12yes—
303C4C2×C3⋊SL(2,ℤ/4)12yes—
304C3—43no—
305C3#8551911yescharacteristic, complemented
306C2—22no—
307C2—62no—
308C2—62no—
309C2#1059658612yes—
310C2C22×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C22×C3⋊S4
FittingC22×C3×#41D3
socleC22×C3C22×S4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC22×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #10593660C2semidirect
3. #10593658C2semidirect
4. #10593654C2semidirect
7. C2×C3×#41⋊C3C4semidirect
12. C3⋊#859C22semidirect
28. C3×#41⋊C3C2×C4semidirect
29. C2×C3×SL(2,𝔽3)C2×C4semidirect
38. C2×C3×#41Dic3semidirect
44. C2×#41⋊C3Dic3semidirect
109. #41⋊C3C2×Dic3semidirect
110. C2×SL(2,𝔽3)C2×Dic3semidirect
134. C2×#41C3⋊Dic3semidirect
305. C3#85519semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
#105936602
#105936582
#105936542
#855193
#840723
#641354