Finite Group Databasebeta

← #11923079

Subgroups — #11923079

Order 1152.

768 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 81 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×#994C211yesmaximal, characteristic, complemented
2C2×#64267C211yesmaximal, characteristic, complemented
3#10593661C214yesmaximal, complemented
4#10593658C211yesmaximal, characteristic, complemented
5#85530—33nomaximal
6#84101—31nomaximal
7C3×#994—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2212yescomplemented
10#64267—22no—
11C2×C3⋊GL(2,𝔽3)C2212yescomplemented
12#64156—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
15C2×#994D313yescomplemented
16C2×#986—33no—
17#5830—31no—
18C2×#949—31no—
19C22×#213—31no—
20#5427—312no—
21#5424—33no—
22#5238—34no—
23#5234—31no—
24#5233—31no—
25#5217—34no—
26#5215—31no—
27C2×C32×Q8—41no—
28C2×C4⋊(C3⋊D3)—41no—
29C3×#41⋊C3—22no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊GL(2,𝔽3)—22no—
33#3752—44no—
34C4×C3⋊Dic3—41no—
35#3008—91noSylow
36C3×H(5,𝔽2)—22no—
37C3×H(5,𝔽2)—62no—
38C2×C3×#41—31no—
39C2×C3×#41C2×D312yes—
40C2×C3×#41C2×D311yescharacteristic
41C22×C3×D4—31no—
42C22×C3×D4—32no—
43C22×D12—31no—
44#994—26no—
45C2×#41⋊C3C2×D313yes—
46C2×#41⋊C3C2×D316yescomplemented
47#986—66no—
48C2×GL(2,𝔽3)—36no—
49#949—62no—
50#949—62no—
51#947—61no—
52#947—123no—
53C2×#215—32no—
54#940—61no—
55#937—62no—
56C2×#213—32no—
57C2×#213—34no—
58C2×#213—64no—
59#930—34no—
60#926—31no—
61C2×#208—31no—
62C22×C3⋊C8—31no—
63#859—36no—
64#835—32no—
65#832—34no—
66#832—32no—
67#830—34no—
68C22×C3⋊D3—42no—
69C32×Q8—82no—
70C2×C32×C4—41no—
71C2×C32×C4—42no—
72C2×C3⋊Dic3—42no—
73C4⋊(C3⋊D3)—82no—
74C3×SL(2,𝔽3)C2×D411yescharacteristic
75C2×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
76C2×AGL(1,ℤ/8)—91no—
77C22×D8—91no—
78#535—91no—
79#449—94no—
80#447—94no—
81#418—91no—
82#416—91no—
83#408—91no—
84C24×C3—61no—
85C23×D3—61no—
86C3×#41—22no—
87C3×#41—32no—
88C3×#41—62no—
89C3×#41—62no—
90C3×#41—62no—
91C3×#41#6712yes—
92C2×C3×Q8—43no—
93C2×C3×Q8S411yescharacteristic
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—31no—
96C2×C3×D4—32no—
97C2×C3×D4—34no—
98C2×C3×D4—32no—
99C2×C3×D4—31no—
100C2×C3×D4—62no—
101C2×C3×D4—64no—
102C2×C3×D4—64no—
103C22×C3×C4—31no—
104C22×C3×C4—32no—
105C2×#67—64no—
106C2×D12—32no—
107C2×D12—62no—
108C2×D12—62no—
109C2×D12—61no—
110C2×D12—123no—
111#41⋊C3—26no—
112#41⋊C3#6716yescomplemented
113C2×SL(2,𝔽3)C22×D313yes—
114GL(2,𝔽3)—66no—
115#217—61no—
116#215—62no—
117#215—62no—
118#213—68no—
119#213—64no—
120(C2×D3)⋊C4—64no—
121(C2×D3)⋊C4—1212no—
122C4⋊Dic3—32no—
123C4×Dic3—61no—
124C4×Dic3—123no—
125#208—62no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—62no—
129C22×C32—41no—
130C2×C3⋊D3—82no—
131C2×C3⋊D3—84no—
132C32×C4—42no—
133C32×C4—82no—
134C3⋊Dic3—82no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)—62no—
137C2×#41—31no—
138C2×#41C2×C3⋊D312yes—
139C2×#41C2×C3⋊D311yescharacteristic
140C22×D4—31no—
141C22×D4—32no—
142C22×D4—91no—
143AGL(1,ℤ/8)—182no—
144AGL(1,ℤ/8)—182no—
145C2×SD16—92no—
146C2×D8—94no—
147C2×D8—92no—
148C2×D8—184no—
149C2×M16—91no—
150C22×C8—91no—
151#127—181no—
152#124—181no—
153#121—94no—
154#120—182no—
155#117—91no—
156#103—92no—
157#102—92no—
158#102—94no—
159C22⋊C8—94no—
160C23×C3—31no—
161C23×C3—32no—
162C23×C3—64no—
163C23×C3—242no—
164C22×D3—62no—
165C22×D3—126no—
166C22×D3—122no—
167C22×D3—242no—
168C3×Q8—22no—
169C3×Q8—31no—
170C3×Q8—86no—
171C3×Q8C2×#6711yescharacteristic
172C3×D4—32no—
173C3×D4—34no—
174C3×D4—64no—
175C3×D4—61no—
176C3×D4—62no—
177C3×D4—62no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—43no—
182C2×C3×C4—46no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4C2×S412yes—
186C2×C3×C4C2×S411yescharacteristic
187#67—244no—
188C2×Dic3—62no—
189C2×Dic3—126no—
190D12—62no—
191D12—62no—
192D12—122no—
193D12—246no—
194SL(2,𝔽3)C2×#6713yes—
195C3⋊C8—62no—
196C3⋊C8—62no—
197C2×C32—42no—
198C2×C32—41no—
199C3⋊D3—162no—
200C24—61no—
201C24—181no—
202#41—22no—
203#41—32no—
204#41—62no—
205#41—62no—
206#41—62no—
207#41#63812yes—
208C2×Q8C3⋊S411yescharacteristic
209C2×Q8C22×C3⋊D311yescharacteristic
210C2×D4—31no—
211C2×D4—32no—
212C2×D4—34no—
213C2×D4—32no—
214C2×D4—31no—
215C2×D4—62no—
216C2×D4—64no—
217C2×D4—64no—
218C2×D4—92no—
219C2×D4—182no—
220C2×D4—182no—
221C2×D4—184no—
222C2×D4—181no—
223C22×C4—31no—
224C22×C4—32no—
225SD16—182no—
226SD16—182no—
227D8—188no—
228D8—184no—
229M16—182no—
230C2×C8—92no—
231C2×C8—92no—
232C2×C8—182no—
233C4⋊C4—92no—
234C22⋊C4—181no—
235C22⋊C4—184no—
236C42—181no—
237C22×C3—32no—
238C22×C3—43no—
239C22×C3—62no—
240C22×C3—64no—
241C22×C3—62no—
242C22×C3—244no—
243C22×C3C22×S411yescharacteristic
244C2×D3—124no—
245C2×D3—122no—
246C2×D3—246no—
247C2×D3—2412no—
248C2×D3—244no—
249C3×C4—22no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—46no—
253C3×C4—86no—
254C3×C4GL(2,ℤ/4)12yes—
255Dic3—122no—
256Dic3—246no—
257C32—41noSylow
258C23—31no—
259C23—32no—
260C23—64no—
261C23—182no—
262C23—242no—
263C23—362no—
264C23—722no—
265Q8—22no—
266Q8—31no—
267Q8C2×#63811yescharacteristic
268D4—32no—
269D4—34no—
270D4—64no—
271D4—61no—
272D4—62no—
273D4—62no—
274D4—182no—
275D4—182no—
276D4—362no—
277D4—724no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—182no—
284C2×C4C2×C3⋊S412yes—
285C2×C4C2×C3⋊S411yescharacteristic
286C8—182no—
287C8—182no—
288C2×C3—46no—
289C2×C3—43no—
290C2×C3—62no—
291C2×C3—122no—
292C2×C3#592512yes—
293C2×C3C2×GL(2,ℤ/4)11yescharacteristic
294D3—242no—
295D3—486no—
296C22—32no—
297C22—62no—
298C22—64no—
299C22—62no—
300C22—244no—
301C22—364no—
302C22—362no—
303C22—724no—
304C22C22×C3⋊S411yescharacteristic
305C4—22no—
306C4—31no—
307C4—31no—
308C4—362no—
309C4#6426912yes—
310C3—43no—
311C3#8553011yescharacteristic, complemented
312C2—62no—
313C2—122no—
314C2—722no—
315C2#1059658812yes—
316C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×H(5,𝔽2)D3
socleC22×C3C22×S4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C2×#64267C2semidirect
3. #10593661C2semidirect
4. #10593658C2semidirect
9. C2×C3×#41⋊C3C22semidirect
11. C2×C3⋊GL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C2×C3×H(5,𝔽2)D3semidirect
15. C2×#994D3semidirect
30. C3×#41⋊C3D4semidirect
46. C2×#41⋊C3C2×D3semidirect
75. C2×H(5,𝔽2)C3⋊D3semidirect
112. #41⋊C3#67semidirect
311. C3#85530semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C2×#642672
#105936612
#105936582
#855303
#841013
#641564