Finite Group Databasebeta

← #11923085

Subgroups — #11923085

Order 1152.

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

normal subgroups only

This group has 820 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#64267C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593655C214yesmaximal, complemented
5#85540—33nomaximal
6#84113—31nomaximal
7C3×2−1+4⋊C3—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yescomplemented
10#64267—22no—
11C2×C3⋊GL(2,𝔽3)C2212yes—
12#64149—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×C3×2−1+4D311yescharacteristic, complemented
15C2×2−1+4⋊C3D313yescomplemented
16C2×#986—33no—
17#5830—31no—
18C2×#949—31no—
19C22×#215—31no—
20#5424—33no—
21#5421—312no—
22#5238—34no—
23#5234—31no—
24#5233—31no—
25#5227—34no—
26#5225—31no—
27C2×C32×D4—41no—
28C2×#638—44no—
29C2×C4⋊(C3⋊D3)—41no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)—22no—
34C4×C3⋊Dic3—41no—
35#3010—91noSylow
36C3×2−1+4—22no—
37C3×2−1+4—62no—
38C2×C3×#41—31no—
39C2×C3×#41—32no—
40C2×C3×#41C2×D311yescharacteristic
41C22×C3×Q8—31no—
42C22×C3×Q8C2×D312yes—
43C22×D12—31no—
442−1+4⋊C3—26no—
45C2×#41⋊C3C2×D313yes—
46C22×SL(2,𝔽3)C2×D316yescomplemented
47#986—66no—
48C2×GL(2,𝔽3)—36no—
49#949—62no—
50#949—62no—
51#947—61no—
52C2×#215—34no—
53C2×#215—32no—
54C2×#215—64no—
55#940—61no—
56#940—123no—
57#937—62no—
58C2×#213—32no—
59#930—34no—
60#926—31no—
61C2×#208—31no—
62C22×C3⋊C8—31no—
63#859—36no—
64#835—34no—
65#835—32no—
66#832—32no—
67#830—34no—
68C23×C32—42no—
69C22×C3⋊D3—42no—
70C32×D4—82no—
71C2×C32×C4—41no—
72#638—88no—
73C2×C3⋊Dic3—42no—
74C4⋊(C3⋊D3)—82no—
75C3×SL(2,𝔽3)C2×D411yescharacteristic
76C2×2−1+4C3⋊D311yescharacteristic, complemented
77C2×AGL(1,ℤ/8)—91no—
78C22×SD16—91no—
79#535—91no—
80#449—94no—
81#448—94no—
82#419—91no—
83#416—91no—
84#408—91no—
85C23×D3—61no—
86C3×#41—32no—
87C3×#41—62no—
88C3×#41—62no—
89C3×#41—64no—
90C3×#41—64no—
91C3×#41#6712yescomplemented
92C2×C3×Q8—22no—
93C2×C3×Q8—31no—
94C2×C3×Q8—32no—
95C2×C3×Q8—34no—
96C2×C3×Q8—62no—
97C2×C3×Q8—62no—
98C2×C3×Q8C22×D311yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4—32no—
101C2×C3×D4—43no—
102C2×C3×D4S411yescharacteristic
103C22×C3×C4—31no—
104C22×C3×C4—32no—
105C22×C3×C4—61no—
106C2×#67—64no—
107C2×#67—1212no—
108C2×D12—32no—
109C2×D12—62no—
110C2×D12—62no—
111C2×D12—61no—
112C2×D12—123no—
113#41⋊C3#6716yescomplemented
114C2×SL(2,𝔽3)—26no—
115C2×SL(2,𝔽3)C22×D313yes—
116GL(2,𝔽3)—66no—
117#217—61no—
118#215—68no—
119#215—62no—
120#215—62no—
121#213—64no—
122(C2×D3)⋊C4—64no—
123C4⋊Dic3—32no—
124C4×Dic3—61no—
125C4×Dic3—123no—
126#208—62no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—62no—
130C22×C32—41no—
131C22×C32—84no—
132C22×C32—82no—
133C2×C3⋊D3—82no—
134C2×C3⋊D3—84no—
135C32×C4—42no—
136C3⋊Dic3—44no—
1372−1+4—22no—
1382−1+4—62no—
139C2×#41—31no—
140C2×#41—32no—
141C2×#41C2×C3⋊D311yescharacteristic
142C22×Q8—31no—
143C22×Q8C2×C3⋊D312yes—
144C22×D4—91no—
145AGL(1,ℤ/8)—182no—
146AGL(1,ℤ/8)—182no—
147C2×SD16—94no—
148C2×SD16—92no—
149C2×SD16—184no—
150C2×D8—92no—
151C2×M16—91no—
152C22×C8—91no—
153#127—181no—
154#124—181no—
155#121—94no—
156#120—182no—
157#117—91no—
158#103—94no—
159#103—92no—
160#102—92no—
161C22⋊C8—94no—
162C23×C3—31no—
163C23×C3—46no—
164C23×C3C2×S412yes—
165C22×D3—62no—
166C22×D3—126no—
167C22×D3—122no—
168C22×D3—242no—
169C3×Q8—31no—
170C3×Q8—34no—
171C3×Q8—62no—
172C3×Q8—62no—
173C3×Q8C2×#6711yescharacteristic
174C3×D4—22no—
175C3×D4—32no—
176C3×D4—61no—
177C3×D4—64no—
178C3×D4—86no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—43no—
184C2×C3×C4—62no—
185C2×C3×C4—62no—
186C2×C3×C4—64no—
187C2×C3×C4C2×S411yescharacteristic
188#67—244no—
189#67—2424no—
190C2×Dic3—62no—
191C2×Dic3—126no—
192D12—62no—
193D12—62no—
194D12—122no—
195D12—246no—
196SL(2,𝔽3)C2×#6713yes—
197C3⋊C8—62no—
198C3⋊C8—62no—
199C2×C32—42no—
200C2×C32—41no—
201C2×C32—162no—
202C3⋊D3—162no—
203C24—181no—
204#41—32no—
205#41—62no—
206#41—62no—
207#41—64no—
208#41—64no—
209#41#63812yescomplemented
210C2×Q8—22no—
211C2×Q8—31no—
212C2×Q8—32no—
213C2×Q8—34no—
214C2×Q8—62no—
215C2×Q8—62no—
216C2×Q8C22×C3⋊D311yescharacteristic
217C2×D4—31no—
218C2×D4—32no—
219C2×D4—92no—
220C2×D4—182no—
221C2×D4—182no—
222C2×D4—184no—
223C2×D4—181no—
224C2×D4C3⋊S411yescharacteristic
225C22×C4—31no—
226C22×C4—32no—
227C22×C4—61no—
228SD16—188no—
229SD16—182no—
230SD16—182no—
231D8—184no—
232M16—182no—
233C2×C8—92no—
234C2×C8—92no—
235C2×C8—182no—
236C4⋊C4—92no—
237C22⋊C4—184no—
238C22⋊C4—181no—
239C42—181no—
240C22×C3—24no—
241C22×C3—22no—
242C22×C3—32no—
243C22×C3—43no—
244C22×C3—62no—
245C22×C3—812no—
246C22×C3—86no—
247C22×C3C22×S411yescharacteristic
248C2×D3—124no—
249C2×D3—122no—
250C2×D3—246no—
251C2×D3—2412no—
252C2×D3—244no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—46no—
256C3×C4—62no—
257C3×C4GL(2,ℤ/4)12yes—
258Dic3—1212no—
259Dic3—122no—
260C32—41noSylow
261C23—31no—
262C23—182no—
263C23—362no—
264C23—722no—
265C23C2×C3⋊S412yes—
266Q8—31no—
267Q8—34no—
268Q8—62no—
269Q8—62no—
270Q8C2×#63811yescharacteristic
271D4—22no—
272D4—32no—
273D4—61no—
274D4—64no—
275D4—182no—
276D4—182no—
277D4—362no—
278D4—724no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—64no—
286C2×C4—182no—
287C2×C4C2×C3⋊S411yescharacteristic
288C8—182no—
289C8—182no—
290C2×C3—42no—
291C2×C3—46no—
292C2×C3—43no—
293C2×C3—62no—
294C2×C3—166no—
295C2×C3#592612yes—
296C2×C3C2×GL(2,ℤ/4)11yescharacteristic
297D3—242no—
298D3—486no—
299C22—24no—
300C22—22no—
301C22—32no—
302C22—62no—
303C22—364no—
304C22—362no—
305C22—724no—
306C22C22×C3⋊S411yescharacteristic
307C4—31no—
308C4—31no—
309C4—62no—
310C4—362no—
311C4#6426912yes—
312C3—43no—
313C3#8554011yescharacteristic, complemented
314C2—42no—
315C2—62no—
316C2—722no—
317C2#1059658912yes—
318C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×#64267C2semidirect
3. #10593658C2semidirect
4. #10593655C2semidirect
9. C22×C3×SL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C2×C3×2−1+4D3semidirect
15. C2×2−1+4⋊C3D3semidirect
30. C3×#41⋊C3D4semidirect
46. C22×SL(2,𝔽3)C2×D3semidirect
76. C2×2−1+4C3⋊D3semidirect
91. C3×#41#67semidirect
113. #41⋊C3#67semidirect
209. #41#638semidirect
313. C3#85540semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×#642672
#105936582
#105936552
#855403
#841133
#641494