Finite Group Databasebeta

← #11922314

Subgroups — #11922314

Order 1152.

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

normal subgroups only

This group has 661 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×C3⋊#994C211yesmaximal, characteristic, complemented
2C2×C3×#986C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593203C214yesmaximal, complemented
5#85530—31nomaximal
6#83947—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8C3⋊#994—22no—
9C2×C3⋊(#41⋊C3)C2212yescomplemented
10C3×#986—22no—
11C2×C3×GL(2,𝔽3)C2212yescomplemented
12#63863—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×#1009D311yescharacteristic, complemented
15C2×#994—31no—
16C2×#986D311yescharacteristic, complemented
17C2×C3×AGL(1,ℤ/8)—31no—
18#5805—31no—
19C22×#213—31no—
20#5427—34no—
21#5424—31no—
22#5424—61no—
23#5233—31no—
24#5172—34no—
25#5156—34no—
26#5131—31no—
27#5129—31no—
28C2×C3×D12—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊(#41⋊C3)—22no—
33C3×GL(2,𝔽3)—22no—
34C4×C3⋊Dic3—41no—
35(C2×D3)⋊Dic3—44no—
36#3008—91noSylow
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×D4—31no—
39#1009—22no—
40#1009—62no—
41C2×#239C2×D312yes—
42C2×D3×D4—32no—
43C2×#235—31no—
44C22×D12—31no—
45#994—62no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—32no—
48C2×#41⋊C3C2×D311yescharacteristic
49#986—22no—
50C2×GL(2,𝔽3)C2×D312yescomplemented
51C3×AGL(1,ℤ/8)—62no—
52C3×AGL(1,ℤ/8)—62no—
53C2×C3×SD16—32no—
54C2×C3×D8—32no—
55C2×C3×M16—31no—
56#953—62no—
57#947—121no—
58#940—61no—
59#938—34no—
60#936—61no—
61C2×#213—34no—
62C2×#213—64no—
63#926—31no—
64C22×C3⋊C8—31no—
65#859—32no—
66#859—62no—
67#835—32no—
68#832—32no—
69#821—34no—
70(C2×D3)⋊C8—34no—
71C22×C3×D3—42no—
72C2×C32×C4—41no—
73C2×C3⋊Dic3—42no—
74C2×C3×Dic3—42no—
75C3×D12—82no—
76C3×Dic6—82no—
77C3×SL(2,𝔽3)C2×D411yescharacteristic
78C2×H(5,𝔽2)—31no—
79C2×AGL(1,ℤ/8)—31no—
80C22×D8—91no—
81#535—91no—
82#449—94no—
83#447—94no—
84#418—91no—
85#416—91no—
86#408—91no—
87C24×C3—61no—
88C23×D3—61no—
89C3×#41—32no—
90C3×#41#6712yes—
91C2×C3×Q8—121no—
92C2×C3×Q8C22×D311yescharacteristic
93C2×C3×D4—31no—
94C2×C3×D4—32no—
95C2×C3×D4—62no—
96C2×C3×D4—62no—
97C2×C3×D4—61no—
98C22×C3×C4—31no—
99C2×#67—32no—
100C2×#67—64no—
101#239—22no—
102#239—62no—
103D3×D4—64no—
104D3×D4—64no—
105#235—62no—
106#235—62no—
107C2×D12—32no—
108C2×D12—34no—
109C2×D12—31no—
110C2×D12—41no—
111C2×D12—62no—
112C2×C4×D3—32no—
113C2×Dic6S411yescharacteristic
114#41⋊C3—22no—
115#41⋊C3—62no—
116#41⋊C3#6712yescomplemented
117C2×SL(2,𝔽3)—21no—
118C2×SL(2,𝔽3)C22×D311yescharacteristic
119GL(2,𝔽3)—22no—
120C3×SD16—62no—
121C3×SD16—62no—
122C3×D8—64no—
123C3×M16—62no—
124C2×C3×C8—32no—
125#217—61no—
126#217—64no—
127#213—68no—
128(C2×D3)⋊C4—124no—
129C4⋊Dic3—32no—
130C4×Dic3—61no—
131C4×Dic3—121no—
132C4×Dic3—241no—
133C2×C3⋊C8—32no—
134C2×C3⋊C8—62no—
135C22×C32—41no—
136C2×C3×D3—82no—
137C2×C3×D3—84no—
138C32×C4—42no—
139C3⋊Dic3—82no—
140C3×Dic3—82no—
141H(5,𝔽2)—62no—
142H(5,𝔽2)—182no—
143C2×#41—32no—
144C2×#41—91no—
145C2×#41D3211yescharacteristic
146C22×D4—31no—
147C22×D4—91no—
148C22×D4—92no—
149AGL(1,ℤ/8)—62no—
150AGL(1,ℤ/8)—62no—
151C2×SD16—32no—
152C2×D8—32no—
153C2×D8—94no—
154C2×D8—184no—
155C2×M16—31no—
156C22×C8—91no—
157#127—181no—
158#124—181no—
159#121—94no—
160#120—182no—
161#117—91no—
162#103—92no—
163#102—94no—
164#102—92no—
165C22⋊C8—94no—
166C23×C3—31no—
167C23×C3—62no—
168C23×C3—122no—
169C23×C3—242no—
170C22×D3—32no—
171C22×D3—42no—
172C22×D3—64no—
173C22×D3—242no—
174C3×Q8—31no—
175C3×Q8—242no—
176C3×Q8C2×#6711yescharacteristic
177C3×D4—32no—
178C3×D4—61no—
179C3×D4—62no—
180C3×D4—62no—
181C3×D4—122no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—41no—
186C2×C3×C4—81no—
187C2×C3×C4—122no—
188C2×C3×C4C2×S411yescharacteristic
189#67—64no—
190#67—244no—
191C2×Dic3—62no—
192C2×Dic3—122no—
193C2×Dic3—242no—
194C2×Dic3C2×S412yes—
195D12—34no—
196D12—62no—
197D12—62no—
198D12—82no—
199C4×D3—62no—
200C4×D3—62no—
201Dic6—22no—
202SL(2,𝔽3)—21no—
203SL(2,𝔽3)C2×#6711yescharacteristic
204C3×C8—62no—
205C3⋊C8—62no—
206C2×C32—42no—
207C2×C32—41no—
208C3×D3—162no—
209C24—61no—
210C24—181no—
211#41—32no—
212#41—62no—
213#41—182no—
214#41—182no—
215#41—182no—
216#41#62512yes—
217C2×Q8—31no—
218C2×Q8C2×D3211yescharacteristic
219C2×D4—31no—
220C2×D4—32no—
221C2×D4—62no—
222C2×D4—62no—
223C2×D4—61no—
224C2×D4—92no—
225C2×D4—94no—
226C2×D4—92no—
227C2×D4—91no—
228C2×D4—182no—
229C2×D4—184no—
230C2×D4—184no—
231C2×D4—184no—
232C22×C4—31no—
233C22×C4—92no—
234SD16—62no—
235SD16—62no—
236D8—64no—
237D8—188no—
238M16—62no—
239C2×C8—32no—
240C2×C8—92no—
241C2×C8—182no—
242C4⋊C4—92no—
243C22⋊C4—184no—
244C22⋊C4—181no—
245C42—181no—
246C22×C3—32no—
247C22×C3—41no—
248C22×C3—62no—
249C22×C3—81no—
250C22×C3—124no—
251C22×C3—122no—
252C22×C3—244no—
253C22×C3C22×S411yescharacteristic
254C2×D3—64no—
255C2×D3—62no—
256C2×D3—82no—
257C2×D3—84no—
258C2×D3—244no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—42no—
262C3×C4—82no—
263C3×C4—242no—
264C3×C4GL(2,ℤ/4)12yes—
265Dic3—22no—
266Dic3—122no—
267Dic3—242no—
268Dic3—244no—
269C32—41noSylow
270C23—31no—
271C23—62no—
272C23—92no—
273C23—122no—
274C23—184no—
275C23—242no—
276C23—722no—
277Q8—31no—
278Q8—62no—
279Q8C2×#62511yescharacteristic
280D4—32no—
281D4—61no—
282D4—62no—
283D4—62no—
284D4—94no—
285D4—122no—
286D4—182no—
287D4—184no—
288D4—182no—
289D4—724no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4D3×S411yescharacteristic
298C8—62no—
299C8—182no—
300C2×C3—42no—
301C2×C3—41no—
302C2×C3—62no—
303C2×C3—82no—
304C2×C3—81no—
305C2×C3—242no—
306C2×C3#592512yes—
307C2×C3C2×GL(2,ℤ/4)11yescharacteristic
308D3—122no—
309D3—162no—
310C22—32no—
311C22—62no—
312C22—124no—
313C22—122no—
314C22—184no—
315C22—182no—
316C22—244no—
317C22—724no—
318C22C2×D3×S411yescharacteristic
319C4—31no—
320C4—31no—
321C4—62no—
322C4—362no—
323C4#6420912yes—
324C3—41no—
325C3—81no—
326C3#8553011yescharacteristic, complemented
327C2—62no—
328C2—242no—
329C2—362no—
330C2#1059644812yes—
331C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#994C2semidirect
2. C2×C3×#986C2semidirect
3. #10593658C2semidirect
4. #10593203C2semidirect
9. C2×C3⋊(#41⋊C3)C22semidirect
11. C2×C3×GL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C2×#1009D3semidirect
16. C2×#986D3semidirect
30. C3×#41⋊C3D4semidirect
50. C2×GL(2,𝔽3)C2×D3semidirect
116. #41⋊C3#67semidirect
326. 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×C3×#9862
#105936582
#105932032
#855303
#839473
#638634