Finite Group Databasebeta

← #11920690

Subgroups — #11920690

Order 1152.

389 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 47 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×#986C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73078—31nomaximal
5#72088—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#986—21no—
8C3×#986C412yescomplemented
9C2×C3×GL(2,𝔽3)—21no—
10Dic3×SL(2,𝔽3)—21no—
11C3⋊GU(2,𝔽3)—21no—
12#63562—41nomaximal
13C2×#986D311yescharacteristic
14C2×C3×AGL(1,ℤ/8)—31no—
15#5827D311yescharacteristic, complemented
16#520⋊C3—31no—
17C2×GU(2,𝔽3)—31no—
18C2×GU(2,𝔽3)—61no—
19C2×#837—31no—
20#5221—31no—
21#5215—31no—
22#4555—31no—
23#4554—31no—
24C2×C3×D12—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C3×GL(2,𝔽3)—22no—
29C2×C3⋊(C3⋊C8)—41no—
30C3×C4⋊Dic3—41no—
31#1883—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33C22×C3×D4—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3C2×D311yescharacteristic
36#986—21no—
37#986Dic312yescomplemented
38C2×GL(2,𝔽3)—21no—
39C3×AGL(1,ℤ/8)—32no—
40C3×AGL(1,ℤ/8)—32no—
41C3×AGL(1,ℤ/8)—61no—
42C3×AGL(1,ℤ/8)—61no—
43C2×C3×SD16—61no—
44C2×C3×D8—61no—
45C2×C3×M16—31no—
46C2×#217—62no—
47Q8×Dic3—21no—
48D4×Dic3—61no—
49#926—31no—
50C2×C4⋊Dic3—31no—
51C2×C4×Dic3—31no—
52C2×#208—31no—
53C4×SL(2,𝔽3)—61no—
54GU(2,𝔽3)—62no—
55GU(2,𝔽3)—61no—
56#837—61no—
57#837—61no—
58#832—61no—
59#832—61no—
60#809—121no—
61C22×C3×D3—81no—
62C2×C32×C4—41no—
63C2×C3×Dic3—81no—
64C3×D12—42no—
65C3×D12—81no—
66C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
67C3⋊(C3⋊C8)—81no—
68C2×AGL(1,ℤ/8)—31no—
69#520—31no—
70C2×#104—91no—
71#418—91no—
72#390—91no—
73#343—91no—
74#339—91no—
75C24×C3—61no—
76C3×#41—31no—
77C3×#41—31no—
78C3×#41C4×D311yescharacteristic
79C3×#41D1211yescharacteristic
80C2×C3×Q8#6711yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—32no—
83C2×C3×D4—61no—
84C2×C3×D4—61no—
85C2×C3×D4—61no—
86C2×C3×D4—61no—
87C2×C3×D4—61no—
88C22×C3×C4—31no—
89C22×Dic3—32no—
90C22×Dic3—61no—
91C2×D12—41no—
92#41⋊C3—21no—
93#41⋊C3—21no—
94#41⋊C3C2×Dic311yescharacteristic
95#41⋊C3#6711yescharacteristic
96C2×SL(2,𝔽3)—21no—
97C2×SL(2,𝔽3)#6711yescharacteristic
98GL(2,𝔽3)—22no—
99C3×SD16—62no—
100C3×SD16—62no—
101C3×D8—62no—
102C3×D8—62no—
103C3×M16—32no—
104C3×M16—61no—
105C2×C3×C8—61no—
106C3×C4⋊C4—121no—
107#217—61no—
108#217—124no—
109C4⋊Dic3—31no—
110C4⋊Dic3—61no—
111C4⋊Dic3S411yescharacteristic
112C4×Dic3—61no—
113C4×Dic3—61no—
114C4×Dic3—61no—
115#208—61no—
116#208—61no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—121no—
119C2×C3⋊C8—241no—
120C22×C32—41no—
121C2×C3×D3—82no—
122C2×C3×D3—161no—
123C2×C3×D3—161no—
124C32×C4—41no—
125C32×C4—41no—
126C3×Dic3—161no—
127C2×#41D3211yescharacteristic
128C22×D4—31no—
129AGL(1,ℤ/8)—32no—
130AGL(1,ℤ/8)—32no—
131AGL(1,ℤ/8)—61no—
132AGL(1,ℤ/8)—61no—
133C2×SD16—61no—
134C2×D8—61no—
135C2×M16—31no—
136C2×M16—91no—
137C4×Q8—61no—
138C4×D4—181no—
139#117—91no—
140C2×C4⋊C4—91no—
141C2×C22⋊C4—182no—
142C2×C42—91no—
143#104—181no—
144#104—181no—
145#102—181no—
146#102—181no—
147C23×C3—31no—
148C23×C3—62no—
149C23×C3—121no—
150C23×C3—122no—
151C23×C3—122no—
152C23×C3—121no—
153C22×D3—81no—
154C3×Q8—31no—
155C3×Q8(C2×D3)⋊C411yescharacteristic
156C3×D4—31no—
157C3×D4—31no—
158C3×D4—61no—
159C3×D4—62no—
160C3×D4—62no—
161C3×D4—121no—
162C3×D4—121no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4—241no—
171C2×C3×C4C2×S411yescharacteristic
172C2×Dic3—21no—
173C2×Dic3—62no—
174C2×Dic3—61no—
175C2×Dic3—121no—
176C2×Dic3—121no—
177C2×Dic3—122no—
178D12—42no—
179D12—81no—
180SL(2,𝔽3)—21no—
181SL(2,𝔽3)#21711yescharacteristic, complemented
182C3×C8—62no—
183C3⋊C8—121no—
184C3⋊C8—241no—
185C3⋊C8—242no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C3×D3—162no—
190C24—61no—
191#41—31no—
192#41—31no—
193#41D3×Dic311yescharacteristic
194#41#62611yescharacteristic
195C2×Q8#62511yescharacteristic
196C2×D4—31no—
197C2×D4—32no—
198C2×D4—61no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C2×D4—61no—
203C22×C4—31no—
204C22×C4—92no—
205C22×C4—181no—
206SD16—62no—
207SD16—62no—
208D8—62no—
209D8—62no—
210M16—32no—
211M16—61no—
212M16—181no—
213M16—181no—
214C2×C8—61no—
215C2×C8—181no—
216C4⋊C4—31no—
217C4⋊C4—91no—
218C4⋊C4—181no—
219C22⋊C4—181no—
220C22⋊C4—364no—
221C42—181no—
222C42—181no—
223C42—181no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—41no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—81no—
230C22×C3—124no—
231C22×C3—122no—
232C22×C3—242no—
233C22×C3—241no—
234C22×C3—241no—
235C22×C3GL(2,ℤ/4)11yescharacteristic
236C2×D3—82no—
237C2×D3—161no—
238C2×D3—161no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—81no—
244C3×C4—81no—
245C3×C4—481no—
246C3×C4C4×S411yescharacteristic
247C3×C4C4⋊S411yescharacteristic
248Dic3—41no—
249Dic3—122no—
250Dic3—121no—
251C32—41noSylow
252C23—31no—
253C23—62no—
254C23—121no—
255C23—122no—
256C23—122no—
257C23—121no—
258Q8—31no—
259Q8(C2×D3)⋊Dic311yescharacteristic
260D4—31no—
261D4—31no—
262D4—61no—
263D4—62no—
264D4—62no—
265D4—121no—
266D4—121no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—61no—
273C2×C4—182no—
274C2×C4—181no—
275C2×C4—361no—
276C2×C4—361no—
277C2×C4—362no—
278C2×C4D3×S411yescharacteristic
279C8—62no—
280C8—361no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—81no—
287C2×C3—81no—
288C2×C3—81no—
289C2×C3—242no—
290C2×C3(C2×S4)⋊C411yescharacteristic
291C2×C3#542911yescharacteristic
292C2×C3#543111yescharacteristic
293D3—162no—
294C22—31no—
295C22—31no—
296C22—61no—
297C22—61no—
298C22—124no—
299C22—122no—
300C22—242no—
301C22—241no—
302C22—241no—
303C22#6420911yescharacteristic
304C4—31no—
305C4—31no—
306C4—121no—
307C4—362no—
308C4—361no—
309C4Dic3×S411yescharacteristic
310C4#6420711yescharacteristic
311C3—41no—
312C3—81no—
313C3#7307811yescharacteristic, complemented
314C2—61no—
315C2—61no—
316C2—242no—
317C2#1059322911yescharacteristic
318C2#1059323311yescharacteristic
319C2(C2×S4)⋊Dic311yescharacteristic
subgroupG/H
centerC22#64209
commutatorC3×#41⋊C3C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×#986C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. C2×C3⋊GU(2,𝔽3)C2semidirect
8. C3×#986C4semidirect
15. #5827D3semidirect
37. #986Dic3semidirect
66. C3×SL(2,𝔽3)C22⋊C4semidirect
181. SL(2,𝔽3)#217semidirect
313. C3#73078semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9862
#5827⋊C32
C2×C3⋊GU(2,𝔽3)2
#730783
#720883
#635624