Finite Group Databasebeta

← #11923775

Subgroups — #11923775

Order 1152.

652 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 35 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
1C3×#5948C211yesmaximal, characteristic, complemented
2C3⋊#5931C212yesmaximal, complemented
3#87510—33nomaximal
4#84082—31nomaximal
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6#64267—42nomaximal
7C4⋊(C3⋊S4)—41nomaximal
8C22×C3×#41D311yescharacteristic, complemented
9#5948D313yescomplemented
10#5931—36no—
11C2×#949—32no—
12C2×#930—31no—
13#5238—34no—
14#5232—32no—
15#5227—32no—
16#5217—32no—
17#5207—31no—
18C2×C3⋊S4—42no—
19C3×#41⋊C3—42no—
20C3×#41⋊C3—81no—
21C3×C4×A4—41no—
22C3⋊GL(2,𝔽3)—44no—
23#3006—91noSylow
24C2×C3×#41—32no—
25C2×C3×#41—32no—
26C2×C3×#41—61no—
27C2×C3×#41—61no—
28C22×C3×Q8C2×D311yescharacteristic
29C22×C3×D4—31no—
30C23×C3×C4—31no—
31C22×#67—61no—
32C22×D12—31no—
33C22⋊SL(2,𝔽3)C2×D313yescomplemented
34#986—126no—
35C4⋊S4—123no—
36#949—62no—
37#949—62no—
38#949—62no—
39#949—62no—
40C2×#215—34no—
41C2×#213—34no—
42#930—32no—
43#930—32no—
44#930—61no—
45#930—61no—
46C2×(C2×D3)⋊C4—61no—
47C2×C4⋊Dic3—31no—
48C2×#208—32no—
49#835—34no—
50#832—34no—
51#830—32no—
52#830—32no—
53C2×C3×A4—41no—
54C3⋊S4—82no—
55C4⋊(C3⋊D3)—161no—
56C3×SL(2,𝔽3)—42no—
57C3×SL(2,𝔽3)—81no—
58C22×#41C3⋊D311yescharacteristic, complemented
59C2×AGL(1,ℤ/8)—92no—
60C2×#121—91no—
61#449—94no—
62#448—92no—
63#447—92no—
64#417—92no—
65#407—91no—
66C24×C3—31no—
67C23×D3—61no—
68C2×S4—126no—
69C3×#41—21no—
70C3×#41—32no—
71C3×#41—32no—
72C3×#41—32no—
73C3×#41—61no—
74C3×#41—62no—
75C3×#41—61no—
76C3×#41—61no—
77C3×#41—62no—
78C3×#41S412yescomplemented
79C2×C3×Q8—32no—
80C2×C3×Q8—61no—
81C2×C3×D4—32no—
82C2×C3×D4—61no—
83C2×C3×D4—62no—
84C2×C3×D4—62no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—62no—
91C22×C3×C4—62no—
92C22×C3×C4S411yescharacteristic
93C2×#67—62no—
94C2×#67—62no—
95C2×#67—122no—
96C2×#67—122no—
97C22×Dic3—61no—
98C2×D12—32no—
99C2×D12—62no—
100C2×D12—62no—
101C2×D12—61no—
102#41⋊C3—46no—
103#41⋊C3—83no—
104C4×A4—43no—
105GL(2,𝔽3)—1212no—
106#215—64no—
107#215—64no—
108#213—64no—
109#213—64no—
110(C2×D3)⋊C4—62no—
111(C2×D3)⋊C4—62no—
112C4⋊Dic3—32no—
113C4⋊Dic3—61no—
114#208—62no—
115#208—62no—
116C2×C3⋊C8—34no—
117C2×C3⋊D3—162no—
118C3×A4—41no—
119C32×C4—161no—
120C2×#41—32no—
121C2×#41—32no—
122C2×#41—61no—
123C2×#41—61no—
124C22×Q8C2×C3⋊D311yescharacteristic
125C22×D4—31no—
126C22×D4—91no—
127C22×D4—181no—
128C23×C4—31no—
129AGL(1,ℤ/8)—182no—
130AGL(1,ℤ/8)—182no—
131AGL(1,ℤ/8)—182no—
132AGL(1,ℤ/8)—182no—
133C2×SD16—94no—
134C2×D8—94no—
135C2×M16—92no—
136#121—92no—
137#121—92no—
138#121—181no—
139#121—181no—
140C2×C4⋊C4—91no—
141C2×C22⋊C4—181no—
142#103—94no—
143#102—94no—
144C22⋊C8—92no—
145C22⋊C8—92no—
146C23×C3—31no—
147C23×C3—31no—
148C23×C3—62no—
149C23×C3—61no—
150C23×C3—61no—
151C23×C3—121no—
152C23×C3C2×S411yescharacteristic
153C22×D3—62no—
154C22×D3—122no—
155C22×D3—122no—
156C22×D3—122no—
157C2×A4—43no—
158S4—246no—
159C3×Q8—21no—
160C3×Q8—32no—
161C3×Q8—61no—
162C3×Q8C2×S412yes—
163C3×D4—32no—
164C3×D4—32no—
165C3×D4—62no—
166C3×D4—61no—
167C3×D4—62no—
168C3×D4—61no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180#67—122no—
181#67—124no—
182#67—122no—
183C2×Dic3—62no—
184C2×Dic3—122no—
185D12—64no—
186D12—121no—
187D12—121no—
188D12—483no—
189SL(2,𝔽3)—46no—
190SL(2,𝔽3)—83no—
191C3⋊C8—64no—
192C2×C32—161no—
193C3⋊D3—322no—
194C24—31no—
195C24—181no—
196#41—21no—
197#41—32no—
198#41—32no—
199#41—32no—
200#41—61no—
201#41—62no—
202#41—61no—
203#41—61no—
204#41—62no—
205#41C3⋊S412yescomplemented
206C2×Q8—32no—
207C2×Q8—61no—
208C2×D4—32no—
209C2×D4—61no—
210C2×D4—62no—
211C2×D4—62no—
212C2×D4—92no—
213C2×D4—182no—
214C2×D4—182no—
215C2×D4—181no—
216C2×D4—182no—
217C2×D4—182no—
218C2×D4—362no—
219C2×D4—362no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—181no—
228C22×C4C3⋊S411yescharacteristic
229SD16—184no—
230SD16—184no—
231D8—184no—
232D8—184no—
233M16—182no—
234M16—182no—
235C2×C8—94no—
236C4⋊C4—92no—
237C4⋊C4—181no—
238C22⋊C4—182no—
239C22⋊C4—182no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—32no—
243C22×C3—61no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—122no—
249C22×C3—122no—
250C22×C3#98611yescharacteristic, complemented
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—122no—
254C2×D3—244no—
255C2×D3—486no—
256A4—43no—
257C3×C4—32no—
258C3×C4—31no—
259C3×C4—61no—
260C3×C4—163no—
261C3×C4#102011yescharacteristic
262Dic3—122no—
263C32—161noSylow
264C23—31no—
265C23—31no—
266C23—62no—
267C23—61no—
268C23—61no—
269C23—121no—
270C23—182no—
271C23—362no—
272C23—362no—
273C23—362no—
274C23C2×C3⋊S411yescharacteristic
275Q8—21no—
276Q8—32no—
277Q8—61no—
278Q8C2×C3⋊S412yes—
279D4—32no—
280D4—32no—
281D4—62no—
282D4—61no—
283D4—62no—
284D4—61no—
285D4—184no—
286D4—361no—
287D4—362no—
288D4—364no—
289D4—362no—
290D4—361no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—182no—
303C2×C4—362no—
304C8—184no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—62no—
308C2×C3—121no—
309C2×C3—163no—
310C2×C3C2×#102011yescharacteristic
311D3—242no—
312D3—966no—
313C22—31no—
314C22—31no—
315C22—32no—
316C22—61no—
317C22—61no—
318C22—61no—
319C22—61no—
320C22—61no—
321C22—122no—
322C22—122no—
323C22—362no—
324C22—362no—
325C22—362no—
326C22—724no—
327C22#6426711yescharacteristic, complemented
328C4—32no—
329C4—31no—
330C4—61no—
331C4—362no—
332C4C3⋊#102011yescharacteristic
333C3—163no—
334C3#8751011yescharacteristic, complemented
335C2—31no—
336C2—31no—
337C2—62no—
338C2—121no—
339C2—722no—
340C2C2×C3⋊#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊#1020
commutator, cosocle, hypocenterC3×C22⋊SL(2,𝔽3)C22
FittingC22×C3×#41D3
socleC23×C3C2×S4
hypercenterC4C3⋊#1020
LCSG ▷ C3×C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#5948C2semidirect
2. C3⋊#5931C2semidirect
5. C3×C22⋊SL(2,𝔽3)C22semidirect
8. C22×C3×#41D3semidirect
9. #5948D3semidirect
33. C22⋊SL(2,𝔽3)C2×D3semidirect
58. C22×#41C3⋊D3semidirect
78. C3×#41S4semidirect
205. #41C3⋊S4semidirect
250. C22×C3#986semidirect
327. C22#64267semidirect
334. C3#87510semidirect

Maximal subgroups

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

subgroupindex
C3×#59482
C3⋊#59312
#875103
#840823
#642674
C4⋊(C3⋊S4)4