Finite Group Databasebeta

← #11920723

Subgroups — #11920723

Order 1152.

402 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 45 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
2#10593657C211yesmaximal, characteristic, complemented
3C3×#5423C211yesmaximal, characteristic, complemented
4#73152—31nomaximal
5#70414—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊#994—21no—
8C3⋊#994—21no—
9C2×C3⋊(#41⋊C3)C412yescomplemented
10C3⋊GU(2,𝔽3)—21no—
11C3×GU(2,𝔽3)—21no—
12#63560—41nomaximal
13C2×#1009D311yescharacteristic
14C2×#994—31no—
15#5423—31no—
16#5423—61no—
17#5423D311yescharacteristic, complemented
18C3×#421—31no—
19#5236—31no—
20#5132—31no—
21#5131—31no—
22#5006—31no—
23#5001—31no—
24C2×C3×Dic6—41no—
25C3×#41⋊C3D411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C3⋊(#41⋊C3)—22no—
29#3748—41no—
30C3×#208—41no—
31#1795—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33#1009—21no—
34#1009—21no—
35#1009—61no—
36#1009—61no—
37C2×#239Dic312yes—
38C2×D3×D4—32no—
39C2×#235—31no—
40C22×D12—31no—
41#994—61no—
42#994—61no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3—32no—
45C2×#41⋊C3C2×D311yescharacteristic
46C2×C3×M16—31no—
47C3×#117—31no—
48#926—31no—
49C2×#208—31no—
50GU(2,𝔽3)—21no—
51GU(2,𝔽3)—62no—
52GU(2,𝔽3)—61no—
53C3×#104—61no—
54C3×#104—61no—
55#837—61no—
56#837—61no—
57#836—121no—
58#824—61no—
59#823—61no—
60#821—62no—
61#809—62no—
62#806—62no—
63C2×C32×C4—41no—
64C2×C3×Dic3—42no—
65C3×Dic6—81no—
66C3×Dic6—81no—
67C3×SL(2,𝔽3)C22⋊C411yescharacteristic
68C3⋊(C3⋊C8)—81no—
69C3×C3⋊C8—81no—
70C2×H(5,𝔽2)—31no—
71#421—31no—
72#421—91no—
73#418—91no—
74#418—91no—
75#413—91no—
76#410—91no—
77C23×D3—61no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41#6711yescharacteristic
81C3×#41#6711yescharacteristic
82C2×C3×Q8—121no—
83C2×C3×Q8C2×Dic311yescharacteristic
84C2×C3×D4—31no—
85C22×C3×C4—31no—
86C2×#67—32no—
87#239—22no—
88#239—62no—
89D3×D4—64no—
90D3×D4—62no—
91D3×D4—62no—
92#235—61no—
93#235—61no—
94#235—61no—
95#235—61no—
96C2×D12—32no—
97C2×D12—32no—
98C2×D12—32no—
99C2×D12—31no—
100C2×D12—61no—
101C2×D12—61no—
102C2×C4×D3—32no—
103C2×Dic6S411yescharacteristic
104#41⋊C3—21no—
105#41⋊C3—21no—
106#41⋊C3—62no—
107#41⋊C3D1211yescharacteristic
108#41⋊C3#6711yescharacteristic
109C2×SL(2,𝔽3)—21no—
110C2×SL(2,𝔽3)C4×D311yescharacteristic
111C3×M16—61no—
112C3×M16—61no—
113C2×C3×C8—61no—
114C3×C4⋊C4—61no—
115C3×C22⋊C4—61no—
116C3×C42—61no—
117#217—61no—
118C4⋊Dic3—61no—
119C4×Dic3—61no—
120#208—41no—
121#208—61no—
122#208—61no—
123#208—121no—
124#208—241no—
125C2×C3⋊C8—61no—
126C22×C32—41no—
127C32×C4—41no—
128C32×C4—41no—
129C3×Dic3—82no—
130H(5,𝔽2)—61no—
131H(5,𝔽2)—61no—
132H(5,𝔽2)—181no—
133H(5,𝔽2)—181no—
134C2×#41—32no—
135C2×#41—91no—
136C2×#41D3211yescharacteristic
137C22×D4—91no—
138C22×D4—92no—
139C2×M16—31no—
140C2×M16—91no—
141#117—31no—
142#117—91no—
143#104—61no—
144#104—61no—
145#104—181no—
146#104—181no—
147#102—182no—
148#102—182no—
149#101—181no—
150#100—181no—
151#99—182no—
152C23×C3—31no—
153C22×D3—32no—
154C22×D3—62no—
155C22×D3—62no—
156C22×D3—242no—
157C3×Q8—31no—
158C3×Q8—241no—
159C3×Q8—241no—
160C3×Q8#21711yescharacteristic
161C3×D4—31no—
162C3×D4—31no—
163C3×D4—61no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—41no—
170C2×C3×C4—81no—
171C2×C3×C4—122no—
172C2×C3×C4—121no—
173C2×C3×C4C2×S411yescharacteristic
174#67—62no—
175#67—62no—
176C2×Dic3—121no—
177C2×Dic3SL(2,ℤ/4)12yes—
178D12—34no—
179D12—62no—
180D12—61no—
181D12—61no—
182C4×D3—62no—
183C4×D3—62no—
184Dic6—21no—
185Dic6—21no—
186SL(2,𝔽3)—21no—
187SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
188C3×C8—121no—
189C3⋊C8—81no—
190C3⋊C8—121no—
191C3⋊C8—241no—
192C3⋊C8—242no—
193C2×C32—41no—
194C2×C32—81no—
195C24—181no—
196#41—31no—
197#41—31no—
198#41—62no—
199#41—181no—
200#41—181no—
201#41—181no—
202#41—181no—
203#41—182no—
204#41#62511yescharacteristic
205#41#62611yescharacteristic
206C2×Q8—31no—
207C2×Q8D3×Dic311yescharacteristic
208C2×D4—31no—
209C2×D4—92no—
210C2×D4—92no—
211C2×D4—92no—
212C2×D4—92no—
213C2×D4—91no—
214C2×D4—181no—
215C2×D4—181no—
216C2×D4—182no—
217C2×D4—182no—
218C2×D4—184no—
219C22×C4—31no—
220C22×C4—92no—
221M16—61no—
222M16—61no—
223M16—181no—
224M16—181no—
225C2×C8—61no—
226C2×C8—181no—
227C4⋊C4—61no—
228C4⋊C4—181no—
229C22⋊C4—61no—
230C22⋊C4—181no—
231C42—61no—
232C42—181no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—41no—
236C22×C3—81no—
237C22×C3—121no—
238C22×C3C2×SL(2,ℤ/4)11yescharacteristic
239C2×D3—62no—
240C2×D3—122no—
241C2×D3—242no—
242C2×D3—242no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—81no—
248C3×C4—81no—
249C3×C4—242no—
250C3×C4—241no—
251C3×C4GL(2,ℤ/4)11yescharacteristic
252C3×C4GL(2,ℤ/4)11yescharacteristic
253Dic3—22no—
254Dic3—241no—
255C32—41noSylow
256C23—31no—
257C23—92no—
258C23—182no—
259C23—182no—
260C23—722no—
261Q8—31no—
262Q8—61no—
263Q8—61no—
264Q8(C2×D3)⋊Dic311yescharacteristic
265D4—31no—
266D4—31no—
267D4—61no—
268D4—94no—
269D4—182no—
270D4—182no—
271D4—182no—
272D4—181no—
273D4—181no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—121no—
281C2×C4—182no—
282C2×C4—182no—
283C2×C4—361no—
284C2×C4D3×S411yescharacteristic
285C8—121no—
286C8—361no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—161no—
294C2×C3#543211yescharacteristic
295D3—122no—
296C22—31no—
297C22—31no—
298C22—121no—
299C22—182no—
300C22—362no—
301C22—722no—
302C22—722no—
303C22D3×SL(2,ℤ/4)11yescharacteristic
304C4—31no—
305C4—31no—
306C4—62no—
307C4—241no—
308C4—721no—
309C4#6420911yescharacteristic
310C4#6421011yescharacteristic
311C3—41no—
312C3—81no—
313C3#7315211yescharacteristic, complemented
314C2—21no—
315C2—61no—
316C2—61no—
317C2—362no—
318C2(C2×D3)⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2(C2×D3)⋊SL(2,ℤ/4)
commutatorC2×C3×SL(2,𝔽3)C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC2×C3#5432
prime radicalC2×C3⋊#994C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #10593657C2semidirect
3. C3×#5423C2semidirect
9. C2×C3⋊(#41⋊C3)C4semidirect
17. #5423D3semidirect
313. C3#73152semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
#105936572
C3×#54232
#731523
#704143
#635604