Finite Group Databasebeta

← #11921030

Subgroups — #11921030

Order 1152.

410 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 55 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×#64264C211yesmaximal, characteristic
2C3×#519⋊C3C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4#73074—33nomaximal
5#71662—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7#64264—21no—
8#64264—21no—
9C2×C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
10C2×C3⋊#226C411yescharacteristic, complemented
11C3×C4×SL(2,𝔽3)—21no—
12C3⋊#859—21no—
13#63633—41nomaximal
14C2×#983—33no—
15C3×#519D311yescharacteristic, complemented
16C2×#942—31no—
17#519⋊C3D313yescomplemented
18#5420—33no—
19#5225—31no—
20#5006—31no—
21#5005—31no—
22#4533—31no—
23#4531—31no—
24C2×#638—41no—
25C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
26C2×C3×SL(2,𝔽3)D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C3⋊GL(2,𝔽3)—21no—
29C3⋊#226—21no—
30C32×C22⋊C4—41no—
31#3757—41no—
32#1905—91noSylow
33C22×C3×Q8C2×D311yescharacteristic
34C2×#235—31no—
35C22×SL(2,𝔽3)C2×D313yes—
36#983—63no—
37#983—63no—
38C2×GL(2,𝔽3)—33no—
39C2×#226—33no—
40C3×C4×Q8—21no—
41C3×C4×Q8—61no—
42C3×#117—31no—
43C3×#117—31no—
44C2×#216—31no—
45C2×#216—31no—
46#942—61no—
47#942—61no—
48#942—61no—
49#942—61no—
50C2×#215—31no—
51C2×#215—31no—
52#926—31no—
53C2×#208—31no—
54C2×#208—31no—
55C4×SL(2,𝔽3)—23no—
56#859—63no—
57#835—61no—
58#835—61no—
59#806—61no—
60#806—61no—
61#806—123no—
62#805—62no—
63C23×C32—41no—
64C22×C3⋊D3—41no—
65C2×C32×C4—81no—
66#638—81no—
67#638—81no—
68C2×C3⋊Dic3—41no—
69C2×C3⋊Dic3—81no—
70C3×SL(2,𝔽3)C22⋊C411yescharacteristic
71C2×#137—91no—
72#519C3⋊D311yescharacteristic, complemented
73#421—91no—
74#419—91no—
75#410—91no—
76#344—91no—
77H(3,ℤ/4)—91no—
78C2×C3×Q8—31no—
79C2×C3×Q8—31no—
80C2×C3×Q8—31no—
81C2×C3×Q8C4×D311yescharacteristic
82C2×C3×Q8D1211yescharacteristic
83C2×C3×Q8#6711yescharacteristic
84C22×C3×C4—31no—
85C2×#67—61no—
86C2×#67—123no—
87#235—61no—
88#235—61no—
89#235—61no—
90#235—61no—
91C2×D12—31no—
92C2×C4×D3—61no—
93C2×Dic6—31no—
94C2×SL(2,𝔽3)C4×D313yes—
95C2×SL(2,𝔽3)D1213yes—
96C2×SL(2,𝔽3)#6713yes—
97GL(2,𝔽3)—63no—
98#226—63no—
99C3×C4⋊C4—61no—
100C3×C4⋊C4—61no—
101C3×C22⋊C4—31no—
102C3×C22⋊C4—43no—
103C3×C22⋊C4S411yescharacteristic
104C3×C42—32no—
105C3×C42—61no—
106#217—61no—
107#217—123no—
108#216—61no—
109#216—61no—
110#216—61no—
111#216—61no—
112#215—61no—
113#215—61no—
114#215—61no—
115#215—61no—
116C4⋊Dic3—61no—
117C4×Dic3—61no—
118#208—61no—
119#208—61no—
120#208—61no—
121#208—61no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—61no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—161no—
129C2×C3⋊D3—81no—
130C2×C3⋊D3—161no—
131C32×C4—161no—
132C3⋊Dic3—81no—
133C3⋊Dic3—161no—
134C2×#41—91no—
135C22×Q8C2×C3⋊D311yescharacteristic
136#137—181no—
137#137—181no—
138#137—181no—
139#137—181no—
140C2×Q16—91no—
141C2×Q16—91no—
142C2×SD16—91no—
143C2×SD16—91no—
144C2×M16—91no—
145C2×M16—91no—
146C4×Q8—21no—
147C4×Q8—61no—
148#117—31no—
149#117—31no—
150#117—91no—
151#104—182no—
152#103—181no—
153#103—181no—
154#99—181no—
155#99—181no—
156C23×C3—43no—
157C23×C3C2×S411yescharacteristic
158C22×D3—61no—
159C22×D3—123no—
160C3×Q8—31no—
161C3×Q8—31no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8(C2×D3)⋊C411yescharacteristic
165C2×C3×C4—21no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—83no—
174#67—121no—
175#67—121no—
176#67—243no—
177#67—243no—
178C2×Dic3—61no—
179C2×Dic3—123no—
180C2×Dic3—121no—
181C2×Dic3—243no—
182D12—61no—
183D12—61no—
184C4×D3—121no—
185C4×D3—121no—
186Dic6—61no—
187Dic6—61no—
188SL(2,𝔽3)(C2×D3)⋊C413yes—
189C3⋊C8—61no—
190C3⋊C8—61no—
191C3⋊C8—121no—
192C2×C32—41no—
193C2×C32—81no—
194C2×C32—81no—
195C2×C32—81no—
196C3⋊D3—161no—
197#41—181no—
198#41—181no—
199#41—181no—
200#41—181no—
201C2×Q8—31no—
202C2×Q8—31no—
203C2×Q8—31no—
204C2×Q8—91no—
205C2×Q8C4×C3⋊D311yescharacteristic
206C2×Q8C4⋊(C3⋊D3)11yescharacteristic
207C2×Q8#63811yescharacteristic
208C2×D4—91no—
209C2×D4—181no—
210C22×C4—31no—
211C22×C4—181no—
212Q16—181no—
213Q16—181no—
214Q16—181no—
215Q16—181no—
216SD16—181no—
217SD16—181no—
218SD16—181no—
219SD16—181no—
220M16—181no—
221M16—181no—
222M16—181no—
223M16—181no—
224C2×C8—91no—
225C2×C8—91no—
226C2×C8—181no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—181no—
230C22⋊C4—31no—
231C22⋊C4—181no—
232C22⋊C4C3⋊S411yescharacteristic
233C42—32no—
234C42—61no—
235C42—181no—
236C22×C3—41no—
237C22×C3—43no—
238C22×C3—43no—
239C22×C3—43no—
240C22×C3—163no—
241C22×C3C4×S411yescharacteristic
242C22×C3C4⋊S411yescharacteristic
243C22×C3GL(2,ℤ/4)11yescharacteristic
244C2×D3—121no—
245C2×D3—243no—
246C2×D3—241no—
247C2×D3—483no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—41no—
253C3×C4—121no—
254C3×C4—163no—
255Dic3—121no—
256Dic3—243no—
257Dic3—241no—
258Dic3—483no—
259C32—41noSylow
260C23—181no—
261C23C2×C3⋊S411yescharacteristic
262Q8—31no—
263Q8—31no—
264Q8—31no—
265Q8—61no—
266Q8—181no—
267Q8—181no—
268Q8#375211yescharacteristic
269D4—181no—
270D4—181no—
271D4—361no—
272D4—361no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—61no—
281C2×C4—181no—
282C2×C4—361no—
283C2×C4—361no—
284C2×C4—361no—
285C8—181no—
286C8—181no—
287C8—361no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—43no—
292C2×C3—83no—
293C2×C3—83no—
294C2×C3—83no—
295C2×C3(C2×S4)⋊C411yescharacteristic
296D3—241no—
297D3—483no—
298C22—41no—
299C22—361no—
300C22—721no—
301C22C4×C3⋊S411yescharacteristic
302C22C4⋊(C3⋊S4)11yescharacteristic
303C22#6426911yescharacteristic
304C4—31no—
305C4—31no—
306C4—31no—
307C4—31no—
308C4—41no—
309C4—121no—
310C4—361no—
311C4—721no—
312C3—43no—
313C3#7307411yescharacteristic, complemented
314C2—21no—
315C2—21no—
316C2—21no—
317C2—721no—
318C2#1059364711yescharacteristic
subgroupG/H
centerC2#10593647
commutatorC2×C3×SL(2,𝔽3)C2×C4
FrattiniC23C2×C3⋊S4
FittingC3×#519D3
socleC2×C3(C2×S4)⋊C4
prime radicalC2×#64264C2
cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2semidirect
3. #10593654C2semidirect
9. C2×C3⋊GL(2,𝔽3)C4semidirect
10. C2×C3⋊#226C4semidirect
15. C3×#519D3semidirect
17. #519⋊C3D3semidirect
72. #519C3⋊D3semidirect
313. C3#73074semidirect

Maximal subgroups

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

subgroupindex
C2×#642642
C3×#519⋊C32
#105936542
#730743
#716623
#636334