Finite Group Databasebeta

← #11921111

Subgroups — #11921111

Order 1152.

484 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 73 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73155—33nomaximal
5#72143—31nomaximal
6C3×2−1+4⋊C3—21no—
7C3×2−1+4⋊C3C412yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C3⋊GU(2,𝔽3)—21no—
11C3⋊#859—21no—
12#63659—41nomaximal
13C2×C3×2−1+4D311yescharacteristic
14C2×2−1+4⋊C3D313yes—
15#5426—33no—
16C2×GU(2,𝔽3)—33no—
17C2×#837—31no—
18C2×#837—31no—
19#5232—31no—
20#5232—31no—
21#5229—31no—
22C2×#836—31no—
23C2×C32×D4—41no—
24C3×#41⋊C3C2×C411yescharacteristic
25C3×#41⋊C3D411yescharacteristic
26C2×C3×SL(2,𝔽3)—22no—
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C4⋊(C3⋊Dic3)—41no—
29C2×C3⋊(C3⋊C8)—41no—
30#1793—91noSylow
31C3×2−1+4—21no—
32C3×2−1+4—32no—
33C3×2−1+4—61no—
34C3×2−1+4Dic312yescomplemented
35C2×C3×#41—31no—
36C2×C3×#41—61no—
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×Q8—21no—
39C22×C3×Q8—31no—
402−1+4⋊C3—23no—
412−1+4⋊C3Dic316yescomplemented
42C2×#41⋊C3C2×D313yes—
43C22×SL(2,𝔽3)—23no—
44C2×C4⋊Dic3—31no—
45C2×C4×Dic3—31no—
46C2×#208—31no—
47C2×#208—31no—
48GU(2,𝔽3)—63no—
49#859—63no—
50#837—61no—
51#837—62no—
52#837—61no—
53#836—62no—
54#835—61no—
55#835—61no—
56#832—61no—
57#832—61no—
58#832—123no—
59C3⋊#95—62no—
60C23×C32—81no—
61C32×D4—42no—
62C32×D4—81no—
63C2×C32×C4—41no—
64C2×C3⋊Dic3—81no—
65C3×SL(2,𝔽3)C22⋊C411yescharacteristic
66C3⋊(C3⋊C8)—81no—
67C2×2−1+4C3⋊D311yescharacteristic
68C2×#104—91no—
69C2×#104—91no—
70#417—91no—
71#417—91no—
72C2×#101—91no—
73#391—91no—
74C3×#41—31no—
75C3×#41—31no—
76C3×#41—32no—
77C3×#41—32no—
78C3×#41—62no—
79C3×#41—61no—
80C3×#41—61no—
81C3×#41—64no—
82C3×#41—62no—
83C3×#41C2×Dic311yescharacteristic
84C3×#41#6711yescharacteristic
85C2×C3×Q8—22no—
86C2×C3×Q8—32no—
87C2×C3×Q8—32no—
88C2×C3×Q8—31no—
89C2×C3×Q8—61no—
90C2×C3×Q8—61no—
91C2×C3×Q8—61no—
92C2×C3×Q8—62no—
93C2×C3×Q8#6711yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—43no—
96C2×C3×D4—61no—
97C2×C3×D4S411yescharacteristic
98C22×C3×C4—31no—
99C22×C3×C4—32no—
100C22×C3×C4—61no—
101C22×Dic3—61no—
102C22×Dic3—61no—
103#41⋊C3C2×Dic313yes—
104#41⋊C3#6713yes—
105C2×SL(2,𝔽3)—26no—
106C2×SL(2,𝔽3)#6713yes—
107C4⋊Dic3—61no—
108C4⋊Dic3—61no—
109C4⋊Dic3—123no—
110C4×Dic3—61no—
111C4×Dic3—61no—
112#208—61no—
113#208—61no—
114#208—61no—
115#208—61no—
116C2×C3⋊C8—61no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—123no—
119C22×C32—41no—
120C22×C32—82no—
121C22×C32—161no—
122C22×C32—161no—
123C32×C4—41no—
124C32×C4—41no—
125C3⋊Dic3—161no—
1262−1+4—21no—
1272−1+4—32no—
1282−1+4—61no—
1292−1+4C3⋊Dic312yescomplemented
130C2×#41—31no—
131C2×#41—61no—
132C2×#41C2×C3⋊D311yescharacteristic
133C22×Q8—21no—
134C22×Q8—31no—
135C2×M16—91no—
136C2×M16—91no—
137C2×C4⋊C4—91no—
138C2×C42—91no—
139#104—181no—
140#104—182no—
141#104—181no—
142#103—181no—
143#103—181no—
144#102—181no—
145#102—181no—
146#101—182no—
147#95—182no—
148C23×C3—21no—
149C23×C3—31no—
150C23×C3—83no—
151C3×Q8—31no—
152C3×Q8—32no—
153C3×Q8—62no—
154C3×Q8—62no—
155C3×Q8—61no—
156C3×Q8#21711yescharacteristic
157C3×D4—21no—
158C3×D4—31no—
159C3×D4—31no—
160C3×D4—46no—
161C3×D4—61no—
162C3×D4—62no—
163C3×D4—62no—
164C3×D4—83no—
165C3×D4SL(2,ℤ/4)12yes—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—34no—
172C2×C3×C4—43no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—61no—
177C2×C3×C4C2×S411yescharacteristic
178C2×Dic3—62no—
179C2×Dic3—121no—
180C2×Dic3—121no—
181C2×Dic3—122no—
182C2×Dic3—121no—
183C2×Dic3—243no—
184SL(2,𝔽3)#21713yes—
185C3⋊C8—121no—
186C3⋊C8—121no—
187C3⋊C8—243no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—162no—
192#41—31no—
193#41—31no—
194#41—32no—
195#41—32no—
196#41—62no—
197#41—61no—
198#41—61no—
199#41—64no—
200#41—62no—
201#41C2×C3⋊Dic311yescharacteristic
202#41#63811yescharacteristic
203C2×Q8—22no—
204C2×Q8—32no—
205C2×Q8—32no—
206C2×Q8—31no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—62no—
211C2×Q8#63811yescharacteristic
212C2×D4—31no—
213C2×D4—61no—
214C2×D4C3⋊S411yescharacteristic
215C22×C4—31no—
216C22×C4—32no—
217C22×C4—61no—
218C22×C4—181no—
219C22×C4—181no—
220M16—181no—
221M16—181no—
222M16—181no—
223M16—181no—
224C2×C8—181no—
225C2×C8—181no—
226C4⋊C4—181no—
227C4⋊C4—181no—
228C42—181no—
229C42—181no—
230C22×C3—22no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—43no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3—86no—
239C22×C3—163no—
240C22×C3—163no—
241C22×C3GL(2,ℤ/4)11yescharacteristic
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—43no—
245C3×C4—43no—
246C3×C4—62no—
247C3×C4C2×SL(2,ℤ/4)11yescharacteristic
248C3×C4GL(2,ℤ/4)11yescharacteristic
249Dic3—122no—
250Dic3—241no—
251Dic3—483no—
252C32—41noSylow
253C23—21no—
254C23—31no—
255Q8—31no—
256Q8—32no—
257Q8—62no—
258Q8—62no—
259Q8—61no—
260Q8#375711yescharacteristic
261D4—21no—
262D4—31no—
263D4—31no—
264D4—61no—
265D4—62no—
266D4—62no—
267D4C3⋊SL(2,ℤ/4)12yes—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—34no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—182no—
279C2×C4—361no—
280C2×C4—361no—
281C2×C4—362no—
282C2×C4—361no—
283C2×C4C2×C3⋊S411yescharacteristic
284C8—361no—
285C8—361no—
286C2×C3—42no—
287C2×C3—43no—
288C2×C3—43no—
289C2×C3—43no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—166no—
293C2×C3#543011yescharacteristic
294C2×C3#543111yescharacteristic
295C2×C3#543211yescharacteristic
296C22—22no—
297C22—31no—
298C22—31no—
299C22—41no—
300C22—41no—
301C22—61no—
302C22—61no—
303C22#6426911yescharacteristic
304C4—31no—
305C4—31no—
306C4—62no—
307C4—362no—
308C4—721no—
309C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
310C4#6426911yescharacteristic
311C3—43no—
312C3#7315511yescharacteristic, complemented
313C2—42no—
314C2—61no—
315C2—61no—
316C2#1059366411yescharacteristic
317C2#1059366511yescharacteristic
318C2#1059366611yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC2×C3×2−1+4D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×C3×#41⋊C3C22
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593660C2semidirect
3. C2×C3⋊GU(2,𝔽3)C2semidirect
7. C3×2−1+4⋊C3C4semidirect
34. C3×2−1+4Dic3semidirect
41. 2−1+4⋊C3Dic3semidirect
129. 2−1+4C3⋊Dic3semidirect
312. C3#73155semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
#105936602
C2×C3⋊GU(2,𝔽3)2
#731553
#721433
#636594