Finite Group Databasebeta

← #11921109

Subgroups — #11921109

Order 1152.

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

normal subgroups only

This group has 496 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×#994C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73153—33nomaximal
5#72140—31nomaximal
6C3×#994—21no—
7C3×#994C412yescomplemented
8C2×C3×#41⋊C3—21no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C3⋊GU(2,𝔽3)—21no—
11C3⋊#859—21no—
12#63662—41nomaximal
13C2×C3×H(5,𝔽2)D311yescharacteristic
14C2×#994D313yes—
15#5426—33no—
16C2×GU(2,𝔽3)—33no—
17C2×#837—31no—
18C2×#837—31no—
19#5232—31no—
20#5232—31no—
21#5221—31no—
22C2×#833—31no—
23C2×C32×Q8—41no—
24C3×#41⋊C3—22no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C4⋊(C3⋊Dic3)—41no—
29C2×C3⋊(C3⋊C8)—41no—
30#1792—91noSylow
31C3×H(5,𝔽2)—21no—
32C3×H(5,𝔽2)—32no—
33C3×H(5,𝔽2)—61no—
34C3×H(5,𝔽2)Dic312yescomplemented
35C2×C3×#41—21no—
36C2×C3×#41—31no—
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×D4—31no—
39C22×C3×D4—61no—
40#994—23no—
41#994Dic316yescomplemented
42C2×#41⋊C3—23no—
43C2×#41⋊C3C2×D313yes—
44C2×#217—62no—
45C2×C4⋊Dic3—31no—
46C2×C4×Dic3—31no—
47C2×#208—31no—
48C2×#208—31no—
49GU(2,𝔽3)—63no—
50#859—63no—
51#837—61no—
52#837—62no—
53#837—61no—
54#835—61no—
55#835—61no—
56#835—123no—
57#833—62no—
58#832—61no—
59#832—61no—
60C32×Q8—42no—
61C32×Q8—81no—
62C2×C32×C4—41no—
63C2×C32×C4—81no—
64C2×C3⋊Dic3—81no—
65C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
66C3⋊(C3⋊C8)—81no—
67C2×H(5,𝔽2)C3⋊D311yescharacteristic
68C2×#104—91no—
69C2×#104—91no—
70#417—91no—
71#417—91no—
72C2×#100—91no—
73#390—91no—
74C24×C3—32no—
75C3×#41—22no—
76C3×#41—31no—
77C3×#41—31no—
78C3×#41—32no—
79C3×#41—32no—
80C3×#41—61no—
81C3×#41—61no—
82C3×#41—62no—
83C3×#41C2×Dic311yescharacteristic
84C3×#41#6711yescharacteristic
85C2×C3×Q8—43no—
86C2×C3×Q8#6711yescharacteristic
87C2×C3×Q8S411yescharacteristic
88C2×C3×D4—31no—
89C2×C3×D4—32no—
90C2×C3×D4—32no—
91C2×C3×D4—31no—
92C2×C3×D4—61no—
93C2×C3×D4—61no—
94C2×C3×D4—61no—
95C2×C3×D4—61no—
96C2×C3×D4—62no—
97C2×C3×D4—62no—
98C2×C3×D4—64no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C22×Dic3—61no—
102C22×Dic3—61no—
103#41⋊C3—26no—
104#41⋊C3C2×Dic313yes—
105#41⋊C3#6713yes—
106C2×SL(2,𝔽3)#6713yes—
107#217—124no—
108C4⋊Dic3—61no—
109C4⋊Dic3—61no—
110C4⋊Dic3—123no—
111C4×Dic3—61no—
112C4×Dic3—61no—
113#208—61no—
114#208—61no—
115#208—61no—
116#208—61no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—123no—
120C22×C32—41no—
121C32×C4—41no—
122C32×C4—41no—
123C32×C4—82no—
124C3⋊Dic3—161no—
125H(5,𝔽2)—21no—
126H(5,𝔽2)—32no—
127H(5,𝔽2)—61no—
128H(5,𝔽2)C3⋊Dic312yescomplemented
129C2×#41—21no—
130C2×#41—31no—
131C2×#41C2×C3⋊D311yescharacteristic
132C22×D4—31no—
133C22×D4—61no—
134C2×M16—91no—
135C2×M16—91no—
136C2×C4⋊C4—91no—
137C2×C22⋊C4—182no—
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#100—182no—
147C23×C3—31no—
148C23×C3—34no—
149C23×C3—62no—
150C23×C3—61no—
151C23×C3—122no—
152C23×C3—122no—
153C3×Q8—21no—
154C3×Q8—31no—
155C3×Q8—46no—
156C3×Q8—83no—
157C3×Q8#21711yescharacteristic
158C3×Q8SL(2,ℤ/4)12yes—
159C3×D4—31no—
160C3×D4—31no—
161C3×D4—32no—
162C3×D4—62no—
163C3×D4—62no—
164C3×D4—61no—
165C3×D4—62no—
166C3×D4—62no—
167C3×D4—61no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—43no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—83no—
178C2×C3×C4C2×S411yescharacteristic
179C2×Dic3—62no—
180C2×Dic3—121no—
181C2×Dic3—121no—
182C2×Dic3—122no—
183C2×Dic3—121no—
184C2×Dic3—243no—
185SL(2,𝔽3)#21713yescomplemented
186C3⋊C8—121no—
187C3⋊C8—121no—
188C3⋊C8—243no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C24—32no—
193#41—22no—
194#41—31no—
195#41—31no—
196#41—32no—
197#41—32no—
198#41—61no—
199#41—61no—
200#41—62no—
201#41C2×C3⋊Dic311yescharacteristic
202#41#63811yescharacteristic
203C2×Q8#63811yescharacteristic
204C2×Q8C3⋊S411yescharacteristic
205C2×D4—31no—
206C2×D4—32no—
207C2×D4—32no—
208C2×D4—31no—
209C2×D4—61no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—62no—
214C2×D4—62no—
215C2×D4—64no—
216C22×C4—31no—
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—
228C22⋊C4—364no—
229C42—181no—
230C42—181no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—43no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—62no—
237C22×C3—124no—
238C22×C3—121no—
239C22×C3—121no—
240C22×C3—242no—
241C22×C3GL(2,ℤ/4)11yescharacteristic
242C3×C4—22no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—43no—
246C3×C4—43no—
247C3×C4—86no—
248C3×C4C2×SL(2,ℤ/4)11yescharacteristic
249C3×C4GL(2,ℤ/4)11yescharacteristic
250Dic3—122no—
251Dic3—241no—
252Dic3—483no—
253C32—41noSylow
254C23—31no—
255C23—34no—
256C23—62no—
257C23—61no—
258C23—122no—
259C23—122no—
260Q8—21no—
261Q8—31no—
262Q8#375711yescharacteristic
263Q8C3⋊SL(2,ℤ/4)12yes—
264D4—31no—
265D4—31no—
266D4—32no—
267D4—62no—
268D4—62no—
269D4—61no—
270D4—62no—
271D4—62no—
272D4—61no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—182no—
282C2×C4—361no—
283C2×C4—361no—
284C2×C4—362no—
285C2×C4—361no—
286C2×C4C2×C3⋊S411yescharacteristic
287C8—361no—
288C8—361no—
289C2×C3—43no—
290C2×C3—43no—
291C2×C3—43no—
292C2×C3—61no—
293C2×C3—61no—
294C2×C3—122no—
295C2×C3#542811yescharacteristic
296C2×C3#542911yescharacteristic
297C2×C3#543211yescharacteristic
298C22—31no—
299C22—31no—
300C22—61no—
301C22—61no—
302C22—62no—
303C22—124no—
304C22—121no—
305C22—121no—
306C22—242no—
307C22#6426911yescharacteristic
308C4—22no—
309C4—31no—
310C4—31no—
311C4—362no—
312C4—721no—
313C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
314C4#6426911yescharacteristic
315C3—43no—
316C3#7315311yescharacteristic, complemented
317C2—61no—
318C2—61no—
319C2—122no—
320C2#1059366211yescharacteristic
321C2#1059366311yescharacteristic
322C2#1059366611yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC2×C3×H(5,𝔽2)D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×#994C2
cosocleC2×C3×#41⋊C3C22
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

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×#994C4semidirect
34. C3×H(5,𝔽2)Dic3semidirect
41. #994Dic3semidirect
65. C3×SL(2,𝔽3)C22⋊C4semidirect
128. H(5,𝔽2)C3⋊Dic3semidirect
185. SL(2,𝔽3)#217semidirect
316. C3#73153semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
#105936602
C2×C3⋊GU(2,𝔽3)2
#731533
#721403
#636624