Finite Group Databasebeta

← #11922963

Subgroups — #11922963

Order 1152.

486 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 71 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×#557⋊C3C211yesmaximal, characteristic
2#10593623C212yesmaximal
3C4×C3⋊#226C212yesmaximal
4C4×C3⋊#226C211yesmaximal, characteristic
5#10593620C211yesmaximal, characteristic
6#85414—33nomaximal
7#83682—31nomaximal
8C2×C3⋊#226C2211yescharacteristic
9#64096—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2212yes—
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#557D311yescharacteristic
15#5567—31no—
16#557⋊C3D313yes—
17#5389—36no—
18C4×#226—36no—
19C4×#226—33no—
20#5386—33no—
21#5094—31no—
22#5078—31no—
23#5065—32no—
24#5063—32no—
25C4×#216—32no—
26C4×#216—31no—
27Q8×C3⋊C8—31no—
28#5022—32no—
29#5020—31no—
30C2×#634—41no—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊#226Q812yescomplemented
33C32×C4⋊C4—41no—
34#3750—42no—
35C4×C3⋊Dic3—42no—
36C4×C3⋊Dic3—41no—
37#3383—91noSylow
38C2×#226—33no—
39C3×C4⋊Q8—31no—
40C3×#125—32no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×D311yescharacteristic
43C3×C4×Q8C2×D312yes—
44#944—61no—
45C2×#216—31no—
46#889—62no—
47#887—61no—
48#887—123no—
49#870—32no—
50#869—31no—
51C4×Dic6—32no—
52C4×Dic6—31no—
53C4×SL(2,𝔽3)C2×D313yes—
54C4×SL(2,𝔽3)C2×D316yes—
55#859—36no—
56#859—33no—
57#835—32no—
58#835—31no—
59#810—31no—
60#810—32no—
61#808—32no—
62#808—32no—
63#807—32no—
64#807—31no—
65#807—32no—
66C4⋊(C3⋊C8)—31no—
67C4⋊(C3⋊C8)—32no—
68C4×C3⋊C8—32no—
69C4×C3⋊C8—31no—
70C2×C32×C4—41no—
71C2×C32×C4—42no—
72C2×C3⋊Dic3—41no—
73C2×C3⋊Dic3—42no—
74C2×C3⋊Dic3—41no—
75#634—44no—
76C3×SL(2,𝔽3)C2×Q811yescharacteristic
77#557—91no—
78#557C3⋊D311yescharacteristic
79#500—91no—
80#499—92no—
81#479—92no—
82#479—92no—
83#475—91no—
84#475—91no—
85C8×Q8—91no—
86C4×Q16—92no—
87C4×Q16—91no—
88C2×C3×Q8—31no—
89C2×C3×Q8C22×D311yescharacteristic
90C2×Dic6—31no—
91C2×Dic6—61no—
92C2×Dic6—123no—
93C2×SL(2,𝔽3)C22×D313yes—
94#226—36no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—43no—
99C3×C4⋊C4—61no—
100C3×C4⋊C4S411yescharacteristic
101C3×C42—31no—
102C3×C42—32no—
103#216—32no—
104#216—32no—
105C4⋊Dic3—32no—
106C4⋊Dic3—31no—
107C4⋊Dic3—61no—
108Dic3⋊C4—61no—
109Dic3⋊C4—62no—
110Dic3⋊C4—62no—
111Dic3⋊C4—62no—
112Dic3⋊C4—126no—
113C4×Dic3—62no—
114C4×Dic3—61no—
115C4×Dic3—126no—
116C4×Dic3—123no—
117C2×C3⋊C8—31no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—32no—
120C22×C32—41no—
121C32×C4—42no—
122C32×C4—82no—
123C3⋊Dic3—42no—
124C3⋊Dic3—44no—
125C3⋊Dic3—81no—
126C2×Q16—91no—
127C4⋊Q8—31no—
128C4⋊Q8—91no—
129C4⋊Q8—181no—
130#125—32no—
131#125—92no—
132#125—182no—
133C4×Q8—31no—
134C4×Q8—91no—
135C4×Q8—92no—
136C4×Q8—181no—
137C4×Q8C2×C3⋊D311yescharacteristic
138C4×Q8C2×C3⋊D312yes—
139#107—91no—
140#107—92no—
141#107—92no—
142#106—92no—
143#106—92no—
144C4⋊C8—91no—
145C4⋊C8—92no—
146#103—92no—
147#103—91no—
148#103—92no—
149#103—91no—
150C4×C8—92no—
151C4×C8—91no—
152C3×Q8—31no—
153C3×Q8—62no—
154C3×Q8D3×Q811yescharacteristic
155C2×C3×C4—31no—
156C2×C3×C4—32no—
157C2×C3×C4—31no—
158C2×C3×C4—43no—
159C2×C3×C4—46no—
160C2×C3×C4C2×S412yes—
161C2×C3×C4C2×S411yescharacteristic
162C2×Dic3—61no—
163C2×Dic3—61no—
164C2×Dic3—62no—
165C2×Dic3—126no—
166C2×Dic3—123no—
167C2×Dic3—123no—
168Dic6—32no—
169Dic6—61no—
170Dic6—1212no—
171Dic6—122no—
172SL(2,𝔽3)D3×Q813yes—
173C3⋊C8—32no—
174C3⋊C8—61no—
175C3⋊C8—62no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×Q8—31no—
180C2×Q8—91no—
181C2×Q8—181no—
182C2×Q8C22×C3⋊D311yescharacteristic
183Q16—92no—
184Q16—92no—
185C2×C8—91no—
186C2×C8—92no—
187C2×C8—91no—
188C4⋊C4—31no—
189C4⋊C4—32no—
190C4⋊C4—32no—
191C4⋊C4—61no—
192C4⋊C4—91no—
193C4⋊C4—92no—
194C4⋊C4—181no—
195C4⋊C4—182no—
196C4⋊C4—182no—
197C4⋊C4—181no—
198C4⋊C4—182no—
199C4⋊C4C3⋊S411yescharacteristic
200C42—32no—
201C42—31no—
202C42—182no—
203C42—181no—
204C22×C3—43no—
205C22×C3C22×S411yescharacteristic
206C3×C4—22no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—46no—
210C3×C4—62no—
211C3×C4—61no—
212C3×C4—86no—
213C3×C4#98512yes—
214Dic3—62no—
215Dic3—126no—
216Dic3—1212no—
217Dic3—122no—
218Dic3—121no—
219Dic3—243no—
220C32—41noSylow
221Q8—31no—
222Q8—62no—
223Q8—92no—
224Q8—181no—
225Q8—362no—
226Q8Q8×C3⋊D311yescharacteristic
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—32no—
230C2×C4—181no—
231C2×C4—181no—
232C2×C4—182no—
233C2×C4C2×C3⋊S412yes—
234C2×C4C2×C3⋊S411yescharacteristic
235C8—92no—
236C8—182no—
237C8—181no—
238C2×C3—43no—
239C2×C3—43no—
240C2×C3—43no—
241C2×C3Q8×S411yescharacteristic
242C2×C3C2×#98511yescharacteristic
243C2×C3#592711yescharacteristic
244C22C22×C3⋊S411yescharacteristic
245C4—22no—
246C4—31no—
247C4—31no—
248C4—62no—
249C4—61no—
250C4—182no—
251C4—362no—
252C4—361no—
253C4#6426612yes—
254C3—43no—
255C3#8541411yescharacteristic, complemented
256C2Q8×C3⋊S411yescharacteristic
257C2C2×#6426611yescharacteristic
258C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
32. C3⋊#226Q8semidirect
255. C3#85414semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936232
C4×C3⋊#2262
#105936202
#854143
#836823
#640964