Finite Group Databasebeta

← #11922965

Subgroups — #11922965

Order 1152.

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

normal subgroups only

This group has 484 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#10593625C211yesmaximal, characteristic
3#10593622C212yesmaximal
4C4×C3⋊#226C211yesmaximal, characteristic
5#10593620C212yesmaximal
6#85415—33nomaximal
7#83683—31nomaximal
8C2×C3⋊#226C2211yescharacteristic
9#64097—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2211yescharacteristic
13C3⋊#859C2212yes—
14C3×#557D311yescharacteristic
15Q8×Dic6—31no—
16#557⋊C3D313yes—
17#5391—33no—
18#5388—36no—
19C4×#226—33no—
20#5386—36no—
21#5094—31no—
22#5080—31no—
23#5064—32no—
24#5063—32no—
25#5030—32no—
26C4×#216—31no—
27#5024—31no—
28#5021—31no—
29#5020—32no—
30C2×#634—42no—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊#226Q812yescomplemented
33C32×C4⋊C4—41no—
34C4⋊(C3⋊Dic3)—41no—
35#3750—42no—
36C4×C3⋊Dic3—41no—
37#3387—91noSylow
38C2×#226—33no—
39C3×C4⋊Q8—31no—
40C3×#125—32no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×D312yes—
43C3×C4×Q8C2×D311yescharacteristic
44Q8×Dic3—61no—
45C2×#216—31no—
46#888—62no—
47#888—61no—
48#888—123no—
49#869—32no—
50#869—31no—
51C4×Dic6—31no—
52C4×Dic6—32no—
53C4×SL(2,𝔽3)C2×D316yes—
54C4×SL(2,𝔽3)C2×D313yes—
55#859—33no—
56#859—36no—
57#835—31no—
58#835—32no—
59#810—31no—
60#810—32no—
61#808—32no—
62#808—32no—
63#808—32no—
64#807—32no—
65#807—31no—
66C4⋊(C3⋊C8)—32no—
67C4⋊(C3⋊C8)—31no—
68#803—32no—
69C4×C3⋊C8—31no—
70C2×C32×C4—41no—
71C2×C32×C4—42no—
72C2×C3⋊Dic3—41no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—42no—
75#634—84no—
76C3×SL(2,𝔽3)C2×Q811yescharacteristic
77Q82—91no—
78#557C3⋊D311yescharacteristic
79#501—92no—
80#498—91no—
81#479—92no—
82#477—91no—
83#475—91no—
84#475—92no—
85#446—91no—
86#441—92no—
87C4×Q16—91no—
88C2×C3×Q8—31no—
89C2×C3×Q8C22×D311yescharacteristic
90C2×Dic6—31no—
91C2×Dic6—61no—
92C2×Dic6—62no—
93C2×Dic6—126no—
94C2×SL(2,𝔽3)C22×D313yes—
95#226—36no—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—43no—
100C3×C4⋊C4—61no—
101C3×C4⋊C4S411yescharacteristic
102C3×C42—31no—
103C3×C42—32no—
104#216—32no—
105#216—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—31no—
108C4⋊Dic3—62no—
109C4⋊Dic3—61no—
110C4⋊Dic3—123no—
111Dic3⋊C4—61no—
112Dic3⋊C4—62no—
113Dic3⋊C4—126no—
114C4×Dic3—62no—
115C4×Dic3—61no—
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—42no—
125C3⋊Dic3—82no—
126C2×Q16—91no—
127C4⋊Q8—31no—
128C4⋊Q8—92no—
129C4⋊Q8—91no—
130C4⋊Q8—182no—
131C4⋊Q8—181no—
132#125—32no—
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#106—92no—
142#106—92no—
143#106—92no—
144C4⋊C8—91no—
145C4⋊C8—92no—
146#103—91no—
147#103—92no—
148#103—91no—
149#103—92no—
150#97—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—123no—
166C2×Dic3—123no—
167C2×Dic3—126no—
168Dic6—32no—
169Dic6—61no—
170Dic6—242no—
171Dic6—241no—
172Dic6—2412no—
173SL(2,𝔽3)D3×Q813yes—
174C3⋊C8—32no—
175C3⋊C8—61no—
176C3⋊C8—62no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—41no—
180C2×Q8—31no—
181C2×Q8—91no—
182C2×Q8—181no—
183C2×Q8—182no—
184C2×Q8C22×C3⋊D311yescharacteristic
185Q16—92no—
186Q16—92no—
187C2×C8—91no—
188C2×C8—91no—
189C2×C8—92no—
190C4⋊C4—32no—
191C4⋊C4—31no—
192C4⋊C4—32no—
193C4⋊C4—61no—
194C4⋊C4—91no—
195C4⋊C4—92no—
196C4⋊C4—181no—
197C4⋊C4—182no—
198C4⋊C4—182no—
199C4⋊C4—181no—
200C4⋊C4C3⋊S411yescharacteristic
201C42—31no—
202C42—32no—
203C42—182no—
204C42—181no—
205C22×C3—43no—
206C22×C3C22×S411yescharacteristic
207C3×C4—22no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—46no—
211C3×C4—62no—
212C3×C4—61no—
213C3×C4—86no—
214C3×C4#98412yes—
215Dic3—62no—
216Dic3—126no—
217Dic3—126no—
218Dic3—121no—
219Dic3—122no—
220Dic3—246no—
221C32—41noSylow
222Q8—31no—
223Q8—62no—
224Q8—92no—
225Q8—181no—
226Q8—722no—
227Q8—721no—
228Q8Q8×C3⋊D311yescharacteristic
229C2×C4—31no—
230C2×C4—32no—
231C2×C4—31no—
232C2×C4—181no—
233C2×C4—181no—
234C2×C4—182no—
235C2×C4C2×C3⋊S412yes—
236C2×C4C2×C3⋊S411yescharacteristic
237C8—92no—
238C8—181no—
239C8—182no—
240C2×C3—43no—
241C2×C3—43no—
242C2×C3—43no—
243C2×C3Q8×S411yescharacteristic
244C2×C3C2×#98411yescharacteristic
245C2×C3#592611yescharacteristic
246C22C22×C3⋊S411yescharacteristic
247C4—22no—
248C4—31no—
249C4—31no—
250C4—62no—
251C4—61no—
252C4—182no—
253C4—361no—
254C4—362no—
255C4#6426512yes—
256C3—43no—
257C3#8541511yescharacteristic, complemented
258C2Q8×C3⋊S411yescharacteristic
259C2C2×#6426511yescharacteristic
260C2#1059658911yescharacteristic
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
257. C3#85415semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936252
#105936222
C4×C3⋊#2262
#105936202
#854153
#836833
#640974