Finite Group Databasebeta

← #11922973

Subgroups — #11922973

Order 1152.

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

normal subgroups only

This group has 530 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, complemented
2#10593628C212yesmaximal
3C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
4#10593625C212yesmaximal, complemented
5#10593620C211yesmaximal, characteristic, complemented
6#85424—33nomaximal
7#83694—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9#64106—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2212yes—
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#557D311yescharacteristic, complemented
15Q8×D12—31no—
16#557⋊C3D313yescomplemented
17#5394—36no—
18C4×GL(2,𝔽3)—33no—
19#5391—36no—
20#5386—33no—
21#5087—31no—
22#5078—31no—
23#5066—32no—
24#5064—32no—
25#5026—32no—
26C4×#215—31no—
27#5024—31no—
28#5021—32no—
29#5020—31no—
30C2×C4×C3⋊D3—41no—
31C2×#634—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)Q812yescomplemented
34C32×C4⋊C4—41no—
35#3752—42no—
36C4⋊(C3⋊Dic3)—42no—
37#3386—91noSylow
38C2×D3×Q8—61no—
39C2×GL(2,𝔽3)—33no—
40C3×C4⋊Q8—31no—
41C3×#125—32no—
42C3×C4×Q8—31no—
43C3×C4×Q8C2×D311yescharacteristic
44C3×C4×Q8C2×D312yes—
45C2×#215—31no—
46#897—62no—
47#897—61no—
48#897—123no—
49C4×D12—31no—
50C4×D12—32no—
51#869—31no—
52#869—32no—
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#809—31no—
60#809—32no—
61#808—32no—
62#808—31no—
63#807—32no—
64#807—32no—
65#807—32no—
66C4⋊(C3⋊C8)—31no—
67C4⋊(C3⋊C8)—32no—
68#803—32no—
69C4×C3⋊C8—31no—
70C22×C3⋊D3—41no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—42no—
75C4×C3⋊D3—82no—
76#634—82no—
77C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
78#557C3⋊D311yescharacteristic, complemented
79D4×Q8—91no—
80#501—92no—
81#500—91no—
82#478—92no—
83#477—92no—
84#475—91no—
85#474—91no—
86#446—91no—
87#440—92no—
88C4×SD16—91no—
89C2×C3×Q8—31no—
90C2×C3×Q8C22×D311yescharacteristic
91D3×Q8—124no—
92C2×D12—31no—
93C2×C4×D3—62no—
94C2×C4×D3—61no—
95C2×C4×D3—123no—
96C2×Dic6—62no—
97C2×Dic6—61no—
98C2×Dic6—123no—
99C2×SL(2,𝔽3)C22×D313yes—
100GL(2,𝔽3)—36no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—43no—
105C3×C4⋊C4—61no—
106C3×C4⋊C4S411yescharacteristic
107C3×C42—31no—
108C3×C42—32no—
109#215—32no—
110#215—32no—
111(C2×D3)⋊C4—61no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4—126no—
114C4⋊Dic3—31no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3—62no—
118C4⋊Dic3—126no—
119C2×C3⋊C8—31no—
120C2×C3⋊C8—31no—
121C2×C3⋊C8—32no—
122C22×C32—41no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—81no—
125C2×C3⋊D3—81no—
126C32×C4—42no—
127C32×C4—82no—
128C3⋊Dic3—82no—
129C3⋊Dic3—81no—
130C22×Q8—181no—
131C2×SD16—91no—
132C4⋊Q8—31no—
133C4⋊Q8—91no—
134C4⋊Q8—92no—
135#125—32no—
136C22⋊Q8—182no—
137C22⋊Q8—181no—
138C4×Q8—31no—
139C4×Q8C2×C3⋊D311yescharacteristic
140C4×Q8C2×C3⋊D312yes—
141C4×D4—91no—
142C4×D4—92no—
143#107—92no—
144#107—92no—
145#107—92no—
146#106—91no—
147#106—92no—
148C4⋊C8—91no—
149C4⋊C8—92no—
150#103—91no—
151#103—92no—
152#102—91no—
153#102—92no—
154#97—92no—
155C4×C8—91no—
156C22×D3—61no—
157C22×D3—123no—
158C3×Q8—31no—
159C3×Q8—62no—
160C3×Q8D3×Q811yescharacteristic
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—43no—
165C2×C3×C4—46no—
166C2×C3×C4C2×S412yes—
167C2×C3×C4C2×S411yescharacteristic
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—123no—
171C2×Dic3—126no—
172D12—32no—
173D12—61no—
174C4×D3—124no—
175C4×D3—122no—
176C4×D3—246no—
177Dic6—124no—
178Dic6—122no—
179Dic6—246no—
180SL(2,𝔽3)D3×Q813yescomplemented
181C3⋊C8—32no—
182C3⋊C8—61no—
183C3⋊C8—62no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C3⋊D3—82no—
188C2×Q8—31no—
189C2×Q8—182no—
190C2×Q8—181no—
191C2×Q8—364no—
192C2×Q8C22×C3⋊D311yescharacteristic
193C2×D4—91no—
194C22×C4—182no—
195C22×C4—181no—
196SD16—92no—
197SD16—92no—
198C2×C8—91no—
199C2×C8—91no—
200C2×C8—92no—
201C4⋊C4—31no—
202C4⋊C4—32no—
203C4⋊C4—32no—
204C4⋊C4—61no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C4⋊C4—181no—
208C4⋊C4—182no—
209C4⋊C4C3⋊S411yescharacteristic
210C22⋊C4—181no—
211C22⋊C4—182no—
212C42—31no—
213C42—32no—
214C22×C3—43no—
215C22×C3C22×S411yescharacteristic
216C2×D3—62no—
217C2×D3—126no—
218C2×D3—121no—
219C2×D3—121no—
220C2×D3—243no—
221C2×D3—243no—
222C3×C4—22no—
223C3×C4—31no—
224C3×C4—31no—
225C3×C4—46no—
226C3×C4—62no—
227C3×C4—61no—
228C3×C4—86no—
229C3×C4#98612yes—
230Dic3—121no—
231Dic3—122no—
232Dic3—246no—
233Dic3—243no—
234C32—41noSylow
235C23—181no—
236Q8—31no—
237Q8—62no—
238Q8—364no—
239Q8—362no—
240Q8Q8×C3⋊D311yescharacteristic
241D4—92no—
242D4—181no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—32no—
246C2×C4—181no—
247C2×C4—182no—
248C2×C4—364no—
249C2×C4—362no—
250C2×C4C2×C3⋊S412yes—
251C2×C4C2×C3⋊S411yescharacteristic
252C8—92no—
253C8—181no—
254C8—182no—
255C2×C3—43no—
256C2×C3—43no—
257C2×C3—43no—
258C2×C3Q8×S411yescharacteristic
259C2×C3C2×#98611yescharacteristic
260C2×C3#592711yescharacteristic
261D3—122no—
262D3—246no—
263C22—182no—
264C22—361no—
265C22—361no—
266C22C22×C3⋊S411yescharacteristic
267C4—22no—
268C4—31no—
269C4—31no—
270C4—62no—
271C4—61no—
272C4—362no—
273C4—361no—
274C4#6426712yes—
275C3—43no—
276C3#8542411yescharacteristic, complemented
277C2—362no—
278C2Q8×C3⋊S411yescharacteristic
279C2C2×#6426711yescharacteristic
280C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
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
1. C3×#557⋊C3C2semidirect
4. #10593625C2semidirect
5. #10593620C2semidirect
14. C3×#557D3semidirect
16. #557⋊C3D3semidirect
33. C3⋊GL(2,𝔽3)Q8semidirect
77. C3×SL(2,𝔽3)C2×Q8semidirect
78. #557C3⋊D3semidirect
180. SL(2,𝔽3)D3×Q8semidirect
276. C3#85424semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936282
C4×C3⋊GL(2,𝔽3)2
#105936252
#105936202
#854243
#836943
#641064