Finite Group Databasebeta

← #11922972

Subgroups — #11922972

Order 1152.

596 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 69 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, complemented
3C4⋊(C3⋊GL(2,𝔽3))C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#85422—33nomaximal
7#83690—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊GL(2,𝔽3)C2212yes—
10#64104—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#557D311yescharacteristic, complemented
15#5577—31no—
16#557⋊C3D313yescomplemented
17#5394—36no—
18C4⋊GL(2,𝔽3)—33no—
19C4⋊GL(2,𝔽3)—36no—
20C4×GL(2,𝔽3)—33no—
21#5088—31no—
22#5087—31no—
23#5067—32no—
24#5066—32no—
25#5027—32no—
26#5027—31no—
27#5026—32no—
28C4×#215—31no—
29#5024—31no—
30C2×C4⋊(C3⋊D3)—41no—
31C2×C4⋊(C3⋊D3)—42no—
32C2×C4×C3⋊D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊GL(2,𝔽3)—21no—
35C3⋊GL(2,𝔽3)—22no—
36C32×C4⋊C4—41no—
37#3752—42no—
38#3368—91noSylow
39C2×#239—61no—
40C2×GL(2,𝔽3)—33no—
41C2×GL(2,𝔽3)—36no—
42C3×C4⋊Q8—31no—
43C3×#125—32no—
44C3×C4×Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46C3×C4×Q8C2×D312yes—
47C2×#215—31no—
48C2×#215—32no—
49#895—62no—
50#895—61no—
51#895—123no—
52#874—32no—
53#874—31no—
54C4×D12—31no—
55C4×D12—32no—
56C4×SL(2,𝔽3)C2×D313yes—
57C4×SL(2,𝔽3)C2×D316yes—
58#859—33no—
59#835—31no—
60#809—31no—
61#809—32no—
62#809—32no—
63#809—32no—
64#809—32no—
65#808—31no—
66#807—32no—
67C4⋊(C3⋊C8)—32no—
68C4⋊(C3⋊C8)—31no—
69#803—32no—
70C4×C3⋊C8—31no—
71C22×C3⋊D3—41no—
72C22×C3⋊D3—42no—
73C2×C32×C4—41no—
74C2×C32×C4—42no—
75C2×C3⋊Dic3—41no—
76C4⋊(C3⋊D3)—82no—
77C4⋊(C3⋊D3)—84no—
78C4×C3⋊D3—82no—
79C3×SL(2,𝔽3)#4111yescharacteristic, complemented
80#557C3⋊D311yescharacteristic, complemented
81#550—91no—
82#490—92no—
83#486—91no—
84#478—92no—
85#474—91no—
86#460—92no—
87#460—91no—
88#446—91no—
89#440—92no—
90C4×SD16—91no—
91C2×C3×Q8—31no—
92C2×C3×Q8C22×D311yescharacteristic
93#239—124no—
94C2×D12—31no—
95C2×D12—32no—
96C2×D12—62no—
97C2×D12—61no—
98C2×D12—61no—
99C2×D12—62no—
100C2×D12—123no—
101C2×D12—126no—
102C2×C4×D3—62no—
103C2×C4×D3—61no—
104C2×C4×D3—123no—
105C2×SL(2,𝔽3)C22×D313yes—
106GL(2,𝔽3)—63no—
107GL(2,𝔽3)—66no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—43no—
112C3×C4⋊C4—61no—
113C3×C4⋊C4S411yescharacteristic
114C3×C42—31no—
115C3×C42—32no—
116#215—61no—
117#215—62no—
118#215—61no—
119#215—62no—
120(C2×D3)⋊C4—61no—
121(C2×D3)⋊C4—62no—
122(C2×D3)⋊C4—126no—
123C4⋊Dic3—31no—
124C2×C3⋊C8—31no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—32no—
127C22×C32—41no—
128C2×C3⋊D3—42no—
129C2×C3⋊D3—81no—
130C2×C3⋊D3—81no—
131C2×C3⋊D3—82no—
132C2×C3⋊D3—82no—
133C2×C3⋊D3—82no—
134C32×C4—42no—
135C32×C4—82no—
136C3⋊Dic3—81no—
137C2×#41—181no—
138C2×SD16—91no—
139C2×SD16—92no—
140C4⋊Q8—31no—
141#127—91no—
142#127—92no—
143#125—32no—
144#121—182no—
145#121—181no—
146C4×Q8—31no—
147C4×Q8C2×C3⋊D311yescharacteristic
148C4×Q8C2×C3⋊D312yes—
149C4×D4—91no—
150C4×D4—92no—
151#107—92no—
152#106—91no—
153C4⋊C8—91no—
154C4⋊C8—92no—
155#103—91no—
156#102—91no—
157#102—92no—
158#102—92no—
159#102—92no—
160#102—92no—
161#97—92no—
162C4×C8—91no—
163C22×D3—61no—
164C22×D3—62no—
165C22×D3—123no—
166C22×D3—126no—
167C3×Q8—31no—
168C3×Q8—62no—
169C3×Q8#23911yescharacteristic
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—32no—
173C2×C3×C4—43no—
174C2×C3×C4—46no—
175C2×C3×C4C2×S412yes—
176C2×C3×C4C2×S411yescharacteristic
177C2×Dic3—61no—
178C2×Dic3—123no—
179D12—61no—
180D12—62no—
181D12—61no—
182D12—62no—
183D12—124no—
184D12—122no—
185D12—246no—
186D12—242no—
187D12—241no—
188D12—2412no—
189C4×D3—124no—
190C4×D3—122no—
191C4×D3—246no—
192SL(2,𝔽3)#23913yescomplemented
193C3⋊C8—61no—
194C3⋊C8—61no—
195C3⋊C8—62no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C3⋊D3—82no—
200C3⋊D3—162no—
201#41—364no—
202C2×Q8—31no—
203C2×Q8C22×C3⋊D311yescharacteristic
204C2×D4—91no—
205C2×D4—92no—
206C2×D4—182no—
207C2×D4—181no—
208C2×D4—181no—
209C2×D4—182no—
210C22×C4—182no—
211C22×C4—181no—
212SD16—181no—
213SD16—182no—
214SD16—181no—
215SD16—182no—
216C2×C8—91no—
217C2×C8—91no—
218C2×C8—92no—
219C4⋊C4—31no—
220C4⋊C4—32no—
221C4⋊C4—32no—
222C4⋊C4—61no—
223C4⋊C4—91no—
224C4⋊C4C3⋊S411yescharacteristic
225C22⋊C4—181no—
226C22⋊C4—182no—
227C42—31no—
228C42—32no—
229C22×C3—43no—
230C22×C3C22×S411yescharacteristic
231C2×D3—126no—
232C2×D3—121no—
233C2×D3—122no—
234C2×D3—121no—
235C2×D3—122no—
236C2×D3—121no—
237C2×D3—122no—
238C2×D3—243no—
239C2×D3—243no—
240C2×D3—246no—
241C2×D3—246no—
242C2×D3—246no—
243C3×C4—22no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—46no—
247C3×C4—62no—
248C3×C4—61no—
249C3×C4—86no—
250C3×C4#98612yes—
251Dic3—121no—
252Dic3—243no—
253C32—41noSylow
254C23—181no—
255C23—182no—
256Q8—31no—
257Q8—62no—
258Q8#383211yescharacteristic
259D4—181no—
260D4—182no—
261D4—181no—
262D4—182no—
263D4—364no—
264D4—362no—
265D4—722no—
266D4—721no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—181no—
271C2×C4—364no—
272C2×C4—362no—
273C2×C4C2×C3⋊S412yes—
274C2×C4C2×C3⋊S411yescharacteristic
275C8—181no—
276C8—182no—
277C8—181no—
278C2×C3—43no—
279C2×C3—43no—
280C2×C3—43no—
281C2×C3#591911yescharacteristic
282C2×C3C2×#98611yescharacteristic
283C2×C3#592611yescharacteristic
284D3—246no—
285D3—241no—
286D3—242no—
287D3—486no—
288C22—361no—
289C22—362no—
290C22—361no—
291C22—362no—
292C22—361no—
293C22—362no—
294C22C22×C3⋊S411yescharacteristic
295C4—22no—
296C4—31no—
297C4—31no—
298C4—62no—
299C4—61no—
300C4—361no—
301C4#6426712yes—
302C3—43no—
303C3#8542211yescharacteristic, complemented
304C2—721no—
305C2—722no—
306C2#1059658211yescharacteristic
307C2C2×#6426711yescharacteristic
308C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
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
2. #10593628C2semidirect
3. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
4. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
14. C3×#557D3semidirect
16. #557⋊C3D3semidirect
79. C3×SL(2,𝔽3)#41semidirect
80. #557C3⋊D3semidirect
192. SL(2,𝔽3)#239semidirect
303. C3#85422semidirect

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
C4×C3⋊GL(2,𝔽3)2
#854223
#836903
#641044