Finite Group Databasebeta

← #11922957

Subgroups — #11922957

Order 1152.

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

normal subgroups only

This group has 550 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#10593661C212yesmaximal, complemented
3#10593658C212yesmaximal, complemented
4C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#10593625C211yesmaximal, characteristic, complemented
6#85408—33nomaximal
7#83571—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
10#64101—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#550D311yescharacteristic, complemented
15#5563—31no—
16C4⋊(#41⋊C3)D313yescomplemented
17#5427—36no—
18#5424—36no—
19C4×GL(2,𝔽3)—33no—
20#5391—33no—
21#5238—32no—
22#5233—32no—
23#5072—31no—
24#5070—31no—
25#5038—32no—
26#5035—32no—
27C4×#215—31no—
28#5021—31no—
29D4×C3⋊C8—31no—
30C2×C4×C3⋊D3—41no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)—21no—
34C32×C4⋊C4—41no—
35#3752—42no—
36C4⋊(C3⋊Dic3)—41no—
37C4×C3⋊Dic3—42no—
38#3305—91noSylow
39C2×C3×#41C2×D312yes—
40C2×#41⋊C3C2×D316yes—
41C2×GL(2,𝔽3)—33no—
42C3×#127—31no—
43C3×#121—32no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×D4—31no—
46C2×#215—31no—
47#938—61no—
48C2×#213—32no—
49#930—32no—
50#926—32no—
51C22×C3⋊C8—32no—
52#892—61no—
53#892—123no—
54#885—62no—
55C4×D12—31no—
56#869—31no—
57C4×SL(2,𝔽3)C2×D313yes—
58#859—36no—
59#859—33no—
60#835—31no—
61#835—32no—
62#832—32no—
63#832—32no—
64#832—32no—
65#830—32no—
66#809—31no—
67#808—31no—
68#807—32no—
69C4⋊(C3⋊C8)—31no—
70C4×C3⋊C8—31no—
71C22×C3⋊D3—41no—
72C2×C32×C4—41no—
73C2×C32×C4—42no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—42no—
76C4×C3⋊D3—44no—
77C3×SL(2,𝔽3)#4111yescharacteristic
78#555—91no—
79#550C3⋊D311yescharacteristic, complemented
80#486—91no—
81#482—92no—
82#477—91no—
83#466—92no—
84#459—91no—
85#449—92no—
86C4×SD16—91no—
87C8×D4—91no—
88#416—92no—
89C3×#41—22no—
90C3×#41—62no—
91C2×C3×Q8C22×D311yescharacteristic
92C2×C3×D4—32no—
93C2×C3×D4—31no—
94C2×C3×D4—61no—
95C22×C3×C4—32no—
96C2×#67—62no—
97C2×D12—31no—
98C2×C4×D3—61no—
99C2×C4×D3—123no—
100C2×Dic6—61no—
101#41⋊C3—26no—
102C2×SL(2,𝔽3)C22×D313yes—
103GL(2,𝔽3)—63no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—43no—
106C3×C4⋊C4S411yescharacteristic
107C3×C22⋊C4—32no—
108C3×C42—31no—
109#217—62no—
110#215—61no—
111#215—61no—
112#213—122no—
113(C2×D3)⋊C4—61no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—126no—
116C4⋊Dic3—32no—
117C4⋊Dic3—31no—
118C4⋊Dic3—61no—
119C4⋊Dic3—123no—
120C4×Dic3—62no—
121C4×Dic3—126no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—122no—
126C22×C32—41no—
127C2×C3⋊D3—42no—
128C2×C3⋊D3—81no—
129C2×C3⋊D3—81no—
130C32×C4—42no—
131C32×C4—82no—
132C3⋊Dic3—42no—
133C3⋊Dic3—82no—
134C2×#41C2×C3⋊D312yes—
135C2×SD16—91no—
136C2×D8—92no—
137C22×C8—92no—
138C4⋊Q8—91no—
139#127—31no—
140#124—182no—
141#121—32no—
142#121—92no—
143#121—181no—
144C4×Q8C2×C3⋊D311yescharacteristic
145C4×D4—31no—
146C4×D4—91no—
147#117—92no—
148#117—181no—
149#107—92no—
150#106—91no—
151C4⋊C8—91no—
152#103—91no—
153#103—92no—
154#102—91no—
155#102—92no—
156#102—92no—
157#102—92no—
158C22⋊C8—92no—
159C4×C8—91no—
160C23×C3—32no—
161C22×D3—61no—
162C22×D3—123no—
163C3×Q8—31no—
164C3×Q8#23711yescharacteristic
165C3×D4—62no—
166C3×D4—62no—
167C3×D4—62no—
168C3×D4—241no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—43no—
172C2×C3×C4—46no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4C2×S412yes—
176C2×C3×C4C2×S411yescharacteristic
177#67—242no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—123no—
181C2×Dic3—126no—
182D12—61no—
183D12—61no—
184C4×D3—1212no—
185C4×D3—122no—
186Dic6—241no—
187SL(2,𝔽3)#23713yes—
188C3⋊C8—61no—
189C3⋊C8—62no—
190C3⋊C8—61no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C3⋊D3—82no—
195#41—22no—
196#41—62no—
197C2×Q8—181no—
198C2×Q8C22×C3⋊D311yescharacteristic
199C2×D4—32no—
200C2×D4—31no—
201C2×D4—61no—
202C2×D4—91no—
203C2×D4—182no—
204C22×C4—32no—
205C22×C4—181no—
206SD16—181no—
207SD16—181no—
208D8—362no—
209C2×C8—91no—
210C2×C8—91no—
211C2×C8—92no—
212C2×C8—362no—
213C4⋊C4—31no—
214C4⋊C4—91no—
215C4⋊C4—92no—
216C4⋊C4—181no—
217C4⋊C4C3⋊S411yescharacteristic
218C22⋊C4—32no—
219C22⋊C4—181no—
220C22⋊C4—182no—
221C22⋊C4—182no—
222C42—31no—
223C42—182no—
224C22×C3—43no—
225C22×C3—62no—
226C22×C3—62no—
227C22×C3—62no—
228C22×C3C22×S411yescharacteristic
229C2×D3—126no—
230C2×D3—121no—
231C2×D3—121no—
232C2×D3—121no—
233C2×D3—243no—
234C2×D3—243no—
235C3×C4—22no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—46no—
239C3×C4—61no—
240C3×C4—86no—
241C3×C4#98512yes—
242Dic3—126no—
243Dic3—121no—
244Dic3—122no—
245Dic3—246no—
246C32—41noSylow
247C23—32no—
248C23—181no—
249Q8—31no—
250Q8—721no—
251Q8#383011yescharacteristic
252D4—62no—
253D4—62no—
254D4—62no—
255D4—181no—
256D4—181no—
257D4—241no—
258D4—722no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—181no—
264C2×C4—182no—
265C2×C4—362no—
266C2×C4C2×C3⋊S412yes—
267C2×C4C2×C3⋊S411yescharacteristic
268C8—181no—
269C8—182no—
270C8—181no—
271C2×C3—43no—
272C2×C3—43no—
273C2×C3—43no—
274C2×C3—122no—
275C2×C3#591411yescharacteristic
276C2×C3C2×#98511yescharacteristic
277C2×C3#592511yescharacteristic
278D3—246no—
279D3—241no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—361no—
284C22—361no—
285C22—361no—
286C22C22×C3⋊S411yescharacteristic
287C4—22no—
288C4—31no—
289C4—31no—
290C4—61no—
291C4—361no—
292C4—362no—
293C4#6426612yes—
294C3—43no—
295C3#8540811yescharacteristic, complemented
296C2—122no—
297C2—721no—
298C2#1059657711yescharacteristic
299C2C2×#6426611yescharacteristic
300C2#1059658811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#550D3
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×C4⋊(#41⋊C3)C2semidirect
2. #10593661C2semidirect
3. #10593658C2semidirect
4. C4×C3⋊GL(2,𝔽3)C2semidirect
5. #10593625C2semidirect
14. C3×#550D3semidirect
16. C4⋊(#41⋊C3)D3semidirect
79. #550C3⋊D3semidirect
295. C3#85408semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(#41⋊C3)2
#105936612
#105936582
C4×C3⋊GL(2,𝔽3)2
#105936252
#854083
#835713
#641014