Finite Group Databasebeta

← #11922961

Subgroups — #11922961

Order 1152.

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

normal subgroups only

This group has 568 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#10593660C212yesmaximal, complemented
4C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#10593620C211yesmaximal, characteristic, complemented
6#85412—33nomaximal
7#83574—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
10#64106—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#550D311yescharacteristic, complemented
15#5555—31no—
16C4⋊(#41⋊C3)D313yescomplemented
17#5427—36no—
18#5426—36no—
19C4×GL(2,𝔽3)—33no—
20#5386—33no—
21#5238—32no—
22#5232—32no—
23#5072—31no—
24#5070—31no—
25#5039—32no—
26#5034—32no—
27C4×#215—31no—
28#5020—31no—
29#5012—31no—
30C2×C4×C3⋊D3—41no—
31C2×#634—41no—
32C3×#41⋊C3—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊GL(2,𝔽3)—21no—
35C32×C4⋊C4—41no—
36#3752—42no—
37C4⋊(C3⋊Dic3)—42no—
38#3332—91noSylow
39C2×C3×#41C2×D312yes—
40C2×#237—61no—
41C2×#41⋊C3C2×D316yes—
42C2×GL(2,𝔽3)—33no—
43C3×#127—31no—
44C3×#121—32no—
45C3×C4×Q8C2×D311yescharacteristic
46C3×C4×D4—31no—
47C2×#215—31no—
48C2×#213—32no—
49#930—32no—
50C2×C4⋊Dic3—32no—
51C2×#208—32no—
52#897—61no—
53#897—123no—
54#886—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—32no—
68#808—31no—
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—82no—
77#634—82no—
78C3×SL(2,𝔽3)#4111yescharacteristic, complemented
79#550C3⋊D311yescharacteristic, complemented
80#547—91no—
81#486—91no—
82#481—92no—
83#475—91no—
84#469—92no—
85#459—91no—
86#449—92no—
87C4×SD16—91no—
88#436—91no—
89#417—92no—
90C3×#41—22no—
91C3×#41—62no—
92C2×C3×Q8C22×D311yescharacteristic
93C2×C3×D4—32no—
94C2×C3×D4—31no—
95C2×C3×D4—61no—
96C22×C3×C4—32no—
97C2×#67—62no—
98C22×Dic3—62no—
99#237—124no—
100C2×D12—31no—
101C2×C4×D3—61no—
102C2×C4×D3—123no—
103C2×Dic6—61no—
104C2×Dic6—123no—
105#41⋊C3—26no—
106C2×SL(2,𝔽3)C22×D313yes—
107GL(2,𝔽3)—63no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—43no—
110C3×C4⋊C4S411yescharacteristic
111C3×C22⋊C4—32no—
112C3×C42—31no—
113#215—61no—
114#215—61no—
115#213—122no—
116(C2×D3)⋊C4—61no—
117(C2×D3)⋊C4—62no—
118(C2×D3)⋊C4—126no—
119C4⋊Dic3—32no—
120C4⋊Dic3—31no—
121C4⋊Dic3—61no—
122C4⋊Dic3—62no—
123C4⋊Dic3—126no—
124#208—122no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—32no—
128C22×C32—41no—
129C2×C3⋊D3—42no—
130C2×C3⋊D3—81no—
131C2×C3⋊D3—81no—
132C32×C4—42no—
133C32×C4—82no—
134C3⋊Dic3—82no—
135C3⋊Dic3—81no—
136C2×#41—181no—
137C2×#41C2×C3⋊D312yes—
138C2×SD16—91no—
139C2×D8—92no—
140C2×M16—92no—
141C4⋊Q8—91no—
142#127—31no—
143#123—182no—
144C22⋊Q8—181no—
145#121—32no—
146#121—92no—
147C4×Q8C2×C3⋊D311yescharacteristic
148C4×D4—31no—
149C4×D4—91no—
150C2×C4⋊C4—92no—
151#106—91no—
152#106—92no—
153C4⋊C8—91no—
154#103—91no—
155#103—92no—
156#102—91no—
157#102—92no—
158#102—92no—
159#102—92no—
160C22⋊C8—92no—
161C4×C8—91no—
162C23×C3—32no—
163C22×D3—61no—
164C22×D3—123no—
165C3×Q8—31no—
166C3×Q8#23711yescharacteristic
167C3×D4—62no—
168C3×D4—62no—
169C3×D4—62no—
170C3×D4—241no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—43no—
174C2×C3×C4—46no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4C2×S412yes—
178C2×C3×C4C2×S411yescharacteristic
179#67—124no—
180C2×Dic3—62no—
181C2×Dic3—61no—
182C2×Dic3—123no—
183C2×Dic3—126no—
184C2×Dic3—124no—
185D12—61no—
186D12—61no—
187C4×D3—122no—
188C4×D3—246no—
189Dic6—122no—
190Dic6—246no—
191SL(2,𝔽3)#23713yescomplemented
192C3⋊C8—61no—
193C3⋊C8—62no—
194C3⋊C8—61no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C3⋊D3—82no—
199#41—22no—
200#41—62no—
201#41—364no—
202C2×Q8—181no—
203C2×Q8C22×C3⋊D311yescharacteristic
204C2×D4—32no—
205C2×D4—31no—
206C2×D4—61no—
207C2×D4—91no—
208C2×D4—182no—
209C22×C4—32no—
210C22×C4—182no—
211C22×C4—181no—
212SD16—181no—
213SD16—181no—
214D8—362no—
215M16—362no—
216C2×C8—91no—
217C2×C8—91no—
218C2×C8—92no—
219C4⋊C4—31no—
220C4⋊C4—91no—
221C4⋊C4—92no—
222C4⋊C4—181no—
223C4⋊C4—182no—
224C4⋊C4C3⋊S411yescharacteristic
225C22⋊C4—32no—
226C22⋊C4—181no—
227C22⋊C4—182no—
228C42—31no—
229C22×C3—43no—
230C22×C3—62no—
231C22×C3—62no—
232C22×C3—62no—
233C22×C3C22×S411yescharacteristic
234C2×D3—126no—
235C2×D3—121no—
236C2×D3—121no—
237C2×D3—121no—
238C2×D3—243no—
239C2×D3—243no—
240C3×C4—22no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—46no—
244C3×C4—61no—
245C3×C4—86no—
246C3×C4#98612yes—
247Dic3—122no—
248Dic3—121no—
249Dic3—246no—
250Dic3—243no—
251C32—41noSylow
252C23—32no—
253C23—181no—
254Q8—31no—
255Q8—362no—
256Q8#383011yescharacteristic
257D4—62no—
258D4—62no—
259D4—62no—
260D4—181no—
261D4—181no—
262D4—241no—
263D4—364no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—181no—
269C2×C4—182no—
270C2×C4—364no—
271C2×C4—362no—
272C2×C4C2×C3⋊S412yes—
273C2×C4C2×C3⋊S411yescharacteristic
274C8—181no—
275C8—182no—
276C8—181no—
277C2×C3—43no—
278C2×C3—43no—
279C2×C3—43no—
280C2×C3—122no—
281C2×C3#591411yescharacteristic
282C2×C3C2×#98611yescharacteristic
283C2×C3#592411yescharacteristic
284D3—246no—
285D3—241no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—361no—
290C22—361no—
291C22—361no—
292C22C22×C3⋊S411yescharacteristic
293C4—22no—
294C4—31no—
295C4—31no—
296C4—61no—
297C4—361no—
298C4—362no—
299C4#6426712yes—
300C3—43no—
301C3#8541211yescharacteristic, complemented
302C2—122no—
303C2—721no—
304C2#1059657711yescharacteristic
305C2C2×#6426711yescharacteristic
306C2#1059658711yescharacteristic
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. #10593660C2semidirect
4. C4×C3⋊GL(2,𝔽3)C2semidirect
5. #10593620C2semidirect
14. C3×#550D3semidirect
16. C4⋊(#41⋊C3)D3semidirect
78. C3×SL(2,𝔽3)#41semidirect
79. #550C3⋊D3semidirect
191. SL(2,𝔽3)#237semidirect
301. C3#85412semidirect

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
#105936602
C4×C3⋊GL(2,𝔽3)2
#105936202
#854123
#835743
#641064