Finite Group Databasebeta

← #11922896

Subgroups — #11922896

Order 1152.

672 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#64267C211yesmaximal, characteristic
2C2×#64265C211yesmaximal, characteristic, complemented
3C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
4#10593629C214yesmaximal, complemented
5#85347—33nomaximal
6#82527—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8#64267—22no—
9#64265—22no—
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C2×C3⋊#226C2212yescomplemented
12#64084—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C2×#986—33no—
15C2×#984—33no—
16C3×C4×#41D311yescharacteristic, complemented
17C2×#951—31no—
18C2×#949—31no—
19C2×#875—31no—
20C4×#41⋊C3D313yescomplemented
21#5395—312no—
22#5028—34no—
23#5016—34no—
24#5004—31no—
25#5001—31no—
26#4924—31no—
27C2×C4⋊(C3⋊D3)—41no—
28C2×#634—41no—
29C3×#41⋊C3D412yes—
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31C3⋊GL(2,𝔽3)—22no—
32C3⋊#226—22no—
33C32×C42—41no—
34#3752—44no—
35#3040—91noSylow
36C2×C3×#41C2×D311yescharacteristic
37C22×D12—31no—
38C22×Dic6—31no—
39C2×#41⋊C3C2×D313yes—
40#986—66no—
41#984—66no—
42C2×GL(2,𝔽3)—36no—
43C2×#226—36no—
44C3×C4×Q8C2×D312yes—
45C3×C4×D4—32no—
46C3×#117—31no—
47C2×C3×C42—31no—
48#951—62no—
49#951—62no—
50#949—62no—
51#949—62no—
52C2×#216—32no—
53C2×#215—32no—
54C2×#214—32no—
55C2×#213—32no—
56C2×(C2×D3)⋊C4—62no—
57C2×#208—31no—
58C2×#208—31no—
59#875—34no—
60#875—61no—
61#875—61no—
62#875—123no—
63C4×SL(2,𝔽3)C2×D316yescomplemented
64#810—34no—
65#809—34no—
66C4⋊(C3⋊C8)—34no—
67C22×C3⋊D3—42no—
68C2×C32×C4—41no—
69C2×C32×C4—42no—
70C2×C3⋊Dic3—42no—
71C4⋊(C3⋊D3)—82no—
72#634—82no—
73C3×SL(2,𝔽3)C2×D411yescharacteristic
74C2×#137—91no—
75C2×AGL(1,ℤ/8)—91no—
76C2×#124—91no—
77C4×#41C3⋊D311yescharacteristic, complemented
78#464—94no—
79#463—94no—
80#423—91no—
81#419—91no—
82#418—91no—
83C23×D3—61no—
84C3×#41—32no—
85C3×#41D1212yes—
86C2×C3×Q8C22×D311yescharacteristic
87C2×C3×D4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—32no—
90C22×Dic3—61no—
91C2×D12—32no—
92C2×D12—62no—
93C2×D12—62no—
94C2×D12—61no—
95C2×D12—123no—
96C2×Dic6—32no—
97C2×Dic6—62no—
98C2×Dic6—62no—
99C2×Dic6—61no—
100C2×Dic6—123no—
101#41⋊C3D1216yes—
102C2×SL(2,𝔽3)C22×D313yes—
103GL(2,𝔽3)—66no—
104#226—66no—
105C3×C4⋊C4—32no—
106C3×C22⋊C4—61no—
107C3×C42—32no—
108C3×C42—31no—
109C3×C42—43no—
110C3×C42S411yescharacteristic
111#216—62no—
112#216—62no—
113#215—62no—
114#215—62no—
115#214—64no—
116#213—64no—
117(C2×D3)⋊C4—64no—
118(C2×D3)⋊C4—64no—
119(C2×D3)⋊C4—1212no—
120#208—62no—
121#208—62no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—32no—
124C22×C32—41no—
125C2×C3⋊D3—82no—
126C2×C3⋊D3—84no—
127C32×C4—42no—
128C32×C4—82no—
129C3⋊Dic3—82no—
130C2×#41C2×C3⋊D311yescharacteristic
131C22×Q8—91no—
132C22×D4—91no—
133#137—182no—
134#137—182no—
135AGL(1,ℤ/8)—182no—
136AGL(1,ℤ/8)—182no—
137C2×Q16—92no—
138C2×SD16—92no—
139C2×SD16—92no—
140C2×D8—92no—
141C2×M16—91no—
142C2×M16—91no—
143#124—94no—
144#124—181no—
145#124—181no—
146C4×Q8C2×C3⋊D312yes—
147C4×D4—32no—
148#117—31no—
149C2×C22⋊C4—182no—
150C2×C42—31no—
151C4⋊C8—94no—
152#103—94no—
153#102—94no—
154C23×C3—31no—
155C22×D3—62no—
156C22×D3—126no—
157C22×D3—122no—
158C22×D3—124no—
159C3×Q8—31no—
160C3×Q8C2×D1211yescharacteristic
161C3×D4—32no—
162C3×D4—61no—
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—32no—
166C2×C3×C4—32no—
167C2×C3×C4—43no—
168C2×C3×C4—46no—
169C2×C3×C4—122no—
170C2×C3×C4C2×S412yes—
171C2×C3×C4C2×S411yescharacteristic
172C2×Dic3—62no—
173C2×Dic3—126no—
174C2×Dic3—122no—
175D12—62no—
176D12—62no—
177D12—122no—
178D12—246no—
179Dic6—62no—
180Dic6—62no—
181Dic6—122no—
182Dic6—246no—
183SL(2,𝔽3)C2×D1213yes—
184C3⋊C8—62no—
185C3⋊C8—62no—
186C2×C32—41no—
187C2×C32—42no—
188C3⋊D3—162no—
189C24—181no—
190#41—32no—
191#41C4⋊(C3⋊D3)12yes—
192C2×Q8—92no—
193C2×Q8—182no—
194C2×Q8—182no—
195C2×Q8—181no—
196C2×Q8C22×C3⋊D311yescharacteristic
197C2×D4—31no—
198C2×D4—92no—
199C2×D4—182no—
200C2×D4—182no—
201C2×D4—181no—
202C22×C4—31no—
203C22×C4—32no—
204C22×C4—181no—
205Q16—182no—
206Q16—182no—
207SD16—184no—
208SD16—182no—
209SD16—182no—
210D8—184no—
211M16—182no—
212M16—182no—
213C2×C8—92no—
214C2×C8—92no—
215C4⋊C4—32no—
216C22⋊C4—61no—
217C22⋊C4—184no—
218C22⋊C4—184no—
219C42—32no—
220C42—31no—
221C42C3⋊S411yescharacteristic
222C22×C3—32no—
223C22×C3—43no—
224C22×C3—62no—
225C22×C3C22×S411yescharacteristic
226C2×D3—122no—
227C2×D3—124no—
228C2×D3—246no—
229C2×D3—2412no—
230C2×D3—244no—
231C3×C4—22no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—46no—
235C3×C4—62no—
236C3×C4—86no—
237C3×C4C4⋊S412yes—
238Dic3—122no—
239Dic3—246no—
240C32—41noSylow
241C23—31no—
242C23—182no—
243C23—362no—
244C23—364no—
245Q8—31no—
246Q8—182no—
247Q8—182no—
248Q8—362no—
249Q8C2×C4⋊(C3⋊D3)11yescharacteristic
250D4—32no—
251D4—61no—
252D4—182no—
253D4—182no—
254D4—362no—
255C2×C4—31no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—122no—
260C2×C4—182no—
261C2×C4—362no—
262C2×C4C2×C3⋊S412yes—
263C2×C4C2×C3⋊S411yescharacteristic
264C8—182no—
265C8—182no—
266C2×C3—43no—
267C2×C3—46no—
268C2×C3—62no—
269C2×C3C2×C4⋊S411yescharacteristic
270C2×C3#592312yes—
271D3—242no—
272D3—486no—
273C22—32no—
274C22—62no—
275C22—362no—
276C22—364no—
277C22—724no—
278C22C22×C3⋊S411yescharacteristic
279C4—22no—
280C4—31no—
281C4—31no—
282C4—62no—
283C4—362no—
284C4C4⋊(C3⋊S4)12yes—
285C3—43no—
286C3#8534711yescharacteristic, complemented
287C2—62no—
288C2—722no—
289C2C2×C4⋊(C3⋊S4)11yescharacteristic
290C2#1059658612yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×#41D3
socleC22×C3C22×S4
prime radicalC2×#64267C2
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#64265C2semidirect
3. C3×C4×#41⋊C3C2semidirect
4. #10593629C2semidirect
11. C2×C3⋊#226C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
16. C3×C4×#41D3semidirect
20. C4×#41⋊C3D3semidirect
63. C4×SL(2,𝔽3)C2×D3semidirect
77. C4×#41C3⋊D3semidirect
286. C3#85347semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C2×#642652
C3×C4×#41⋊C32
#105936292
#853473
#825273
#640844