Finite Group Databasebeta

← #11922319

Subgroups — #11922319

Order 1152.

585 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 59 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×C3×#984C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4#10593208C214yesmaximal, complemented
5#85347—31nomaximal
6#83968—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8#64267—22no—
9C2×C3⋊GL(2,𝔽3)C2212yes—
10C3×#984—22no—
11C2×C3×#226C2212yescomplemented
12#63864—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C2×#986—31no—
15C2×#986—61no—
16C2×#984D311yescharacteristic, complemented
17C2×C3×#137—31no—
18Dic3×#41D311yescharacteristic, complemented
19C2×#949—31no—
20C2×#947—31no—
21C4×#41⋊C3—31no—
22#5395—34no—
23#5225—31no—
24#5185—34no—
25#5165—34no—
26#5131—31no—
27#5118—31no—
28C2×C4⋊(C3⋊D3)—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3D412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊GL(2,𝔽3)—22no—
33C3×#226—22no—
34C3×C4×Dic3—41no—
35#3722—44no—
36#3040—91noSylow
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×Q8—31no—
39C22×D12—31no—
40C2×#41⋊C3—21no—
41C2×#41⋊C3C2×D311yescharacteristic
42#986—64no—
43#986—62no—
44#984—22no—
45C2×GL(2,𝔽3)—32no—
46C2×GL(2,𝔽3)—62no—
47C2×#226C2×D312yescomplemented
48C3×#137—62no—
49C3×#137—62no—
50C2×C3×Q16—32no—
51C2×C3×SD16—32no—
52C2×C3×M16—31no—
53#949—62no—
54#949—62no—
55#947—34no—
56#947—61no—
57#947—61no—
58Q8×Dic3C2×D312yes—
59C2×#215—32no—
60D4×Dic3—32no—
61C2×#213—32no—
62C2×(C2×D3)⋊C4—62no—
63#926—31no—
64C2×C4×Dic3—31no—
65C2×#208—31no—
66#875—121no—
67C4×SL(2,𝔽3)—32no—
68#835—34no—
69#821—34no—
70Dic3⋊C8—34no—
71C22×C3⋊D3—42no—
72C2×C32×C4—41no—
73C4⋊(C3⋊D3)—82no—
74C2×C3×Dic3—42no—
75C2×C3×Dic3—42no—
76C3×Dic6—82no—
77C3×SL(2,𝔽3)C2×D411yescharacteristic
78C2×#137—31no—
79C2×AGL(1,ℤ/8)—91no—
80C2×#124—91no—
81C4×#41—31no—
82#464—94no—
83#463—94no—
84#423—91no—
85#419—91no—
86#418—91no—
87C23×D3—61no—
88C3×#41—32no—
89C3×#41D1212yes—
90C2×C3×Q8—32no—
91C2×C3×Q8—62no—
92C2×C3×Q8—62no—
93C2×C3×Q8—61no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—61no—
98C22×Dic3—32no—
99C2×D12—32no—
100C2×D12—62no—
101C2×D12—62no—
102C2×D12—61no—
103C2×D12—121no—
104C2×D12—241no—
105C2×Dic6—41no—
106#41⋊C3—22no—
107#41⋊C3#6712yes—
108C2×SL(2,𝔽3)—21no—
109C2×SL(2,𝔽3)C22×D311yescharacteristic
110GL(2,𝔽3)—64no—
111GL(2,𝔽3)—62no—
112#226—22no—
113C3×Q16—62no—
114C3×Q16—62no—
115C3×SD16—64no—
116C3×M16—62no—
117C2×C3×C8—32no—
118C3×C42—121no—
119#217—61no—
120#215—62no—
121#215—62no—
122#213—64no—
123(C2×D3)⋊C4—64no—
124(C2×D3)⋊C4—64no—
125(C2×D3)⋊C4—124no—
126C4⋊Dic3—32no—
127C4×Dic3—31no—
128C4×Dic3—32no—
129C4×Dic3S411yescharacteristic
130#208—62no—
131C2×C3⋊C8—32no—
132C22×C32—41no—
133C2×C3⋊D3—82no—
134C2×C3⋊D3—84no—
135C32×C4—42no—
136C3×Dic3—82no—
137C3×Dic3—82no—
138C2×#41D3211yescharacteristic
139C22×Q8—31no—
140C22×D4—91no—
141#137—62no—
142#137—62no—
143AGL(1,ℤ/8)—182no—
144AGL(1,ℤ/8)—182no—
145C2×Q16—32no—
146C2×SD16—32no—
147C2×SD16—92no—
148C2×D8—92no—
149C2×M16—31no—
150C2×M16—91no—
151#124—94no—
152#124—181no—
153#124—181no—
154C4×Q8—32no—
155C4×D4—92no—
156#117—91no—
157C2×C22⋊C4—182no—
158C2×C42—91no—
159C4⋊C8—94no—
160#103—94no—
161#102—94no—
162C23×C3—31no—
163C22×D3—62no—
164C22×D3—122no—
165C22×D3—122no—
166C22×D3—124no—
167C22×D3—242no—
168C3×Q8—31no—
169C3×Q8—62no—
170C3×Q8—62no—
171C3×Q8—122no—
172C3×Q8C2×D1211yescharacteristic
173C3×D4—32no—
174C3×D4—61no—
175C2×C3×C4—31no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—41no—
179C2×C3×C4—62no—
180C2×C3×C4—81no—
181C2×C3×C4—122no—
182C2×C3×C4—122no—
183C2×C3×C4C2×S411yescharacteristic
184C2×Dic3—32no—
185C2×Dic3—42no—
186C2×Dic3—122no—
187C2×Dic3C2×S412yes—
188D12—62no—
189D12—62no—
190D12—122no—
191D12—242no—
192D12—244no—
193Dic6—82no—
194SL(2,𝔽3)—21no—
195SL(2,𝔽3)C2×#6711yescharacteristic
196C3×C8—62no—
197C3⋊C8—62no—
198C2×C32—41no—
199C2×C32—42no—
200C3⋊D3—162no—
201C24—181no—
202#41—32no—
203#41#62612yes—
204C2×Q8—32no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—61no—
208C2×Q8C2×D3211yescharacteristic
209C2×D4—31no—
210C2×D4—92no—
211C2×D4—182no—
212C2×D4—182no—
213C2×D4—181no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—92no—
217Q16—62no—
218Q16—62no—
219SD16—64no—
220SD16—182no—
221SD16—182no—
222D8—184no—
223M16—62no—
224M16—182no—
225C2×C8—32no—
226C2×C8—92no—
227C4⋊C4—92no—
228C22⋊C4—184no—
229C22⋊C4—184no—
230C22⋊C4—181no—
231C42—31no—
232C42—92no—
233C42—91no—
234C22×C3—32no—
235C22×C3—41no—
236C22×C3—62no—
237C22×C3—81no—
238C22×C3C22×S411yescharacteristic
239C2×D3—122no—
240C2×D3—124no—
241C2×D3—242no—
242C2×D3—244no—
243C2×D3—244no—
244C2×D3—244no—
245C2×D3—484no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—42no—
249C3×C4—82no—
250C3×C4—122no—
251C3×C4—242no—
252C3×C4C4⋊S412yes—
253Dic3—22no—
254Dic3—62no—
255Dic3—82no—
256C32—41noSylow
257C23—31no—
258C23—182no—
259C23—362no—
260C23—364no—
261Q8—31no—
262Q8—62no—
263Q8—62no—
264Q8—122no—
265Q8C2×#62611yescharacteristic
266D4—32no—
267D4—61no—
268D4—182no—
269D4—182no—
270D4—362no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—62no—
276C2×C4—92no—
277C2×C4—122no—
278C2×C4—362no—
279C2×C4D3×S411yescharacteristic
280C8—62no—
281C8—182no—
282C2×C3—42no—
283C2×C3—41no—
284C2×C3—62no—
285C2×C3—82no—
286C2×C3—81no—
287C2×C3C2×C4⋊S411yescharacteristic
288C2×C3#592312yes—
289D3—242no—
290D3—482no—
291D3—484no—
292C22—32no—
293C22—62no—
294C22—364no—
295C22—362no—
296C22—724no—
297C22C2×D3×S411yescharacteristic
298C4—31no—
299C4—31no—
300C4—62no—
301C4—122no—
302C4—182no—
303C4#6420712yes—
304C3—41no—
305C3—81no—
306C3#8534711yescharacteristic, complemented
307C2—62no—
308C2—722no—
309C2#1059645612yes—
310C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×#64267C2
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#984C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. #10593208C2semidirect
11. C2×C3×#226C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
16. C2×#984D3semidirect
18. Dic3×#41D3semidirect
47. C2×#226C2×D3semidirect
306. 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×C3×#9842
Dic3×#41⋊C32
#105932082
#853473
#839683
#638644