Finite Group Databasebeta

← #11922913

Subgroups — #11922913

Order 1152.

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

normal subgroups only

This group has 480 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
1C3×#520⋊C3C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic
3#10593658C211yesmaximal, characteristic
4#10593625C212yesmaximal, complemented
5#10593623C212yesmaximal, complemented
6#85363—33nomaximal
7#82559—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64099—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3⋊#859C2211yescharacteristic
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#520D311yescharacteristic
15#5519—31no—
16#520⋊C3D313yes—
17#5426—33no—
18#5424—33no—
19#5391—36no—
20#5389—36no—
21#5233—31no—
22#5232—31no—
23#5022—32no—
24#5021—32no—
25#5009—32no—
26#5007—32no—
27#4999—31no—
28#4964—31no—
29C2×#807—31no—
30C3×#41⋊C3Q812yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C32×C4⋊C4—41no—
33C4⋊(C3⋊Dic3)—42no—
34C4⋊(C3⋊Dic3)—41no—
35#3750—42no—
36C4×C3⋊Dic3—41no—
37#3083—91noSylow
38C2×C3×#41C2×D311yescharacteristic
39C2×#41⋊C3C2×D313yes—
40C3×C4×Q8C2×D312yes—
41C3×C4×D4—32no—
42C3×#117—31no—
43C2×C3×C4⋊C4—31no—
44#926—31no—
45C2×C4⋊Dic3—31no—
46C2×#208—31no—
47C22×C3⋊C8—31no—
48#890—61no—
49#890—123no—
50#888—61no—
51#878—62no—
52#870—32no—
53#869—32no—
54C4×SL(2,𝔽3)C2×D316yes—
55#859—33no—
56#859—36no—
57#859—33no—
58#835—32no—
59#835—31no—
60#835—31no—
61#832—32no—
62#832—31no—
63#832—31no—
64#808—32no—
65#807—32no—
66#807—32no—
67#807—32no—
68C4⋊(C3⋊C8)—32no—
69C4⋊(C3⋊C8)—32no—
70C2×C32×C4—41no—
71C2×C32×C4—42no—
72C2×C3⋊Dic3—41no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—42no—
75C3×SL(2,𝔽3)C2×Q811yescharacteristic
76#544—91no—
77#520C3⋊D311yescharacteristic
78#479—92no—
79#478—92no—
80#477—92no—
81#474—92no—
82#428—91no—
83C2×#107—91no—
84#424—91no—
85#417—91no—
86#416—91no—
87C3×#41—32no—
88C3×#41Dic612yes—
89C2×C3×Q8C22×D311yescharacteristic
90C2×C3×D4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—61no—
93C22×Dic3—61no—
94C2×Dic6—62no—
95#41⋊C3Dic616yescomplemented
96C2×SL(2,𝔽3)C22×D313yes—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—31no—
99C3×C4⋊C4—43no—
100C3×C4⋊C4S411yescharacteristic
101C3×C22⋊C4—32no—
102C3×C42—32no—
103#217—61no—
104C4⋊Dic3—31no—
105C4⋊Dic3—32no—
106C4⋊Dic3—31no—
107C4⋊Dic3—62no—
108C4⋊Dic3—61no—
109C4⋊Dic3—126no—
110C4⋊Dic3—123no—
111Dic3⋊C4—62no—
112Dic3⋊C4—62no—
113Dic3⋊C4—126no—
114C4×Dic3—61no—
115C4×Dic3—123no—
116#208—62no—
117C2×C3⋊C8—31no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—62no—
121C22×C32—41no—
122C32×C4—42no—
123C32×C4—82no—
124C3⋊Dic3—81no—
125C3⋊Dic3—82no—
126C3⋊Dic3—81no—
127C2×#41C2×C3⋊D311yescharacteristic
128C2×M16—91no—
129C22×C8—91no—
130C4⋊Q8—92no—
131C4⋊Q8—181no—
132#125—92no—
133#125—181no—
134C22⋊Q8—182no—
135C4×Q8C2×C3⋊D312yes—
136C4×D4—32no—
137#117—31no—
138#117—91no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—91no—
141#107—92no—
142#107—92no—
143#107—92no—
144#106—92no—
145C4⋊C8—92no—
146C4⋊C8—92no—
147#103—91no—
148#103—91no—
149#103—92no—
150#102—91no—
151#102—91no—
152#102—92no—
153C23×C3—31no—
154C3×Q8—31no—
155C3×Q8C2×Dic611yescharacteristic
156C3×D4—32no—
157C3×D4—61no—
158C2×C3×C4—31no—
159C2×C3×C4—32no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—43no—
163C2×C3×C4—46no—
164C2×C3×C4—241no—
165C2×C3×C4C2×S412yes—
166C2×C3×C4C2×S411yescharacteristic
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—123no—
171C2×Dic3—123no—
172C2×Dic3—126no—
173C2×Dic3—241no—
174Dic6—242no—
175SL(2,𝔽3)C2×Dic613yes—
176C3⋊C8—61no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182#41—32no—
183#41#63412yes—
184C2×Q8—182no—
185C2×Q8C22×C3⋊D311yescharacteristic
186C2×D4—31no—
187C22×C4—31no—
188C22×C4—61no—
189C22×C4—181no—
190M16—182no—
191C2×C8—92no—
192C2×C8—91no—
193C2×C8—91no—
194C2×C8—182no—
195C4⋊C4—32no—
196C4⋊C4—31no—
197C4⋊C4—91no—
198C4⋊C4—92no—
199C4⋊C4—91no—
200C4⋊C4—182no—
201C4⋊C4—181no—
202C4⋊C4—182no—
203C4⋊C4—182no—
204C4⋊C4C3⋊S411yescharacteristic
205C22⋊C4—32no—
206C22⋊C4—181no—
207C42—32no—
208C42—181no—
209C22×C3—32no—
210C22×C3—43no—
211C22×C3—61no—
212C22×C3—61no—
213C22×C3C22×S411yescharacteristic
214C3×C4—22no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—46no—
218C3×C4—62no—
219C3×C4—86no—
220C3×C4A4⋊Q812yes—
221Dic3—121no—
222Dic3—122no—
223Dic3—121no—
224Dic3—246no—
225Dic3—243no—
226Dic3—243no—
227C32—41noSylow
228C23—31no—
229Q8—31no—
230Q8—722no—
231Q8C2×#63411yescharacteristic
232D4—32no—
233D4—61no—
234C2×C4—31no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—181no—
239C2×C4—181no—
240C2×C4—182no—
241C2×C4—241no—
242C2×C4—721no—
243C2×C4C2×C3⋊S412yes—
244C2×C4C2×C3⋊S411yescharacteristic
245C8—181no—
246C8—181no—
247C8—182no—
248C2×C3—43no—
249C2×C3—43no—
250C2×C3—43no—
251C2×C3—62no—
252C2×C3C2×A4⋊Q811yescharacteristic
253C2×C3#592511yescharacteristic
254C2×C3#592711yescharacteristic
255C22—32no—
256C22—61no—
257C22—61no—
258C22C22×C3⋊S411yescharacteristic
259C4—22no—
260C4—31no—
261C4—31no—
262C4—62no—
263C4—362no—
264C4—361no—
265C4—361no—
266C4#6425912yes—
267C3—43no—
268C3#8536311yescharacteristic, complemented
269C2—62no—
270C2C2×#6425911yescharacteristic
271C2#1059658811yescharacteristic
272C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#520D3
socleC22×C3C22×S4
prime radicalC2×C3×#41⋊C3C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
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
4. #10593625C2semidirect
5. #10593623C2semidirect
30. C3×#41⋊C3Q8semidirect
95. #41⋊C3Dic6semidirect
268. C3#85363semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#105936602
#105936582
#105936252
#105936232
#853633
#825593
#640994