Finite Group Databasebeta

← #11922885

Subgroups — #11922885

Order 1152.

636 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 89 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×#64265C212yesmaximal
2C3×C4×#41⋊C3C211yesmaximal, characteristic
3C4⋊(C3⋊#226)C214yesmaximal, complemented
4#85336—33nomaximal
5#82533—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64265—24no—
8C2×C3⋊#226C2214yes—
9#64078—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C2×#984—36no—
12C3×C4×#41D311yescharacteristic
13C2×#951—32no—
14C2×#869—31no—
15C4×#41⋊C3D313yes—
16C4⋊#226—312no—
17#5031—34no—
18#5019—34no—
19#5004—32no—
20#4924—31no—
21C2×#634—42no—
22C3×#41⋊C3D412yes—
23C2×C3×SL(2,𝔽3)C2311yescharacteristic
24C3⋊#226—24no—
25C32×C42—41no—
26C4⋊(C3⋊Dic3)—44no—
27#3039—91noSylow
28C2×C3×#41C2×D311yescharacteristic
29C22×Dic6—32no—
30C2×#41⋊C3C2×D313yes—
31#984—612no—
32C2×#226—312no—
33C3×C4×Q8C2×D312yes—
34C3×C4×D4—32no—
35C3×#117—31no—
36C2×C3×C42—31no—
37#951—64no—
38#951—64no—
39C2×#216—34no—
40C2×#214—34no—
41C2×C4⋊Dic3—62no—
42C2×#208—32no—
43#869—34no—
44#869—61no—
45#869—61no—
46#869—123no—
47C4×SL(2,𝔽3)C2×D316yes—
48#810—38no—
49C4⋊(C3⋊C8)—34no—
50C2×C32×C4—41no—
51C2×C32×C4—42no—
52C2×C3⋊Dic3—44no—
53#634—84no—
54C3×SL(2,𝔽3)C2×D411yescharacteristic
55C2×#137—92no—
56C2×C4⋊Q8—91no—
57C4×#41C3⋊D311yescharacteristic
58#462—94no—
59#461—94no—
60#423—91no—
61#419—92no—
62C3×#41—32no—
63C3×#41D1212yes—
64C2×C3×Q8C22×D311yescharacteristic
65C2×C3×D4—31no—
66C22×C3×C4—31no—
67C22×C3×C4—32no—
68C22×Dic3—62no—
69C2×Dic6—34no—
70C2×Dic6—64no—
71C2×Dic6—64no—
72C2×Dic6—62no—
73C2×Dic6—126no—
74#41⋊C3D1216yes—
75C2×SL(2,𝔽3)C22×D313yes—
76#226—612no—
77C3×C4⋊C4—32no—
78C3×C22⋊C4—61no—
79C3×C42—32no—
80C3×C42—31no—
81C3×C42—43no—
82C3×C42S411yescharacteristic
83#216—64no—
84#216—64no—
85#214—68no—
86C4⋊Dic3—64no—
87C4⋊Dic3—64no—
88C4⋊Dic3—1212no—
89#208—64no—
90C2×C3⋊C8—34no—
91C22×C32—41no—
92C32×C4—42no—
93C32×C4—44no—
94C3⋊Dic3—84no—
95C2×#41C2×C3⋊D311yescharacteristic
96C22×Q8—92no—
97#137—184no—
98#137—184no—
99C2×Q16—94no—
100C2×SD16—94no—
101C2×M16—92no—
102C4⋊Q8—94no—
103C4⋊Q8—181no—
104C4⋊Q8—181no—
105C4×Q8C2×C3⋊D312yes—
106C4×D4—32no—
107#117—31no—
108C2×C4⋊C4—182no—
109C2×C42—31no—
110C4⋊C8—94no—
111#103—98no—
112C23×C3—31no—
113C3×Q8—31no—
114C3×Q8C2×D1211yescharacteristic
115C3×D4—32no—
116C3×D4—61no—
117C2×C3×C4—31no—
118C2×C3×C4—32no—
119C2×C3×C4—32no—
120C2×C3×C4—32no—
121C2×C3×C4—46no—
122C2×C3×C4—43no—
123C2×C3×C4—64no—
124C2×C3×C4C2×S411yescharacteristic
125C2×C3×C4C2×S412yes—
126C2×Dic3—64no—
127C2×Dic3—1212no—
128C2×Dic3—124no—
129Dic6—64no—
130Dic6—64no—
131Dic6—124no—
132Dic6—2412no—
133SL(2,𝔽3)C2×D1213yes—
134C3⋊C8—64no—
135C2×C32—41no—
136C2×C32—42no—
137#41—32no—
138#41C4⋊(C3⋊D3)12yes—
139C2×Q8—94no—
140C2×Q8—184no—
141C2×Q8—184no—
142C2×Q8—182no—
143C2×Q8C22×C3⋊D311yescharacteristic
144C2×D4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—182no—
148Q16—184no—
149Q16—184no—
150SD16—188no—
151M16—184no—
152C2×C8—94no—
153C4⋊C4—32no—
154C4⋊C4—184no—
155C4⋊C4—184no—
156C22⋊C4—61no—
157C42—32no—
158C42—31no—
159C42C3⋊S411yescharacteristic
160C22×C3—32no—
161C22×C3—43no—
162C22×C3—62no—
163C22×C3C22×S411yescharacteristic
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—46no—
167C3×C4—412no—
168C3×C4—62no—
169C3×C4C4⋊S412yes—
170C3×C4#98414yescomplemented
171Dic3—124no—
172Dic3—2412no—
173C32—41noSylow
174C23—31no—
175Q8—31no—
176Q8—184no—
177Q8—184no—
178Q8—364no—
179Q8C2×C4⋊(C3⋊D3)11yescharacteristic
180D4—32no—
181D4—61no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—64no—
187C2×C4—184no—
188C2×C4—364no—
189C2×C4C2×C3⋊S411yescharacteristic
190C2×C4C2×C3⋊S412yes—
191C8—184no—
192C2×C3—43no—
193C2×C3—46no—
194C2×C3—62no—
195C2×C3C2×C4⋊S411yescharacteristic
196C2×C3C2×#98412yes—
197C22—32no—
198C22—62no—
199C22C22×C3⋊S411yescharacteristic
200C4—31no—
201C4—31no—
202C4—62no—
203C4—364no—
204C4C4⋊(C3⋊S4)12yes—
205C4#6426514yescomplemented
206C3—43no—
207C3#8533611yescharacteristic, complemented
208C2—62no—
209C2C2×C4⋊(C3⋊S4)11yescharacteristic
210C2C2×#6426512yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×#41D3
socleC22×C3C22×S4
prime radicalC2×C3×#41⋊C3C22
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
3. C4⋊(C3⋊#226)C2semidirect
170. C3×C4#984semidirect
205. C4#64265semidirect
207. C3#85336semidirect

Maximal subgroups

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

subgroupindex
C2×#642652
C3×C4×#41⋊C32
C4⋊(C3⋊#226)2
#853363
#825333
#640784