Finite Group Databasebeta

← #11921018

Subgroups — #11921018

Order 1152.

460 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 93 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×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C212yesmaximal
3#73063—33nomaximal
4#71631—31nomaximal
5C22×C3×SL(2,𝔽3)C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C412yescomplemented
7C3⋊#859C414yescomplemented
8#63658—41nomaximal
9C2×C3×C4×Q8D311yescharacteristic
10C2×C4×SL(2,𝔽3)D313yes—
11C2×#859—36no—
12C2×#835—32no—
13C4×C4⋊Dic3—31no—
14C2×#803—31no—
15#4529—32no—
16C22×C32×C4—41no—
17C22×C3⋊Dic3—42no—
18C2×C3×SL(2,𝔽3)C2×C412yes—
19C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
20#1763—91noSylow
21C22×C3×Q8C2×D311yescharacteristic
22C22×SL(2,𝔽3)C2×D313yes—
23C3×C4×Q8—32no—
24C3×C4×Q8Dic312yescomplemented
25C2×C3×C4⋊C4—31no—
26C2×C3×C42—31no—
27C2×C4⋊Dic3—32no—
28C2×C4×Dic3—61no—
29C22×C3⋊C8—32no—
30C4×SL(2,𝔽3)Dic316yescomplemented
31#859—312no—
32#835—34no—
33#835—34no—
34C3⋊#95—61no—
35C3⋊#95—123no—
36#803—62no—
37C23×C32—41no—
38C2×C32×C4—42no—
39C2×C32×C4—82no—
40C2×C3⋊Dic3—44no—
41C2×C3⋊Dic3—82no—
42C2×C3⋊Dic3—82no—
43C3×SL(2,𝔽3)C4211yescharacteristic, complemented
44C2×C4×Q8C3⋊D311yescharacteristic
45C2×#103—92no—
46C2×#97—91no—
47C4×C4⋊C4—91no—
48#340—92no—
49C2×C3×Q8—32no—
50C2×C3×Q8—31no—
51C2×C3×Q8C4×D312yes—
52C2×C3×Q8C2×Dic311yescharacteristic
53C22×C3×C4—31no—
54C22×C3×C4—31no—
55C22×C3×C4—43no—
56C22×C3×C4S411yescharacteristic
57C22×Dic3—62no—
58C22×Dic3—126no—
59C2×SL(2,𝔽3)C4×D316yes—
60C2×SL(2,𝔽3)C2×Dic313yes—
61C3×C4⋊C4—32no—
62C3×C4⋊C4—61no—
63C3×C42—32no—
64C3×C42—32no—
65C4⋊Dic3—34no—
66C4⋊Dic3—62no—
67C4×Dic3—122no—
68C2×C3⋊C8—34no—
69C2×C3⋊C8—62no—
70C2×C3⋊C8—62no—
71C22×C32—42no—
72C22×C32—42no—
73C22×C32—41no—
74C22×C32—41no—
75C22×C32—41no—
76C32×C4—82no—
77C3⋊Dic3—84no—
78C22×Q8C2×C3⋊D311yescharacteristic
79C22×C8—92no—
80C4×Q8—32no—
81C4×Q8C3⋊Dic312yescomplemented
82C2×C4⋊C4—31no—
83C2×C4⋊C4—92no—
84C2×C42—31no—
85C2×C42—181no—
86#103—94no—
87#103—94no—
88#97—182no—
89#95—181no—
90C23×C3—43no—
91C23×C3C2×S411yescharacteristic
92C3×Q8—32no—
93C3×Q8—31no—
94C3×Q8—61no—
95C3×Q8C4×Dic311yescharacteristic
96C2×C3×C4—22no—
97C2×C3×C4—32no—
98C2×C3×C4—32no—
99C2×C3×C4—31no—
100C2×C3×C4—31no—
101C2×C3×C4—32no—
102C2×C3×C4—46no—
103C2×C3×C4—61no—
104C2×C3×C4—61no—
105C2×C3×C4—86no—
106C2×C3×C4SL(2,ℤ/4)12yes—
107C2×Dic3—64no—
108C2×Dic3—1212no—
109C2×Dic3—122no—
110C2×Dic3—122no—
111C2×Dic3—246no—
112C2×Dic3—246no—
113SL(2,𝔽3)C4×Dic313yescomplemented
114C3⋊C8—64no—
115C2×C32—42no—
116C2×C32—42no—
117C2×C32—41no—
118C2×C32—41no—
119C2×C32—41no—
120C2×Q8—32no—
121C2×Q8—31no—
122C2×Q8C4×C3⋊D312yes—
123C2×Q8C2×C3⋊Dic311yescharacteristic
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—182no—
127C22×C4C3⋊S411yescharacteristic
128C2×C8—94no—
129C2×C8—182no—
130C2×C8—182no—
131C4⋊C4—32no—
132C4⋊C4—61no—
133C4⋊C4—94no—
134C4⋊C4—182no—
135C42—32no—
136C42—32no—
137C42—362no—
138C22×C3—46no—
139C22×C3—46no—
140C22×C3—43no—
141C22×C3—43no—
142C22×C3—43no—
143C22×C3C4×S412yes—
144C22×C3#98312yes—
145C22×C3#98411yescharacteristic
146C22×C3#98611yescharacteristic
147C22×C3C2×SL(2,ℤ/4)11yescharacteristic
148C3×C4—22no—
149C3×C4—32no—
150C3×C4—31no—
151C3×C4—31no—
152C3×C4—62no—
153C3×C4—86no—
154Dic3—124no—
155Dic3—2412no—
156C32—41noSylow
157C23C2×C3⋊S411yescharacteristic
158Q8—32no—
159Q8—31no—
160Q8—61no—
161Q8C4×C3⋊Dic311yescharacteristic
162C2×C4—22no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—32no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—184no—
171C2×C4—362no—
172C2×C4—362no—
173C2×C4C3⋊SL(2,ℤ/4)12yes—
174C8—184no—
175C2×C3—46no—
176C2×C3—46no—
177C2×C3—43no—
178C2×C3—43no—
179C2×C3—43no—
180C2×C3#538812yes—
181C2×C3#539412yes—
182C2×C3C4×SL(2,ℤ/4)11yescharacteristic
183C2×C3#542011yescharacteristic
184C2×C3#542611yescharacteristic
185C22C4×C3⋊S412yes—
186C22#6426412yes—
187C22#6426511yescharacteristic
188C22#6426711yescharacteristic
189C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
190C4—22no—
191C4—32no—
192C4—31no—
193C4—31no—
194C4—62no—
195C4—364no—
196C3—43no—
197C3#7306311yescharacteristic, complemented
198C2#1059362212yes—
199C2#1059362812yes—
200C2C4×C3⋊SL(2,ℤ/4)11yescharacteristic
201C2#1059365411yescharacteristic
202C2#1059366011yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C42
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radical, cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. C3×C4×SL(2,𝔽3)C4semidirect
7. C3⋊#859C4semidirect
24. C3×C4×Q8Dic3semidirect
30. C4×SL(2,𝔽3)Dic3semidirect
43. C3×SL(2,𝔽3)C42semidirect
81. C4×Q8C3⋊Dic3semidirect
113. SL(2,𝔽3)C4×Dic3semidirect
197. C3#73063semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
#730633
#716313
#636584