Finite Group Databasebeta

← #11921053

Subgroups — #11921053

Order 1152.

270 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 49 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2C4⋊(C3⋊#226)C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73097—33nomaximal
5#71941—31nomaximal
6C2×C3⋊#226—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9#63662—41nomaximal
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)D313yes—
12C4⋊#226—33no—
13#5079—31no—
14#5031—31no—
15#5020—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4546—31no—
18#4535—31no—
19#4535—31no—
20C2×C32×Q8—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊#226—41no—
23C4⋊(C3⋊Dic3)—41no—
24C2×C3⋊(C3⋊C8)—41no—
25#1645—91noSylow
26C2×#226—63no—
27C3×C4⋊Q8—31no—
28C3×C4⋊Q8—61no—
29C3×C4×Q8—21no—
30C3×C4×Q8—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#216—61no—
33#869—31no—
34C4×SL(2,𝔽3)—23no—
35C4×SL(2,𝔽3)C2×D313yes—
36#835—62no—
37#835—61no—
38#835—123no—
39#810—61no—
40#808—61no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C4×C3⋊C8—31no—
44C32×Q8—42no—
45C32×Q8—81no—
46C2×C32×C4—41no—
47C2×C32×C4—81no—
48C2×C3⋊Dic3—81no—
49C3×SL(2,𝔽3)SD1611yescharacteristic
50C3⋊(C3⋊C8)—81no—
51Q82C3⋊D311yescharacteristic
52#487—91no—
53#475—91no—
54#462—91no—
55#333—91no—
56Q8⋊C8—91no—
57Q8⋊C8—91no—
58C2×C3×Q8—32no—
59C2×C3×Q8—43no—
60C2×C3×Q8#6711yescharacteristic
61C2×C3×Q8S411yescharacteristic
62C2×Dic6—61no—
63C2×SL(2,𝔽3)#6713yes—
64#226—123no—
65C3×C4⋊C4—31no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—61no—
68C3×C4⋊C4—61no—
69C3×C42—31no—
70C3×C42—61no—
71#216—121no—
72#216—121no—
73C4⋊Dic3—61no—
74C4⋊Dic3—61no—
75C4⋊Dic3—123no—
76C2×C3⋊C8—61no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8—61no—
79C2×C3⋊C8—123no—
80C22×C32—41no—
81C32×C4—41no—
82C32×C4—41no—
83C32×C4—82no—
84C3⋊Dic3—161no—
85C2×Q16—181no—
86C4⋊Q8—31no—
87C4⋊Q8—61no—
88C4⋊Q8—91no—
89C4×Q8—21no—
90C4×Q8—31no—
91C4×Q8C2×C3⋊D311yescharacteristic
92#106—181no—
93C4⋊C8—91no—
94C4⋊C8—91no—
95#103—181no—
96#103—182no—
97#103—181no—
98C4×C8—91no—
99C3×Q8—21no—
100C3×Q8—31no—
101C3×Q8—46no—
102C3×Q8—83no—
103C3×Q8—122no—
104C3×Q8#21511yescharacteristic
105C3×Q8#22612yescomplemented
106C2×C3×C4—21no—
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—43no—
110C2×C3×C4—61no—
111C2×C3×C4—83no—
112C2×C3×C4C2×S411yescharacteristic
113C2×Dic3—121no—
114C2×Dic3—243no—
115Dic6—121no—
116Dic6—121no—
117SL(2,𝔽3)#21513yes—
118C3⋊C8—121no—
119C3⋊C8—121no—
120C3⋊C8—121no—
121C3⋊C8—243no—
122C2×C32—41no—
123C2×C32—41no—
124C2×C32—41no—
125C2×Q8—32no—
126C2×Q8—181no—
127C2×Q8#63811yescharacteristic
128C2×Q8C3⋊S411yescharacteristic
129Q16—361no—
130Q16—361no—
131C2×C8—181no—
132C2×C8—181no—
133C2×C8—181no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—181no—
139C4⋊C4—181no—
140C42—31no—
141C42—61no—
142C22×C3—43no—
143C22×C3GL(2,ℤ/4)11yescharacteristic
144C3×C4—22no—
145C3×C4—31no—
146C3×C4—31no—
147C3×C4—43no—
148C3×C4—43no—
149C3×C4—61no—
150C3×C4—86no—
151C3×C4—121no—
152C3×C4C2×#22611yescharacteristic
153C3×C4#98311yescharacteristic
154Dic3—241no—
155Dic3—483no—
156C32—41noSylow
157Q8—21no—
158Q8—31no—
159Q8—122no—
160Q8—361no—
161Q8—361no—
162Q8#375511yescharacteristic
163Q8C3⋊#22612yescomplemented
164C2×C4—21no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—61no—
168C2×C4—361no—
169C2×C4C2×C3⋊S411yescharacteristic
170C8—361no—
171C8—361no—
172C8—361no—
173C2×C3—43no—
174C2×C3—43no—
175C2×C3—43no—
176C2×C3#541711yescharacteristic
177C2×C3#541911yescharacteristic
178C2×C3#542811yescharacteristic
179C22#6426911yescharacteristic
180C4—22no—
181C4—31no—
182C4—31no—
183C4—61no—
184C4—121no—
185C4—721no—
186C4C2×C3⋊#22611yescharacteristic
187C4#6426411yescharacteristic
188C3—43no—
189C3#7309711yescharacteristic, complemented
190C2#1059365111yescharacteristic
191C2#1059365311yescharacteristic
192C2#1059366211yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×Q82D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
105. C3×Q8#226semidirect
163. Q8C3⋊#226semidirect
189. C3#73097semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C4⋊(C3⋊#226)2
C3⋊(SL(2,𝔽3)⋊C8)2
#730973
#719413
#636624