Finite Group Databasebeta

← C3×#73009 #11920787

Subgroups — C3×#73009 #11920787

Order 1152.

201 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 44 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×C4×#226C211yesmaximal, characteristic
2C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73009C311yesmaximal, characteristic, direct product factor
5C3×#1587—31nomaximal
6C2×C3×#226C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#859C411yescharacteristic
9C3×Dic3⋊C8—41nomaximal
10C4×#226C2×C311yescharacteristic
11C3×#446—31no—
12C3×C8×Q8D311yescharacteristic
13C3×C4×Q16—31no—
14C8×SL(2,𝔽3)—21no—
15C8×SL(2,𝔽3)C2×C311yescharacteristic
16SL(2,𝔽3)⋊C8C2×C311yescharacteristic
17C3×#334—31no—
18C3×Q8⋊C8—31no—
19C3×Q8⋊C8—31no—
20C3×#322—31no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3×#226—21no—
23C2×C32×C8—41no—
24C3×C4×Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#1587—31noSylow
27C2×#226C3×C411yescharacteristic
28C2×C3×Q16—31no—
29C3×C4×Q8—31no—
30C3×C4×Q8C2×D311yescharacteristic
31C4×SL(2,𝔽3)—21no—
32C4×SL(2,𝔽3)C22×C311yescharacteristic
33#859C3×C411yescharacteristic
34C3×#107—31no—
35C3×C4⋊C8—31no—
36C3×C4⋊C8—31no—
37C3×C4⋊C8—61no—
38C3×#103—31no—
39C3×#103—31no—
40C3×#97—61no—
41C3×C4×C8—31no—
42C3×C4×C8—31no—
43C3×C4×C8—31no—
44Dic3⋊C8—41no—
45C2×C32×C4—41no—
46C2×C3×Dic3—41no—
47C2×C3×Dic3—41no—
48C3×SL(2,𝔽3)M1611yescharacteristic
49C32×C8—81no—
50C3×C3⋊C8—81no—
51#446—31no—
52C8×Q8C3×D311yescharacteristic
53C4×Q16—31no—
54#334—31no—
55Q8⋊C8—31no—
56Q8⋊C8—31no—
57#322—31no—
58C2×C3×Q8—31no—
59C2×C3×Q8C4×D311yescharacteristic
60C2×SL(2,𝔽3)—21no—
61C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
62#226—21no—
63C3×Q16—61no—
64C3×Q16—61no—
65C2×C3×C8—31no—
66C2×C3×C8—31no—
67C2×C3×C8—31no—
68C2×C3×C8—41no—
69C2×C3×C8—61no—
70C2×C3×C8—61no—
71C2×C3×C8—81no—
72C2×C3×C8S411yescharacteristic
73C3×C4⋊C4—31no—
74C3×C4⋊C4—31no—
75C3×C4⋊C4—61no—
76C3×C42—31no—
77C3×C42—61no—
78C4×Dic3—41no—
79C2×C3⋊C8—41no—
80C22×C32—41no—
81C32×C4—41no—
82C32×C4—41no—
83C3×Dic3—42no—
84C3×Dic3—81no—
85C2×Q16—31no—
86C4×Q8—31no—
87C4×Q8C2×C3×D311yescharacteristic
88#107—31no—
89C4⋊C8—31no—
90C4⋊C8—31no—
91C4⋊C8—61no—
92#103—31no—
93#103—31no—
94#97—61no—
95C4×C8—31no—
96C4×C8—31no—
97C4×C8—31no—
98C3×Q8—31no—
99C3×Q8—61no—
100C3×Q8—61no—
101C3×Q8ΓL(1,𝔽25)11yescharacteristic
102C2×C3×C4—31no—
103C2×C3×C4—31no—
104C2×C3×C4—41no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—81no—
108C2×C3×C4C2×S411yescharacteristic
109C2×Dic3—41no—
110C2×Dic3—41no—
111SL(2,𝔽3)—21no—
112SL(2,𝔽3)C3×M1611yescharacteristic
113C3×C8—21no—
114C3×C8—61no—
115C3×C8—61no—
116C3×C8—62no—
117C3×C8—61no—
118C3×C8—81no—
119C3×C8—82no—
120C3×C8—121no—
121C3⋊C8—81no—
122C2×C32—41no—
123C2×C32—41no—
124C2×C32—41no—
125C2×Q8—31no—
126C2×Q8C3×C4×D311yescharacteristic
127Q16—61no—
128Q16—61no—
129C2×C8—31no—
130C2×C8—31no—
131C2×C8—31no—
132C2×C8—61no—
133C2×C8—61no—
134C2×C8C3×S411yescharacteristic
135C4⋊C4—31no—
136C4⋊C4—31no—
137C4⋊C4—61no—
138C42—31no—
139C42—61no—
140C22×C3—41no—
141C22×C3—81no—
142C22×C3C4×S411yescharacteristic
143C3×C4—31no—
144C3×C4—31no—
145C3×C4—41no—
146C3×C4—41no—
147C3×C4—61no—
148C3×C4—81no—
149C3×C4—81no—
150C3×C4—121no—
151C3×C4—121no—
152C3×C4#98311yescharacteristic
153C3×C4#98411yescharacteristic
154Dic3—42no—
155Dic3—81no—
156C32—41noSylow
157Q8—31no—
158Q8—61no—
159Q8—61no—
160Q8C3×ΓL(1,𝔽25)11yescharacteristic
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4C2×C3×S411yescharacteristic
166C8—21no—
167C8—61no—
168C8—61no—
169C8—62no—
170C8—61no—
171C8—121no—
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—81no—
176C2×C3—81no—
177C2×C3—81no—
178C2×C3#538811yescharacteristic
179C2×C3#540011yescharacteristic
180C2×C3#540411yescharacteristic
181C22C3×C4×S411yescharacteristic
182C4—31no—
183C4—31no—
184C4—61no—
185C4—121no—
186C4—121no—
187C4C3×#98311yescharacteristic
188C4C3×#98411yescharacteristic
189C3—41no—
190C3—81no—
191C3#7300911yescharacteristic, direct product factor
192C2C3×#538811yescharacteristic
193C2C3×#540011yescharacteristic
194C2C3×#540411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC3×C8×Q8D3
socleC22×C3C4×S4
prime radicalC2×C3×SL(2,𝔽3)C2×C4
cosocleC4×SL(2,𝔽3)C22×C3
hypercenterC2×C3×C8S4
hypocenterSL(2,𝔽3)C3×M16
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. #73009C3direct (N × H)
191. C3#73009direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#2262
C3×C8×SL(2,𝔽3)2
C3×SL(2,𝔽3)⋊C82
#730093
C3×#15873
C3×Dic3⋊C84