Finite Group Databasebeta

← C3×#73098 #11920876

Subgroups — C3×#73098 #11920876

Order 1152.

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

normal subgroups only

This group has 237 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5391C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73098C311yesmaximal, characteristic, direct product factor
5C3×#1633—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#859—21no—
9C3×#835—41nomaximal
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)—21no—
12Q8×SL(2,𝔽3)C2×C311yescharacteristic
13#5391C2×C311yescharacteristic
14C3×#494—31no—
15C3×#477—31no—
16C3×#477—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic
18C3×#333—31no—
19C3×Q8⋊C8—31no—
20C3×Q8⋊C8—31no—
21C2×C32×Q8—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3×C4⋊Dic3—41no—
24C2×C3×C3⋊C8—41no—
25#1633—31noSylow
26C2×C3×Q16—62no—
27C3×C4⋊Q8—31no—
28C3×C4⋊Q8—31no—
29C3×C4⋊Q8—61no—
30C3×C4×Q8—21no—
31C3×C4×Q8—31no—
32C3×C4×Q8C2×D311yescharacteristic
33C4×SL(2,𝔽3)—22no—
34C4×SL(2,𝔽3)—21no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)C22×C311yescharacteristic
37#859—21no—
38C3×#107—61no—
39C3×#107—61no—
40C3×C4⋊C8—31no—
41C3×C4⋊C8—31no—
42C3×#103—61no—
43C3×#103—61no—
44C3×C4×C8—31no—
45#835—41no—
46C32×Q8—42no—
47C32×Q8—81no—
48C2×C32×C4—41no—
49C2×C32×C4—81no—
50C2×C3×Dic3—81no—
51C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
52C3×C3⋊C8—81no—
53Q82C3×D311yescharacteristic
54#494—31no—
55#477—31no—
56#477—31no—
57#333—31no—
58Q8⋊C8—31no—
59Q8⋊C8—31no—
60C2×C3×Q8—32no—
61C2×C3×Q8—41no—
62C2×C3×Q8—61no—
63C2×C3×Q8—81no—
64C2×C3×Q8#6711yescharacteristic
65C2×C3×Q8S411yescharacteristic
66C2×SL(2,𝔽3)—21no—
67C2×SL(2,𝔽3)C3×D411yescharacteristic
68C3×Q16—124no—
69C2×C3×C8—61no—
70C2×C3×C8—61no—
71C2×C3×C8—61no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—61no—
75C3×C4⋊C4—61no—
76C3×C4⋊C4—61no—
77C3×C4⋊C4—61no—
78C3×C42—31no—
79C3×C42—61no—
80C4⋊Dic3—41no—
81C2×C3⋊C8—41no—
82C22×C32—41no—
83C32×C4—41no—
84C32×C4—41no—
85C32×C4—82no—
86C3×Dic3—161no—
87C2×Q16—62no—
88C4⋊Q8—31no—
89C4⋊Q8—31no—
90C4⋊Q8—61no—
91C4×Q8—21no—
92C4×Q8—31no—
93C4×Q8C2×C3×D311yescharacteristic
94#107—61no—
95#107—61no—
96C4⋊C8—31no—
97C4⋊C8—31no—
98#103—61no—
99#103—61no—
100C4×C8—31no—
101C3×Q8—21no—
102C3×Q8—31no—
103C3×Q8—42no—
104C3×Q8—81no—
105C3×Q8—82no—
106C3×Q8—82no—
107C3×Q8—122no—
108C3×Q8—122no—
109C3×Q8#21611yescharacteristic
110C3×Q8#22612yes—
111C2×C3×C4—21no—
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—41no—
115C2×C3×C4—61no—
116C2×C3×C4—81no—
117C2×C3×C4—82no—
118C2×C3×C4—81no—
119C2×C3×C4—121no—
120C2×C3×C4C2×S411yescharacteristic
121C2×Dic3—81no—
122SL(2,𝔽3)—21no—
123SL(2,𝔽3)C3×Q1611yescharacteristic, complemented
124C3×C8—62no—
125C3×C8—121no—
126C3×C8—121no—
127C3⋊C8—81no—
128C2×C32—41no—
129C2×C32—41no—
130C2×C32—41no—
131C2×Q8—32no—
132C2×Q8—61no—
133C2×Q8C3×#6711yescharacteristic
134C2×Q8C3×S411yescharacteristic
135Q16—124no—
136C2×C8—61no—
137C2×C8—61no—
138C2×C8—61no—
139C4⋊C4—31no—
140C4⋊C4—31no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C42—31no—
146C42—61no—
147C22×C3—41no—
148C22×C3—81no—
149C22×C3GL(2,ℤ/4)11yescharacteristic
150C3×C4—22no—
151C3×C4—31no—
152C3×C4—31no—
153C3×C4—41no—
154C3×C4—41no—
155C3×C4—61no—
156C3×C4—82no—
157C3×C4—84no—
158C3×C4—81no—
159C3×C4—81no—
160C3×C4—121no—
161C3×C4—241no—
162C3×C4C2×#22611yescharacteristic
163C3×C4#98311yescharacteristic
164Dic3—161no—
165C32—41noSylow
166Q8—21no—
167Q8—31no—
168Q8—122no—
169Q8—122no—
170Q8C3×#21611yescharacteristic
171Q8C3×#22612yes—
172C2×C4—21no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—61no—
176C2×C4—121no—
177C2×C4C2×C3×S411yescharacteristic
178C8—62no—
179C8—121no—
180C8—121no—
181C2×C3—41no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—81no—
185C2×C3—81no—
186C2×C3—81no—
187C2×C3#541611yescharacteristic
188C2×C3#541911yescharacteristic
189C2×C3#542911yescharacteristic
190C22C3×GL(2,ℤ/4)11yescharacteristic
191C4—22no—
192C4—31no—
193C4—31no—
194C4—61no—
195C4—121no—
196C4—241no—
197C4C2×C3×#22611yescharacteristic
198C4C3×#98311yescharacteristic
199C3—41no—
200C3—81no—
201C3#7309811yescharacteristic, direct product factor
202C2C3×#541611yescharacteristic
203C2C3×#541911yescharacteristic
204C2C3×#542911yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×Q82D3
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C3×Q16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. #73098C3direct (N × H)
51. C3×SL(2,𝔽3)Q16semidirect
123. SL(2,𝔽3)C3×Q16semidirect
201. C3#73098direct (N × H)

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
C3×#53912
C3×SL(2,𝔽3)⋊C82
#730983
C3×#16333
C3×#8354