Finite Group Databasebeta

← C3×#73097 #11920875

Subgroups — C3×#73097 #11920875

Order 1152.

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

normal subgroups only

This group has 233 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×C4⋊#226C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73097C311yesmaximal, characteristic, direct product factor
5C3×#1645—31nomaximal
6C2×C3×#226—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#835—41nomaximal
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)—21no—
12Q8×SL(2,𝔽3)C2×C311yescharacteristic
13C4⋊#226C2×C311yescharacteristic
14C3×#487—31no—
15C3×#475—31no—
16C3×#462—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×#226—41no—
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#1645—31noSylow
27C2×#226—21no—
28C2×C3×Q16—61no—
29C3×C4⋊Q8—31no—
30C3×C4⋊Q8—31no—
31C3×C4⋊Q8—61no—
32C3×C4×Q8—21no—
33C3×C4×Q8—31no—
34C3×C4×Q8C2×D311yescharacteristic
35C4×SL(2,𝔽3)—22no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C22×C311yescharacteristic
39C3×#106—61no—
40C3×C4⋊C8—31no—
41C3×C4⋊C8—31no—
42C3×#103—61no—
43C3×#103—62no—
44C3×#103—61no—
45C3×C4×C8—31no—
46#835—41no—
47C32×Q8—42no—
48C32×Q8—81no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3×Dic3—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic
53C3×C3⋊C8—81no—
54Q82C3×D311yescharacteristic
55#487—31no—
56#475—31no—
57#462—31no—
58#333—31no—
59Q8⋊C8—31no—
60Q8⋊C8—31no—
61C2×C3×Q8—32no—
62C2×C3×Q8—41no—
63C2×C3×Q8—61no—
64C2×C3×Q8—81no—
65C2×C3×Q8#6711yescharacteristic
66C2×C3×Q8S411yescharacteristic
67C2×SL(2,𝔽3)—21no—
68C2×SL(2,𝔽3)C3×D411yescharacteristic
69#226—41no—
70C3×Q16—121no—
71C3×Q16—121no—
72C2×C3×C8—61no—
73C2×C3×C8—61no—
74C2×C3×C8—61no—
75C3×C4⋊C4—31no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C42—31no—
82C3×C42—61no—
83C4⋊Dic3—41no—
84C2×C3⋊C8—41no—
85C22×C32—41no—
86C32×C4—41no—
87C32×C4—41no—
88C32×C4—82no—
89C3×Dic3—161no—
90C2×Q16—61no—
91C4⋊Q8—31no—
92C4⋊Q8—31no—
93C4⋊Q8—61no—
94C4×Q8—21no—
95C4×Q8—31no—
96C4×Q8C2×C3×D311yescharacteristic
97#106—61no—
98C4⋊C8—31no—
99C4⋊C8—31no—
100#103—61no—
101#103—62no—
102#103—61no—
103C4×C8—31no—
104C3×Q8—21no—
105C3×Q8—31no—
106C3×Q8—42no—
107C3×Q8—81no—
108C3×Q8—82no—
109C3×Q8—82no—
110C3×Q8—121no—
111C3×Q8—121no—
112C3×Q8—122no—
113C3×Q8#21511yescharacteristic
114C3×Q8#22612yescomplemented
115C2×C3×C4—21no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—61no—
120C2×C3×C4—81no—
121C2×C3×C4—82no—
122C2×C3×C4—81no—
123C2×C3×C4—121no—
124C2×C3×C4C2×S411yescharacteristic
125C2×Dic3—81no—
126SL(2,𝔽3)—21no—
127SL(2,𝔽3)C3×SD1611yescharacteristic
128C3×C8—121no—
129C3×C8—121no—
130C3×C8—121no—
131C3⋊C8—81no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×Q8—32no—
136C2×Q8—61no—
137C2×Q8C3×#6711yescharacteristic
138C2×Q8C3×S411yescharacteristic
139Q16—121no—
140Q16—121no—
141C2×C8—61no—
142C2×C8—61no—
143C2×C8—61no—
144C4⋊C4—31no—
145C4⋊C4—31no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C42—31no—
151C42—61no—
152C22×C3—41no—
153C22×C3—81no—
154C22×C3GL(2,ℤ/4)11yescharacteristic
155C3×C4—22no—
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—61no—
161C3×C4—82no—
162C3×C4—84no—
163C3×C4—81no—
164C3×C4—81no—
165C3×C4—121no—
166C3×C4—241no—
167C3×C4C2×#22611yescharacteristic
168C3×C4#98311yescharacteristic
169Dic3—161no—
170C32—41noSylow
171Q8—21no—
172Q8—31no—
173Q8—121no—
174Q8—121no—
175Q8—122no—
176Q8C3×#21511yescharacteristic
177Q8C3×#22612yescomplemented
178C2×C4—21no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—121no—
183C2×C4C2×C3×S411yescharacteristic
184C8—121no—
185C8—121no—
186C8—121no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—81no—
191C2×C3—81no—
192C2×C3—81no—
193C2×C3#541711yescharacteristic
194C2×C3#541911yescharacteristic
195C2×C3#542811yescharacteristic
196C22C3×GL(2,ℤ/4)11yescharacteristic
197C4—22no—
198C4—31no—
199C4—31no—
200C4—61no—
201C4—121no—
202C4—241no—
203C4C2×C3×#22611yescharacteristic
204C4C3×#98311yescharacteristic
205C3—41no—
206C3—81no—
207C3#7309711yescharacteristic, direct product factor
208C2C3×#541711yescharacteristic
209C2C3×#541911yescharacteristic
210C2C3×#542811yescharacteristic
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×SD16
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. #73097C3direct (N × H)
114. C3×Q8#226semidirect
177. Q8C3×#226semidirect
207. C3#73097direct (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×C4⋊#2262
C3×SL(2,𝔽3)⋊C82
#730973
C3×#16453
C3×#8354