Finite Group Databasebeta

← C3×#73154 #11920932

Subgroups — C3×#73154 #11920932

Order 1152.

423 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C3×#5423C212yesmaximal, complemented
3#73154C311yesmaximal, characteristic, direct product factor
4C3×#1796—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C2211yescharacteristic
7C22×C3×SL(2,𝔽3)C412yescomplemented
8C3×GU(2,𝔽3)—22no—
9C3×#833—41nomaximal
10C2×C3×2−1+4D311yescharacteristic
11C2×2−1+4⋊C3—21no—
12C2×2−1+4⋊C3C2×C311yescharacteristic
13#5423C2×C312yescomplemented
14C3×#421—32no—
15C3×#419—32no—
16C3×#413—31no—
17C3×#410—31no—
18C2×C32×D4—41no—
19C3×#41⋊C3D412yes—
20C2×C3×SL(2,𝔽3)—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3×#208—42no—
23#1796—31noSylow
24C3×2−1+4—22no—
25C3×2−1+4—62no—
26C2×C3×#41—31no—
27C2×C3×#41—32no—
28C2×C3×#41C2×D311yescharacteristic
29C22×C3×Q8—31no—
30C22×C3×Q8Dic312yes—
312−1+4⋊C3—24no—
322−1+4⋊C3—22no—
33C2×#41⋊C3—21no—
34C2×#41⋊C3C22×C311yescharacteristic
35C22×SL(2,𝔽3)—22no—
36C22×SL(2,𝔽3)C3×C412yescomplemented
37C2×C3×M16—32no—
38C3×#117—32no—
39GU(2,𝔽3)—22no—
40C3×#104—62no—
41C3×#104—62no—
42C3×#103—64no—
43C3×#101—61no—
44C3×#100—61no—
45C3×#99—62no—
46#833—41no—
47C23×C32—42no—
48C32×D4—82no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C22⋊C411yescharacteristic
51C3×C3⋊C8—82no—
52C2×2−1+4C3×D311yescharacteristic
53#421—32no—
54#419—32no—
55#413—31no—
56#410—31no—
57C3×#41—32no—
58C3×#41—62no—
59C3×#41—62no—
60C3×#41—64no—
61C3×#41—64no—
62C3×#41#6712yes—
63C2×C3×Q8—22no—
64C2×C3×Q8—31no—
65C2×C3×Q8—32no—
66C2×C3×Q8—34no—
67C2×C3×Q8—62no—
68C2×C3×Q8—62no—
69C2×C3×Q8C2×Dic311yescharacteristic
70C2×C3×D4—31no—
71C2×C3×D4—32no—
72C2×C3×D4—41no—
73C2×C3×D4—81no—
74C2×C3×D4S411yescharacteristic
75C22×C3×C4—31no—
76C22×C3×C4—32no—
77C22×C3×C4—61no—
78#41⋊C3—22no—
79#41⋊C3C3×D412yes—
80C2×SL(2,𝔽3)—24no—
81C2×SL(2,𝔽3)—22no—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
84C3×M16—62no—
85C3×M16—62no—
86C2×C3×C8—62no—
87C3×C4⋊C4—62no—
88C3×C22⋊C4—62no—
89C3×C42—62no—
90#208—42no—
91C22×C32—41no—
92C22×C32—82no—
93C22×C32—162no—
94C32×C4—42no—
952−1+4—22no—
962−1+4—62no—
97C2×#41—31no—
98C2×#41—32no—
99C2×#41C2×C3×D311yescharacteristic
100C22×Q8—31no—
101C22×Q8C3×Dic312yes—
102C2×M16—32no—
103#117—32no—
104#104—62no—
105#104—62no—
106#103—64no—
107#101—61no—
108#100—61no—
109#99—62no—
110C23×C3—31no—
111C23×C3—42no—
112C23×C3—82no—
113C23×C3SL(2,ℤ/4)12yes—
114C3×Q8—31no—
115C3×Q8—34no—
116C3×Q8—62no—
117C3×Q8—62no—
118C3×Q8#21711yescharacteristic
119C3×D4—22no—
120C3×D4—32no—
121C3×D4—61no—
122C3×D4—64no—
123C3×D4—82no—
124C3×D4—84no—
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—41no—
130C2×C3×C4—62no—
131C2×C3×C4—62no—
132C2×C3×C4—64no—
133C2×C3×C4—81no—
134C2×C3×C4—122no—
135C2×C3×C4C2×S411yescharacteristic
136SL(2,𝔽3)—21no—
137SL(2,𝔽3)C3×C22⋊C411yescharacteristic
138C3×C8—122no—
139C3⋊C8—82no—
140C2×C32—41no—
141C2×C32—81no—
142C2×C32—162no—
143#41—32no—
144#41—62no—
145#41—62no—
146#41—64no—
147#41—64no—
148#41C3×#6712yes—
149C2×Q8—22no—
150C2×Q8—31no—
151C2×Q8—32no—
152C2×Q8—34no—
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8C2×C3×Dic311yescharacteristic
156C2×D4—31no—
157C2×D4—32no—
158C2×D4C3×S411yescharacteristic
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—61no—
162M16—62no—
163M16—62no—
164C2×C8—62no—
165C4⋊C4—62no—
166C22⋊C4—62no—
167C42—62no—
168C22×C3—22no—
169C22×C3—32no—
170C22×C3—42no—
171C22×C3—41no—
172C22×C3—82no—
173C22×C3—84no—
174C22×C3—81no—
175C22×C3—121no—
176C22×C3—162no—
177C22×C3—164no—
178C22×C3C2×SL(2,ℤ/4)11yescharacteristic
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—42no—
182C3×C4—62no—
183C3×C4—82no—
184C3×C4—242no—
185C3×C4GL(2,ℤ/4)12yes—
186C32—41noSylow
187C23—31no—
188C23C3×SL(2,ℤ/4)12yes—
189Q8—31no—
190Q8—34no—
191Q8—62no—
192Q8—62no—
193Q8C3×#21711yescharacteristic
194D4—22no—
195D4—32no—
196D4—61no—
197D4—64no—
198C2×C4—31no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—64no—
205C2×C4—122no—
206C2×C4C2×C3×S411yescharacteristic
207C8—122no—
208C2×C3—21no—
209C2×C3—42no—
210C2×C3—41no—
211C2×C3—62no—
212C2×C3—81no—
213C2×C3—82no—
214C2×C3—81no—
215C2×C3—162no—
216C2×C3—164no—
217C2×C3#543211yescharacteristic
218C22—22no—
219C22—32no—
220C22—42no—
221C22—121no—
222C22C2×C3×SL(2,ℤ/4)11yescharacteristic
223C4—31no—
224C4—31no—
225C4—62no—
226C4—242no—
227C4C3×GL(2,ℤ/4)12yes—
228C3—41no—
229C3—81no—
230C3#7315411yescharacteristic, direct product factor
231C2—21no—
232C2—42no—
233C2—62no—
234C2C3×#543211yescharacteristic
subgroupG/H
center, socleC2×C3#5432
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×#41⋊C3C22×C3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5423C2semidirect
3. #73154C3direct (N × H)
7. C22×C3×SL(2,𝔽3)C4semidirect
13. #5423C2×C3semidirect
36. C22×SL(2,𝔽3)C3×C4semidirect
230. C3#73154direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C3×#54232
#731543
C3×#17963
C3×#8334