Finite Group Databasebeta

← #84100⋊C3 #11921374

Subgroups — #84100⋊C3 #11921374

Order 1152.

407 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 59 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×#994C211yesmaximal, characteristic
2#5819⋊C3C212yesmaximal, complemented
3#84100C311yesmaximal, characteristic, complemented
4#2888⋊C3—31nomaximal
5C3×#994—22no—
6C2×C3×#41⋊C3C412yes—
7C2×C3×#41⋊C3C2211yescharacteristic
8C3⋊(#131⋊C3)—22no—
9C3×#836—41nomaximal
10C2×C3×H(5,𝔽2)C2×C311yescharacteristic
11C2×#994—21no—
12C2×#994D311yescharacteristic
13#5819C2×C312yes—
14#568⋊C3—32no—
15#5237—31no—
16#5237—31no—
17C2×#833—32no—
18C2×C32×Q8—41no—
19C3×#41⋊C3—22no—
20C3×#41⋊C3D412yes—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3×#208—42no—
23#2888—31noSylow
24C3×H(5,𝔽2)—22no—
25C3×H(5,𝔽2)—62no—
26C2×C3×#41—31no—
27C2×C3×#41C3×C412yes—
28C2×C3×#41C22×C311yescharacteristic
29C22×C3×D4—32no—
30C22×C3×D4—31no—
31#994—24no—
32#994—22no—
33C2×#41⋊C3—22no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3Dic312yes—
36C2×#41⋊C3C2×D311yescharacteristic
37#948—22no—
38#948—62no—
39C2×#208—32no—
40C2×#208—32no—
41#131⋊C3—62no—
42C3×#101—121no—
43#836—31no—
44#836A411yescharacteristic
45#833—32no—
46#833—32no—
47C32×Q8—82no—
48C2×C32×C4—42no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C22⋊C411yescharacteristic
51C3×C3⋊C8—82no—
52C2×H(5,𝔽2)C3×D311yescharacteristic
53#568—32no—
54#413—91no—
55#413—91no—
56C2×#100—92no—
57C24×C3—32no—
58C3×#41—22no—
59C3×#41—32no—
60C3×#41—62no—
61C3×#41—62no—
62C3×#41—62no—
63C3×#41C3×D412yes—
64C2×C3×Q8—41no—
65C2×C3×Q8—81no—
66C2×C3×Q8C2×C3×C411yescharacteristic
67C2×C3×Q8C2×A411yescharacteristic
68C2×C3×D4—31no—
69C2×C3×D4—32no—
70C2×C3×D4—32no—
71C2×C3×D4—31no—
72C2×C3×D4—62no—
73C2×C3×D4—62no—
74C2×C3×D4—64no—
75C2×C3×D4—64no—
76C22×C3×C4—31no—
77C22×C3×C4—32no—
78#41⋊C3—24no—
79#41⋊C3—22no—
80#41⋊C3—22no—
81#41⋊C3#6712yes—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)C2×Dic311yescharacteristic
84C3×M16—122no—
85#208—32no—
86#208—62no—
87#208—62no—
88#208C2×A412yes—
89C2×C3⋊C8—62no—
90C2×C3⋊C8—62no—
91C22×C32—41no—
92C32×C4—42no—
93C32×C4—82no—
94H(5,𝔽2)—22no—
95H(5,𝔽2)—62no—
96C2×#41—31no—
97C2×#41C3×Dic312yes—
98C2×#41C2×C3×D311yescharacteristic
99C22×D4—31no—
100C22×D4—32no—
101#131—62no—
102#131—182no—
103C2×M16—92no—
104C2×M16—92no—
105#101—31no—
106#101—91no—
107#100—92no—
108#100—92no—
109C23×C3—31no—
110C23×C3—32no—
111C23×C3—64no—
112C23×C3—242no—
113C3×Q8—22no—
114C3×Q8—31no—
115C3×Q8—82no—
116C3×Q8—84no—
117C3×Q8C3×C22⋊C411yescharacteristic
118C3×D4—32no—
119C3×D4—32no—
120C3×D4—64no—
121C3×D4—62no—
122C3×D4—62no—
123C3×D4—62no—
124C2×C3×C4—31no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4—62no—
130C2×C3×C4—62no—
131C2×C3×C4—81no—
132C2×C3×C4—82no—
133C2×C3×C4C4×A412yes—
134C2×C3×C4C22×A411yescharacteristic
135SL(2,𝔽3)—21no—
136SL(2,𝔽3)#21711yescharacteristic
137C3×C8—242no—
138C3⋊C8—22no—
139C3⋊C8—62no—
140C2×C32—41no—
141C2×C32—81no—
142C24—32no—
143#41—22no—
144#41—32no—
145#41—62no—
146#41—62no—
147#41—62no—
148#41C3×#6712yes—
149C2×Q8C2×C3×Dic311yescharacteristic
150C2×Q8D3×A411yescharacteristic
151C2×D4—31no—
152C2×D4—32no—
153C2×D4—31no—
154C2×D4—32no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4—64no—
158C2×D4—64no—
159C22×C4—31no—
160C22×C4—32no—
161M16—32no—
162M16—92no—
163M16—182no—
164M16—182no—
165C2×C8—182no—
166C2×C8—182no—
167C22×C3—32no—
168C22×C3—41no—
169C22×C3—62no—
170C22×C3—81no—
171C22×C3—121no—
172C22×C3—122no—
173C22×C3—244no—
174C22×C3C2×C4×A411yescharacteristic
175C3×C4—22no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—42no—
179C3×C4—82no—
180C3×C4—84no—
181C3×C4—82no—
182C3×C4D4×A412yes—
183C32—41noSylow
184C23—31no—
185C23—32no—
186C23—64no—
187C23—242no—
188Q8—22no—
189Q8—31no—
190Q8C3×#21711yescharacteristic
191D4—32no—
192D4—32no—
193D4—64no—
194D4—62no—
195D4—62no—
196D4—62no—
197C2×C4—31no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4Dic3×A412yes—
203C2×C4C2×D3×A411yescharacteristic
204C8—62no—
205C8—182no—
206C2×C3—21no—
207C2×C3—41no—
208C2×C3—62no—
209C2×C3—81no—
210C2×C3—82no—
211C2×C3—81no—
212C2×C3—122no—
213C2×C3A4×C22⋊C411yescharacteristic
214C22—32no—
215C22—62no—
216C22—121no—
217C22—122no—
218C22—244no—
219C22C2×Dic3×A411yescharacteristic
220C4—22no—
221C4—31no—
222C4—31no—
223C4A4×#6712yes—
224C3—41no—
225C3—81no—
226C3#2888⋊C311yescharacteristic, complemented
227C2—21no—
228C2—62no—
229C2—122no—
230C2A4×#21711yescharacteristic
subgroupG/H
centerC2A4×#217
commutatorC2×C3×Q8C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC2×C3×H(5,𝔽2)C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC2×C3×#994C2
cosocleC2×C3×#41C22×C3
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #5819⋊C3C2semidirect
3. #84100C3semidirect
226. C3#2888⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
#5819⋊C32
#841003
#2888⋊C33
C3×#8364