Finite Group Databasebeta

← #11921110

Subgroups — #11921110

Order 1152.

460 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 69 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
2#10593657C212yesmaximal, complemented
3#73154—33nomaximal
4#72144—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C2211yescharacteristic
7C22×C3×SL(2,𝔽3)C412yescomplemented
8C3⋊GU(2,𝔽3)—22no—
9#63660—41nomaximal
10C2×C3×2−1+4D311yescharacteristic
11C2×2−1+4⋊C3D313yes—
12#5423—36no—
13#5240—31no—
14#5237—31no—
15#5236—32no—
16#5225—32no—
17C2×C32×D4—41no—
18C3×#41⋊C3D412yes—
19C2×C3×SL(2,𝔽3)—22no—
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21#3748—42no—
22#1796—91noSylow
23C3×2−1+4—22no—
24C3×2−1+4—62no—
25C2×C3×#41—31no—
26C2×C3×#41—32no—
27C2×C3×#41C2×D311yescharacteristic
28C22×C3×Q8—31no—
29C22×C3×Q8Dic312yes—
302−1+4⋊C3—26no—
31C2×#41⋊C3C2×D313yes—
32C22×SL(2,𝔽3)Dic316yescomplemented
33#926—32no—
34C2×#208—32no—
35GU(2,𝔽3)—66no—
36#837—62no—
37#837—62no—
38#836—61no—
39#835—64no—
40#834—62no—
41#833—61no—
42#833—123no—
43C23×C32—42no—
44C32×D4—82no—
45C2×C32×C4—41no—
46C3×SL(2,𝔽3)C22⋊C411yescharacteristic
47C3⋊(C3⋊C8)—82no—
48C2×2−1+4C3⋊D311yescharacteristic
49#421—92no—
50#419—92no—
51#413—91no—
52#410—91no—
53C3×#41—32no—
54C3×#41—62no—
55C3×#41—62no—
56C3×#41—64no—
57C3×#41—64no—
58C3×#41#6712yes—
59C2×C3×Q8—22no—
60C2×C3×Q8—31no—
61C2×C3×Q8—32no—
62C2×C3×Q8—34no—
63C2×C3×Q8—62no—
64C2×C3×Q8—62no—
65C2×C3×Q8C2×Dic311yescharacteristic
66C2×C3×D4—31no—
67C2×C3×D4—32no—
68C2×C3×D4—43no—
69C2×C3×D4S411yescharacteristic
70C22×C3×C4—31no—
71C22×C3×C4—32no—
72C22×C3×C4—61no—
73#41⋊C3#6716yes—
74C2×SL(2,𝔽3)—26no—
75C2×SL(2,𝔽3)C2×Dic313yes—
76#217—62no—
77C4⋊Dic3—62no—
78C4×Dic3—62no—
79#208—62no—
80#208—62no—
81#208—126no—
82C2×C3⋊C8—62no—
83C22×C32—41no—
84C22×C32—82no—
85C22×C32—162no—
86C32×C4—42no—
872−1+4—22no—
882−1+4—62no—
89C2×#41—31no—
90C2×#41—32no—
91C2×#41C2×C3⋊D311yescharacteristic
92C22×Q8—31no—
93C22×Q8C3⋊Dic312yes—
94C2×M16—92no—
95#117—92no—
96#104—182no—
97#104—182no—
98#103—184no—
99#101—181no—
100#100—181no—
101#99—182no—
102C23×C3—31no—
103C23×C3—46no—
104C23×C3SL(2,ℤ/4)12yes—
105C3×Q8—31no—
106C3×Q8—34no—
107C3×Q8—62no—
108C3×Q8—62no—
109C3×Q8#21711yescharacteristic
110C3×D4—22no—
111C3×D4—32no—
112C3×D4—61no—
113C3×D4—64no—
114C3×D4—86no—
115C2×C3×C4—31no—
116C2×C3×C4—32no—
117C2×C3×C4—32no—
118C2×C3×C4—32no—
119C2×C3×C4—43no—
120C2×C3×C4—62no—
121C2×C3×C4—62no—
122C2×C3×C4—64no—
123C2×C3×C4C2×S411yescharacteristic
124C2×Dic3—122no—
125SL(2,𝔽3)#21713yes—
126C3⋊C8—122no—
127C3⋊C8—246no—
128C2×C32—41no—
129C2×C32—81no—
130C2×C32—162no—
131#41—32no—
132#41—62no—
133#41—62no—
134#41—64no—
135#41—64no—
136#41#63812yes—
137C2×Q8—22no—
138C2×Q8—31no—
139C2×Q8—32no—
140C2×Q8—34no—
141C2×Q8—62no—
142C2×Q8—62no—
143C2×Q8C2×C3⋊Dic311yescharacteristic
144C2×D4—31no—
145C2×D4—32no—
146C2×D4C3⋊S411yescharacteristic
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—61no—
150M16—182no—
151M16—182no—
152C2×C8—182no—
153C4⋊C4—182no—
154C22⋊C4—182no—
155C42—182no—
156C22×C3—22no—
157C22×C3—32no—
158C22×C3—42no—
159C22×C3—43no—
160C22×C3—86no—
161C22×C3—121no—
162C22×C3—166no—
163C22×C3C2×SL(2,ℤ/4)11yescharacteristic
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—46no—
167C3×C4—62no—
168C3×C4GL(2,ℤ/4)12yes—
169Dic3—242no—
170C32—41noSylow
171C23—31no—
172C23C3⋊SL(2,ℤ/4)12yes—
173Q8—31no—
174Q8—34no—
175Q8—62no—
176Q8—62no—
177Q8#375711yescharacteristic
178D4—22no—
179D4—32no—
180D4—61no—
181D4—64no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—64no—
189C2×C4—362no—
190C2×C4C2×C3⋊S411yescharacteristic
191C8—362no—
192C2×C3—21no—
193C2×C3—42no—
194C2×C3—43no—
195C2×C3—62no—
196C2×C3—83no—
197C2×C3—166no—
198C2×C3#543211yescharacteristic
199C22—22no—
200C22—32no—
201C22—42no—
202C22—121no—
203C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
204C4—31no—
205C4—31no—
206C4—62no—
207C4—722no—
208C4#6426912yes—
209C3—43no—
210C3#7315411yescharacteristic, complemented
211C2—21no—
212C2—42no—
213C2—62no—
214C2#1059366611yescharacteristic
subgroupG/H
centerC2#10593666
commutatorC2×C3×SL(2,𝔽3)C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC2×C3×2−1+4D3
socleC2×C3#5432
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×C3×#41⋊C3C22
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593657C2semidirect
7. C22×C3×SL(2,𝔽3)C4semidirect
32. C22×SL(2,𝔽3)Dic3semidirect
210. C3#73154semidirect

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
#105936572
#731543
#721443
#636604