Finite Group Databasebeta

← #519⋊H(3,𝔽3) #13152177

Subgroups — #519⋊H(3,𝔽3) #13152177

Order 1728.

491 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C212yesmaximal, complemented
3C32×#519C311yesmaximal, characteristic, complemented
4C3×#519⋊C3C313yesmaximal, complemented
5C2×Q8⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C22⋊C4×H(3,𝔽3)—41nomaximal
8C22×C32×Q8C2×C311yescharacteristic
9C22×C3×SL(2,𝔽3)C2×C313yes—
10C32×C4×Q8—32no—
11C32×C4×Q8C2×C312yescomplemented
12C32×#117—31no—
13C32×#117—31no—
14C3×C4×SL(2,𝔽3)C2×C316yescomplemented
15C23×H(3,𝔽3)—41no—
16C2×C4×H(3,𝔽3)—42no—
17Q8⋊H(3,𝔽3)C2×C411yescharacteristic
18C3×#519—31no—
19C3×#519C3211yescharacteristic
20#519⋊C3—33no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8C3×C412yescomplemented
24C2×C32×Q8C22×C311yescharacteristic
25C22×C32×C4—31no—
26C2×C3×SL(2,𝔽3)C3×C416yescomplemented
27C2×C3×SL(2,𝔽3)C22×C313yes—
28C32×C4⋊C4—32no—
29C32×C4⋊C4—32no—
30C32×C22⋊C4—31no—
31C32×C22⋊C4—43no—
32C32×C22⋊C4A411yescharacteristic
33C32×C42—32no—
34C32×C42—32no—
35C22×H(3,𝔽3)—41no—
36C22×H(3,𝔽3)—42no—
37C22×H(3,𝔽3)—82no—
38C4×H(3,𝔽3)—82no—
39C22×C3×Q8—31no—
40C22×C3×Q8C2×C3211yescharacteristic
41C22×SL(2,𝔽3)—33no—
42C3×C4×Q8—36no—
43C3×C4×Q8—32no—
44C3×C4×Q8—32no—
45C3×C4×Q8C2×C3212yes—
46C3×#117—33no—
47C3×#117—33no—
48C3×#117—31no—
49C3×#117—31no—
50C4×SL(2,𝔽3)—36no—
51C23×C32—43no—
52C23×C32C2×A411yescharacteristic
53C32×Q8—32no—
54C32×Q8—31no—
55C32×Q8—32no—
56C32×Q8C2×C3×C411yescharacteristic
57C2×C32×C4—32no—
58C2×C32×C4—32no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—32no—
62C2×C32×C4—46no—
63C2×C32×C4C2×A412yes—
64C3×SL(2,𝔽3)C2×C3×C413yes—
65#519H(3,𝔽3)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽3)—41no—
67C2×H(3,𝔽3)—42no—
68C2×H(3,𝔽3)—82no—
69C2×C3×Q8—33no—
70C2×C3×Q8—36no—
71C2×C3×Q8—31no—
72C2×C3×Q8—32no—
73C2×C3×Q8—32no—
74C2×C3×Q8—31no—
75C2×C3×Q8C32×C412yes—
76C2×C3×Q8C22×C3211yescharacteristic
77C22×C3×C4—33no—
78C22×C3×C4—31no—
79C2×SL(2,𝔽3)—36no—
80C2×SL(2,𝔽3)—33no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—36no—
83C3×C4⋊C4—36no—
84C3×C4⋊C4—32no—
85C3×C22⋊C4—33no—
86C3×C22⋊C4—31no—
87C3×C22⋊C4—31no—
88C3×C22⋊C4—123no—
89C3×C22⋊C4C3×A411yescharacteristic
90C3×C42—36no—
91C3×C42—32no—
92C3×C42—36no—
93C3×C42—32no—
94C22×C32—22no—
95C22×C32—43no—
96C22×C32—46no—
97C22×C32—86no—
98C22×C32C4×A412yes—
99C22×C32C22×A411yescharacteristic
100C32×C4—22no—
101C32×C4—31no—
102C32×C4—32no—
103C32×C4—31no—
104C32×C4—62no—
105C32×C4—86no—
106C22×Q8C2×H(3,𝔽3)11yescharacteristic
107C4×Q8—32no—
108C4×Q8C2×H(3,𝔽3)12yescomplemented
109#117—31no—
110#117—31no—
111H(3,𝔽3)—41noSylow
112C23×C3—31no—
113C23×C3—123no—
114C23×C3C2×C3×A411yescharacteristic
115C3×Q8—32no—
116C3×Q8—31no—
117C3×Q8—36no—
118C3×Q8—33no—
119C3×Q8—36no—
120C3×Q8—32no—
121C3×Q8—31no—
122C3×Q8C2×C32×C411yescharacteristic
123C2×C3×C4—33no—
124C2×C3×C4—33no—
125C2×C3×C4—36no—
126C2×C3×C4—36no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—36no—
132C2×C3×C4—32no—
133C2×C3×C4—32no—
134C2×C3×C4—126no—
135C2×C3×C4C2×C3×A412yes—
136SL(2,𝔽3)—33no—
137C2×C32—22no—
138C2×C32—43no—
139C2×C32—46no—
140C2×C32—86no—
141C2×C32C2×C4×A411yescharacteristic
142C2×C322−1+4⋊C312yes—
143C2×Q8—31no—
144C2×Q8—32no—
145C2×Q8C4×H(3,𝔽3)12yescomplemented
146C2×Q8C22×H(3,𝔽3)11yescharacteristic
147C22×C4—31no—
148C4⋊C4—32no—
149C4⋊C4—32no—
150C22⋊C4—31no—
151C22⋊C4C22⋊H(3,𝔽3)11yescharacteristic
152C42—32no—
153C42—32no—
154C22×C3—22no—
155C22×C3—31no—
156C22×C3—32no—
157C22×C3—62no—
158C22×C3—123no—
159C22×C3—126no—
160C22×C3—246no—
161C22×C3C3×C4×A412yes—
162C22×C3C22×C3×A411yescharacteristic
163C3×C4—22no—
164C3×C4—33no—
165C3×C4—33no—
166C3×C4—36no—
167C3×C4—31no—
168C3×C4—32no—
169C3×C4—31no—
170C3×C4—62no—
171C3×C4—66no—
172C3×C4—62no—
173C3×C4—246no—
174C32—43no—
175C32#519⋊C311yescharacteristic, complemented
176C23C2×C22⋊H(3,𝔽3)11yescharacteristic
177Q8—32no—
178Q8—31no—
179Q8—32no—
180Q8C2×C4×H(3,𝔽3)11yescharacteristic
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4C2×C22⋊H(3,𝔽3)12yes—
187C2×C3—22no—
188C2×C3—31no—
189C2×C3—32no—
190C2×C3—62no—
191C2×C3—123no—
192C2×C3—126no—
193C2×C3—246no—
194C2×C3C2×C3×C4×A411yescharacteristic
195C2×C3C3×2−1+4⋊C312yes—
196C22—22no—
197C22C4×C22⋊H(3,𝔽3)12yes—
198C22C22×C22⋊H(3,𝔽3)11yescharacteristic
199C4—22no—
200C4—31no—
201C4—31no—
202C4—32no—
203C4—62no—
204C3—31no—
205C3—123no—
206C3C3×#519⋊C311yescharacteristic
207C2—22no—
208C2C2×C4×C22⋊H(3,𝔽3)11yescharacteristic
209C22−1+4⋊H(3,𝔽3)12yes—
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutatorC3×Q8C2×C32×C4
FittingC32×#519C3
prime radicalC22×Q8⋊H(3,𝔽3)C2
cosocleC2×C3×Q8C22×C32
hypercenterC32×C22⋊C4A4
hypocenterQ8C2×C4×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×Q8⋊H(3,𝔽3)C2semidirect
3. C32×#519C3semidirect
4. C3×#519⋊C3C3semidirect
5. C2×Q8⋊H(3,𝔽3)C4semidirect
11. C32×C4×Q8C2×C3semidirect
14. C3×C4×SL(2,𝔽3)C2×C3semidirect
23. C2×C32×Q8C3×C4semidirect
26. C2×C3×SL(2,𝔽3)C3×C4semidirect
65. #519H(3,𝔽3)semidirect
108. C4×Q8C2×H(3,𝔽3)semidirect
145. C2×Q8C4×H(3,𝔽3)semidirect
175. C32#519⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C4×Q8⋊H(3,𝔽3)2
C32×#5193
C3×#519⋊C33
C22⋊C4×H(3,𝔽3)4