Finite Group Databasebeta

← #557⋊H(3,𝔽3) #13152201

Subgroups — #557⋊H(3,𝔽3) #13152201

Order 1728.

371 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C212yesmaximal
3C32×#557C311yesmaximal, characteristic, complemented
4C3×#557⋊C3C313yesmaximal, complemented
5C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
6C4⋊C4×H(3,𝔽3)—41nomaximal
7C32×C4⋊Q8—31no—
8C32×#125—32no—
9C32×C4×Q8—31no—
10C32×C4×Q8C2×C312yes—
11C32×C4×Q8C2×C311yescharacteristic
12C3×C4×SL(2,𝔽3)C2×C313yes—
13C3×C4×SL(2,𝔽3)C2×C316yes—
14C2×C4×H(3,𝔽3)—41no—
15C2×C4×H(3,𝔽3)—42no—
16Q8⋊H(3,𝔽3)Q811yescharacteristic, complemented
17C3×#557—31no—
18C3×#557C3211yescharacteristic
19#557⋊C3—33no—
20C2×C32×Q8—31no—
21C2×C32×Q8C22×C311yescharacteristic
22C2×C3×SL(2,𝔽3)C22×C313yes—
23C32×C4⋊C4—32no—
24C32×C4⋊C4—32no—
25C32×C4⋊C4—31no—
26C32×C4⋊C4—32no—
27C32×C4⋊C4—43no—
28C32×C4⋊C4A411yescharacteristic
29C32×C42—32no—
30C32×C42—31no—
31C22×H(3,𝔽3)—41no—
32C4×H(3,𝔽3)—42no—
33C4×H(3,𝔽3)—82no—
34C3×C4⋊Q8—31no—
35C3×C4⋊Q8—33no—
36C3×#125—32no—
37C3×#125—36no—
38C3×C4×Q8—33no—
39C3×C4×Q8—31no—
40C3×C4×Q8—32no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×C3212yes—
43C3×C4×Q8C2×C3211yescharacteristic
44C4×SL(2,𝔽3)—33no—
45C4×SL(2,𝔽3)—36no—
46C32×Q8—31no—
47C32×Q8—62no—
48C32×Q8C3×Q811yescharacteristic
49C2×C32×C4—31no—
50C2×C32×C4—31no—
51C2×C32×C4—32no—
52C2×C32×C4—43no—
53C2×C32×C4—46no—
54C2×C32×C4C2×A412yes—
55C2×C32×C4C2×A411yescharacteristic
56C3×SL(2,𝔽3)C3×Q813yescomplemented
57#557H(3,𝔽3)11yescharacteristic, complemented, Sylow
58C2×H(3,𝔽3)—41no—
59C2×H(3,𝔽3)—41no—
60C2×H(3,𝔽3)—41no—
61C2×C3×Q8—33no—
62C2×C3×Q8—31no—
63C2×C3×Q8—31no—
64C2×C3×Q8C22×C3211yescharacteristic
65C2×SL(2,𝔽3)—33no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—33no—
69C3×C4⋊C4—36no—
70C3×C4⋊C4—36no—
71C3×C4⋊C4—36no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C4⋊C4—31no—
75C3×C4⋊C4—123no—
76C3×C4⋊C4C3×A411yescharacteristic
77C3×C42—36no—
78C3×C42—33no—
79C3×C42—32no—
80C3×C42—31no—
81C22×C32—43no—
82C22×C32C22×A411yescharacteristic
83C32×C4—22no—
84C32×C4—31no—
85C32×C4—31no—
86C32×C4—46no—
87C32×C4—62no—
88C32×C4—61no—
89C32×C4—86no—
90C32×C4#41⋊C312yes—
91C4⋊Q8—31no—
92#125—32no—
93C4×Q8—31no—
94C4×Q8C2×H(3,𝔽3)12yes—
95C4×Q8C2×H(3,𝔽3)11yescharacteristic
96H(3,𝔽3)—41noSylow
97C3×Q8—31no—
98C3×Q8—33no—
99C3×Q8—31no—
100C3×Q8—66no—
101C3×Q8—62no—
102C3×Q8C32×Q811yescharacteristic
103C2×C3×C4—33no—
104C2×C3×C4—31no—
105C2×C3×C4—33no—
106C2×C3×C4—36no—
107C2×C3×C4—31no—
108C2×C3×C4—32no—
109C2×C3×C4—31no—
110C2×C3×C4—32no—
111C2×C3×C4—123no—
112C2×C3×C4—126no—
113C2×C3×C4C2×C3×A412yes—
114C2×C3×C4C2×C3×A411yescharacteristic
115SL(2,𝔽3)—33no—
116C2×C32—43no—
117C2×C32—43no—
118C2×C32—43no—
119C2×C32Q8×A411yescharacteristic
120C2×C32C2×#41⋊C311yescharacteristic
121C2×C322−1+4⋊C311yescharacteristic
122C2×Q8—31no—
123C2×Q8C22×H(3,𝔽3)11yescharacteristic
124C4⋊C4—31no—
125C4⋊C4—32no—
126C4⋊C4—32no—
127C4⋊C4—32no—
128C4⋊C4C22⋊H(3,𝔽3)11yescharacteristic
129C42—32no—
130C42—31no—
131C22×C3—31no—
132C22×C3—123no—
133C22×C3C22×C3×A411yescharacteristic
134C3×C4—22no—
135C3×C4—33no—
136C3×C4—33no—
137C3×C4—31no—
138C3×C4—31no—
139C3×C4—32no—
140C3×C4—62no—
141C3×C4—66no—
142C3×C4—63no—
143C3×C4—62no—
144C3×C4—61no—
145C3×C4—126no—
146C3×C4—246no—
147C3×C4C3×#41⋊C312yes—
148C32—43no—
149C32#557⋊C311yescharacteristic, complemented
150Q8—31no—
151Q8—62no—
152Q8Q8×H(3,𝔽3)11yescharacteristic
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—32no—
156C2×C4C2×C22⋊H(3,𝔽3)12yes—
157C2×C4C2×C22⋊H(3,𝔽3)11yescharacteristic
158C2×C3—31no—
159C2×C3—31no—
160C2×C3—31no—
161C2×C3—123no—
162C2×C3—123no—
163C2×C3—123no—
164C2×C3C3×Q8×A411yescharacteristic
165C2×C3C2×C3×#41⋊C311yescharacteristic
166C2×C3C3×2−1+4⋊C311yescharacteristic
167C22C22×C22⋊H(3,𝔽3)11yescharacteristic
168C4—22no—
169C4—31no—
170C4—31no—
171C4—62no—
172C4—61no—
173C4#41⋊H(3,𝔽3)12yes—
174C3—31no—
175C3—123no—
176C3C3×#557⋊C311yescharacteristic
177C2Q8×C22⋊H(3,𝔽3)11yescharacteristic
178C2C2×#41⋊H(3,𝔽3)11yescharacteristic
179C22−1+4⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutator, cosocleC2×C3×Q8C22×C32
FittingC32×#557C3
prime radicalC2×Q8⋊H(3,𝔽3)C22
hypercenterC32×C4⋊C4A4
hypocenterQ8Q8×H(3,𝔽3)
LCSG ▷ C2×C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C32×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C32×#557C3semidirect
4. C3×#557⋊C3C3semidirect
16. Q8⋊H(3,𝔽3)Q8semidirect
56. C3×SL(2,𝔽3)C3×Q8semidirect
57. #557H(3,𝔽3)semidirect
149. C32#557⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×Q8⋊H(3,𝔽3)2
C32×#5573
C3×#557⋊C33
C4⋊C4×H(3,𝔽3)4