Finite Group Databasebeta

← H(3,𝔽3)×#474 #13145181

Subgroups — H(3,𝔽3)×#474 #13145181

Order 1728.

594 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 203 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
3H(3,𝔽3)×#107C212yesmaximal, complemented
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C212yesmaximal
6C32×#474C314yesmaximal, complemented
7C2×Q8×H(3,𝔽3)—21no—
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)—21no—
10C2×C8×H(3,𝔽3)C2212yes—
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2212yes—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16C32×C4⋊Q8C2×C314yescomplemented
17C32×C4×D4C2×C314yes—
18C32×#107C2×C318yescomplemented
19C32×C4⋊C8C2×C314yescomplemented
20C32×#102C2×C318yes—
21C23×H(3,𝔽3)—21no—
22Q8×H(3,𝔽3)—42no—
23D4×H(3,𝔽3)—21no—
24D4×H(3,𝔽3)Q812yescomplemented
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)—42no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C8×H(3,𝔽3)—22no—
32C3×#474—34no—
33C3×#474C3211yescharacteristic
34C2×C32×Q8—24no—
35C2×C32×D4C22×C314yes—
36C22×C32×C4—24no—
37C2×C32×C8C22×C318yes—
38C32×C4⋊C4—24no—
39C32×C4⋊C4C22×C314yes—
40C32×C4⋊C4C22×C318yes—
41C32×C22⋊C4—24no—
42C32×C42C22×C314yes—
43C22×H(3,𝔽3)—22no—
44C22×H(3,𝔽3)—41no—
45C22×H(3,𝔽3)—41no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)—21no—
48C4×H(3,𝔽3)—41no—
49C4×H(3,𝔽3)—42no—
50C4×H(3,𝔽3)D812yes—
51C4×H(3,𝔽3)C2×Q811yescharacteristic
52C4×H(3,𝔽3)#4111yescharacteristic
53C3×C4⋊Q8—34no—
54C3×C4⋊Q8C2×C3211yescharacteristic
55C3×C4×D4—34no—
56C3×C4×D4C2×C3211yescharacteristic
57C3×#107—38no—
58C3×#107C2×C3212yes—
59C3×C4⋊C8—34no—
60C3×C4⋊C8C2×C3211yescharacteristic
61C3×#102—38no—
62C3×#102C2×C3212yes—
63C23×C32—24no—
64C32×Q8—48no—
65C32×D4—24no—
66C32×D4C3×Q818yescomplemented
67C2×C32×C4—24no—
68C2×C32×C4—28no—
69C2×C32×C4—48no—
70C2×C32×C4C3×D414yes—
71C2×C32×C4C3×D414yes—
72C2×C32×C4C23×C314yes—
73C32×C8—28no—
74#474H(3,𝔽3)11yescharacteristic, direct product factor, Sylow
75C2×H(3,𝔽3)—42no—
76C2×H(3,𝔽3)C22⋊Q811yescharacteristic
77C2×H(3,𝔽3)C2×D811yescharacteristic
78C2×H(3,𝔽3)#13711yescharacteristic
79C2×C3×Q8—21no—
80C2×C3×Q8—64no—
81C2×C3×D4—34no—
82C2×C3×D4C22×C3211yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4—64no—
85C2×C3×C8—38no—
86C2×C3×C8C22×C3212yes—
87C3×C4⋊C4—21no—
88C3×C4⋊C4—38no—
89C3×C4⋊C4—34no—
90C3×C4⋊C4—64no—
91C3×C4⋊C4C22×C3212yes—
92C3×C4⋊C4C22×C3211yescharacteristic
93C3×C22⋊C4—21no—
94C3×C22⋊C4—64no—
95C3×C42—34no—
96C3×C42C22×C3211yescharacteristic
97C22×C32—28no—
98C22×C32—44no—
99C22×C32—44no—
100C22×C32C2×C3×D414yes—
101C32×C4—24no—
102C32×C4—44no—
103C32×C4—48no—
104C32×C4C3×D818yes—
105C32×C4C2×C3×Q814yes—
106C32×C4C3×#4114yes—
107C4⋊Q8C2×H(3,𝔽3)11yescharacteristic, complemented
108C4×D4C2×H(3,𝔽3)11yescharacteristic
109#107C2×H(3,𝔽3)12yescomplemented
110C4⋊C8C2×H(3,𝔽3)11yescharacteristic, complemented
111#102C2×H(3,𝔽3)12yes—
112H(3,𝔽3)#47411yescharacteristic, direct product factor, Sylow
113C23×C3—21no—
114C23×C3—64no—
115C3×Q8—42no—
116C3×Q8—128no—
117C3×D4—21no—
118C3×D4—38no—
119C3×D4—64no—
120C3×D4C32×Q812yes—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—34no—
124C2×C3×C4—34no—
125C2×C3×C4—34no—
126C2×C3×C4—42no—
127C2×C3×C4—64no—
128C2×C3×C4—68no—
129C2×C3×C4—128no—
130C2×C3×C4C32×D411yescharacteristic
131C2×C3×C4C32×D411yescharacteristic
132C2×C3×C4C23×C3211yescharacteristic
133C3×C8—22no—
134C3×C8—68no—
135C2×C32—48no—
136C2×C32C3×C22⋊Q814yes—
137C2×C32C2×C3×D814yes—
138C2×C32C3×#13714yes—
139C2×Q8—21no—
140C2×D4C22×H(3,𝔽3)11yescharacteristic
141C22×C4—21no—
142C2×C8C22×H(3,𝔽3)12yes—
143C4⋊C4—21no—
144C4⋊C4C22×H(3,𝔽3)12yes—
145C4⋊C4C22×H(3,𝔽3)11yescharacteristic
146C22⋊C4—21no—
147C42C22×H(3,𝔽3)11yescharacteristic
148C22×C3—22no—
149C22×C3—34no—
150C22×C3—41no—
151C22×C3—41no—
152C22×C3—68no—
153C22×C3—124no—
154C22×C3—124no—
155C22×C3C2×C32×D411yescharacteristic
156C3×C4—21no—
157C3×C4—34no—
158C3×C4—34no—
159C3×C4—38no—
160C3×C4—41no—
161C3×C4—42no—
162C3×C4—64no—
163C3×C4—124no—
164C3×C4—128no—
165C3×C4C32×D812yes—
166C3×C4C2×C32×Q811yescharacteristic
167C3×C4C32×#4111yescharacteristic
168C32C3×#47414yescomplemented
169C23—21no—
170Q8—42no—
171D4—21no—
172D4Q8×H(3,𝔽3)12yescomplemented
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4D4×H(3,𝔽3)11yescharacteristic
177C2×C4D4×H(3,𝔽3)11yescharacteristic
178C2×C4C23×H(3,𝔽3)11yescharacteristic
179C8—22no—
180C2×C3—34no—
181C2×C3—34no—
182C2×C3—34no—
183C2×C3—42no—
184C2×C3—128no—
185C2×C3C32×C22⋊Q811yescharacteristic
186C2×C3C2×C32×D811yescharacteristic
187C2×C3C32×#13711yescharacteristic
188C22—22no—
189C22—41no—
190C22—41no—
191C22C2×D4×H(3,𝔽3)11yescharacteristic
192C4—21no—
193C4—41no—
194C4—42no—
195C4D8×H(3,𝔽3)12yes—
196C4C2×Q8×H(3,𝔽3)11yescharacteristic
197C4#41×H(3,𝔽3)11yescharacteristic
198C3—34no—
199C3C32×#47411yescharacteristic
200C2—42no—
201C2H(3,𝔽3)×C22⋊Q811yescharacteristic
202C2C2×D8×H(3,𝔽3)11yescharacteristic
203C2H(3,𝔽3)×#13711yescharacteristic
subgroupG/H
center, socleC22×C3C2×C32×D4
commutator, Frattini, cosocleC2×C3×C4C23×C32
prime radicalC2×D4×H(3,𝔽3)C22
LCSG ▷ C2×C3×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C42×H(3,𝔽3) ◁ G

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. H(3,𝔽3)×#107C2semidirect
4. H(3,𝔽3)×C4⋊C8C2semidirect
6. C32×#474C3semidirect
16. C32×C4⋊Q8C2×C3semidirect
18. C32×#107C2×C3semidirect
19. C32×C4⋊C8C2×C3semidirect
24. D4×H(3,𝔽3)Q8semidirect
66. C32×D4C3×Q8semidirect
74. #474H(3,𝔽3)direct (N × H)
107. C4⋊Q8C2×H(3,𝔽3)semidirect
109. #107C2×H(3,𝔽3)semidirect
110. C4⋊C8C2×H(3,𝔽3)semidirect
112. H(3,𝔽3)#474direct (N × H)
168. C32C3×#474semidirect
172. D4Q8×H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
C4×D4×H(3,𝔽3)2
H(3,𝔽3)×#1072
H(3,𝔽3)×C4⋊C82
H(3,𝔽3)×#1022
C32×#4743